1. 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

2. Syntax of AEL 1 st Order Logic, plus the operator L (applied to formulas) L  means “I know  ” Examples: MScOnSW → L MScSW (or  L MScOnSW →  MScOnSW) young (X)   L  studies (X) → studies (X)

3. 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

4. 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*

5. 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 

6. 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

7. Axioms about beliefs Consistency Axiom  B  Normality Axiom B (F → G) → ( B F → B G) Necessitation rule F B F

8. 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 

9. AELB expansions T* is a static 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

10. The special case of AEB Because of its properties, the case of theories without the knowledge operator is especially interesting Then, 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

11. 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  )

12. 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

13. 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

14. 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.

15. 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?

16. 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   

17. 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   

18. 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   

19. 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.

20. 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

21. 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

22. 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) = 1 - Î(F) –For formulas F and G: Î((F,G)) = min(Î(F), Î(G)) Î(F  G)= 1 if Î(F)  Î(G), and = 0 otherwise I is a model of P iff, for all rule H  B of P: Î(H  B) = 1

23. 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 

24. Minimal Models Semantics D [Truth ordering] For interpretations I and J, I  J iff for all atom A, I(A)  I(J), i.e. 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.

25. T P operator 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 ) T The least model of definite P is T P  ({})

26. 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 !