Definitions and Theorems in Computer Science [Work in Progress]

7/29/2019 Definitions and Theorems in Computer Science [Work in Progress]

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Definitions and Theorems in Computer Science

Willem Van Onsem

https://github.com/KommuSoft/publications

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Contents

I Mathematics, Theoretical Computer Science, Data Structures and Algorithms 1

1 Computation and Language 3

2 Computational Complexity 5

3 Computational Geometry 9

4 Discrete Mathematics 114.1 Operator Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114.2 Algebraic Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

4.3 Probability Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144.4 Number Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154.5 Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

5 Data Structures and Algorithms 175.1 Graph Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

6 Formal Languages and Automata Theory 19

7 Information Theory 21

8 Mathematical Software 23

9 Numerical Analysis 25

II Artificial Intelligence and Machine Learning 27

10 Artificial Intelligence 2910.1 Knowledge Representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2910.2 Optimization Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3010.3 Toxicology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

11 Computer Vision and Pattern Recognition 33

12 Computer Science and Game Theory 35

13 Machine Learning 3713.1 Empirical Law Discovery . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

14 Logic in Computer Science 3914.1 First-Order Logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3914.2 Logic Programming . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4114.3 Theorem Proving . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

14.3.1 Proof Normalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4814.4 Critical Reasoning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4914.5 Argumentation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

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14.6 Explanation Based Learning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5014.7 Transformation Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

14.7.1 Termination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5114.8 Epistemic State . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5214.9 Temporal Reasoning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5214.10Model Ranking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

15 Multiagent Systems 55

16 Neural and Evolutionary Computing 57

17 Programming Languages 59

18 Robotics 61

19 Symbolic Computation 63

III Interaction, Graphics, Sound and Multimedia 65

20 Computers and Society 67

21 Graphics 69

22 Human-Computer Interaction 71

23 Multimedia 73

24 Sound 75

IV Engineering 77

25 Computational Engineering, Finance and Science 79

26 Hardware Architecture 81

27 Operating Systems 83

28 Performance 85

29 Software Engineering 87

30 Systems and Control 89

V Internet, Databases, Networking and Social Media 91

31 Cryptography and Security 93

32 Databases 95

33 Digital Libraries 97

34 Distributed, Parallel and Cluster Computing 99

35 Information Retrieval 101

36 Networking and Internet Architecture 10336.1 Testing Web Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

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37 Social and Information Networks 105

VI Emerging Technologies and Other 107

38 General Literature 109

39 Emerging Technologies 111

40 Other 113

VII Appendices 115

List of Figures 117

List of Tables 119

Bibliography 121

A Index 123

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Introduction

Definition 1 (Definition, Definiendum, Definiens). A definition is a statement that explains the meaning of a term (a word, phrase, or other set of symbols). The term to be defined is the definiendum. The term may have many different senses and multiple meanings. For each meaning, a definiens is a cluster of words that defines that term (and clarifies the speaker’s intention). [Wik13 ]

This publication aims to compile a large number of common definitions and theorems in a single reference guide.

Advantage of a List of Definitions

We expect the number of definitions in computer science to be around 100’000. The major problem with such definitions is

that papers become quite inaccessible due to the fact that they use concepts who are defined elsewhere. This publicationaims to tackle this problem by providing a collection of these definitions. Possible other advantages are a reduction of definition collisions: two meanings for the same term and preventing people from reinventing the wheel over and overagain. A final advantage is that knowledge of a large amount of concepts enriches ones global understanding of computerscience.

How is this Document Compiled?

The definitions, theorems and lemma’s are extracted from a massive amount of papers from conferences, proceedings, journals, etc. Since no individual can read this amount of papers in a lifetime, some intelligent scripts look for patternswho look interesting together with extracting the actual content. The extracted content is then reviewed by the authorand after a minimal amount of modifications added to the right chapter.

The current processing rate is five definitions per day, but we hope by improving the data mining scripts, we will improvethis number. At the moment our scripts already use optical character recognition together with error correction. We arecurrently working on technology that can generate LATEX code for a specific formula.

People who want to help to improve the mining scripts, provide papers or do some post-processing can propose apull-request on http://goo.gl/jZeAG or contact the author at [email protected].

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Chapter 1

Computation and Language

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Chapter 2

Computational Complexity

Definition 2 (Big-Oh notation). If f, g are two functions from N to N, then we

1. say that f = O (g) if there exists a constant c such that f (n) ≤ cg (n) for every sufficiently large n,

2. say that f = Ω (g) if g = O (f ),

3. say that f = Θ (g) is f = O (g) and g = O (f ),

4. say that f = o (g) if for every ε > 0, f (n) ≤ εg (n) for every sufficiently large n, and

5. say that f = ω (g) if g = o (f ).

[AB09]

Definition 3 (Computing a function and running time). Let f ∶ 0, 1∗ → 0, 1∗ and let T ∶ N → N be some functions,and let M be a Turing machine. We say that M computes f if for every x ∈ 0, 1∗, whenever M is initialized to the start configuration on input x, then it halts with f (x) written on its output tape. We say M computes f in T (n)-timeif its computation on every input x requires at most T (x) steps. [AB09]

Definition 4 (The class DTIME). Let T ∶ N → N be some function. A language L is in DTIME (T (n)) if and only f there is a Turing machine that runs in time cT (n) for some constant c > 0 and decides L.

Definition 5 (The class P).

P = ∪c≥1DTIME(nc) . (2.1)

[AB09]

Definition 6 (The class NTIME). For every function T ∶ N → N and L ⊆ 0, 1∗, we say that L ∈ NTIME (T (n))if there is a constant c > 0 and a cT (n)-time non deterministic Turing Machine M such that for every x ∈ 0, 1∗,x ∈ L↔M (x) = 1. [AB09]

Definition 7 (The class NP).NP = ∪c≥1NTIME (nc) . (2.2)

[AB09]

Definition 8 (Reduction, NP-hard and NP-complete). A language L ⊆ 0, 1∗ is polynomial-time Karp reducibleto a language L′ ⊆

0, 1

∗

(sometimes shortened to just “ polynomial-time reducible”), denoted by L ≤ p L′, if there is

a polynomial-time computable function f ∶ 0, 1∗ → 0, 1∗ such that for every x ∈ 0, 1∗, x ∈ L if and only if f (x) ∈ L′.We say that L′ is NP-hard if L ≤ p L′ for every L ∈ NP. We say that L is NP-complete if L′ is NP-hard and L′ ∈ NP. [AB09]

Definition 9 (Complement language). If L ⊆ 0, 1∗ is a language, then we denote by L the complement language of L. That is, L = 0, 1∗ ∖L. [ AB09 ]

Definition 10 (The class coNP).coNP = L ∶ L ∈ NP . (2.3)

[AB09]

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Theorem 1 (Efficient universal Turing machine). There exists a Turing Machine U such that for every x, α ∈ 0, 1∗, U (x, α) = M α (x), where M α denotes the Turing Machine represented by α. Moreover, if M α halts on input x within T steps then U (x, α) halts within CT log T steps, where C is a number independent of x and depending only on M α’s alphabet size, number of tapes, and number of states. [AB09]

Theorem 2. There exists a function UC ∶ 0, 1∗ → 0, 1 that is not computable by any Turing Machine.

Proof. The function U C is defined as follows: For every α ∈ 0, 1∗, if M α (α) = 1, then U C (α) = 0; otherwise (if M α (α) outputs a different value or enters an infinite loop), U C (α) = 1. Suppose for the sake of contradiction that

U C is computable and hence there exists a Turing Machine M such that M (α) = U C (α) for every α ∈ 0, 1∗

. Then,in particular, M (⌊M ⌋) = U C (⌊M ⌋). But this is impossible: By the definition of U C , U C (⌊M ⌋) = 1 ↔ M (⌊M ⌋) = 1.[AB09]

Theorem 3. HALT is not computable by any Turing Machine.

Proof. Suppose, for the sake of contradiction, that there was a Turing Machine M HALT computing HALT . We will useM HALT to show a Turing Machine M UC computing UC , contradicting Theorem 2. The Turing Machine M UC is simple:On input α, M UC runs M HALT (α, α). If the result is 0 (meaning that M α does not halt on α), then M UC outputs 1.Otherwise, M UC uses the universal Turing Machine U to compute b = M α (α). If b = 1, then M UC outputs 0; otherwise,it outputs 1. Under the assumption that M HALT (α, α) outputs HALT (α, α) within a finite number of steps, the TuringMachine M UC (α) will output UC (α). [AB09]

Theorem 4 (Time Hierarchy Theorem). If f, g are time-constructible functions satisfying f

(n

)log f

(n

)= o

(g

(n

)),

then DTIME (f (n)) ⊊ DTIME (g (n)) . (2.4)

Proof. To showcase the essential idea of the proof of Theorem 4 with minimal notation, we prove the simpler statementDTIME (n) ⊊ DTIME n1.5. Consider the following Turing machine D: “On input x, run for x1.4 steps the UniversalTuring Machine U of Theorem 1 to simulate the execution of M x on x. If U outputs some bit b ∈ 0, 1 in this time,then output the opposite answer (i.e., output 1 − b). Else output 0.” Here M x is the machine represented by the stringx. By definition, D halts within n1.4 steps and hence the language L decided by D is in DTIME n1.5. We claim thatL ∉ DTIME (n). For contradiction’s sake, assume that there is some Turing Machine M and constant c such that TuringMachine M , given any input x ∈ 0, 1∗, halts within c x steps and outputs D (x). The time to simulate M by theuniversal Turing machine U on every input x is at most c′c x log x for some number c′ that depends on the alphabet sizeand number of tapes and states of M but is independent of x. There is some number n0 such that n1.4 > c′cn log n forevery n ≥ n0. Let x be a string representing the machine M whose length is at least n0 (such a string exists since M is

represented by infinitely many strings). Then, D (x) will obtain the output b = M (x) within x1.4 steps, but by definitionof D, we have D (x) = 1−b = M (x). Thus we have derived a contradiction. The proof Theorem 4 for general f, g is similarand uses the observation that the slowdown in simulating a machine using U is at most logarithmic. [AB09]

Theorem 5 (Nondeterministic Time Hierarchy Theorem). If f, g are time constructible functions satisfying f (n + 1) = o (g (n)), then

NTIME (f (n)) ⊊ NTIME (g (n)) . (2.5)

Proof. We just showcase the main idea by proving NTIME (n) ⊊ NTIME n1.5. The first instinct is to duplicate theproof of Theorem 4, since there is a universal Turing machine for nondeterministic computation as well. However, thisalone does not suffice because the definition of the new machine D requires the ability to “flip the answer”, in other words,to efficiently compute, given the description of an nondeterminstic Turing machine M and an input x, the value 1−M (x).It is not obvious how to do this using the universal nondeterministic machine: it is unclear how a nondeterministic

machine can just “flip the answer”. Specifically, we do not expect that that the complement of an NTIME (n) languagewill be in NTIME n1.5. Now of course, the complement of every NTIME (n) language is trivially decidable inexponential time (even deterministically) by examining all the possibilities for the machine’s nondeterministic choices, buton first sight this seems to be completely irrelevant to proving NTIME (n) ⊊ NTIME n1.5. Surprisingly, this trivialexponential simulation of a nondeterministic machine does suffice to establish a hierarchy theorem. The key idea will belazy diagonalization, so named because the new machine D is in no hurry to diagonalize and only ensures that it flips theanswer of each linear time nondeterminstic Turing machine M i in only one string out of a sufficiently large (exponentially

large) set of strings. Define the function f ∶ N → N as follows: f (1) = 2 and f (i + 1) = 2f (i)1.2 . Given n, it’s not hard tofind in O n1.5 time the number i such that n is sandwiched between f (i) and f (i + 1). Our diagonalizing machine D

will try to flip the answer of M i on some input in the set 1n ∶ f (i) < n ≤ f (i + 1). D is defined as follows:

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“On input x, if x ∉ 1∗, reject. If x = 1n, then compute i such that f (i) < n ≤ f (i + 1) and

1. If f (i) < n < f (i + 1) then simulate M i on input 1n+1 using nondeterminism in n1.1 time and output itsanswer. (If M i has not halted in this time, then halt and accept.)

2. If n = f (i + 1), accept 1n if and only if M i rejects 1f (i)+1 in (f (i) + 1)1.1 time.”

Part 2 requires going through all possible 2(f (i)+1)1.1 branches of M i on input 1f (i)+1, but that is fine since the input size

f (i + 1) is 2f (i)1.2

. Hence the nondeterminstic Turing machine D runs in O n1.5 time. Let L be the language decided byD. We claim that L ∉ NTIME

(n

). Indeed, suppose for the sake of contradiction that L is decided by an nondeterminstic

Turing machine M running in cn steps (for some constant c). Since each nondeterminstic Turing machine is representedby infinitely many strings, we can find i large enough such that M = M i and on inputs of length n ≥ f (i), M i can besimulated in less than n1.1 steps. This means that the two steps in the description of D ensure, respectively, that

If f (i) < n < f (i + 1) , then D (1n) = M i 1n+1 (2.6)

whereas D 1f (i+1) ≠ M i 1f (i)+1 (2.7)

By our assumption M i and D agree on all inputs 1n for n in the semi-open interval (f (i) , f (i + 1)]. Together with(2.6), this implies that D 1f (i+1) = M i 1f (i)+1, contradicting (2.7).[AB09]

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Chapter 3

Computational Geometry

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Chapter 4

Discrete Mathematics

4.1 Operator Properties

Definition 11 (Commutative operation). A binary operation ⊕ is commutative if a⊕ b = b ⊕ a for all a, b ∈ S .[ Opp11]

Definition 12 (Associative operation). A binary operation ⊕ is associative if a ⊕ (b ⊕ c) = (a⊕ b) ⊕ c for all a,b,c ∈

S .[ Opp11]

Definition 13 (Semi-associative). A binary operator ⊗ is said to be semi-associative if there is an associative operator ⊕ such that for any a, b, c, (a⊗ b)⊗ c = a ⊗ (b ⊕ c).[ MHT03 ]

Definition 14 (Quasi-associative). A binary operator ⊕ is said to be quasi-associative if there is a semi-associative operator ⊗ and a function f such that for any a, b, a⊕ b = a⊗ f b.[ MHT03 ]

Definition 15 (Bi-quasi-associative). A ternary operator f is said to be bi-quasi-associative if there is a semi-associative operator ⊗ and two functions f ′L, f ′R such that for any l, n, r, flnr = l ⊗ f ′Lnr = r ⊗ f ′Rnl. We can fix a bi-quasi-associative operator f by providing ⊗, ⊕ (associative operator for ⊗), f ′L and f ′R, therefore, we will write f with 4-tuple as f ≡ [[⊗,⊕, f ′L, f ′R]].[ MHT03 ]

4.2 Algebraic Structures

Definition 16 (Left identity element). Let S be a set and ⊕ a binary operation on S . An element e ∈ S is called leftidentity element if e⊕ a = a for all a ∈ S .[Opp11 ]

Definition 17 (Right identity element). Let S be a set and ⊕ a binary operation on S . An element e ∈ S is called rightidentity element if a⊕ e = a for all ∈ S .[Opp11 ]

