Auto-Epistemic Logic

• Proposed by Moore (1985)

• Contemplates reflection on self knowledge (auto-epistemic)

• Allows for representing knowledge not just about the external world, but also about the knowledge I have of it

Syntax of AEL

• 1st Order Logic, plus the operator L (applied to formulas)

• L means “I know ”• Examples:

MScOnSW → L MScOnSW

(or L MScOnSW → MScOnSW)

young (X) L studies (X) → studies (X)

Meaning of AEL

• What do I know?– What I can derive (in all models)

• And what do I not know?– What I cannot derive

• But what can be derived depends on what I know– Add knowledge, then test

Semantics of AEL

• T* is an expansion of theory T iff

T* = Th(T{L : T* |= } {L : T* |≠ })

• Assuming the inference rule /L :

T* = CnAEL(T {L : T* |≠ })

• An AEL theory is always two-valued in L, that is, for every expansion:

| L T* L T*

Knowledge vs. Belief

• Belief is a weaker concept– For every formula, I know it or know it not– There may be formulas I do not believe in,

neither their contrary

• The Auto-Epistemic Logic of knowledge and Belief (AELB), introduces also operator B – I believe in

AELB Example

• I rent a film if I believe I’m neither going to baseball nor football games

Bbaseball Bfootball → rent_filme• I don’t buy tickets if I don’t know I’m going to

baseball nor know I’m going to football L baseball L football → buy_tickets

• I’m going to football or baseballbaseball football

• I should not conclude that I rent a film, but do conclude I should not buy tickets

Axioms about beliefs

• Consistency Axiom

B• Normality Axiom

B(F → G) → (B F → B G)

• Necessitation rule

F

B F

Some consequences of the Axioms

• B(F /\ G) ≡ B F /\ B G

• BF → B F

• B(F \/ G) → B F \/ B G

Minimal models

• In what do I believe?– In that which belongs to all preferred models

• Which are the preferred models?– Those that, for one same set of beliefs, have a minimal

number of true things

• A model M is minimal iff there does not exist a smaller model N, coincident with M on B e Latoms

• When is true in all minimal models of T, we write T |=min

AELB expansions

• T* is a static (autoepistemic) expansion of T iff

T* = CnAELB(T {L : T* |≠ }

{B : T* |=min })

where CnAELB denotes closure using the

axioms of AELB plus necessitation for L

Some properties of static autoepistemic expansions

• T* |= L iff T* |=

• T* |= B if T* |=min

• The other direction of the last implication only holds for particular cases (e.g. rational theories – those without positive occurrences of belief atoms).

The special case of AEB

• Because of its properties, the case of theories without the knowledge operator is especially interesting

• The definition of expansion becomes:

T* = (T*)

where (T*) = CnAEB(T {B : T* |=min })

and CnAEB denotes closure using the axioms of AEB

Least expansion

• Theorem: Operator is monotonic, i.e.

T T1 T2 → (T1) (T2)

• Hence, there always exists a minimal expansion of T, obtainable by transfinite induction:– T0 = Cn(T)

– Ti+1 = (Ti)

– T = U T (for limit ordinals )

Consequences

• Every AEB theory has at least one expansion

• If a theory is affirmative (i.e. all clauses have at least a positive literal) then it has at least a consistent expansion

• There is a procedure to compute the semantics

Example

• BEarthquake /\ BFires Calm

• B(Earthquake \/ Fires) Worried

• BEarthquake /\ BFires Panicked

• BCalm CallHome

• Earthquake \/ Fires

Computation of Static Completion

• T0 = CnAEB(T)

– T0 |= Earthquake \/ Fires

– T0 |=min Earthquake \/ Fires

• T1 = (T0)= CnAEB(T0 {B : T0 |=min })

– T1 = CnAEB(T0 {B(Eq \/ Fi), B(Eq \/ Fi),…})

– T1 |= Worried

– T1 |= BEarthquake \/ BFires

– T1 |= B Earthquake \/ B Fires

– T1 |=min Calm and T1 |=min Panicked

Static Completion (cont.)

