LDK R Logics for Data and Knowledge Representation Propositional Logic Originally by Alessandro Agostini and Fausto Giunchiglia Modified by Fausto Giunchiglia,

LLogics for DData and KKnowledgeRRepresentation

Propositional Logic

Originally by Alessandro Agostini and Fausto GiunchigliaModified by Fausto Giunchiglia, Rui Zhang and Vincenzo Maltese

2

Outline Introduction Syntax Semantics

Truth valuation Satisfiability Validity Entailment

Logical implication Reasoning Services Normal Forms

2

Introduction: propositions All knowledge representation languages deal with

sentences

Sentences denote propositions (by definition), namely they express something true or false (the mental image of what you mean when you write the sentence)

Logic languages deal with propositions

Propositional logic is the simplest logic that does this (no individuals, no quantifiers)

3

“All men are mortals” “Obama is the president of the USA”

¬ Mortal ∨ Man President

INTRODUCTION :: SYNTAX :: SEMANTICS :: LOGICAL IMPLICATION :: REASONING SERVICES :: NORMAL FORMS

Introduction: example

4

World

SEMANTICGAP

Monkey

MentalModel

LanguageL

PROPOSITION SENTENCE


Introduction: the famous triangle

5

CONCEPT(what we have in mind)

SYMBOL(what we write)

REFERENT(the real object)

Monkey

stands for

refers to symbolizes


Language (Syntax) Alphabet of symbols: The first step in setting up a

formal language is to list the symbols of the alphabet Propositional Logic (PL) is a symbolic language Σ0: the alphabet of a PL language

6

Σ0

Descriptive Logical, , , …

Constants one proposition

only

A, B, C …

Variables they can be substituted

by any proposition or formula

P, Q, ψ …

NOTE: not only characters but also words (composed by several characters) like “monkey” are descriptive symbols


Additional Symbols Auxiliary symbols

Parentheses: ( )

Defined symbols Logical defined constants are, for all propositions P:

⊥ (falsehood symbol, false, bottom) ⊥ =df P∧¬PT (truth symbol, true, top) T =df ¬⊥

Defined symbols are not strictly necessary However, they increase the usability by reducing the

syntactic complexity of propositions

7


Formation Rules (FR): well formed formulas Well formed formulas (wff) in PL can be described by

the following BNF (*) grammar (codifying the rules):

<Atomic Formula> ::= A | B | ... | P | Q | ... | ⊥ | ⊤

<wff> ::= <Atomic Formula> | ¬<wff> | <wff>∧ <wff> | <wff> ∨ <wff>

Atomic formulas are also called atomic propositions Wff are propositional formulas (or just propositions) A formula is correct if and only if it is a wff

Σ0 + FR define a propositional language

(*) BNF = Backus–Naur form (formal grammar)8

PARSERψ, PLYes, ψ is correct!

No


Example: the Monkey-Banana problem

Significant data: monkey, bananas, box Significant knowledge: Low, High, Climb, GetBanana All other data and knowledge are irrelevant! Both data and knowledge are codified as (atomic or more

complex) formulas in the language

Language (any language you want to define):

MonkeyLow, BananaHigh, MonkeyClimbBox, MonkeyGetBanana

“ There is a monkey in a laboratory with some bananas hanging out of reach from the ceiling. A box is available that will enable the monkey to reach the bananas if he climbs on it. The monkey and box have height Low, but if the monkey climbs onto the box he will have height High, the same as the bananas. At this point the monkey can get a banana [...]”