Definition 18 (Identity element). Let S be a set and ⊕ a binary operation on S . An element e ∈ S is called identityelement (also called neutral element) if it is both a left identity element and a right identity element (i.e., e⊕a = a⊕e = a

for all a ∈ S ). [Opp11 ]

Definition 19 (Inverse element). Let S be a set, ⊕ be a binary operation with an identity element e, and a be an element of S . If there exists an element b ∈ S with a⊕b = b⊕a = e, then a is invertible and b is the inverse element (also called inverse) of a. [ Opp11 ]

Definition 20 (Semigroup). A semigroup is an algebraic structure ⟨S,⊕⟩ that consists of a nonempty set S and an associative binary operation ⊕. The semigroup must be closed (i.e., for al l a, b ∈ S , a ⊕ b must also be an element of S ).[Opp11 ]

Definition 21 (Monoid). A monoid is a semigroup ⟨S,⊕⟩ that has an identity element e ∈ S with respect to ⊕. [ Opp11]

Definition 22 (Group). A group is a monoid ⟨S,⊕⟩ in which every element a ∈ S has an inverse element in S (i.e.,every element a ∈ S is invertible). [ Opp11]

Definition 23 (Commutative group). A group ⟨S,⊕⟩ is a commutative group if the operation ⊕ is commutative (i.e.,a⊕ b = b ⊕ a for all a, b ∈ S ). [Opp11 ]

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Definition 24 (Finite group). A group ⟨S,⊕⟩ is a finite group if it contains only finitely many elements. [Opp11 ]

Definition 25 (Subgroup). A subset H of a group G is a subgroup of G if it is closed under the operation of G and also forms a group. [ Opp11]

Definition 26 (Left coset). Let G be a group and H ⊆ G be a subset of G. For all a ∈ G, the sets a⊕H ∶= a ⊕ hh ∈ H are called left cosets of H . [ Opp11]

Definition 27 (Right coset). Let G be a group and H ⊆ G be a subset of G. For all a ∈ G, the sets H ⊕a ∶= a ⊕ hh ∈ H are called right cosets of H . [ Opp11]

Definition 28 (Coset). Let G be a (commutative) group and H ⊆ G. For all a ∈ G, the sets a ⊕ H and H ⊕ a are equal and are called cosets of H . [ Opp11 ]

Definition 29 (Ring). A ring is an algebraic structure ⟨S,⊕,⊗⟩ with a set S and two associative binary operations ⊕and ⊗ that fulfill the following requirements:

1. ⟨S,⊕⟩ is a commutative group with identity element e1 ;

2. ⟨S,⊗⟩ is a monoid with identity element e2 ;

3. The operation ⊗ is distributive over the operation ⊕. This means that for all a,b,c ∈ S the following two distributive laws must hold:

a ⊗

(b ⊕ c

)=

(a ⊗ b

)⊕

(a⊗ c

)(b ⊕ c)⊗ a = (b ⊗ a)⊕ (c⊗ a) (4.1)

[Opp11 ]

Definition 30 (Field). A ring ⟨S,⊕,⊗⟩ in which ⟨S ∖ e1 ,⊗⟩ is a group is a field. [Opp11 ]

Definition 31 (Subfield). A subset H of a field F is a subfield of F if it closed under the operations of F and also forms a field. [Opp11 ]

Definition 32 (Prime field). A prime field is a field that contains no proper subfield. [Opp11 ]

Definition 33 (Homomorphism). Let A and B be two algebraic structures. A mapping f ∶ A→B is called a homomor-phism of A into B if it preserves the operations of A. That is, if is an operation of A and an operation of B, then f

(x y

)= f

(x

) f

(y

)must hold for all x, y ∈ A. [ Opp11]

Definition 34 (Isomorphism). A homomorphism f ∶ A→ B is an isomorphism if it is injective (“one to one”). In this case, we say that A and B are isomorphic and we write A ≅ B. [Opp11 ]

Definition 35 (Automorphism). An isomorphism f ∶ A→A is an automorphism. [Opp11 ]

Definition 36 (Permutation). Let S be a set. A map f ∶ S → S is a permutation if f is bijective (i.e., injective and surjective). The set of all permutations of S is denoted by Perm S →S , or P (S ) in short. [Opp11 ]

Definition 37 (Common divisors and greatest common divisor). For a, b ∈ Z0, c ∈ Z is a common divisor of a and b if ca and cb. Furthermore, c is the greatest common divisor, denoted gcd (a, b), if it is the largest integer that divides a and b. [Opp11 ]

Definition 38 (Common multiples and least common multiple). For a, b ∈ Z0, c ∈ Z is a common multiple of a and

b if ac and bc. Furthermore, c is the least common multiple, denoted lcm (a, b), if it is the smallest integer that is divided by a and b. [ Opp11]

Definition 39 (Prime number). A natural number 1 < n ∈ N is called a prime number (also called prime) if it divisible only by 1 and itself. [ Opp11]

Definition 40 (Primality decision problem). Given a positive integer n ∈ N, deciding whether n ∈ P (i.e., n is prime) or not (i.e., n is composite) is called the primality decision problem. [ Opp11]

Definition 41 (B-smooth integer). Let B be an integer. An integer n is a B-smooth integer if every prime factor of n is less than B. [ Opp11]

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Definition 42. Let a, b ∈ Z and n ∈ N. a is congruent to b modulo n, denoted a ≡ b ( mod n), if n divides a − b (i.e.,na − b). [Opp11 ]

Definition 43 (Polynomial). Let A be an algebraic structure with addition and multiplication (e.g., a ring or a field). A function p (x) is a polynomial in x over A if it is of the form

p (x) = n

i=0

aixi= a0 + a1x + a2x2 + . . . + anxn (4.2)

where n is a positive integer (i.e., the degree of p (x), denoted as deg ( p)), the coefficients ai ( 0 ≤ i ≤ n) are elements in A, and x is a symbol not belonging to A. [Opp11 ]

Definition 44 (Quadratic residue and square root). An element x ∈ Z∗n is a quadratic residue modulo n if there exists an element y ∈ Z∗n such that x = y2 ( mod n). If such a y exists, then it is called a square root of x modulo n. [ Opp11]

Theorem 6 (Lagrange’s Theorem). If H is a subgroup of G, then H G (i.e., the order of H divides the order of G.

Proof. If H = G, then H G holds trivially. Consequently, we only consider the case in which H ⊂ G. For any a ∈ G∖H ,the coset a ⊕H is a subset of G. The following can be shown:

1. For any a ≠ a, if a ∉ a′ ⊕H then (a⊕H ) ∩ (a′ ⊕H ) = ∅;

2. a⊕H = H .For (1), suppose there exists a b ∈ (a⊕H ) ∩ (a

′

⊕H ). Then there exist c, c′

∈ H such that a ⊕ c = b = a′

⊕ c′

. Applyingvarious group axioms, we have a = a⊕ e = a⊕ c⊕ c−1 = b⊕ c−1 = (a′ ⊕ c′)⊕ c−1 = a′⊕ c′ ⊕ c−1 ∈ a′⊕H . This contradictsour assumption (that a ∉ a′ ⊕ H ). For (2), a⊕H ≤ H holds trivially (by the definition of a coset). Suppose that theinequality is rigorous. This is only possible if there are b, c ∈ H with b ≠ c and a⊕ b = a⊕ c. Applying the inverse elementof a on either side of the equation, we get b = c, contradicting to b ≠ c. In summary, G is partitioned by H and the familyof its mutually disjoint cosets, each has the size H , and hence H G. This proves the theorem. [Opp11]

Theorem 7. For all a,b,c ∈ Z, if ab and bc, then ac.

Proof. If ab and bc, then there exist f, g ∈ Z with b = af and c = bg. Consequently, we can write c = bg = (af )g = a (f g)to express c as a multiple of a. The claim (i.e., ac) follows directly from this equation. [Opp11]

Theorem 8. For all a,b,c ∈ Z, if ab, then acbc for all c.

Proof. If ab, then there exists f ∈ Z with b = af . Consequently, we can write bc = (af ) c = f (ac) to express bc as a multipleof ac. The claim (i.e., acbc) follows directly from this equation. [Opp11]

Theorem 9. For all a,b,c,d,e ∈ Z, if ca and cb, then cda + eb for all d and e.

Proof. If ca and cb, then there exist f, g ∈ Z with a = fc and b = gc. Consequently, we can write da + eb = df c + egc =(df + eg) c to express da+ eb as a multiple of c. The claim (i.e., cda+ eb) follows directly from this equation. [Opp11]

Theorem 10. For all a, b ∈ Z, if ab and b ≠ 0, then a ≤ b.Proof. If ab and b = 0, then there exists 0 ≠ f ∈ Z with b = af . Consequently, b = af ≥ a and the claim (i.e., a ≤ b)follows immediately. [Opp11]

Theorem 11. For all a, b ∈ Z, if ab and ba, then a = b.Proof. Let us assume that ab and ba. If a = 0 then b = 0, and vice versa. If a, b ≠ 0, then it follows from Theorem 10.that a ≤ b and b ≤ a, and hence b = a. [Opp11]

Theorem 12 (Euclid’s division theorem). For all n, d ∈ Z0 there exist unique and efficiently computable q, r ∈ Z such that n = qd + r and 0 ≤ r ≤ d. [ Opp11]

Theorem 13 (Prime density theorem).

limn→∞

π (n) ln (n)n

= 1 (4.3)

[Opp11 ]

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Theorem 14 (Unique factorization). Every natural number n ∈ N can be factored uniquely (up to a permutation of the prime factors):

n = p∈P

pep(n) (4.4)

In this formula, e p (n) refers to the exponent of p in the factorization of n. For almost all p ∈ P this value is zero, and only for finitely many primes p the value e p (n) is greater than zero. [Opp11 ]

Theorem 15 (Chinese remainder theorem). Let

⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩

x ≡ a1 ( mod n1)x ≡ a2 ( mod n2). . .

x ≡ ak ( mod nk)(4.5)

be a system of k congruences with pairwise co-prime moduli n1, . . . , nk. The system has a unique and efficiently computable solution x in Zn with n =∏k

i=1 ni. [Opp11 ]

Theorem 16 (Fermat’s Little Theorem). If p is a prime and a ∈ Z∗ p, then a p−1 ( mod p).

Proof. Because φ ( p) = p − 1 for every prime number p, Fermat’s Little Theorem is just a special case of Euler’sTheorem. [Opp11]

Theorem 17 (Euler’s Theorem). If gcd (a, n) = 1, then aφ(n) ≡ 1 ( mod n).

Proof. Because gcd (a, n) = 1, a ( mod n) must be an element in Z∗n. Also, Z∗n = φ (n). According to a corollary of Lagrange’s Theorem, the order of every element (in a finite group) divides the order of the group. Consequently, the orderof a (i.e., ord (a)) divides φ (n), and hence if we multiply a modulo nφ (n) times we always get a value that is equivalentto 1 modulo n. [Opp11]

4.3 Probability Theory

Definition 45 (Conditional Probability (Bayes’ rule)). The probability of event x conditioned on knowing event y (or more shortly, the conditional probability of x given y) is defined as:

p (xy) ≡ p (x) p (y) (4.6)

If p (y) = 0, then p(xy) is not defined. This definition is also called Bayes’ rule. [Bar11]

Definition 46 (Probability Density Functions). For a single continuous variable x, the probability density function p (x) is a function such that: ⎧⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎩

p(x) ≥ 0

+∞

−∞

p (x) dx = 1

p(a < x < b) = ba

p (x) dx

(4.7)

[Bar11]

Definition 47 (Independence). Events x and y are independent if knowing one event gives no extra information about the other event. Mathematically, this is expressed by:

p (x, y) = p (x) p (y) (4.8)

Provided that p (x) ≠ 0 and p (y) ≠ 0 independence of x and y is equivalent to:

p (xy) = p(x)↔ p (yx) = p (y) (4.9)

If p (xy) = p (x) for all states of x and y, then the variables x and y are said to be independent. If

p (x, y) = kf (x) g (y) (4.10)

for some constant k, and positive functions f and g then x and y are independent. [Bar11]

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Definition 48 (Prior likelihood and Posterior). For data D and variable θ, Bayes’ rule tells us how to update our prior beliefs about the variable θ in light of the data to a posterior belief:

p (θD)posterior

=

p (Dθ)likelihood

p (θ)prior

p (D)evidence

(4.11)

The evidence is also called the marginal likelihood. The term likelihood is used for the probability that a model generates observed data. More fully, if we condition on the model M , we have

p(θD, M ) = p(Dθ, M ) p(θM ) p(DM ) (4.12)

where we see the role of the likelihood p (Dθ, M ) and marginal likelihood p(DM ). likelihood is also called the modellikelihood. The most probable a posteriori (sometimes abbreviated as MAP) setting is that which maximizes the posterior,

θ∗ = argmax θ ( p (θD, M )) (4.13)

[Bar11]

Definition 49 (Conditional Independence).

X Y Z (4.14)denotes that the two sets of variables X and Y are independent of each other provided we know the state of the set of variables Z . For full conditional independence, X and Z must be independent given all states of Z . Formally, this means that

p(X ,Y Z ) = p(X Z ) p (Y Z ) (4.15)

for all states of X , Y , Z . In case the conditioning set is empty we may also write X Y for X Y ∅, in which case X is unconditionally independent of Y . If X and Y are not conditionally independent, they are conditionally dependent.This is written:

X ⊺⊺Y Z (4.16)

[Bar11]

4.4 Number Theory

4.5 Statistics

Definition 50 (Related data). If data d1, d2, . . . , dm describe a common phenomenon altogether, or they refer to the same behavior simultaneously, then they can be treated as related data. [ZN95 ]

Definition 51 (Qualitative correlations among related data). If di and dj are two related data, then the presence of di

somewhat (qualitatively) enhances the presence of dj, and the absence of di somewhat (qualitatively) depresses the presence of dj. The above effects are called qualitative correlations among related data. [ ZN95]

Definition 52 (Support coefficient Function (SCF)). If there are m− 1 data related to di, then the support coefficientfunction (sometimes abbreviated as SCF) of di calculates the total effects from the related data by considering the

qualitative correlations between di and each of its related data. [ZN95 ] Definition 53 (Shift interval). Shift interval is a dynamic region for inaccurate data. Given a standard fuzzy region

for inaccurate di, the shift interval of di varies around the standard fuzzy region on the basis of SC F i. When SC F i shows that the related data support di the shift interval of di becomes wider than the standard fuzzy region. On the other hand,when SC F i shows that the related data do not support di, the shift interval of di becomes narrower than the standard fuzzy region. [ ZN95]

Definition 54 (Evidence based on a Support Coefficient Function (SCF)). SC F i determines the shift interval of di,that is, SC F i determines how widely di is allowed to shift. The wider the shift interval, the more easily di is identified.Therefore, SCFi provides confirmatory or disconfirmatory evidence for identifying dj. [ZN95]

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Definition 55 (Rate of correctness (RC)). the rate that the identified partial component set is exactly the same as the partial component set in the correct solutions. [ ZN95]

Definition 56 (Rate of identification (RI)). The rate that the partial components in the correct solutions are identified.[ZN95]

