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 1 st 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* = Cn AEL (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 B  baseball  B  football → 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 baseball baseball  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 B F →  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 L  atoms 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* = Cn AELB (T  {  L  : T* |≠  }  { B  : T* |= min  }) where Cn AELB 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*) = Cn AEB (T  { B  : T* |= min  }) and Cn AEB denotes closure using the axioms of AEB

Least expansion Theorem: Operator  is monotonic, i.e. T  T 1  T 2 →   (T 1 )    (T 2 ) Hence, there always exists a minimal expansion of T, obtainable by transfinite induction: –T 0 = Cn  (T) –T i+1 =   (T i ) –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 B  Earthquake /\ B  Fires  Calm B  (Earthquake \/ Fires)  Worried B  Earthquake /\ B  Fires  Panicked B  Calm  CallHome Earthquake \/ Fires

Computation of Static Completion T 0 = Cn AEB (T) –T 0 |= Earthquake \/ Fires –T 0 |= min  Earthquake \/  Fires T 1 =   (T 0 )= Cn AEB (T 0  { B  : T 0 |= min  }) –T 1 = Cn AEB (T 0  { B( Eq \/ Fi), B(  Eq \/  Fi),…}) –T 1 |= Worried –T 1 |=  B Earthquake \/  B Fires –T 1 |=  B  Earthquake \/  B  Fires –T 1 |= min  Calm and T 1 |= min  Panicked

Static Completion (cont.) T 2 =   (T 1 )= Cn AEB (T 1  { B  : T 1 |= min  }) –T 2 = Cn AEB (T 1  { B( Eq \/ Fi), B(  Eq \/  Fi), B  Calm, B  Panicked, B Worried,… }) –T 2 |= CallHome T 3 =   (T 2 )= Cn AEB (T 2  { B  : T 2 |= min  }) –T 2 = Cn AEB (T 1  { B( Eq \/ Fi), B(  Eq \/  Fi), B  Calm, B  Panicked, B Worried, B CallHome, … })

LP for Knowledge 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   A 1, …, A n, not B 1, … not B m (n,m  0) where H, A i and B j are atoms Literal not B j are called default literals When no rule in P has default literal, P is called definite The Herbrand base H P 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 H P –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 H P –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: H P = 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:H P → 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 X XaticianableMathem 

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 T I  T J and F I  F J 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.

T P operator (2-valued case) The minimal models of a definite P can be computed (bottom-up) via operator T P D [T P ] Let I be an interpretation of definite P. T P (I) = {H: (H  Body)  P and Body  I} T If P is definite, T P is monotone and continuous. Its minimal fixpoint can be built by:  I 0 = {} and I n = T P (I n-1 ) with n > 0 T The least model of definite P is T P  ({})

T P operator (L-valued case) D [T P ] Let I be an interpretation of definite P. T P (I)(H) = lub { Î( Body ) : (H  Body)  P } T If P is definite, T P is monotone. Its minimal fixpoint can be built by iterating the T P operator:  I 0 = T P  ={}  I α = T P  α = T P (I α -1 ) = T P ( T P  α-1 ), where α is a successor ordinal  I β = T P  = |_| α < β T P  α = |_| α < β I α where β is a limit ordinal

T P operator (L-valued case) T There is a successor ordinal  such that T P  = T P  -1, i.e. there is a least fixpoint of T P. Furthermore, the least model of definite program P coincides with the least fixpoint of T P. 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 ← A 1, …, A m, L, C 1, …, C n has a successor ← (A 1, …, A m, B 1, …, B k, C 1, …, C n ) θ for each rule H :- B 1, …, B k, 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 !