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Logic for Knowledge Representation

Logic for Knowledge Representation. Sindhu Kutty. Why Knowledge Representation?. What else is there?? Lack of motivation for ‘reinventing the wheel’! Declarative semantics. Logical AI. An agent’s knowledge of its world and goals represented by sentences in logic

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Logic for Knowledge Representation

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  1. Logic for Knowledge Representation Sindhu Kutty

  2. Why Knowledge Representation? • What else is there?? • Lack of motivation for ‘reinventing the wheel’! • Declarative semantics

  3. Logical AI • An agent’s knowledge of its world and goals represented by sentences in logic • Sound and complete inference rules lead to useful deductions • Epistemological and heuristic problems

  4. Classical Logic • Logic of truth • Operators • ,,, • (ab) (ab) • Quantifiers • , 

  5. Some examples • Everyone has a father childman father(man,child) • Everyone loves someone personsomeone loves(person,someone) • There is someone who everyone loves someoneperson loves(person,someone)

  6. The truth about implication • The truth table • apple(x)fruit(x) Am I a liar if ‘x’ happens not to be an apple?

  7. Classical Logic - Issues • Truth as opposed to knowledge • What does yx father(x,y) mean? • Is that enough?

  8. Computability Logic - Motivation • The difference between • yx father(x,y) • ㄇyㄩx father(x,y)

  9. Computability Logic • Logic of computability • Logic of interaction • Resource conscious • Some very expressive fragments are recursively enumerable • It captures constructive ability as distinct from truth

  10. Semantics • Computational problems understood as games played by a machine against the environment • Representation:  - environment ㄒ - machine

  11. Elementary Games • Move • An observable action by an agent • Run • Sequence of moves • Legal runs • Even(5) • Who wins this? • What are the moves? • What is the run? Legal runs?

  12. Elementary Games • Even(5)  Even(4) • Who wins this? • What are the moves? • What is the run? Legal runs? • (Even(5)  Even(4)) • Who wins this? Role reversal • What are the moves? • What is the run? Legal runs?

  13. Constant Games • Even(5) ㄩ Even(4) • Choice disjunction • Who wins this? • What are the moves? • What is the run? Legal runs? • (Even(5) ㄇ Even(4)) • Choice conjunction • Who wins this? • What are the moves? • What is the run? Legal runs?

  14. Games • Even(x) ㄩ Even(x) • Who wins this? • What are the moves? • What is the run? Legal runs?

  15. Games • A game is a function from valuations to constant games • A valuation is a function from the set of variables to the set of constants

  16. Games • A = Even(x) ㄩ Even(x) • A(x/5) • Who wins this? • What are the moves? • What is the run? Legal runs?

  17. Choice Existential Quantifier • A = ㄩx (Even(x) ㄩ Even(x)) • <ㄒ5>(ㄩx (Even(x) ㄩ Even(x))) • <ㄒ2>( Even(5) ㄩ Even(5)) • Run • <ㄒ5, ㄒ2 > • ㄒ- won run

  18. Choice Universal Quantifier • A = ㄇx (Even(x) ㄩ Even(x)) • < 5>(ㄇx (Even(x) ㄩ Even(x))) • <ㄒ2>( Even(5) ㄩ Even(5)) • Run • < 5, ㄒ2 > • ㄒ- won run • Run • <> • ㄒ- won run

  19. Blind Quantifiers • A = x (Even(x) ㄩEven(x)) • What run makes this ㄒ- won ? • A = x (Even(x) ㄩEven(x)) • What run makes this ㄒ- won ?

  20. Games • ㄇx ((Odd(x) ㄩ Odd(x)) → (Even(x) ㄩ Even(x))) • Who wins this? • Resource •  x (Even(x) ㄩOdd(x) → ㄇy(Even(x*y) ㄩ Odd(x *y))) • Can this be won?

  21. Interesting Observations! • A A is valid. Why? • For A = Even(5) • A ㄩA is not valid. Why? • The difference between • yx father(x,y) • ㄇyㄩx father(x,y)

  22. CL4 • Operators used so far • CL4├ F iff F is valid • Uniform constructive soundness • Strong completeness • F* not computable for some interpretation

  23. An example • ㄩxㄇy Has(x,y) • ㄇx(ㄩs Symptoms(x,s)  ㄇt (Positive(x,t)ㄩ Positive(x,t))  ㄩy Has(x,y)) • How can ㄒ win this game?

  24. Parallel Recurrence • Parallel Recurrence • A  A  A  … or A • When does ㄒ win this?

  25. Branching vs. Parallel Recurrence • Branching Recurrence • Legal run of A can be thought of as a tree where each branch spells a legal run • ㄒ wins iff it wins in all branches • Root is the empty run • Only  has the capability of making a splitting move - easier for  to win - why? • Which is easier for  to win: • A or A? Why?