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Chapter 5

Data Structures and Algorithms

5.1 Graph Theory

Definition 57 (Graph). A graph G consists of verticesnodes and edges (also called links) between the vertices. Edges may be directed (they have an arrow in a single direction) or undirected. A graph with al l edges directed is called a directedgraph, and one with all edges undirected is called an undirected graph. [Bar11]

Definition 58 (Path, Ancestors, Descendants). A path A ↦ B from node A to node B is a sequence of vertices A =

A0, A1, . . . , An−1, An = B, with (Ai, Ai+1) an edge in the graph, thereby connecting A to B. For a directed graph this means that a path is a sequence of nodes which when we follow the direction of the arrows leads us from A to B. The vertices A

such that A↦ B and B A are the ancestors of B. The vertices B such that AB and B A are the descendants of A. [Bar11]

Definition 59 (Directed Acyclic Graph). A Directed Acyclic Graph (sometimes abbreviated as DAG) is a graph G

with directed edges (arrows on each link) between the vertices (nodes) such that by following a path of vertices from one node to another along the direction of each edge no path will revisit a vertex. In a DAG the ancestors of B are those nodes who have a directed path ending at B. Conversely, the descendants of A are those nodes who have a directed path starting at A. [Bar11]

Definition 60 (Neighbour). For an undirected graph G the neighbours of x, ne (x) are those nodes directly connected to x. [Bar11]

Definition 61 (Clique, Cliquo). Given an undirected graph, a clique is a maximally connected subset of vertices. All the members of the clique are connected to each other; furthermore there is no larger clique that can be made from a clique.A non-maximal clique is sometimes called a cliquo. [Bar11]

Definition 62 (Singly-Connected Graph, Multiply-Connected Graph, Tree, Loopy). A singly-connected graph (alsocalled tree) is a graph where there is only one path from a vertex a to another vertex b. Otherwise the graph is a multiply-connected graph (also called loopy). This definition applies regardless of whether or not the edges in the graph are directed. [ Bar11 ]

Definition 63 (Spanning Tree, Maximum Weight Spanning Tree). A spanning tree of an undirected graph G is a singly-connected subset of the existing edges such that the resulting singly-connected graph covers all vertices of G. Amaximum weight spanning tree is a spanning tree such that the sum of all weights on the edges of the tree is larger than for any other spanning tree of G. [ Bar11 ]

Definition 64 (Best path). A path would be selected as a ”best” path if it satisfies in the two following criteria:

1. Selected path must have the maximum reliability in ab

.

2. Selected path must include the minimum number of ”hops”.

[MCR12]

Definition 65 (Reliability of a path). The reliability of a path is defined by the lowest reliability arc in the path. and

reliability(P ) = min(i,j)∈P

sij (5.1)

[MCR12]

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Definition 66 (Efficient path). A path P ∈ab

is an efficient path if and only if no other path P ′ ∈ab

has a better value

for one criterion and not worse value for the other one. [MCR12]

Definition 67 (Equivalent efficient path). Two efficient paths are equivalent efficient paths if and only if their value agree for both criteria. [MCR12]

Definition 68 (Complete subset of efficient paths). A set C ab ⊆ab

of efficient paths is a complete subset of efficient

paths, if any path P ′ ∉ C ab is either dominated or equivalent to at least one efficient path P ∈ C ab. [ MCR12 ]

Definition 69 (Minimal complete subset of efficient paths). A complete set C ab is a minimal complete subset of efficient paths if and only if no two of its efficient paths are equivalent. [ MCR12 ]

Definition 70 (Minimal node of a Hasse diagram). A minimal node of a Hasse diagram is a node not preceded by any other node of the diagram. [Kul12]

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Chapter 6

Formal Languages and Automata Theory

Definition 71. A nondeterministic finite automaton consists of a set S of states. One of these states, s0 ∈ S , is called the starting state of the automaton and a subset F ⊆ S of the states are accepting states. Additionally, we have a set T of transitions. Each transition t connects a pair of states s1 and s2 and is labeled with a symbol, which is either a character c from the alphabet Σ, or the symbol ε, which indicates an epsilon-transition. A transition from state s tostate t on the symbol c is written as sct.[Mog09]

Definition 72. Given a set M of nondeterministic finite automaton states, we define the ε-closure(M) to be the least

(in terms of the subset relation) solution to the set equation:

ε-closure (M ) = M ∪ ts ∈ ε-closure (M ) and sεt ∈ T (6.1)

Where T is the set of transitions in the non deterministic finite automaton.[ Mog09 ]

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20



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Chapter 7

Information Theory

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22



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Chapter 8

Mathematical Software

23



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24



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Chapter 9

Numerical Analysis

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26



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Part II

Artificial Intelligence and MachineLearning

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Chapter 10

Artificial Intelligence

10.1 Knowledge Representation

Definition 73 (Belief change scenario). A belief change scenario is a triple B = ⟨K,R,C ⟩, where K , R and C are sets of formula over a fixed propositional language LP . Informally, K is a knowledge base which is to be modified in such a way that the resulting knowledge base includes all elements from R and does not include any element from C . The modified knowledge base corresponding to B will be denoted by K +R C . [ KLMB12]

Definition 74 (Belief change extension, Unique (inconsistent) belief change extension). Let B = ⟨K,R,C ⟩ be a belief change scenario over LP . Define a new set P ′ of atoms, isomorphic with P , given by P ′ = p′ ∶ p ∈ P . Let K ′ be a knowledge base obtained from K by replacing any p ∈ P by p′ ∈ P ′. Let EQ be a maximal (with respect to set inclusion)set of equivalences p⇔ p′ p ∈ P such that T h (K ′ ∪ EQ ∪ R) ∩ (C ∪ ) = ∅. The set T h (K ′ ∪ EQ ∪R) ∩LP is called a belief change extension of B. If there is no such set EQ, then B is inconsistent and LP is a unique (inconsistent)belief change extension of B. [ KLMB12 ]

Definition 75 (Class of all belief change extensions). Let E ii∈I be the class of all belief change extensions of B = ⟨K,R,C ⟩. Then

K +R C =⋂i∈I

E i (10.1)

[KLMB12 ]

Definition 76 (Knowledge base, Observation, Defeasible statement, Domain axiom). A knowledge base is a triple KB = ⟨OB,DS,DA⟩, where OB, DS and DA are finite sets of formulas. These sets are referred to as observations,defeasible statements and domain axioms, respectively. [KLMB12 ]

Definition 77 (Revision formula, Revision). Let KB = ⟨OB,DS,DA⟩ be a knowledge base and suppose that α is a revision formula representing a new observation. A revision of KB by α, written KB ∗ α, is a new knowledge base given by ⟨OB1,DS,DA⟩, where OB1 = OB ⊕ α⊖ ¬DA. Here α⊖ ¬DA is a finite representation of the modified knowledge base corresponding to belief change scenario ⟨OB,α , ¬DA⟩. [ KLMB12]

Definition 78 (Belief set corresponding to). Let KB = ⟨OB,DS,DA⟩ be a knowledge base. A belief set correspondingto KB , written BKB , is given by DS + (OB ∪DA). [ KLMB12]

Definition 79 (Prioritized belief revision of KB by α). Let KB = ⟨OB,DS,DA⟩ be a knowledge base and α be a revision formula. Let OB1 = OB ⊕ α ⊖ ¬DA. The prioritized belief revision of KB by α with respect to priorities DS 1 < DS 2 < . . . < DS n, written KB ∗[DS 1<DS 2<...<DS n] α, is the formula

DS 1+ (DS 2 ⊕ . . . (DS n ⊕ (OB1 ∪DA) . . .)) . (10.2)

[]

Definition 80 (Closure of a knowledge base). Let ∗CW A be any closed world assumption policy among (basic) Closed World Assumption (CW A), Generalized Closed World Assumption (GCW A), Extended Generalized Closed World As-sumption (EGCW A), Careful Closed World Assumption (CCW A) and Extended Closed World Assumption (ECW A). Let Σ be a formula from PROP PS and ⟨P,Q,Z ⟩ a partition of Var (Σ). The closure ∗CW A (Σ, ⟨P,Q,Z ⟩) of Σ given ⟨P,Q,Z ⟩with respect to ∗CW A is the formula Σ∪ ¬αα is a ∗CW A-free for negation formula with respect to Σ and ⟨P,Q,Z ⟩.[CMM99 ]

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Definition 81 (Closed World Assumption-free for negation formula). Let Σ and α be two formulas from PROP ps and let ⟨P,Q,Z ⟩ be partition of Var (Σ). α is Closed World Assumption-free (sometimes abbreviated as CW A-free) for negation if and only if α is a positive literal such that Σ ⊭ α holds. [ CMM99]

Definition 82 (Generalized Closed World Assumption-free for negation formula). Let Σ and α be two formulas from PROP ps and let ⟨P,Q,Z ⟩ be partition of Var (Σ). α is Generalized Closed World Assumption-free (sometimes abbreviated as GCW A-free) for negation if and only if α is a positive literal and for each positive clause γ such that Σ ⊭ γ holds, Σ ⊭ α ∨ γ holds. [CMM99]

Definition 83 (Extended Generalized Closed World Assumption-free for negation formula). Let Σ and α be two formulas

from PROP ps and let ⟨P,Q,Z ⟩ be partition of Var (Σ). α is Extended Generalized Closed World Assumption-free(sometimes abbreviated as EGCW A-free) for negation if and only if α is a conjunction of positive literals and for each positive clause γ such that Σ ⊭ γ holds, Σ ⊭ α ∨ γ holds. [CMM99]

Definition 84 (Careful Closed World Assumption-free for negation formula). Let Σ and α be two formulas from PROP ps and let ⟨P,Q,Z ⟩ be partition of Var (Σ). α is Careful Closed World Assumption-free (sometimes abbreviated as CCW A-free) for negation if and only if α is a literal from L+

P and for each clause γ containing only literals from L+

P ∪ LQ and such that Σ ⊭ γ holds, Σ ⊭ α ∨ γ holds. [CMM99 ]

Definition 85 (Extended Closed World Assumption-free for negation formula). Let Σ and α be two formulas from PROP ps and let ⟨P,Q,Z ⟩ be partition of Var (Σ). α is Extended Closed World Assumption-free (sometimes abbreviated as ECW A-free) for negation if and only if Var (α) ∩ Z = ∅ and for each clause γ containing only literals from L+

P ∪ LQ

and such that Σ ⊭ γ holds, Σ ⊭ α ∨ γ holds. [ CMM99]

Definition 86 (*CWA clause inference, *CWA literal inference). Let ∗CW A be any closed world assumption policy among (basic) Closed World Assumption (CW A), Generalized Closed World Assumption (GCW A), Extended Generalized Closed World Assumption (EGCW A), Careful Closed World Assumption (CCW A) and Extended Closed World Assumption (ECW A). *CWA clause inference is the following decision:

1. Input: A formula Σ and clause γ from PROPS PS , a partition ⟨P,Q,Z ⟩ of Var (Σ) and a CWA policy ∗CW A

2. Query: Does ∗CW A (Σ, ⟨P,Q,Z ⟩) ⊧ γ holds?

*CWA literal inference is the restriction of the corresponding ∗CW A clause inference problem where γ is restricted tobe a literal. [CMM99]

Definition 87 (Blake formula). Let Σ be a formula from PROP PS . Σ is a Blake formula if and only if Σ is a CNF formula and for every implicate γ of Σ, there exists a clause π in Σ such that π ⊧ γ holds. [CMM99 ]

Definition 88 (Disjunct normal form formula). Let Σ be a formula from PROP PS . Σ is a Disjunction normal formformula (sometimes abbreviated as DNF formula) if and only if Σ is a finite disjunction of terms. [ CMM99]

Definition 89 (Horn cover formula). Let Σ be a formula from PROP PS . Σ is a Horn cover formula if and only if Σis a finite disjunction of Horn CNF formulas. [ CMM99 ]

Definition 90 (Renamable Horn cover formula). Let Σ be a formula from PROP PS . Σ is a renamable Horn coverformula if and only if Σ is a finite disjunction of renamable Horn CNF formulas. [CMM99]

Definition 91 (Finite representation of the modified knowledge base corresponding to belief change scenario). Let KB = ⟨OB,DS,DA⟩ and suppose that α is a revision formula representing a domain axiom. The revised knowl-edge base is defined by KB ∗ α = ⟨OB1,DS,DA ∪ ⟩, where OB1 = OB ⊕ ⊺ ⊖ ¬ (DA ∪ α) . Here OB ⊕ ⊺ ⊖¬ (DA ∪ α) is a finite representation of the modified knowledge base corresponding to belief changescenario, ⟨OB, ⊺ ,¬ (DA ∪ α) ⟩. [ KLMB12 ]

Definition 92. Let KB = ⟨OB,DS,DA⟩ and suppose that α is a revision formula representing a defeasible statement.The revised knowledge base is defined by KB ∗ α =

⟨OB,DS 1, DA

⟩, where DS 1 = DS ⊕

α

. [ KLMB12]

10.2 Optimization Problems

Definition 93. A neighborhood structure is a function N ∶ S → 2S that assigns to every s ∈ S a set of neighbors N (s) ⊆ S . N (s) is called the neighborhood of s. Often, neighborhood structures are implicitly defined by specifying the changes that must be applied to a solution s in order to generate all its neighbors. The application of such an operator that produces a neighbor s′ ∈ N (s) of a solution s is commonly called a move.[ Alb05 ]

Definition 94. A locally minimal solution (or local minimum) with respect to a neighborhood structure N is a solution s such that ∀s ∈ N (s) ∶ f (s) ≤ f (s). We call s a strict locally minimum if ∀s ∈ N (s) ∶ f (s) < f (s).[ Alb05 ]

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10.3 Toxicology

Definition 95 (Performance comparison). The predictive accuracies of two theories are statistically equivalent then the theory with better explanatory power has better performance. Otherwise the one with higher accuracy has better performance. [ SKMS97]

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Chapter 11

Computer Vision and Pattern Recognition

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Chapter 12

Computer Science and Game Theory

Definition 96. A game A consists of the state of the game objects ( S ), the input of the system (player’s input, M )and the abstract control system (time-independent, F ) which governs the S under the current M :

Game A = ⟨S,M,F ⟩ (12.1)

[KLR+12]

Definition 97. Neutral element of M ( empty input, ) the current state of the game ( s) with input ( p) broken down into two consecutive inputs ( m,n):

F (s, ) = s with s ∈ S

F (s,mn) = F (F (s, m) , n) with s ∈ S and m, n ∈ M (12.2)

[KLR+12]

Definition 98 (Maxset, Minset). The maxset X M is that subset of X at which it is MAX’s turn to make a move. The minset X m is that subset of X at which it is MIN’s turn to make a move. [Bof73 ]

Definition 99 (Evaluation function, Better position). An evaluation function (sometimes abbreviated as EF) over a game graph ⟨X, Γ⟩ is a single valued function f ∶ X → R. If f (x) > f (y) then x is said to be a better position than y

under f . [Bof73 ]

Definition 100 (Strategy S o). S o is that strategy which leads to move xy∗ being made from position x where:

⎧⎪⎪⎪⎨⎪⎪⎪⎩f (x∗) = max

y∈Γxf (y) if x ∈ X M

f (x∗) = miny∈Γx

f (y) if x ∈ X m(12.3)

Ties for y∗ are broken by some subsidiary rule. [Bof73 ]