9


Examples of formulas

10

Formula Correct

inLab-Monkey

inlab-monkey

in-lab-monkey

inLab(Monkey)

(Monkey ∧ GetBanana) ∨ ¬ High

Yes

(wff)

inLab(Monkey)∧

P ¬∧Q

No

NOTE: In PL we do not have individuals (it is not so expressive). Parentheses in inLab(Monkey) are not the auxiliary symbols like in e.g. (Monkey ∧ GetBanana) ∨ ¬ High (problem of ambiguity)


Propositional TheoryPropositional (or sentential) theory

A set of propositions It is a (propositional) knowledge base (containing

true facts) It corresponds to a TBox (terminology) only,

where no meaning is specified yet: it is a syntactic notion

Recall that: knowledge is a set of facts, i.e. statements we assumed statements are propositions

11


Propositional Theories and DatabasesA propositional theory is not a database!

Propositions don’t show data explicitly (no individuals)

In PL the form P(a) must be interpreted as P-a (so the use of form P(a) is ambiguous).

Example: In the sentence ‘inLab(Monkey)’, the string ‘Monkey’ does not refer to an individual from some data domain

12


Example: the Monkey-Banana problem

13

L = {MonkeyLow, BananaHigh, MonkeyClimbBox, MonkeyGetBanana,

, , }

T = { (MonkeyLow BananaHigh MonkeyGetBanana)

(MonkeyLow MonkeyClimbBox)

( MonkeyLow BananaHigh MonkeyGetBanana)}

NOTE: We still need to assign semantics to the symbols in the language.

“ There is a monkey in a laboratory with some bananas hanging out of reach from the ceiling. A box is available that will enable the monkey to reach the bananas if he climbs on it. The monkey and box have height Low, but if the monkey climbs onto the box he will have height High, the same as the bananas. At this point the monkey can get a banana [...]”


Semantics means providing an interpretationSo far the elements of our propositional

language are simply strings of symbols

without formal meaning

The meanings which are intended to be attached to the symbols and propositions form the intended interpretation ν (nu) of the language

14


Semantics: formal model Intensional interpretation

We must make sure to assign the formal meanings out of our intended interpretation to the (symbols of the) language, so that formulas (propositions) really express what we intended.

The mental model: What we have in mind?

In our mind (mental model) we have a set of properties that we associate to propositions. We need to make explicit (as much as possible) what we mean.

The formal modelThis is done by defining a formal model M. Technically: we have to define a pair (M,⊨) for our propositional language

Truth-values

In PL a sentence A is true (false) iff A denotes a formal object which satisfies (does not satisfy) the properties of the object in the real world.

15


Example: Banana (I) Bananas may differ in many ways, but when

using the proposition Banana we have in mind some specific properties: e.g. a given shape, a given color ...

These specific properties determine the intension of the proposition Banana

The intentional interpretation of the proposition Banana determines its truth-value (true or false).

16


Example: Banana (II)

17

World

SEMANTICGAP

Banana

MentalModel

LanguageL

PROPOSITION

What we have in mind?1) There is at least a banana2) The banana is yellow3) The banana has curve shape4) No matter if opened

SENTENCE

Banana (Yellow Curve)

LanguageL’

SENTENCE

OUR MENTAL MODEL MATCHES WITH THE REAL WORLD, SO THE SENTENCES ARE TRUE


Example: Banana (III)

18

World

SEMANTICGAP

Banana

MentalModel

LanguageL

PROPOSITION

What we have in mind?1) There is at least a banana2) The banana is yellow3) The banana has curve shape4) No matter if opened

SENTENCE

Banana (Yellow Curve)

LanguageL’

SENTENCE

OUR MENTAL MODEL DOES NOT MATCH WITH THE REAL WORLD, SO THE SENTENCES ARE FALSE


Example: Banana (IV)

19

I mean banana as something yellow

and curve, no matter if partially

opened

I mean banana as something yellow or red, which can be one or many together, no matter how big they are, and whether they are opened or not

FORMAL MODEL M1 FORMAL MODEL M2

T F T F

MENTAL MODEL MENTAL MODEL

THE WORLD


Truth-values Definition: a truth valuation on a propositional

language L is a mapping ν assigning to each formula A of L a truth value ν(A), namely in the domain D = {T, F}