• T2 = (T1)= CnAEB(T1 {B : T1 |=min })

– T2 = CnAEB(T1 {B(Eq \/ Fi), B(Eq \/ Fi), BCalm, BPanicked, BWorried,… })

– T2 |= CallHome

• T3 = (T2)= CnAEB(T2 {B : T2 |=min })

– T2 = CnAEB(T1 {B(Eq \/ Fi), B(Eq \/ Fi), BCalm, BPanicked, BWorried, BCallHome, … })

LP forKnowledge Representation

• Due to its declarative nature, LP has become a prime candidate for Knowledge Representation and Reasoning

• This has been more noticeable since its relations to other NMR formalisms were established

• For this usage of LP, a precise declarative semantics was in order

Language• A Normal Logic Programs P is a set of rules:

H A1, …, An, not B1, … not Bm (n,m 0)

where H, Ai and Bj are atoms

• Literal not Bj are called default literals

• When no rule in P has default literal, P is called definite

• The Herbrand base HP is the set of all instantiated atoms from program P.

• We will consider programs as possibly infinite sets of instantiated rules.

Declarative Programming

• A logic program can be an executable specification of a problem

member(X,[X|Y]).

member(X,[Y|L]) member(X,L).

• Easier to program, compact code• Adequate for building prototypes• Given efficient implementations, why not use it to

“program” directly?

LP and Deductive Databases

• In a database, tables are viewed as sets of facts:

• Other relations are represented with rules:

),(

).,(

londonlisbonflight

adamlisbonflight

LondonLisbon

AdamLisbon

tofromflight

).,(),(

).,(),,(),(

).,(),(

BAconnectionnotBAherchooseAnot

BCconnectionCAflightBAconnection

BAflightBAconnection

LP and Deductive DBs (cont)

• LP allows to store, besides relations, rules for deducing other relations

• Note that default negation cannot be classical negation in:

• A form of Closed World Assumption (CWA) is needed for inferring non-availability of connections

).,(),(

).,(),,(),(

).,(),(

BAconnectionnotBAherchooseAnot

BCconnectionCAflightBAconnection

BAflightBAconnection

Default Rules

• The representation of default rules, such as

“All birds fly”can be done via the non-monotonic operator not

).(

).(

).()(

).()(

.)(),()(

ppenguin

abird

PpenguinPabnormal

PpenguinPbird

AabnormalnotAbirdAflies

The need for a semantics

• In all the previous examples, classical logic is not an appropriate semantics– In the 1st, it does not derive not member(3,[1,2])

– In the 2nd, it never concludes choosing another company

– In the 3rd, all abnormalities must be expressed

• The precise definition of a declarative semantics for LPs is recognized as an important issue for its use in KRR.

2-valued Interpretations

• A 2-valued interpretation I of P is a subset of HP

– A is true in I (ie. I(A) = 1) iff A I– Otherwise, A is false in I (ie. I(A) = 0)

• Interpretations can be viewed as representing possible states of knowledge.

• If knowledge is incomplete, there might be in some states atoms that are neither true nor false

3-valued Interpretations

• A 3-valued interpretation I of P is a set

I = T U not F

where T and F are disjoint subsets of HP

– A is true in I iff A T– A is false in I iff A F– Otherwise, A is undefined (I(A) = 1/2)

• 2-valued interpretations are a special case, where:

HP = T U F

Lattice-valued interpretations

• We can generalize the previous definition to an arbitrary lattice of truth-values

• Let L be a complete lattice then an interpretation of a program P is a mapping I:HP → L

• Notice that any complete lattice has a least element () and a top element (T) , so a “true” proposition is mapped into T while a “false” proposition is mapped into .