Definition 101 (Locally equivalent evaluation functions, Locally perfect evaluation function). Evaluation functions f

and g are locally equivalent if f (x) > f (y) whenever g (x) > g (y) (12.4)

where x, y ∈ Γw for some w ∈ X . f is locally perfect if it is locally equivalent to λwmm. [Bof73]

Definition 102 (Probabilistic evaluation function). A probabilistic evaluation function (sometimes abbreviated as

PEF) over a game graph ⟨X, Γ⟩ is a bounded function p ∶ X ×R → R with p (x, r) ≥ 0, p (x, r) dr = 1 and p (x, r) = 0

outside some finite range r1 (x) ≤ r ≤ r2 (x). [ Bof73]

Definition 103 (Strategy S ′o). S ′o is that strategy which leads to move xy∗ being made from position x where:

⎧⎪⎪⎪⎨⎪⎪⎪⎩v (x∗) p = max

y∈Γxv (y) p if x ∈ X M

v (x∗) p = miny∈Γx

v (y) p if x ∈ X m(12.5)

[Bof73 ]

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Definition 104 ((Constant) decision rule, Preference function, Preferred move). A (constant) decision rule (also called preference function) for MAX is a single valued function σM ∶ X → X with σM (x) ∈ Γx. x ↦ σM (x) is MAX’s preferred move from x. A preference function σm for MIN can be defined in an analogous way. [ Bof73]

Definition 105 ((Constant) preference pair). If σM and σm are preference functions for MAX and MIN respectively then σ = (σM , σm) is termed a (constant) preference pair on (X, Γ). [ Bof73 ]

Definition 106 (Strategy order relation ≤σ). If u, v ∈ X and σ is a preference pair on (X, Γ) then u ≥σ v if and only if there exists an element t ∈ X such that

1. u, v ∈ Γt

2. u = σM (t) or v = σm (t)[Bof73 ]

Definition 107 (Evaluation function reproduces a preference pair). An evaluation function f over (X, Γ) reproducesa preference pair σ (with respect to a strategy S 0) if f (x) > f (y) whenever x >σ y. [Bof73]

Definition 108 (Strategy order relation ≤∗σ). If u, v ∈ X and σ is a preference pair on (X, Γ) then u ≥∗σ v if and only if there exists is a sequence u = w0, w1, . . . , ws = v such that wi >σ wi−1, i = 1, 2, . . . , s [Bof73]

Definition 109 (Fully consistent strategy). σ is fully consistent if C π1 = C π2 for any pair of paths π1, π2 with the same endpoints. [ Bof73 ]

Definition 110 ((Variable) preference function). A (Variable) preference function for MAX is a single-valued func-tion τ M ∶ X M

→ Γ × X M ∖ I where I is the unit interval I = x0 ≤ x ≤ 1 and

1. τ M (x, u) = 0 if u ∉ Γx

2. τ M (x, u) ≥ 0 if u ∈ Γx

3. u∈Γx

τ M (x, u) = 1

τ M (x, u) can be thought of as the probability that, when in position x, MAX will choose to move to position u. A variable preference rule τ m for MIN can be defined in an analogous way. [ Bof73 ]

Definition 111 ((Variable) preference pair). If τ M and τ m are variable preference functions for MAX and MIN respectively

then τ = (τ M , τ m) is termed a (variable) preference pair on (X, Γ). [ Bof73 ]

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Chapter 13

Machine Learning

Definition 112 (Learning algorithm for a meta-domain). Formally, an algorithm A is a learning algorithm for ameta-domain M in a hypothesis space H with respect to a set of problem distributions T , if for any domain D ∈ M , any choice of a problem distribution P in T , and any target problem solver f ∈ H ,

1. A takes as input the specification of a domain D ∈ M , an error parameter E , and a confidence parameter σ,

2. A may call SolvedProblem, which returns examples (x, f (x)) for D, where x is chosen with probability P (x) from S

×G. The number of oracle calls of A and its running time must be polynomial in the maximum problem size and the length of its input.

3. For all D ∈ M and distributions P ∈ T , with probability at least (1 − δ )A outputs a program f ′ that approximates f

in the sense that

x∈∆

P (x) ≤ (13.1)

where ∆ = xf ′ fails on x while f succeeds .

4. There is a polynomial R such that, for a maximum problem size n, 1

, frac1δ , maximum length I and maximum steplength r of any solution output by SolvedProblem, and an upper bound t on the running times of programs in D

on inputs of size n, if A outputs f ′, the run time of f ′ is bounded by R n,l,r,t, 1

, 1δ

[Tad91]

Definition 113 (Satisfying a spare solution space bias). A problem solver f for a domain D and a problem distribution P satisfies a sparse solution space bias if there is a set of operator sequences mf such that, on any problem x ∈ D

such that P (x) > 0, f (x) ∈ mf and mf is bounded by a polynomial Q in the problem size n. [ Tad91 ]

Definition 114 (Satisfying a macro table bias). A problem solver f satisfies a macro table bias for a domain D in M if there is a feature ordering O = (l , . . . , n) such that,

1. D is serially decomposable for O, and

2. f constructs all its solutions using a macro table M as follows: for each feature i from 1 to n, macros M j,i are successively applied, where j is the value of feature i in the state before applying the macro.

[Tad91]

Definition 115 (Beliefs in Conjoint Analysis). We allow weighted beliefs with a weight parameter coming from

]0, 1

]where 1 means full truth degree (complete certainty, the perfect belief), while a value α ∈ ]0, 1[ describe a regular belief that can be doubted.

1. Regular belief s such as: (A1 (a1) ∧ . . . ∧ At (at)) ∶ α (13.2)

2. Indifference belief s such as: (L↔ R) ∶ 1 (13.3)

Indifference beliefs are always have full truth because we claim that if the respondent would distinguish degrees of truth then she is able to express preference.

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3. Negative belief s such as: (¬F ) ∶ 1 (13.4)

where Ai are attribute predicates and L, R, F are regular atom conjunctions. Again, it is obvious in conjoint to don’t ask user to express thoughts on negative information. As such there are no real negative beliefs such as F ∶ 0. Moreover,the reader may notice that we adopt the intuitionistic logic approach i.e., there is no assumption on any kind of law of excluded middle, as we don’t necessarily assume F ∶ 0⇔ (¬F ) ∶ 1. [GSB12]

13.1 Empirical Law DiscoveryDefinition 116 (X -of-N representation). Let Ai1 ≤ i ≤ MaxAtt be the set of attributes of a domain, and for each Ai,V ij 1 ≤ j ≤ MaxAttVal i be its value set where MaxAtt is the number of attributes, and MaxAttVal i, is the number of differ-ent values of Ai. An X -of-N representation is a set, denoted as X -of-AV kAV k is an attribute-value pair denoted as “ Ai

where N + is the number of attribute-value pairs in the X -of-N representation, called the size of the X -of-N representation,and N is the number of different attributes that appear in the X -of-N representation. The value of an X -of-N can be any number between 0 and N . Given an instance, its value is X if and only if X of the AV k are true. An attribute-value pair AV k (Ai = V ij) is true for an instance if and only if attribute Ai, of the instance has value V ij. [Zhe95 ]

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Chapter 14

Logic in Computer Science

14.1 First-Order Logic

Definition 117 (Signature). S is a signature if S is a four-tuple ⟨P,F,r,C ⟩ where:

1. P is a set of predicate symbols (P 1, P 2, . . . , P n),

2. F is a set of function symbols

(F 1, F 2, . . . , F m

),

3. r is arity or degree of functions and relations. For each P i respectively F j, r (P i) respectively r (F j) is a non-zeronatural number denoting the arity of P i respectively F j,

4. C is a set of constant symbols.

[Tel12]

Definition 118 (Alphabet). An alphabet Σ consists of the following symbols:

1. Signature S = ⟨P,F,r,C ⟩.

2. Collection of variables V .

3. Operators: ¬ ( negation), ∧ ( conjunction), ∨ ( disjunction), → ( implication), ↔ ( equivalence).

4. Quantifiers: ∀ ( forall), ∃ ( exists).

5. Parentheses and punctuation symbols: (, ) and ,.

[Tel12]

Definition 119 (Term). A term is defined inductively as follows:

1. Variable is term.

2. Constant is term.

3. If f is a function symbol ( f ∈ F ) with arity m and t1, t2, . . . , tm are terms of Σ, then f (t1, t2, . . . , tm) is term of Σ.

[Tel12]

Definition 120 (Atom). If p is predicate symbol with arity m and t1, t2, . . . , tm are terms of Σ, then p (t1, t2, . . . , tm) is an atomic formula (also called atom). An atomic formula is a formula and all occurrences of variables in an atomic

formula are free. [Tel12]

Definition 121 (Formula). A formula is defined as follows:

1. An atom is a formula.

2. If H and G are formulas, then:

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(a) ¬H is a formula, the occurrence of variables in ¬H is free respectively bound if it is free respectively bound in H ,

(b) H ∧G is a formula, the occurrence of variables in H ∧G is free respectively bound if it is free respectively bound in H or G,

(c) H ∨G is a formula, the occurrence of variables in H ∨G is free respectively bound if it is free respectively bound in H or G,

(d) H → G is a formula, the occurrence of variables in H → G is free respectively bound if it is free respectively bound in H or G,

(e) H ↔ G is a formula, the occurrence of variables in H ↔ G is free respectively bound if it is free respectively bound in H or G.

3. If H is a formula and x is a variable, then ∀x ∶ H and ∃x ∶ H are formulas. All occurrences of x are bound.

[Tel12]

Definition 122 (Literal). A literal L is an atom or the negation of an atom. [ Tel12 ]

Definition 123 (Clause). A clause is a formula such as ∀x ∶ L1 ∨ L2 ∨ . . . ∨ Lm where each Li is a literal and x =(x1, x2, . . . , xn) are all the variables occurring in L1 ∨L2 ∨ . . . ∨Lm. [ Tel12]

Definition 124 (Horn-clauses). Horn-clauses have the form: ∀x1, x2, . . . , xn ∶ L1 ∧L2 ∧ . . .∧Lm → L where L, L1 ∧L2 ∧

. . . ∧ Lm are a literals and x1, x2, . . . , xn are all variables having free occurrences in L, L1 ∧ L2 ∧ . . . ∧ Lm. [ Tel12]

Definition 125 (Negation of a dilemma, Conjunction of dilemmas, Disjunction of dilemmas). For a given set D of dilemmas it is defined:

1. A negation of a dilemma d = (u¬u) as a dilemma ¬d = (¬uu);

2. A disjunction of the dilemmas d′ = (u′¬u′), d′′ = (u′′¬u′′) as a dilemma d′ ∨ d′′ = [u′ ∨ u′′¬ (u′ ∨ u′′)];

3. A conjunction of the dilemmas d′ = (u′¬u′), d′′ = (u′′¬u′′) as a dilemma d′ ∧ d′′ = [u′ ∧ u′′¬ (u′ ∧ u′′)];

[Kul12]

Definition 126 (Equal certainty relation, Ambivalent dilemma, Equivalent dilemmas, Anti-equivalent dilemmas). Let Dbe a set of dilemmas. Then in D a binary relation ≈c satisfying the conditions of:

1. Reciprocity: for each d ∈ D it holds d ≈c d;

2. Symmetry: for any d′, d′′ ∈ D if d′ ≈c d′′ then also d′′ ≈c d′ holds;

3. Reflexivity: for any d′, d′′ ∈ D if d′ ≈c d′′ then also ¬d′ ≈c ¬d′′ holds;

4. Transitivity: for any d′, d′′, d′′′ ∈ D if d′ ≈c d′′ and d′′ ≈c d′′′ then also d′ ≈c d′′′ holds;

5. Fixation: for any d′, d′′ ∈ D if d′ ≈c ¬d′ and d′′ ≈c ¬d′′ then also d′ ≈c d′′ holds,

will be called an equal certainty relation. Any dilemma satisfying the condition d ≈c ¬d will be called an ambivalentdilemma; any dilemmas such that d′ ≈c d′′ holds will be called equivalent dilemmas; any dilemmas such that d′ ≈c ¬d′′

holds will be called anti-equivalent dilemmas. [ Kul12 ]

Definition 127 (Certainty ranking). Let D be a set of dilemmas with established equal certainty relation. Then a binary relation ⪯c described in D and satisfying the conditions of:

1. reciprocity: for each d ∈ D it holds d ⪯c d;

2. symmetry: for any d′, d′′ ∈ D d′ ⪯c d′′ and d′′ ⪯c d′ hold if and only if d′ ≈c d′′ holds;

3. anti-reflexivity: for any d′, d′′ ∈ D if d′ ⪯c d′′ then ¬d′′ ⪯c ¬d′ holds;

4. transitivity: for any d′, d′′, d′′′ ∈ D if d′ ⪯c d′′ and d′′ ⪯c d′′′ then also d′ ⪯c d′′′ holds,

will be called a certainty ranking. [Kul12]

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Theorem 18. Let D be a set of dilemmas with established equal certainty and certainty ranking relations. Then for any d′, d′′ ∈ D:

1. if d′ ⪯c d′′ and not d′′ ⪯c d′ then d′ ∨ d′′ ⪯c d′′;

2. if d′ ⪯c d′′ and not d′′ ⪯c d′ then d′ ∧ d′′ ⪯c d′;

3. if d′ ≈c d′′ then d′ ∨ d′′ ≈c d′ ∧ d′′ ≈c d′ ≈c d′′;

4. if d′ ≈c d′′ then d′ ∧ d′′ ≈c d′, d′′;

5. if d′ ≈c d′′ then d′, d′′ ≈c d′ ∨ d′′

[Kul12]

14.2 Logic Programming

Definition 128 (Domain declaration for predicate symbol p). A domain declaration for predicate symbol p of arity n is an expression of the following form.

domain p (a1, . . . , an) (14.1)

where ai is either h or d. When a is equal to h, this means that the i-th argument of p ranges over the Herbrand universe.Otherwise, it means that the i-th argument is a list of variables which ranges over d1 In the following, the domains di are

finite and explicit sets of values (i.e constants). [ Hen87]

Definition 129 (Domain set of a logic program). Let dl , . . . , dn the domains appearing in the domain declarations of a logic program P R and different from the Herbrand universe. We note D (P R) the set dd ≠ ∅∧ d ∈ 2di 1 ≤ i ≤ n We call it the domain set of the logic program. The domain set of a logic program contains all domains we possibly need during the computations. [ Hen87]

Definition 130 (Range of a term included in a domain). We say that the range of t is included in a domain dt

denoted t ∈ di if t is a constant ∈ dt or a d-variable xdt such that dt ⊆ d [Hen87 ]

Definition 131 (d-substitution). A d-substitution θ is a finite set of the form v1t1, . . . , vntn where

1. each vi is either a variable or d-variable

2. ti is a term distinct from vi,

3. v1, . . . , vn are all distinct,

4. if vi is a d-variable vdi , ti ∈ di

[Hen87]

Definition 132 (d-substitutions agree on a set of variables and d-variables). We say that two d-substitutions θ and λagree on a set V of variables and d-variables, denoted θ = λ V if and only if xθ = xλ for each x ∈ V where = denotes syntactic equality. [ Hen87 ]

Definition 133 (d-instance). θ is a d-instance of λ in V , denoted λ ≤ θ if and only if xθ = δ λ for some d-substitution δ . [ Hen87 ]

Definition 134 (d-unifier, unifies, more general d-unifier, d-mgu). A d-substitution σ is a d-unifier of some non-empty

and finite subset S = t1, . . . , tn where ti and a literal of a term if and only if t1σ = . . . = tnσ, we also say that σ unifiesS . U N I (S ) is the set of all d-unifiers of S . σ is called the more general d-unifier (also called d-mgu) of S if and only iffor each θ ∈ U N I (S ), θ ≤ σ vars (S ) implies σ ≤ θ vars (S ) where vars(S ) is the set of all variable or d-variable symbols in S . [ Hen87]

Definition 135 (Constraint). Let p be a n-ary predicate symbol, p is a constraint if and only iffor any ground terms either has a successful refutation or has only finitely failed derivations. [ Hen87 ]

Definition 136 (forward checkable literal, forward-variable). A literal p (x1, . . . , xn) in the resolvent is forward check-able if and only if p is a constraint and all it’s arguments are ground but one which is a d-variable. This d-variable is called the forward-variable. [Hen87 ]

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Definition 137 (forward checking inference rule). Let G1 be the goal ← A1, . . . , Am−1, P , Am+1, . . . Ak and P R be a logic program. Gi+1 is derived from Gi using the mgu θi+1 via P R if the following conditions hold:

1. P is forward checkable and xd is the forward variable

2. dnew = a ∈ dP R ⊧ P xa and dnew ≠ ∅

3. θi+1 is

(a)

xd

e

if dnew =

e

(b) xdzdnew where zdnew is a new variable otherwise.