ν(A) = T or F according to the modeler, with A atomic ν(¬A) = T iff ν(A) = F ν(A∧B) = T iff ν(A) = T and ν(B) = T ν(A∨B) = T iff ν(A) = T or ν(B) = T

ν(⊥) = F (since ⊥=df P∧¬P) ν(⊤) = T (since ⊤=df ¬⊥)

20


Example of truth valuation ν

21


, , }

T = { (MonkeyLow BananaHigh MonkeyGetBanana)

(MonkeyLow MonkeyClimbBox)

( MonkeyLow BananaHigh MonkeyGetBanana)}

Informal Semantics:

“If the monkey is low and the banana is high in position, then the monkey cannot get the banana. “

Formal Semantics:

ν(MonkeyLow) = T

ν(BananaHigh) = T

ν(MonkeyClimbBox) = F

ν(MonkeyGetBanana) = F

MODEL


Truth Tables To compute (in polynomial time) truth valuations, the method of truth tables was introduced

(Wittgenstein, 1921).

Truth tables are well-known:

Truth-table (TT) for the

logical constants

¬ , ∧, ∨

22


Truth Relation (Satisfaction Relation) Let ν be a truth valuation on language L, we define

the truth-relation (or satisfaction-relation) ⊨ and write

ν ⊨ A

(read: ν satisfies A) iff ν(A) = True

Given a set of propositions Γ, we define

ν ⊨ Γ

iff if ν ⊨ θ for all formulas θ ∈ Γ 23


Model and Satisfiability

Let ν be a truth valuation on language L. ν is a model of a proposition P (set of propositions Γ) iff ν satisfies P (Γ).

P (Γ) is satisfiable if there is some (at least one) truth valuation ν such that ν ⊨ P (ν ⊨ Γ).

24


Truth and Validity

Let ν be a truth valuation on language L.

P is true under ν if ν ⊨ P

P is valid if ν ⊨ P for all ν (notation: ⊨ P)P is called a tautology

25


Example

26


, , }

Informal Semantics:


Formal Semantics:

ν(MonkeyLow) = T

ν(BananaHigh) = T



MODEL

ν ⊨ MonkeyLow and ν ⊨ BananaHigh [SATISFIABLE]

ν ⊨ MonkeyClimbBox and ν ⊨ MonkeyGetBanana [SATISFIABLE]

MonkeyLow MonkeyLow is a tautology [VALID]


Entailment Propositional entailment

⊨ ψ

where = {θ1, ..., θn} is a finite set of propositions

ν ⊨ θi for all θi in implies ν ⊨ ψ

27


Example

28


, , }

Informal Semantics:


Formal Semantics:

ν(MonkeyLow) = T

ν(BananaHigh) = T



MODEL

= {MonkeyLow, BananaHigh}

ν ⊨ MonkeyLow and ν ⊨ BananaHigh implies ν ⊨ MonkeyGetBanana

therefore ⊨ MonkeyGetBanana [ENTAILMENT]


Entailment Difference in notation

⊨ ψ iff ⊨ ψ

Left side means we have already filtered out all the models that do not satisfy

Right side means that we start from all models.

In any case the final result is the same.

29


Implication: Premise (I) Propositional entailment in the finite, i.e.

⊨ ψ

where = {θ1, ..., θn} is a finite set of propositions

How to embed such a notion directly in the language?