• Some interesting useful complete lattices:– {0,1} with 0 < 1.– {0,1/2,1} with 0 < 1/2 < 1– [0,1]– Belnap’s four valued logic with with 0 <unknown | contradictory < 1

Intermezzo: lattices

• A partially ordered set (poset) is a set equipped with a reflexive, antisymmetric and transitive binary relation ≤:– Reflexivity: a ≤ a– Antisymmetry: if a ≤ b and b ≤ a then a=b– Transitivity: if a ≤ b and b ≤ c then a ≤ c

• A lattice is a poset such that for any two elements x and y the set {x,y} has both a least upper bound (join or supremum - \/) and a greatest lower bound (meet or infimum - /\). The join and meet obey to the following properties:– Commutative laws: a \/ b = b \/ a, and a /\ b = b /\ a– Associative laws: a \/ (b \/ c)= (a \/ b) \/ c, and a /\ (b /\ c)= (a /\ b) /\ c– Absorption laws: a \/ (a /\ b)= a, and a /\ (a \/ b)= a

• Notice that x ≤ y iff x = x /\ y, or equivalently y = x \/ y. • A complete lattice is a lattice where all subsets have a join and a meet.

Models

• Models can be defined via an evaluation function Î:– For an atom A, Î(A) = I(A)– For a formula F, Î(not F) = T - Î(F) (for lattices with

complement)– For formulas F and G:

• Î((F,G)) = glb(Î(F), Î(G))• Î((F;G)) = lub(Î(F), Î(G))• Î(F G)= T iff Î(F) ≥ Î(G).

• I is a model of P iff, for all rule H B of P:

Î(H B) = T

Minimal Models Semantics• The idea of this semantics is to minimize positive

information. What is implied as true by the program is true; everything else is false.

• {pr(c),pr(e),ph(s),ph(e),aM(c),aM(e)} is a model• Lack of information that cavaco is a physicist, should indicate that he isn’t• The minimal model is: {pr(c),ph(e),aM(e)}

)(

)(

)()(

cavacopresident

einsteinphysicist

XphysicistXaticianableMathem

Minimal Models Semantics

D[Truth ordering] For interpretations I and J, I J iff for all atom A, I(A) I(J), i.e. for the case of 2-valued interpretations TI TJ and FI FJ

T Every definite logic program has a least (truth ordering) model.

D[minimal models semantics] An atom A is true in (definite) P iff A belongs to its least model. Otherwise, A is false in P.

TP operator (2-valued case)• The minimal models of a definite P can be

computed (bottom-up) via operator TP

D [TP] Let I be an interpretation of definite P.

TP(I) = {H: (H Body) P and Body I}

T If P is definite, TP is monotone and continuous. Its minimal fixpoint can be built by:

I0 = {} and In = TP(In-1) with n > 0

T The least model of definite P is TP({})

TP operator (L-valued case)

D[TP] Let I be an interpretation of definite P.

TP(I)(H) = lub{Î(Body): (H Body) P}

T If P is definite, TP is monotone. Its minimal fixpoint can be built by iterating the TP operator:

I0 = TP={}

Iα = TPα = TP(Iα -1) = TP (TP

α-1), where α is a successor ordinal

Iβ = TP = |_| α < β TP

α = |_| α < β Iα where β is a limit ordinal

TP operator (L-valued case)

T There is a successor ordinal such that TP = TP

-

1 , i.e. there is a least fixpoint of TP.

Furthermore, the least model of definite program P coincides with the least fixpoint of TP.

In general, more than iterations might be needed to reach the least fixpoint. However, if lattice L is finite then at most iterations are enough.

Computation of minimal models

• For the 2-valued case there is a complete method: SLD resolution (Linear resolution with a selection function for definite sentences).

• A SLD-goal of the form← A1, …, Am, L, C1, …, Cn

has a successor ← (A1, …, Am, B1, …, Bk, C1, …, Cn ) θ

for each rule H :- B1, …, Bk, belonging to the program such that L and H unify with mgu θ.

• A SLD-derivation is a sequence of applications of SLD-resolution, and a SLD-refutation is a SLD-derivation which ends in the empty clause, i.e. no goals after ←.

• For the lattice-valued case, there are proof procedures based on tabulation methods, which we will not present.

On Minimal Models

• SLD can be used as a proof procedure for the minimal models semantics:– If the is a SLD-derivation for A, then A is true– Otherwise, A is false

• The semantics does not apply to normal programs:– p not q has two minimal models:

{p} and {q}

There is no least model !