4. Gi+1 is the goal ← (A1, . . . , Am−1, Am+1, . . . Ak)θi+1

Such a derivation rule is called the forward checking inference rule (sometimes abbreviated as FCIR). [Hen87 ]

Definition 138 (Efficient computation rule with respect to forward declarations). A computation rule is efficient withrespect to the forward declarations, if it selects only a predicate submitted to forward declaration when it is ground or forward checkable and if, whenever the resolvent contains literals submitted to a forward declaration which are either

forward-checkable or ground, it selects one of them. [ Hen87 ]

Definition 139 (lookahead checkable literal, lookahead-variable). A literal p (t1, . . . , tn) in the resolvent is lookaheadcheckable if and only if p is a constraint and their exists at least one ti which is a domain-variable and each other argument is either ground or a domain-variable. The domain-variables in t1, . . . , tn are called the lookahead-variables.[Hen87]

Definition 140 (lookahead inference rule). Let G1 be the goal ←A1, . . . , Am−1, P , Am+1, . . . Ak and P R be a logic program.Gi+1 is derived from Gi using the mgu θi+1 via P R if the following conditions hold:

1. P is lookahead checkable and x1, . . . , xn are the lookahead variables whose domains are dx1 , . . . dxn.

2. For each xdxjj let:

(a) dzj = yj ∈ dxj ∃y1 ∈ dx1 , . . . , yj−1 ∈ dxj−1 , yj+1 ∈ dxj+1 , . . . , yn ∈ dxn such that P R ⊧ P θ with θ = x1y1, . . . , xnynand dxj ≠ ∅.

(b) ej as

i. a new variable of domain dzj if dzj =

e1, . . . , el

> 1.

ii. the constant e if dzj if dzj = e.

3. θi+1 = x1z1, . . . , xnzn4. Gi+1 is the goal

(a) ← (A1, . . . , Am−1, Am+1, . . . Ak) θi+1 if at most one zi is a d-variable.

(b) ← (A1, . . . , Am−1, P , Am+1, . . . Ak)θi+1 otherwise.

Such a derivation rule is called the lookahead inference rule (sometimes abbreviated as LAIR). [Hen87 ]

Definition 141 (Efficient computation rule with respect to the lookahead declarations). A computation rule is efficientwith respect to the lookahead declarations, if a literal in the resolvent submitted to a lookahead declaration is only selected if either it is lookahead checkable or all its arguments are ground. [ Hen87]

Definition 142 (Revise(M, a)). Let a be a formula and M a model. Revise(M, a) is the set of models M ′ such that

1. M ′ and M have the same universe and agree on all functions.

2. a and the protected formulas of T are true in M ′.

3. There is no other model M ′′ such that for some 1 < i < 1,

(a) M ′′ satisfies ( 1) and ( 2 );

(b) M ′′ and M ′. agree on all predicates in strata 1 through i − 1; and

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(c) the differences between M” and M on predicates in stratum i are a proper subset of the differences between M ′

and M on those predicates.

[Win89]

Definition 143 (Preferred model at the i-th stratum). A model M1 is preferred to model M2 at the i-th stratum(written M1 <i M2) if and only if

1. M1 and M2 have identical universes;

2. M1 and M2 agree on all predicates and functions, except possibly those of S i and V i;

3. For all predicates P in stratum S t, and for all x such that P (x) is true in M1, P (x) is true in M2; and

4. For some predicate P in stratum S i and some x, P (x) is false in M1 and true in M2.

[Win89]

Definition 144 (Abductive problem, Background theory of the abductive problem, Abducible set of the abductive prob-lem, Goal of the abductive problem, Solution of an abductive problem, Minimal solution of an abductive problem). The triple ⟨Π, A , g⟩ is an abductive problem if and only if Π is a set of propositional Horn Clauses (called the backgroundtheory of the abductive problem), A a set of propositions (called the abducible set of the abductive problem)and g is a proposition (called the goal of the abductive problem). The set of propositions ∆ is a solution of theabductive problem

⟨Π, A , g

⟩if and only if

1. ∆ ⊆ A

2. ∆ ∪Π ⊢ g

3. ∆ ∪Π is consistent

4. (a ∈ A ∧∆ +Π ⊢ a)→ a ∈ ∆

∆ is a minimal solution of ⟨Π,A,g⟩ if and only if it is a solution of ⟨Π,A,g⟩ and no subset of ∆ is a solution of ⟨Π, A , g⟩. [ Esh93 ]

Definition 145 (Only-if set, only−if (T, S ), only−if (T ), props(T )). Let the clauses p←Q1, p←Q2, . . . , p ←Qk. where Q1, Q2, . . . , Ok are conjunctions of propositions, be all the clauses in Π which have p at their head. Let nc1 , nc2 ,...,nck be the

names of these clauses. Then the only-if set of p with respect to Π is ¬ p ∨ nc1 ∨ nc2 ∨ . . . ∨nck , nc1 →Q1, nc2 →Q2, . . . , nck

For at set of propositions S and the Horn clause theory T , we use only − if (T, S ) to denote the union of only-if sets of all the propositions in S with respect to T . We use only − if (T ) to denote only − if (T,props (T )) where props (T ) is the set of all propositions in T . [ Esh93 ]

Definition 146 (Truth-assignment, Model). Let C be a propositional clausal theory. S the set of propositions in C , and M a set of propositions. Then the truth-assignment induced by M is the assignment of true to all propositions in S

which are in M . and false to those propositions in S which are not in M . M is a model of C if and only if the truth assignment induced by M satisfies all the clauses in C . [ Esh93]

Definition 147 (Model which minimizes, Minimization with respect to). Let C be a propositional clausal theory. and A

a set of propositions. Then M is a model of C which minimizes A if and only if

1. M is a model of C

2. There is no other model M ′ of C such that M ′ ∩A ⊆ M ∩ A

We say that ∆ is a minimization of A with respect to C if and only if there is a model M of C which minimizes A

and ∆ = M ∩A [ Esh93]

Definition 148 (Unit refutable). A propositional clausal theory C is unit refutable if and only if, for every set of unit clauses U , if C ∪U is inconsistent. then the empty clause is unit-derivable from C ∪ U . [Esh93 ]

Definition 149 (Connection graph). Given a set of clauses C . the connection graph of C is the graph obtained by drawing a link between each complimentary pair of literals in the set. [ Esh93]

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Definition 150 (Chain). Given the set of clauses C , the sequence [(x1, e1, y1) , (x2, e2, y2) , . . . , (xn, en, yn)] of c-triples is a chain in C if and only if for all k, ek is a clause in C and yk = ¬xk+1. [ Esh93]

Definition 151 (Tied chain). [(x1, e1, y1) , (x2, e2, y2) , . . . , (xn, en, yn)] is a tied chain in C if and only if it is a chain in C and x1 = yn. [ Esh93]

Definition 152 (Subgoal clause). Let C be a set of clauses. Let T be a connection tableau for C . Let s1, s2, . . . , sn, n ≥ 1be the subgoals of T Then we call s1 ∨ s2 ∨ . . . ∨ sn the subgoal clause of T . [Fuc99 ]

Definition 153 (Query tableau, Query clause). Let C be a set of clauses. Let T be a connection tableau for C . Let S ⊆ C be a set of start clauses. Let S be the clause below the unlabeled root of T . If S is an instance of a clause from S we call T

a query tableau (with respect to S ) and the subgoal clause of T a query clause (with respect to S ). [ Fuc99]

Definition 154 (Lemma tableau, Lemma clause). Let C be a clause set. Let T be a connection tableau for C . Let C = s1 ∨ s2 ∨ . . . ∨ sn be the subgoal clause of T . Let H be the set of subgoals which are immediate successors of the root.If H ≠ ∅ we call T a lemma tableau. Then, let s+ i, 1 ≤ i ≤ n, be the element of H which is left-most in T . We call the contrapositive si ← ¬s1 ∧¬s2 ∧ . . . ∧ ¬si−1 ∧ ¬si+1 ∧ . . . ∧ ¬sn of C the lemma clause of T . [ Fuc99]

Definition 155 (Legal global instances). Given an open global system G = ⟨ϕ1, v1⟩ , ⟨ϕ2, v2⟩ , . . . , ⟨ϕn, vn⟩, the set of legal global instances is Linst (G) = D instance of Rvi ⊆ ϕi (D) , i = 1, . . . , n. [ BB03 ]

Definition 156 (Certain answer). Given an open global system G and a query Q (X ) to the system, a tuple t is a certainanswer to Q in G if for every global instance D ∈ Linst (G), it holds D ⊧Q [t]. We denote with CertainG (Q) the set of

certain answers to Q in G

. [BB03] Definition 157 (Minimal legal global instance, Minimal legal global instances). Given a global system, G, an instance D is minimal if D ∈ Linst (G) and is minimal with respect to set inclusion, i.e. there is no other instance in Linst (G)that is a proper subset of D (as a set of atoms). We denote by Mininst (G) the set of minimal legal global instancesof G with respect to set inclusion. [BB03]

Definition 158 (Consistent global system). A global system G is consistent with respect to IC . if for all DMininst (G),D ⊧ IC . [BB03 ]

Definition 159 (Minimal answer, Minimal answers). The ground tuple a is a minimal answer to a query Q posed toG if for every D ∈ Miminst (G), a ∈ Q (D), where Q (D) is the answer set for Q in D. The set of minimal answers is denoted by MinimalG (Q). [ BB03]

Definition 160 (Database distance). Let D, D′ be database instances over the same schema and domain. The distance∆ (D, D′), between D and D′ is the symmetric difference ∆ (D, D′) = (Σ (D) ∖Σ(D′)) ∪ (Σ (D′) ∖Σ (D)) [ BB03 ]

Definition 161 (Database order relation). For database instances D, D′, D′′, we define D′≤DD′′ if ∆ (D, D′) ⊆ ∆ (D, D′′)[BB03 ]

Definition 162 (Repair of a global system). Let G be a global system and IC a set of global IC ’s. A repair of G with respect to IC is a global database instance D′. such that D′

⊧ IC and D′ is ≤D-minimal for some D ∈ Mininst (G).[BB03 ]

Definition 163 (Consistent answer). Given a global system G, a set of global integrity constraints IC . and a global first-order query Q (X ). we say that a (ground) tuple t. is a consistent answer to Q with respect to IC if and only if for every repair D of G. D ⊧Q [t]. [BB03 ]

Definition 164 (Consistent answers). We denote by ConsisG

(Q

)the set of consistent answers to Q in G. [ BB03 ]

Definition 165 (Program of an open global system). Given an open global system G, the program, Π (G), contains the following clauses:

1. Fact dom (a) for every constant a ∈ U : and the fact V (a) whenever V (a) ∈ vi for some source extension vi in G.

2. For every View (source) predicate V in the system with description V X ← P 1 (X 1) ∧ P 2 (X 2) ∧ . . . ∧ P n (X n), the

rules P j (X j) ∧ ⋀Xi∈Xj∖X

F i X, X i, j = 1, . . . , n

3. For every predicate F i X, X i introduced in 2 , the rule F i X, X i← V X ∧ dom X ∧ choiceX, X i.

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[BB03 ]

Definition 166 (Instance associated to a stable model). The instance associated to a stable model M of Π (G) is DM = P (a) P ∈ R ∧ p (a) ∈M. [ BB03]

Definition 167 (Repair program). The repair program Π (G, IC ), of G with respect to IC contains the following clauses:

1. Facts as in definition 165 (item 1).

2. Each of the rules of item 2 in definition 165 is replaced by P j (X j , td)← V X ∧ ⋀Xi∈Xj∖XF i X, X i

3. Exactly the same rules as in item 3 in definition 165 .

4. For every predicate P ∈R, the clauses:

⎧⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎩

P X, t∗ ← P X, td ∧ dom X P X, t∗ ← P X, ta ∧ dom X P X, f ∗← P X, f a ∧ dom X P X, t∗ ← dom X ∧¬P X, td

(14.2)

5. For every first-order global universal IC of the form

∀ Q1 Y 1 ∨ Q2 Y 2 ∨ . . . ∨Qn Y n← P 1 X 1 ∧ P 2 X 2 ∧ . . . ∧ P m X m ∧ ϕ (14.3)

where P i, Qj ∈R, and ϕ is a conjunction of built-in atoms. The clause n

⋁i=1

P i (X i, f a) m

⋁j=1

Qj (Y j , ta)← n

⋀i=1

P i (X i, t∗) m

⋀j=1

Q

dom X ∧ ϕ where X is the tuple of all variables appearing in database atoms in the rule.

6. For every referential IC of the form ∀x (P (x)→ ∃yQ (x′, y)). with x′ ⊆, the clauses:

⎧⎪⎪⎪⎨⎪⎪⎪⎩P (X, f a) ∨ Q (X ′,null, ta)← P (X, t∗) ∧ aux X ′ ∧ ¬Q X ′,null, td ∧ dom X aux X ′←Q X ′, Y , td ∧¬Q (X ′, Y , f a) ∧ dom (X ′, Y )aux X ′←Q X ′, Y , ta ∧ dom (X ′, Y ) (14.4)

7. For every predicate P ∈R, the interpretation clauses:

⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩

P (a, f ∗ ∗)← P (a, f a)P (a, f ∗ ∗)← ¬P (a, td) ∧¬P (a, ta)P (a, t ∗ ∗)← ¬P (a, ta)P (a, t ∗ ∗)← ¬P (a, td) ∧ ¬P (a, f a)

(14.5)

[BB03 ]

Definition 168 (Instance associated to a choice model). The instance associated to a choice model M of Π (G, IC )is DM = P (a) P (a, t ∗ ∗) ∈M. [BB03 ]

Definition 169 (Normal logic program). A normal logic program P is a set of rules of the form:

a←

b1 ∧

b2 ∧

. . .∧

bn (14.6)Where a is an atom and the bi are in Lit ∼. [ BK93]

Definition 170 (Closure of a set of negative literals of a logic program). Let L ⊆ NEG (P ) be a set of negative literals,P a normal logic program. The closure of L under P , C P (L), is the smallest set such that:

1. L ⊆ C P (L),

2. if a← b1, b2, . . . bn ∈ P and b1, b2, . . . , bn ∈ C P (L) then a ∈ C P (L).