As we will see, this would allow us to use truth tables and directly reformulate entailment in terms of the other reasoning services

30


Implication: Premise (II)

Entailment can be viewed as the logical implication

(θ1 ∧ θ2 ∧ ... ∧ θn) → ψ

to be read θ1 ∧ θ2 ∧ ... ∧ θn logically implies ψ

→ is a new symbol that we add to the language

31


We extend our alphabet of symbols with the following defined logical constants:→ (implication)↔ (double implication or equivalence)

<Atomic Formula> ::= A | B | ... | P | Q | ... | ⊥ | ⊤

<wff> ::= <Atomic Formula> | ¬<wff> | <wff>∧ <wff> | <wff>∨ <wff> |

<wff> → <wff> | <wff> ↔ <wff> (new rules) The operator precedence is (from higher to lower): ¬, ∧, ∨, →,

↔ Let propositions ψ, θ, and finite set {θ1,...,θn} of propositions

be given. We define: ⊨ θ → ψ iff θ ⊨ ψ ⊨ (θ1∧...∧θn) → ψ iff {θ1,...,θn} ⊨ ψ ⊨ θ ↔ ψ iff θ → ψ and ψ → θ

Implication and equivalence

32


Truth Table of → By defining the semantics of → in terms of ⊨

we have the following truth-table for the logical implication P→Q:

REMEMBER: Given ν, P ⊨ Q iff ν ⊨ P then ν ⊨ Q.

33


Three Properties of Logical Implication Property 1:

For all ψ and θ: ψ → θ iff ¬ψ ∨ θ

Property 2: For all ψ: ¬ψ iff ψ →⊥

Property 3: For all ψ: ⊥ ⊨ ψ(inconsistent theories imply any proposition)

34


Examples (tautologies) Laws for ∧, ¬ and →:

(A ∧ ¬B) → ¬(A → B) ¬(A → B) → (A ∧ ¬B)

Pierce’s law: ((A → B) → A) → A

De Morgan’s laws: ¬(A ∨ B) → (¬A ∧ ¬B)

(¬A ∧ ¬B) → ¬(A ∨ B) ¬(A ∧ B) → (¬A ∨ ¬B)

(¬A ∨ ¬B) → ¬(A ∧ B)

35


Exercise (tautologies)Prove the following double implications (i.e.,

logical equivalences) by using truth tables.

36

(α ∧ β) ↔ (β ∧ α) Commutativity of ∧

(α ∨ β) ↔ (β ∨ α) Commutativity of ∨

((α ∧ β) ∧ γ) ↔ (α ∧ (β ∧ γ)) Associativity of ∧

((α ∨ β) ∨ γ) ↔ (α ∨ (β ∨ γ)): Associativity of ∨

¬(¬α) ↔ α Double-negation elimination

(α →β) ↔ (¬β →¬α) Contraposition


Exercise (tautologies)Prove the following double implications (i.e.,

logical equivalences) by using truth tables.

37

(α → β) ↔ (¬α ∨ β) →-elimination

(α ↔ β) ↔ ((α → β) ∧ (β → α)) ↔-elimination

¬(α ∧ β) ↔ (¬α ∨ ¬β) DeMorgan Law for ∧

¬(α ∨ β) ↔ (¬α ∧ ¬β) DeMorgan Law for ∨

(α ∧ (β ∨ γ)) ↔ ((α ∧ β) ∨ (α ∧ γ)) Distributivity of ∧ over ∨

(α ∨ (β ∧ γ)) ↔ ((α ∨ β) ∧ (α ∨ γ)) Distributivity of ∨ over ∧


Reasoning ServicesThe basic reasoning tasks (or “services”) we

can represent (and compute) using ⊨ are: Model Checking (EVAL) Satisfiability (SAT) Unsatisfiability (UnSAT) Validity (VAL) Entailment (ENT)

38


Reasoning Services: EVALModel Checking (EVAL)

Is a proposition P true under a truth-valuation ν?

Check ν ⊨ P

EVALP , νYes

No

Does ν ⊨ MonkeyLow MonkeyGetBanana ?


39

Reasoning Services: SATSatisfiability (SAT)

Is there a truth-valuation ν where P is true?

find ν such that ν ⊨ P

SATPν

No

Find a ν that ν ⊨ MonkeyLow GetBanana

For instance, the truth-valuation ν where both are true


40

Reasoning Services (unSAT)Unsatisfiability is the impossibility to find a

truth-valuation ν

41

Monkey is satisfiable.Monkey ∨¬ Monkey is satisfiable.