  26. Some Examples • Potential knowledge of everyone’s gender • ㄇx(Female(x) ㄩ Female(x)) • Any difference if we say • ㄇx(Female(x) ㄩ Female(x))

  27. Back to the Game… • ㄇx(ㄩs Symptoms(x,s)  ㄇt (Positive(x,t)ㄩ Positive(x,t))  ㄩy Has(x,y))

  28. Resource Consciousness • What if n tests were needed? • -conjunction of n identical conjuncts • What if an unbounded number of tests was needed? • ㄇt (Positive(x,t)ㄩ Positive(x,t))

  29. Another Example • x (Red(x) Acid(x)) • x (Acid(x)  Red(x)) • ㄇx(Red(x)ㄩRed(x)) • Does KB  ㄇx(Acid(x)ㄩAcid(x))?

  30. Possible Worlds Analysis • TRUE(KNOW(A,IMP(ACID,RES(DO(A,TEST),AND(ACID,RED))))) • TRUE(KNOW(A,IMP(NOT(ACID),RES(DO(A,TEST),AND(NOT(ACID),NOT(RED)))))) • TRUE(ACID) • Prove: TRUE(RES(DO(A,TEST),KNOW(A,ACID)))

  31. Computability Logic • Epistemic variants of classical logic get messy • They are also non-semidecidable • The KNOW operator is not constructive

  32. An Epistemic Variant • Axioms • M1 P, s.t. P is an axiom of ordinary propositional logic • M2 KNOW(A,P)  P (knowledge axiom) • M3 KNOW(A,P)  KNOW(A,KNOW(A,P)) (positive introspection axiom) • M4 KNOW(A,(PQ))  (KNOW(A,P)  KNOW(A,Q)) (logical omniscience) • M5 If P is an axiom, then KNOW(A,P) is an axiom

  33. Possible World Analysis • Individual propositions known by an agent • State of affairs compatible with what agent knows • Model theory

  34. Accessibility Relation • K(A, W1, W2) Possible world W2 is compatible or consistent with what A knows in possible world W1 • For the following illustrations Agent A knows PW0 is the actual world • Goal: Capture axioms M1-M5 model-theoretically

  35. Accessibility Relation - Definition • If agent A knows P in W0, then P is true in every accessible world P KA W1 P KA P W0 W2 KA P Wn

  36. Accessibility Relation - Properties • M2 KNOW(A,P)  P • If agent A knows P, then P is true a1 K(a1, W0, W0) KA P W0

  37. Accessibility Relation - Properties • M4 KNOW(A,(PQ)) (KNOW(A,P)  KNOW(A,Q)) • If P is true in every world W1 s.t. K(A,W0,W1) then A knows that P in the actual world P KA KA W1 P KA P W2 W0 KA P Wn

  38. Accessibility Relation - Properties • M3 KNOW(A,P)  KNOW(A,KNOW(A,P)) • a1 w1 w2 (K(a1, W0, w1)  (K(a1, w1, w2)  K(a1, W0, w2))) KA P KA P KA KA P KA KA P KA KA W0 P KA P KA

  39. Accessibility Relation - Properties • M5 If P is an axiom, then KNOW(A,P) is an axiom • Generalize these constraints to hold not just for the actual world but for all possible worlds and for all agents

  40. The Example Revisited • x (Red(x) Acid(x)) • x (Acid(x)  Red(x)) • ㄇx(Red(x)ㄩRed(x)) • Then CL4├ KB  ㄇx(Acid(x)ㄩAcid(x))

  41. The Example Revisited • TRUE(KNOW(A,IMP(ACID,RES(DO(A,TEST),AND(ACID,RED))))) • TRUE(KNOW(A,IMP(NOT(ACID),RES(DO(A,TEST),AND(NOT(ACID),NOT(RED)))))) • TRUE(ACID) • Prove: TRUE(RES(DO(A,TEST),KNOW(A,ACID)))

  42. To Sum Up • Classical Logic • Querying unnatural • Epistemic variants • not constructive • messy • Advantages of Computability Logic • Simplicity • Resource consciousness • Uniform Constructive Soundness of CL4

  43. References • Japaridze, G. Introduction to computability logic. Annals of Pure and Applied Logic, 123 (2003), pp.1-99. • Japaridze, G. Computability logic: a formal theory of interaction, 2004. • Moore, R. A formal theory of knowledge and action, Formal Theories of Commonsense Worlds(1985), pp. 319-358. • McCarthy, J. Concepts of Logical AI, in progress.

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