[BK93]

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Definition 171 (Normal logic program-consistent). Let L ⊆ NEG (P ) be it set of negative literals, P a normal logic program. L is P -consistent if and only if C P (L) is consistent. [BK93 ]

Definition 172 (Consistent normal logic program). Let P be a normal logic program. P is consistent if and only if ∅is P -consistent. [ BK93 ]

Definition 173 (Logic program cover of a closure). Let P be a logic program, H ⊆ NEG (P ), and C a P -closure of H .The P -cover of C , COV P (C ), is the set

H ∪ ¬a ∈ NEG (P ) a ∈ C (14.7)

[BK93]

Definition 174 (Extension of a logic program, Extension base of a logic program). Let P be a logic program, H ⊆ NEG (P ),and C a P -closure of H . C is an extension of P if and only if

1. C is consistent,

2. there is no H ′ ⊆ NEG (P ) with consistent closure C ′ such that COV P (C ) ⊊ COV P (C ′).

if C is an extension, H is called an extension base of P . [BK93 ]

Definition 175 (General logic program). A general logic program P is a set of rules of the form:

c1 ∨ c2 ∨ . . . ∨ cm ← b1 ∧ b2 ∧ . . . ∧ bn (14.8)

where all the literals ci and bj are in Lit +. [ BK93 ]

Definition 176 (General logic program closure of a set of literals). Let L be a set of literals from NEG (P ). C is a P -closure of L if it is a smallest set such that:

1. L ⊆ C ,

2. if l and ¬l or a and ¬a are in C , then C = Lit +

3. if c1 ∨ c2 ∨ . . . ∨ cm ← b1 ∧ b2 ∧ . . . ∧ bn ∈ P and b1, b2, . . . , bn ∈ C then for at least one ci, ci ∈ C .

[BK93]

Definition 177 (Extension of a default theory). Let Γ (S ) be any least set such that:

1. W ⊆ Γ (S )2. Γ (S ) is deductively closed

3. If α ∈ Γ (S ) and all β i are consistent with S , then one of γ i is in Γ (S ).

An extension of ⟨W, D⟩ is any set S that is equal to some Γ (S ). [ BK93]

Definition 178 (Cover with respect to a default theory). Let C be some Γ (S ) the cover COV∆ (C ) with respect to∆ = ⟨W, D⟩ is the set:

S ∪ C (14.9)

Where S are all the sentences not in S . [ BK93]

Definition 179 (Abductive extension of a default theory). Let C be some Γ (S ). C is an abductive extension of ∆if and only if

1. C ⊆ S ,2. COV ∆ (S ) is maximal.

[BK93]

Definition 180 (Moderately-grounded extension of a set of set of sentences). Let A be a set of sentences of L. Let U be a subset of L0 and U its complement L0 ∖ U . U is a moderately-grounded extension of A if and only if it satisfies the equation:

U = φ ∈ L0A ∪¬LU ⊢K45 φ (14.10)

[BK93]

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Definition 181 (Cover of an A theory under hypotheses). The cover of an AE theory A under hypotheses S ,COV A (C A (S )), is given by:

S ∪ ¬Lφφ ∈ C A (S ) (14.11)

[BK93]

Definition 182 (Abductive extension of an AE theory). The closure of an AE theory A, C A (S ), is an abductiveextension of A if and only if

1. C A

(S

)is consistent

2. COV A (C A (S )) is maximal.

[BK93]

14.3 Theorem Proving

Definition 183 (Loop in computation paths, Diamond in computation paths). A loop consists of two different compu-tation paths going from a term t to a term t′. A diamond is a loop which has either form I or II in figure 14.1. We illustrate this using ⋅ as the main function symbol. [Gei75]

s ⋅ r

s′ ⋅ r′

(a) Form I

s ⋅ rl

s′ ⋅ r′ + s′

s′ ⋅ r′l s ⋅ r + s

(b) Form II

Figure 14.1: Different forms of a diamond in computation paths

Definition 184 (Simple variant computation paths, Homologous computation paths). Two computation paths are simplevariants if they differ by a diamond, i. e. they look like figure 14.2. Two paths are homologous if there is a sequence of computation paths P 1, P 2, . . . , P n such that P 1 and P n are two paths in question and for i = 1, 2, . . . , n − 1, P i and P i+1

are simple variants. Note that homologous paths assign the same value to a term. [Gei75]

t

s

r

t′

Figure 14.2: Simple variant computation paths

Definition 185 (Abstract ingredients of a term, Simple ingredients of a term, Raw material in an ingredient, Controlelement in an ingredient). Let q 1, q 2, . . . , q n denote the occurrences of strokes in the term t. Ing (t) denotes the set of list expressions obtained from q 1, q 2, . . . , q n, and ⋅,exp, according to the following rules.

1. q 1, q 2, . . . , q n are in Ing (t)2. If i and j are in Ing (t) and f is ⋅ or exp, then i,j,f is in Ing (t).

The members of Ing (t) are called the abstract ingredients of t and are called the simple ingredients of t. Ing 0 (t) is the set of simple ingredients of t. In i,j,f , i is the raw material and j is called the control element. [Gei75]

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Definition 186 (Homorphism). A mapping H ∶ Ing (t)→ Ing (t) is a homomorphism (sometimes abbreviated as hom)if and only if for all i, j, f in Ing (t):

H i, j, f = H (i) , H j , f (14.12)

[Gei75]

Definition 187 (Real ingredient with respect to an evaluation path). Let P = t1, t2, . . . , tn be an evaluation path for t;thus t = tn. Define the homomorphism H [P ] = H (t1 → t2) H (t2 → t3) . . . H (tn−1 → tn) the set of real ingredientswith respect to P is the set H [P ] (Ing 0 (tn)) and is denoted Ing (t; P ). [ Gei75]

Definition 188 (Equivalence relation for ingredients). Let i and j be ingredients in Ing (t). We define i ≈ j if and only if there is a sequence of pairs i

1, j1 , i

2, j2 , . . . , in, j

n of ingredients of Ing (t) such that i = in, j = j

nand for

k = 1, 2, . . . , n either:

1. ik = jk

2. there exists ingredients u, v, w and function symbols f , g such that ik, jk equals ⟨⟨u,v,f ⟩ , w , g⟩ , ⟨⟨u,w,g⟩ , ⟨v,w,g⟩ ,

or

3. there exists k′, k′′ less than k such that ik = ⟨ik′ , ik′′ , f ⟩ and jk= j

k′, j

k′′, f .

[Gei75]

Definition 189 (Skeptical explanation of a literal, Minimal explanation of a literal) . Let P be a consistent Extended Logic Program, and O a literal. Suppose that Ψ is a set of candidate priorities on L p. A set ψ of priorities is a (skeptical)

explanation of O (with respect to P, Ψ) if

1. ψ ⊆ Ψ, and

2. P ⊧ψ O.

Also, ψ is a minimal explanation of O if no ψ′ ⊂ ψ is an explanation of O. [ IS99]

Definition 190 (Preference abduction framework, Skeptical explanation of a literal, Minimal explanation of a literal) .A preference abduction framework is a triple ⟨P, Γ, Ψ⟩ where P is an Extended Logic Program, Γ ⊆ LP is a set of abducibles, and Ψ is a set of candidate priorities on LP . A pair ⟨A, ψ⟩ is a (skeptical) explanation of a literal O

(with respect to ⟨P, Γ, Ψ⟩) if

1. A ⊆ Γ,

2. ψ ⊆ Ψ,

3. P ∪ A is consistent, and

4. P ∪ A ⊧ψ O

Also ⟨A, ψ⟩ is a minimal explanation of O if for any explanation ⟨A′, ψ′⟩ of O, A′ ⊆ A and ψ′ ⊆ ψ imply A′ = A and ψ′ = ψ. [ IS99 ]

14.3.1 Proof Normalization

Definition 191 (Discharged assumptions by the rule). At applications of ⊃ I , ∃E and IND rules, certain assumptions are said to be “ discharged” by the rule. The conclusion of a proof is said to depend on the assumptions that have not been discharged. [Got85 ]

Definition 192 (Proper parameter of the inference rule). In applications of the rule ∀I , ∃E and IND, the variable “a”

(see figure 14.3 ) is called the proper parameter of the inference rule. [ Got85 ]

Definition 193 (Normal proof). A proof Π is said to be normal, if no reduction rule is applicable to Π. [ Got85 ]

Definition 194 (Harrop formula class). The class of Harrop formulas is defined inductively as follows:

1. Every atomic formula is a Harrop formula

2. If A and B are Harrop formula’s, then A ∧ B and ∀x (A (x)) are Harrop formulas.

3. If B is a Harrop formula, then A ⊃ B is a Harrop formula, regardless of the form of A.

[Got85 ]

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A B∧I

A ∧B

A ∧ B∧E

A

B

[A]A ⊃ B

⊃ I A A ⊃ B

⊃ E B

A (a)∀x (A(x)) ∀I

∀x(A (x))∀E

A (a)A (t)

∃x (A (x)) ∃I ∃x(A (x)) C

[A (a)]∃E

C

A (0) A(s (a))[A (a)]IND

A (t)Figure 14.3: Inference rules in natural deduction.

14.4 Critical Reasoning

Definition 195 (Critical Environment). The critical environment for node n relative to S ⊆ A is:

S

(n

)=⋂

E ∈ E

(n

) E ⊆ S

(14.13)

[RdKS93 ]

Definition 196 (Critical Conflict). The critical conflict for S ⊆ A is:

S () =⋂ C a conflict C ⊆ S (14.14)

[RdKS93 ]

Definition 197 (Critical abstraction of a theory). The critical abstraction of a theory T relative to S ⊆ A is the theory:

S (T ) = T ∪ S (n)→ nn ∈ L ∪ (14.15)

[RdKS93 ]

Definition 198 (Critical cover). A critical cover relative to a subset S of A is a set S 1, S 2, . . . , S k of subsets of S such that:

1. Critical consistency: S i () is non-empty, for all i.

2. Covering: Every conflict C ⊆ S is subsumed by S i (), for some i.

3. Disjointness S i () is disjoint from S j, far every distinct i and j.

[RdKS93 ]

Definition 199 (Critical diagnosis). A critical diagnosis of T relative to a S ⊆ A is a set ∆∩ S such that:

1. ∆ is a diagnosis of S (T ).

2. There is no other diagnosis ∆′ of S

(T

), such that ∆′ ∩ S is a strict subset of S ⊆ A.

[RdKS93 ]

14.5 Argumentation

Definition 200 (Argumentation framework, Arguments in an argumentation framework). An argumentation frame-work is a pair AF = ⟨A,R⟩ where A is a set of arguments, and R is a binary relation representing a defeat relationshipbetween arguments, i.e. R ⊆ A ×A. (A, B) ∈ R or equivalently ”ARB” means that argument A defeats argument B. We also say ”A attacks B”. Here an argument is an abstract entity whose role is only determined by its relation to other arguments. [Dun93]

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D R A F T

Definition 201 (Acceptable argument with respect to a set of arguments,Admissible conflict-free set of arguments). 1An argument A is acceptable with respect to a set S of arguments if and only if for each argument B: if B

attacks A then B is attacked by some argument in S .

2. A conflict-free set of arguments S is admissible if and only if each argument in S is acceptable with respect to S .[ Dun93]

Definition 202 (Maximal preferred extension of an argumentation framework). A preferred extension of an argu-mentation framework AF is a maximal (with respect to set inclusion) admissible set of arguments of AF . [ Dun93 ]

Definition 203 (Stable extension). A conflict-free set of arguments S is called a stable extension if and only if S attacks each argument which does not belong to S . [ Dun93 ]

Definition 204 (Characteristic function of an argumentation framework). The characteristic function of an ar-gumentation framework AF = ⟨A,R⟩ denoted by F AF , is defined as follows: F AF ∶ 2A → 2A where F AF (S ) =AA is acceptable with respect to S . [Dun93 ]

Definition 205 (Grounded extension of an argumentation framework). The grounded extension of an argumenta-tion framework AF , denoted by GAF , is the least fixed point of F AF . [ Dun93]

Definition 206 (Complete extension). An admissible set of arguments S is called a complete extension if and only if each argument which is acceptable with respect to S , belongs to S . [Dun93 ]

14.6 Explanation Based LearningDefinition 207 (Explanation structure, Internal structure of an explanation structure, External structure of an explanat

For a horn clause C = A ← B1 ∧ B2 ∧ . . . ∧ Bn let it’s equivalent copy be C ′ = A′← B′

1∧ B′

2∧ . . . ∧ B′

n. Then the following expression is an explanation structure.

A′∶ (A← (B1 ∶ B′

1) ∧ (B2 ∶ B′

2) ∧ . . . ∧ (Bn ∶ B′

n)) (14.16)

C and C ′ are called its internal and external structure respectively.

2. For an explanation structure T , let C ′ be its external structure and σ be a substitution. Then, the expression T σ,which has the same internal structure as T and has the external structure C ′σ is called an instantiation of T .Inversely, T is called an uninstantiation of T σ.

3. For two explanation structures S and T , let A′

← B′

1 ∧ B′

2 ∧ . . . ∧ B′

n and B′

i ← C ′

1 ∧ C ′

2 ∧ . . . ∧ C ′

m be their external structures respectively. Then, the expression U connected S and T at B′

i is called a (binary) resolvent of S andT . Inversely S is called the unresolution of U and T is called the remainder.

4. An explanation structure S is more general than T denoted by S ⪯ T if and only if there exists a sequence of uninstantiations and unresolutions from T to S .

[ YK91 ]

Definition 208 (Obliged generalization of an explanation structure, EBG, EBG Macro). 1. An explanation T is an obliged generalization, if and only if the internal structure of T does not include any facts of a certain example and its external structure is a definite clause.

2. Obliged generalizations of an explanation for some example are called EBG. For an EBG, its external structure is called an EBG macro.

[ YK91 ]

Definition 209 (More general explanation, Generalization space). 1. For an explanation T ∈ T , let T 0 be an unin-stantiation of T . Then T ′ = (T ∖ T ) ∪ T 0 is an uninstantiation of T . Similarly let T 0 be an unresolution and T 0, T 1, . . . , T n are its remainders. Then T ′ = (T ∖ T ) ∪ T 1, T 2, . . . , T n is an unresolution of T .

2. An explanation S is more general than T , denoted by S < T if and only if there exists a sequence of uninstantiations and unresolutions from T to S .