Monkey ∧¬ Monkey is unsatisfiable (inconsistent). In fact, for every truth-valuation ν, either

ν(A) = T, so ν(¬A) = F, hence ν(P) = F; or ν(A) = F, so ν(¬A) = T, hence ν(P) = T.


Reasoning Services: VALValidity (VAL)

Is P true according to all possible truth-valuation ν?

Check whether for all ν, ν ⊨ P

VALPYes

No

Is P = MonkeyLow ∨ MonkeyLow valid? Yes!


42

Reasoning Services (Entailment) A set Γ (eventually empty) of propositions θ

entails a proposition ψ, written: Γ ⊨ ψ

if ν ⊨ θ for all θ ∈ Γ, then ν ⊨ ψ (for a given ν or for all ν)

If Γ ⊨ ψ, then we say that ψ is a logical consequence of Γ, and that Γ logically implies ψ.

NOTE: ⊨ P is the same as {} ⊨ P 43


Reasoning Services: ENTEntailment (ENT)

All θ ∈ Γ true in ν (in all ν) implies ψ true in ν (in all ν)

check Γ ⊨ ψ in ν (in all ν ) by checking that: given that ν ⊨ θ for all θ ∈ Γ implies ν ⊨ ψ

ENTΓ , ψ, ν Yes

No

Does MonkeyLow ⊨ MonkeyGetBanana ? No!

Does MonkeyLow BananaHigh ⊨ MonkeyGetBanana? Yes!


44

Reasoning Services (Entailment)Exercises:

Demonstrate that {today} ⊨ today ∨ tomorrow

Demonstrate that {today} ⊭ tomorrow

Suppose P = today ∨ tomorrow. Then: Define Γ such that Γ ⊨ P Define Γ such that Γ ⊭ P

45


Reasoning Services: properties (I) EVAL is the easiest task. We just test one assignment.

SAT is NP complete. We need to test in the worst case all the assignments. We stop when we find one which is true.

UnSAT is CO-NP. We need to test in the worst case all the assignments. We stop when we find one which is true.

VAL is CO-NP. We need to test all the assignments and verify that they are all true. We stop when we find one which is false.

ENT is CO-NP. It can be computed using VAL (see next slide)

46


Reasoning Services: properties (II) VAL(ψ) iff UnSAT( ψ)

SAT(ψ) VAL(ψ) iff SAT( ψ) VAL( ψ)

⊨ ψ, where = {θ1, …, θn} iff VAL(θ1 … θn ψ)

47

ψ = (A A) is valid, while ( A A) is unsatisfiable

ψ = A is satisfiable in all models where ν(A) = T. Informally, the left side says that there is at least an assignment which makes A true and there is at least an assignment which makes A false; the right side says that there is at least an assignment which makes A true (in other words it makes A false) and there is at least an assignment which makes A false (in other words it makes A true). This means that not all assignments make A true (false).

Take A ⊨ A B. the formula (A A B) is valid.


Reasoning Services: properties (III) EVAL is easy.

ENT can be computed using VAL

VAL can be computed using UnSAT

UnSAT is the opposite of SAT

All reasoning tasks can be reduced to SAT!!!SAT is the most important reasoning service.

48


Normal Forms: Literals A literal is either an atomic proposition or the

negation of an atomic proposition, i.e., for all propositional variables P

P and ¬P are literals.