3. A class of all generalizations of the obliged general of an example set is called a generalization space.[ YK91 ]

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Definition 210 (Common EGB, Least EBG). For EBGs T 1, . . . , T n, T is a common EBG if and only if T ⪯ T 1, T ⪯T 2, . . . , T ⪯ T n. The least generalization in common EBGs is called the least EBG. [YK91]

Definition 211 (Least EGB generalization of an explanation, More general by least EBG, Least EBG space). 1. For S ⊆ T , let T 0 be the least EBG of S and T 1, T 2, . . . , T n be its remainder. Then T ′ = (T ⊆ S ) ∪ T 0, T 1, . . . , T n is called a least EBG generalization of T .

2. S is more general by least EBG than T if and only if there exists a sequence of least EBG generalizations from T to S .

3. A class of all least EBG generalizations of the obliged generalization of an example set is called its least EBGspace.

[ YK91 ]

Definition 212 (Usage degree, Backtrack number B (T ,E ), Operationally criteria for generalizations). 1. For a gen-eralization S = (T ∖ T ) ∪ T 0, T 1, . . . , T n, its usage degree is a function such that

U (S ,T ) = n

i=0

T i if T i ∈ T

0 otherwise (14.17)

Where T i denotes the number of clauses used in T i. In general, when S ⪯ T , the usage degree sums up all usage degrees through some generalization path.

2. For an example set E and its generalization T , the backtrack number B (T ,E ) is the minimum of the sum of goal failures which occur during a macrotable of T reconstructs all explanations of E .

3. For two generalizations S and T of an example set E , S is more operational than T if and only if and U (S ,E ) ≥ U (T .E ) and B (S ,E ) ≥ B (T .E ).

[ YK91 ]

14.7 Transformation Problems

14.7.1 Termination

Definition 213 (type-1-separated subset). S 1 is a type-1-separated subset of S if and only if for all f ∈ F and all s ∈ S 1 ∩S f , f

(s

)∈ S 1. [ Flo75 ]

Definition 214 (type-2-separated subset). S 1 is a type-2-separated subset of S if and only if for all f ∈ F and all s ∈ (S ∖S 1) ∩ S f , f (s) ∈ S ∖S 1. [ Flo75 ]

Definition 215 (type-3-separated subset). S 1 is a type-3-separated subset of S if and only if S 1 is both a type-1-separated subset and type-2-separated subset of S . [Flo75]

Definition 216 (Consistent type-k-system of subsets). γ is a consistent type-k-system of subsets of S if and only if for all s1 ∈ S , s2 ∈ S the statement defined by decision rule k is true. [Flo75 ]

Definition 217 (admissible type-k-system of subsets). γ is an admissible type-k-system of subsets of S if and only if for all s1, s2 ∈ S , the following is true: (s1, s2) is solvable if and only if for all S i ∈ γ holds condition C k where C 1, C 2and C 3 are defined as follows:

C 1 s1 ∈ S i implies s2 ∈ S i.

C 2 s2 ∈ S i implies s1 ∈ S i.

C 3 s1 ∈ S i if and only if s2 ∈ S i.

[Flo75 ]

Definition 218 (Cycle within an operator set of a conjunctive problem space). Within the operator set F of a conjunctive problem space, there is a cycle with respect to the relation ≻ if and only if there are operators f i1 , f i2 , . . . , f il such that f i1 ≻ f i2 ≻ . . . ≻ f il . [Flo75]

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14.8 Epistemic State

Definition 219 (Epistemic state, Admissible belief states). An epistemic state £ is a triple ⟨S,≺, l⟩, where S is a set of objects called admissible belief states, ≺ is a preference relation on S , while l is a labeling function assigning each admissible state a deductively closed theory. [Boc99]

Definition 220 (Sceptically valid in an epistemic state). A conditional A ∼ B will be said to be sceptically valid in anepistemic state £ if all preferred states in (A) support A→B. [Boc99 ]

Definition 221 (Credulously valid in an epistemic state). A conditional A

∼ B

will be said to be credulously valid inan epistemic state £ if either (A) is empty or at least one preferred state in (A) supports A→B. [Boc99]

Definition 222 (Credulous nonmonotonic inference relation, Rational monotony). A nonmonotonic inference relation will be called credulous if it is regular and satisfies Rational Monotony:

(RM ) If A ∼ B and A ∼ C , then A ∧ C ∼ B (14.18)

[Boc99 ]

Definition 223 (Permissive inference relation, Cut rule, Or rule). An inference relation will be called permissive if it satisfies the basic postulates and the Cut rule:

(Cut ) If A ∼ B and A ∧ B ∼ C , then A ∼ C (14.19)

It can be shown that, in the context of B, Cut implies the rule Or:

(Or ) If A ∼ C and B ∼ C , then A ∨ B ∼ C (14.20)

[Boc99 ]

Definition 224 (Permissively valid conditional in an epistemic state). A conditional A ∼ B will be said to be permissivelyvalid in an epistemic state £ if any preferred state in (A) is consistent with A ∧ B. [ Boc99 ]

14.9 Temporal Reasoning

Definition 225 (Specification of a document). A specification s = ⟨O, C ⟩ of a document is made of a set of objects O

and a set of constraints C between these objects (i.e., a relation between several objects). The set of all specifications will be noted S . [ELD03 ]

Definition 226 (Relation graph of a specification). A specification s = ⟨O, C ⟩ can be represented as a complete direct labeled graph called a relation graph gs = ⟨N, E , λ⟩ such that the elements of O are in bijection with those of N and λ ∶ E → 2A13 is a total function from the arcs to temporal relations such that for each x r y ∈ C , λ (⟨x, y⟩) ⊆ r. [ ELD03]

Definition 227 (Resolved relation graph of a specification). . A relation graph is resolved if and only if all the labels are singletons. [ELD03]

Definition 228 (Interpretation of a specification). An interpretation of a specification is a pair ⟨I, D⟩ such that D

is the domain of interpretation and I is a function from O to D and from C to D×D (i.e., such that a constraint applied to two elements of the domain of interpretation is either true of false). [ELD03]

Definition 229 (Temporal interpretation of a specification). in order to interpret the temporal aspects of multimedia documents, we consider the interpretations such that the objects in O are interpreted as intervals of the positive real numbers and the constraints are interpreted as the corresponding relations in the temporal interval algebra. [ELD03]

Definition 230 (Model of a specification). A model of a specification ⟨O, C ⟩ is an interpretation ⟨I, D⟩ such that for each o r o′ ∈ C . ⟨I (o) , I (o′)⟩ ∈ I (r) is true. The set of models of a specification s is noted Ms. [ ELD03]

Definition 231 (Qualitative representation of a model of a specification). . The qualitative representation of amodel ⟨I, D⟩ of a specification ⟨O, C ⟩ is a complete direct labeled graph ⟨N, E , λ⟩ such that the elements of O are in bijection with those of N and λ ∶ E → 2A13 is a total function from the arcs to temporal relations such that for each I (x) r I (y), λ (⟨x, y⟩) = r. [ ELD03]

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Definition 232 (Adaptation constraint). . An adaptation constraint a determines a set of possible executions M a.The set of adaptation constraints will be noted A. [ ELD03 ]

Definition 233 (Classification of adaptation, Compliant specification, Refining adaption, Transgressive adaption). Three types of adaptation can be identified in function of the value of Ms∩M p (inducing three different constraints on the model selection function α):

Compliant specification Ms ∩M p =Ms,: the source document satisfies the adaptation constraints (all models of Ms,satisfy the adaptation constraints. so α is identity).

Refining adaptation ∅ ⊂Ms∩M p ⊂Ms: there exists some models of s satisfying the adaptation constraints ( α (Ms) =Ms ∩M p).

Transgressive adaptation Ms ∩M p = ∅: no model of s satisfies the adaptation constraints ( α will then select some models of M p, closest to those of the specification s).

[ELD03]

Definition 234 (Distance between sets of models). The distance between sets of models is defined as:

∆ (M,M′) = F m∈M,m′∈M′d (m, m′) (14.21)

[ELD03]

Definition 235 (Distance between resolved relation graphs). The distance between resolved relation graphs is defined as:

d (λ, λ′) = n,n′∈N

1 if λ (⟨n, n′⟩) ≠ λ′ (⟨n, n′⟩)0 otherwise

(14.22)

[ELD03]

Definition 236 (Distance between interval relations based on endpoints). The distance between interval relationsbased on endpoints is defined as:

δ

(r, r′

)=

4

i=11 if γ

(r

) [i

]≠ γ

(r′

) [i

]0 otherwise (14.23)

[ELD03]

Definition 237 (Distance between models based on endpoints). The distance between models based on endpointsis defined as:

d (λ, λ′) = n,n′∈N

δ (λ (⟨n, n′⟩) , λ′ (⟨n, n′⟩)) (14.24)

[ELD03]

Definition 238 (Conceptual neighborhood relation). The conceptual neighborhood relation is a binary relation N XΓ

between elements of a set of relations Γ such that N XΓ

(r, r′) if and only if the continuous transformation X of a situation

involving two individuals x and y can transform r (x, y) into r

′

(x, y) without transiting by a third relation. [ELD03]

Definition 239 (Conceptual distance between two relations). The conceptual distance δ ′ between two relations is the length of the shortest path between r and r′ in the graph of N X

Γ. [ELD03 ]

Definition 240 (Conceptual distance between models). The conceptual distance between models is defined as:

d (λ, λ′) = n,n′∈N

δ ′ (λ (⟨n, n′⟩) , λ′ (⟨n, n′⟩)) (14.25)

[ELD03]

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14.10 Model Ranking

Definition 241 (Robust with respect to a set of defaults). An integer ranking κ, over models is said to be robust withrespect to a set of defaults ri, with associated strengths si, if no two defaults share a common minimal falsifying model in κ. [BP99 ]

Definition 242 (Me-valid integer ranking). An integer ranking κ, over α set of defaults ri, with associated strengths si, is said to be me-valid with respect to that set if it satisfies

κ (m) = rim⊧ai∧¬bi

κ (ri) (14.26)

and for all r

κ (vr) + sr = κ (f r) (14.27)

[BP99 ]

1

Definition 243 (Distinct me-valid integer rankings). Two me-valid rankings, κ and κ′ are said to be distinct if and only if κ (r) ≠ κ′ (r) for some default r. Such a default is said to be distinctly ranked. [ BP99]

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Chapter 15

Multiagent Systems

Definition 244 (Sequence of decision elements, decision element, participation, positive set, neutral set, negative set,rate of return, degree of secure of rate Z , date of knowledge). The structure decision P of finite set of decision elements E = e1, e2, . . . , eY is called as sequence of decision elements: P = ⟨EW + ,EW ± ,EW − ,Z,SP,DT ⟩, Where:

1. EW + = ⟨e0, pe0⟩ , ⟨eq, peq⟩ , . . . , ⟨e p, pe p⟩; couple ⟨ex, pex⟩, where ex ∈ E and pex ∈ [0, 1]. Denote a decision elementand participation this element in set EW +; decision element ex ∈ EW + will be denoted as e+x; The set EW + is called positive set, in other words it is a set of decision elements, about which the agent knows that these elements

are in the environment.

2. EW ± = ⟨er, per⟩ , ⟨es, pes⟩ , . . . , ⟨et, pet⟩; couple ⟨ex, pex⟩, where ex ∈ E and pex ∈ [0, 1]. Denote a decision element and participation this element in set EW +; decision element ex ∈ EW + will be denoted as e±x; The set EW ± is called neutral set, in other words it is a set of decision elements, about which the agent does not know that these elements are in the environment.

3. EW − = ⟨er, per⟩ , ⟨es, pes⟩ , . . . , ⟨et, pet⟩; couple ⟨ex, pex⟩, where ex ∈ E and pex ∈ [0, 1]. Denote a decision element and participation this element in set EW −; decision element ex ∈ EW + will be denoted as e−x; The set EW ± is called a negative set, in other words it is a set of decision elements, about which the agent knows that these elements are not in the environment.

4. Z ∈ [0, 1] – rate of return

5. SP ∈ [0, 1] – degree of secure of rate Z

6. DT – date of knowledge.

[SKH12]

Definition 245 (Profile). Set of decision elements E = e1, e2, . . . , eY is given. A profile A = A(1), A(2), . . . , A(M ) is called set of M decisions of finite set of decision elements E , such that:

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

A(1) = EW +(1) ,EW ±(1) ,EW −(1) , Z (1), SP (1), DT (1)A(2) = EW +(2) ,EW ±(2) ,EW −(2) , Z (2), SP (2), DT (2)

⋮

A(M ) = EW +(M ),EW ±(M )

,EW −(M ), Z (M ), SP (M ), DT (M )

(15.1)

[SKH12]

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Chapter 16

Neural and Evolutionary Computing

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58



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Chapter 17

Programming Languages

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60



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Chapter 18

Robotics

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62



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Chapter 19

Symbolic Computation

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64



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Part III

Interaction, Graphics, Sound andMultimedia

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Chapter 20

Computers and Society

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68



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Chapter 21

Graphics

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Chapter 22

Human-Computer Interaction

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Chapter 23

Multimedia

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Chapter 24

Sound

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Part IV

Engineering

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Chapter 25

Computational Engineering, Finance and

Science

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Chapter 26

Hardware Architecture

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82



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Chapter 27

Operating Systems

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84



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Chapter 28

Performance

85



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86



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Chapter 29

Software Engineering

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Chapter 30

Systems and Control

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Part V

Internet, Databases, Networking andSocial Media

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Chapter 31

Cryptography and Security

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Chapter 32

Databases

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Chapter 33

Digital Libraries

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Chapter 34

Distributed, Parallel and Cluster

Computing

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Chapter 35

Information Retrieval

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Chapter 36

Networking and Internet Architecture

36.1 Testing Web Applications

Definition 246 (Web application). A web application A is a tuple ⟨W,S,I,A,W 0, W , M , T ⟩, where:

1. S , I , A are disjoint finite sets of state, input, and action values. Constant values can be shared among these sets.

2. W is finite set of the representations of web pages,

3. W 0 ∈ W is the representation of home page, W ∉ W is the the representation of error page,

4. M is a set of business metadata values,

5. T is a set of technical (non-model) metadata values

In the testing process tester has to pass through all the test case scenario steps defined in selected test case scenario. Aspecification of an interaction of the tester with system under test is defined in each of test case scenario step. Instructions describing the test case scenario step are defined clearly without ambiguities. For brevity we will limit the user interaction with the SUT to only one action that is captured in the test scenario step. [ FBJ12]

Definition 247 (Test case scenario). A test case scenario is a finite sequence ⟨V i, S i, I i, Ai, Ri, T Di⟩i≥0 where V i is a web page page representation used in step i, S i ⊆ S is set of state values used in step i, I i ⊆ I is set of values of input

elements used in step i, Ai ∈ A is an action taken in step i, Ri is set of restricting constraints rules, T Di ⊆ S i∪

I i∪

M ∪

T is finite set of state, input, business and technical metadata values. [FBJ12 ]

Definition 248 (Inbound client-side constraint rule). An inbound client-side constraint rule is a function φin ∶ I →0, 1. [ FBJ12]

Definition 249 (Inbound server-side constraint rule). An inbound server-side constraint rule is a function Φin ∶S ∪ I ∪M ∪T → 0, 1. The outbound server-side constraint rules are the rules validating the state on the server side, any metadata and database right after the server action is executed and before the data is sent back to client. [ FBJ12 ]

Definition 250 (Outbound server-side constraint rule). An outbound server-side constraint rule is a function Φout ∶ S ∪ I ∪ M ∪ T → 0, 1. [FBJ12 ]

Definition 251 (Passed test case scenario). Test case scenario step i with configuration ⟨V i, S i, I i, Ai, Ri, T Di⟩ where V iis a web page page representation used in step i, S

i⊆ S is set of state values used in step i, I

i⊆ I is set of values of input

elements used in step i, Ai ∈ A is an action taken in step i, Ri is set of restricting constraints rules, T Di ⊆ S i ∪ I i ∪M ∪T

is finite set of state, input, business and technical metadata values used in step i, is called passed when for all rule constraints f ∈ Ri:

1. if f is client-side inbound rule, then f ( j) = 1 where j ∈ I i

2. if f is client-side outbound rule or server-side inbound rule, then f (k) = 1 where k ∈ T Di

3. V i+1 is not error web page W

4. execution of action Ai results in success

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[FBJ12 ]

Definition 252 (Test scenario). Let W = ⟨W,S,I,A,W 0, W , M , T ⟩ be a web application. A test scenario of W is an finite sequence of test case scenario step configurations ⟨V i, S i, I i, Ai, Ri, T Di⟩i≥0 where V i is a web page page represen-tation used in step i, S i ⊆ S is set of state values used in step i, I i ⊆ I is set of values of input elements used in step i,Ai ∈ A is an action taken in step i, Ri is set of restricting constraints rules, T Di ⊆ Si ∪ I i ∪ M ∪ T is finite set of state,input, business and technical metadata values used in step i, we say that given test case scenario is passed when each test scenario step K i is passed for each i ≥ 0. Our target is to find errors left in the system that were not discovered by the unit testing. To have higher confidence about the correctness of the implementation of the system the SUT must be

covered by our test case scenarios–every single action the user can execute must be used in at least one test case scenariostep and this test case scenario must be successfully completed by the tester without problems. [ FBJ12]

Definition 253 (Covered test scenario step). A test scenario step K i with test case scenario step configuration ⟨V i, S i, I i, Ai,

where V i is a web page page representation used in step i, S i ⊆ S is set of state values used in step i, I i ⊆ I is set of val-ues of input elements used in step i, Ai ∈ A is an action taken in step i, Ri is set of restricting constraints rules,T Di ⊆ Si ∪ I i ∪ M ∪ T is finite set of state, input, business and technical metadata values used in step i, then action Ai ,respectively input element I i , respectively action Ai , is covered by K i when K i is passed. [ FBJ12]

Definition 254 (Test case action verification ratio). Given the finite set A of action symbols of the web application A,then

averify =Averified

A

(36.1)

is called test case action verification ratio of the SUT and

Averified = ⋃T ∈E

⋃τ ∈T

aa ∈ Ai ∧ τ is passed (36.2)

is the finite set of actions covered by all passed test scenarios. E is the set of executed test case scenarios, T denotes test case scenario test, is i-th single step in test case scenario with configuration ⟨V i, S i, I i, Ai, Ri, T Di⟩. We then use Ra = 100% ⋅ averify to express the SUT test case action verification percentage. The test case state, respectively

input, verification ratio of the SUT are defined in a similar way. [ FBJ12]

Definition 255 (Test case state verification ratio). Given the finite set S of state symbols of the web application A, then

sverify

=S verified

S (36.3)

is called test case state verification ratio of the SUT and

S verified = ⋃T ∈E

⋃τ ∈T

ss ∈ S i ∧ τ is passed (36.4)

is the finite set of states covered by all passed test scenarios. E , T and τ are defined as in Definition 254. [FBJ12 ]

Definition 256 (Test case input verification ratio). Given the finite set I of input symbols of the web application A, then

iverify =I verified

I (36.5)

is called test case input verification ratio of the SUT and

I verified = ⋃T ∈E

⋃τ ∈T

ii ∈ S i ∧ τ is passed (36.6)

is the finite set of states covered by all passed test scenarios. E , T and τ are defined as in Definition 254. [FBJ12 ]

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Chapter 37

Social and Information Networks

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Part VI

Emerging Technologies and Other

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Chapter 38

General Literature

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Chapter 39

Emerging Technologies

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112



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Chapter 40

Other

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114



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Part VII

Appendices

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List of Figures

14.1 Different forms of a diamond in computation paths . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4714.2 Simple variant computation paths . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4714.3 Inference rules in natural deduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

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List of Tables

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Index

X -of-N representation, 38ε-closure(M), 19d-instance, 41d-mgu, see more general d-unifierd-substitutions agree on a set of variables and d-variables,

41d-unifier, 41”best” path, 17(Variable) preference function, 36(variable) preference pair, 36*CWA clause inference, 30

*CWA literal inference, 30O (g), 5Revise (M, a), 42U N I (S ), 41Ω (g), 5Θ (g), 5ω (g), 5o (g), 5only − if (T, S ), 43only − if (T ), 43

props (T ), 43DTIME, 5

NP-complete, 5NP-hard, 5NP, 5NTIME, 5P, 5coNP, 5

abducible set of the abductive problem, 43Abductive extension of a default theory, 46Abductive extension of an AE theory, 47abductive problem, 43Abstract ingredients of a term, 47Acceptable argument with respect to a set of arguments, 50

accepting states, 19adaptation constraint, 53admissible belief states, 52Admissible conflict-free set of arguments, 50admissible type-k-system of subsets, 51alphabet, 39ambivalent dilemma, 40ancestors, 17anti-equivalent dilemmas, 40argumentation framework, 49

Arguments in an argumentation framework, 49arity, 39associative, 11atom, see atomic formulaatomic formula, 39automorphism, 12

B-smooth integer, 12background theory of the abductive problem, 43backtrack number B (T ,E ), 51Bayes’ rule, 14

belief change extension, 29belief change scenario, 29belief set corresponding to, 29better position, 35bi-quasi-associative, 11Binary resolvent of two explanation structures, 50Blake formula, 30business metadata, 103

Careful Closed World Assumption-free, 30CCW A-free, see Careful Closed World Assumption-freecertain answer, 44certainty ranking, 40

chain, 44characteristic function of an argumentation framework, 50class of all belief change extensions, 29clause, 40clique, 17cliquo, 17Closed World Assumption-free, 30Closure of a knowledge base, 29Closure of a set of negative literals of a logic program, 45coefficient, 13Collection of variables, 39common divisor, 12common EBG, 51

common multiple, 12commutative, 11commutative group, 11complement language, 5complete extension, 50complete subset of efficient paths, 18Compliant specification, 53computes f , 5computes f in T (n)-time, 5conceptual distance between models, 53

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Conceptual distance between two relations, 53conceptual neighborhood relation, 53conditional independence, 15conditional probability, 14conditionally dependent, 15congruent to b modulo n, 13conjunction, 39conjunction of the dilemmas, 40connection graph, 43consistent answer, 44consistent answers, 44Consistent global system, 44Consistent normal logic program, 46consistent type-k-system of subsets, 51constant symbols, 39constraint, 41coset, 12Cover of an A theory under hypotheses, 47Cover with respect to a default theory, 46covered, 104Covering in critical cover, 49

Credulous nonmonotonic inference relation, 52credulously valid in an epistemic state, 52critical abstraction of a theory, 49critical conflict, 49Critical consistency in critical cover, 49critical cover, 49critical diagnosis, 49critical environment, 49Cut rule, 52CW A-free, see Closed World Assumption-freeCycle within an operator set of a conjunctive problem space,

51

DAG, see Directed Acyclic Graphdatabase distance, 44Database order relation ≤D, 44date of knowledge, 55decision element, 55decision rule, 36defeasible statement, 29definiendum, viidefiniens, viidefinition, viidegree, 13degree of functions, 39degree of secure of rate Z , 55

descendants, 17Diamond in computation paths, 47Directed Acyclic Graph, 17directed graph, 17Discharged assumptions by the rule, 48Disjointness in critical cover, 49disjunction, 39Disjunction normal form formula, 30disjunction of the dilemmas, 40distance between interval relations based on endpoints, 53

distance between models based on endpoints, 53distance between resolved relation graphs, 53distance between sets of models, 53Distinct me-valid integer rankings, 54DNF formula, see Disjunction normal form formuladomain axiom, 29domain declaration for predicate symbol p, 41domain set of the logic program, 41

EBG, 50EBG macro, 50ECW A-free, see Extended Closed World Assumption-freeedges, 17EF, see evaluation functionEfficient computation rule with respect to forward declara-

tions, 42efficient computation rule with respect to the lookahead dec-

larations, 42efficient path, 18EGCW A-free, see Extended Generalized Closed World Assum

free

empty input, 35epistemic state, 52epsilon-transition, 19equal certainty relation, 40equivalence, 39Equivalence relation for ingredients, 48equivalent dilemmas, 40equivalent efficient paths, 18error page, 103evaluation function, 35Evaluation function reproduces, 36evidence, 15Evidence based on a Support Coefficient Function, 15

executed test case scenarios, 104exists, 39explanation structure, 50Extended Closed World Assumption-free, 30Extended Generalized Closed World Assumption-free, 30Extension base of a logic program, 46Extension of a default theory, 46Extension of a logic program, 46External structure of an explanation structure, 50

FCIR, see forward checking inference ruleFermat’s Little Theorem, 14field, 12

finite group, 12finite representation of the modified knowledge base corre-

sponding to belief change scenario, 30forall, 39formula, 39forward checkable literal, 41forward checking inference rule, 42forward-variable, 41Fully consistent strategy, 36function symbols, 39

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game, 35game objects, 35GCW A-free, see Generalized Closed World Assumption-

freegeneral logic program, 46General logic program closure, 46Generalization of an explanation structure, 50generalization space, 50Generalized Closed World Assumption-free, 30goal of the abductive problem, 43graph, 17greatest common divisor, 12grounded extension of an argumentation framework, 50group, 11

Harrop formula class, 48hom, see homomorphismhome page, 103Homologous computation paths, 47homomorphism, 12, 48Horn cover formula, 30Horn-clause, 40

identity element, 11implication, 39inbound client-side constraint rule, 103inbound server-side constraint rule, 103independence, 14Indifference belief, 37instance associated to a choice model, 45instance associated to a stable model, 45Instantiation of an explanation structure, 50Internal structure of an explanation structure, 50interpretation of a specification, 52inverse, see inverse elementinverse element, 11invertible, 11isomorphism, 12

knowledge base, 29

LAIR, see lookahead inference rulelearning algorithm for a meta-domain, 37least common multiple, 12least EBG, 51least EBG space, 51Least EGB generalization of an explanation, 51left coset, 12

left identity element, 11legal global instances, 44lemma clause, 44lemma tableau, 44likelihood, 15links, see edgesliteral, 40local minimum, 30locally equivalent evaluation functions, 35locally minimal solution, 30

locally perfect evaluation function, 35Logic program cover of a closure, 46lookahead checkable literal, 42lookahead inference rule, 42lookahead-variable, 42Loop in computation paths, 47loopy, see multiply-connected graph

MAP, see most probable a posteriorimarginal likelihood, 15maximum weight spanning tree, 17maxset, 35Me-valid integer ranking, 54minimal answer, 44minimal answers, 44minimal complete subset of efficient paths, 18Minimal explanation of a literal, 48Minimal legal global instance, 44Minimal legal global instances, 44minimal node of a Hasse diagram, 18Minimal solution of an abductive problem, 43

minimization with respect to, 43minset, 35model, 43model likelihood, 15model which minimizes, 43Moderately-grounded extension of a set of set of sentences,

46monoid, 11more general d-unifier, 41more general by least EBG than, 51More general explanation, 50most probable a posteriori, 15move, 30

multiply-connected graph, 17

negation, 39negation of a dilemma, 40Negative belief, 38negative set, 55neighborhood structure, 30neighbour, 17Neutral element, 35neutral element, see identity elementneutral set, 55nondeterministic finite automaton, 19Nondeterministic Time Hierarchy Theorem, 6

normal logic program, 45Normal logic program-consistent, 46Normal proof, 48

obliged generalization, 50observation, 29only-if set, 43Operationally criteria for generalizations, 51Operators, 39Or rule, 52

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outbound server-side constraint rule, 103

Parentheses, 39participation, 55passed, 103, 104path, 17PEF, see probabilistic evaluation functionPerformance comparison, 31Permissive inference relation, 52permissively valid conditional in an epistemic state, 52permutation, 12polynomial, 13polynomial-time Karp reducible, 5polynomial-time reducible, 5positive set, 55posterior, 15predicate symbols, 39preference abduction framework, 48preference function, see decision rulepreference pair, 36preferred extension of an argumentation framework, 50

Preferred model at the i-th stratum, 43preferred move, 36primality decision problem, 12prime, see prime numberprime field, 12prime number, 12prior, 15prioritized belief revision of KB by α, 29probabilistic evaluation function, 35probability density function, 14profile, 55Program of an open global system, 44proper parameter of the inference rule, 48

punctuation symbols, 39

quadratic residue, 13qualitative correlations among related data, 15Qualitative representation of a model of a specification, 52Quantifiers, 39quasi-associative, 11query clause, 44query tableau, 44

range of t is included in a domain dt, 41Rate of correctness, 16Rate of identification, 16

rate of return, 55Rational Monotony, 52Real ingredient with respect to an evaluation path, 48Refining adaptation, 53Regular belief, 37related data, 15Relation graph of a specification, 52reliability of a path, 17Remainder of an explanation structure, 50renamable Horn cover formula, 30

repair of a global system, 44repair program, 45Resolved relation graph of a specification, 52revision, 29revision formula, 29right coset, 12right identity element, 11ring, 12robust with respect to a set of defaults, 54

Satisfying a macro table bias, 37Satisfying a sparse solution space bias, 37sceptically valid in an epistemic state, 52SCF, see support coefficient functionsemi-associative, 11semigroup, 11sequence of decision elements, 55Shift interval, 15signature, 39Simple ingredients of a term, 47Simple variant computation paths, 47

singly-connected graph, 17Skeptical explanation of a literal, 48solution of an abductive problem, 43spanning tree, 17Specification of a document, 52square root, 13stable extension, 50starting state, 19Strategy S o, 35Strategy S ′o, 35Strategy order relation ≤σ, 36Strategy order relation ≤∗σ, 36strict locally minimum, 30

subfield, 12subgoal, 44subgroup, 12support coefficient function, 15

technical (non-model) metadata, 103Temporal interpretation of a specification, 52term, 39test case action verification percentage, 104test case action verification ratio, 104test case input verification ratio, 104test case scenario, 103test case scenario test, 104

test case state verification ratio, 104test scenario, 104tied chain, 44Time Hierarchy Theorem, 6Transgressive adaptation, 53transitions, 19tree, see singly-connected graphtruth-assignment, 43type-1-separated subset, 51type-2-separated subset, 51

126



RA F T

type-3-separated subset, 51

unconditionally independent, 15undirected graph, 17unifies, 41Uninstanstiation of an explanation structure, 50unique (inconsistent) belief change extension, 29unit refutable, 43Unresolution of an explanation structure, 50

usage degree, 51

vertices, 17

web application, 103web pages, 103

Documents

Definitions and Theorems in Computer Science [Work in Progress]