P is a positive literal¬P is a negative literal

49

Monkey HighBanana ¬GetBanana ¬ClimbBox


Normal Forms: Clauses A (disjunctive) clause is a disjunction of literals α1

… αn

A unit clause is a clause that contains only a single literal

An empty clause does not contain literals

50

B ∨ ¬C ∨ ¬D Monkey ∨ ¬Monkey

B and ¬C are unit clausesB ∨ ¬C is not


Conjunctive Normal Form (CNF)A proposition is in conjunctive normal form (CNF)

if it is a (finite) conjunction of (disjunctive) clauses

51

Examples of formulas in CNF:

¬A ∧ B

(A ∨ ¬B) ∧ (B ∨ ¬C ∨ ¬D)

A ∧ (B ∨ C ∨ ¬D)

A B (¬ A C) (B ¬ D E)

Examples of formulas NOT in CNF:¬(A ∧ B) ¬ is the outmost operatorA ∧ (B ∨ C ∧ ¬D) ∧ is nested within ∨


Conversion of a formula to CNF Theorem (e.g. Mendelson, 1987 [Prop. 1.4, Ex. 1.41(a)])

Every formula P has an equivalent CNF, namely P can be converted (in polynomial time) into a proposition Q such that: Q is in CNF P and Q are equivalent (i.e., have the same truth table)

A conversion to a CNF that operates as in the theorem above is said to be complete

The algorithm implementing the conversion to a CNF is based on the tautologies about logical equivalence (or double implication): Double-negation law (i.e. elimination of ¬¬) De Morgan’s laws Distributive laws Elimination of → and

52


Conversion to CNF: Basic Tautologies

53

(α → β) ↔ (¬α ∨ β)c →-elimination

(α ↔ β) ↔ ((α → β) ∧ (β → α)) ↔-elimination

¬(α ∧ β) ↔ (¬α ∨ ¬β) DeMorgan Law for ∧

¬(α ∨ β) ↔ (¬α ∧ ¬β) DeMorgan Law for ∨

(α ∧ (β ∨ γ)) ↔ ((α ∧ β) ∨ (α ∧ γ))

Distributivity of ∧ over ∨

(α ∨ (β ∧ γ)) ↔ ((α ∨ β) ∧ (α ∨ γ))

Distributivity of ∨ over ∧

¬(¬α) ↔ α Double-negation elimination


Example: Conversion to CNF of A ↔ (B ∨ C)

1. Eliminate ↔:

(A→(B ∨C)) ∧ ((B ∨C) →A)

2. Eliminate →:

(¬A∨B ∨C) ∧ (¬(B ∨C) ∨A) Move ¬ inwards using de Morgan’s laws:

(¬A∨B ∨C) ∧ ((¬B ∧¬C) ∨A) Apply distributivity of ∨ over ∧ and flatten:

(¬A∨B ∨C) ∧ (¬B ∨A) ∧ (¬C ∨A) (*)

(*) in CNF and equivalent to ‘A↔(B∨C)’.

54


Normal Forms: Conjunctive clauses A conjunctive clause is a conjunction of literals α1

… αn

NOTE: a conjunctive clause IS NOT a clause!

55

B ¬C ¬D Monkey ¬Monkey


Disjunctive Normal Form (DNF)A proposition is in disjunctive normal form (DNF) if

it is a (finite) disjunction of conjunctive clauses

56

Examples of formulas in DNF:

¬A ∨ B

(A ∧ ¬B) ∨ (B ∧ ¬C ∧ ¬D)

A ∨ (B ∧ C ∧ ¬D)

A ∨ B ∨ (¬ A ∧ C) ∨ (B ∧ ¬ D ∧ E)

Examples of formulas NOT in DNF:¬(A ∨ B) ¬ is the outmost operatorA ∨ (B ∨ C ∧ ¬D) ∧ is nested within ∨


Horn clauses A Horn clause is a (disjunctive) clause with at most one

positive literal.

NOTE: ¬ A ¬ B D is equivalent to A B D

57

¬ AA ¬ B ¬ D ¬ A B ¬ D


Documents

LDK R Logics for Data and Knowledge Representation Propositional Logic Originally by Alessandro Agostini and Fausto Giunchiglia Modified by Fausto Giunchiglia,