Logic and Artificial Intelligence Lecture 22

Transcription

1 Logic and Artificial Intelligence Lecture 22 Eric Pacuit Currently Visiting the Center for Formal Epistemology, CMU Center for Logic and Philosophy of Science Tilburg University ai.stanford.edu/ epacuit November 23, 2011 Logic and Artificial Intelligence 1/22

2 Preference (Modal) Logics Modal betterness model M = W,, V Logic and Artificial Intelligence 2/22

3 Preference (Modal) Logics Modal betterness model M = W,, V Preference Modalities ϕ: there is a world at least as good (as the current world) satisfying ϕ M, w = ϕ iff there is a v w such that M, v = ϕ Logic and Artificial Intelligence 2/22

4 Preference (Modal) Logics Modal betterness model M = W,, V Preference Modalities ϕ: there is a world at least as good (as the current world) satisfying ϕ M, w = ϕ iff there is a v w such that M, v = ϕ M, w = ϕ iff there is v w and w v such that M, v = ϕ Logic and Artificial Intelligence 2/22

5 Preference Lifting, I Given a preference ordering over a set of objects X, we want to lift this to an ordering ˆ over (X ). Given, what reasonable properties can we infer about ˆ? S. Barberá, W. Bossert, and P.K. Pattanaik. Ranking sets of objects. In Handbook of Utility Theory, volume 2. Kluwer Academic Publishers, Logic and Artificial Intelligence 3/22

6 Preference Lifting, II You know that x y z Can you infer that {x, y} ˆ {z}? Logic and Artificial Intelligence 4/22

7 Preference Lifting, II You know that x y z Can you infer that {x, y} ˆ {z}? You know that x y z Can you infer anything about {y} and {x, z}? Logic and Artificial Intelligence 4/22

8 Preference Lifting, II You know that x y z Can you infer that {x, y} ˆ {z}? You know that x y z Can you infer anything about {y} and {x, z}? You know that w x y z Can you infer that {w, x, y} ˆ {w, y, z}? Logic and Artificial Intelligence 4/22

9 Preference Lifting, II You know that x y z Can you infer that {x, y} ˆ {z}? You know that x y z Can you infer anything about {y} and {x, z}? You know that w x y z Can you infer that {w, x, y} ˆ {w, y, z}? You know that w x y z Can you infer that {w, x} ˆ {y, z}? Logic and Artificial Intelligence 4/22

10 Preference Lifting, III There are different interpretations of X ˆ Y : You will get one of the elements, but cannot control which. You can choose one of the elements. You will get the full set. Logic and Artificial Intelligence 5/22

11 Preference Lifting, IV Kelly Principle (EXT) {x} ˆ {y} provided x y (MAX) A ˆ Max(A) (MIN) Min(A) ˆ A J.S. Kelly. Strategy-Proofness and Social Choice Functions without Single- Valuedness. Econometrica, 45(2), pp , Logic and Artificial Intelligence 6/22

12 Preference Lifting, IV Gärdenfors Principle (G1) A ˆ A {x} if a x for all a A (G2) A {x} ˆ A if x a for all a A P. Gärdenfors. Manipulation of Social Choice Functions. Journal of Economic Theory. 13:2, , Logic and Artificial Intelligence 7/22

13 Preference Lifting, IV Gärdenfors Principle (G1) A ˆ A {x} if a x for all a A (G2) A {x} ˆ A if x a for all a A P. Gärdenfors. Manipulation of Social Choice Functions. Journal of Economic Theory. 13:2, , Independence (IND) A {x} ˆ B {x} if A ˆ B and x A B Logic and Artificial Intelligence 7/22

14 Preference Lifting, V Theorem (Kannai and Peleg). If X 6, then no weak order satisfies both the Gärdenfors principle and independence. Y. Kannai and B. Peleg. A Note on the Extension of an Order on a Set to the Power Set. Journal of Economic Theory, 32(1), pp , Logic and Artificial Intelligence 8/22

15 From Worlds to Sets, I M, w = ϕ ψ iff there is s, t such that M, s = ϕ and M, t = ψ and s t Logic and Artificial Intelligence 9/22

16 From Worlds to Sets, I M, w = ϕ ψ iff there is s, t such that M, s = ϕ and M, t = ψ and s t M, w = ϕ ψ iff there is s, t such that M, s = ϕ and M, t = ψ and s t Logic and Artificial Intelligence 9/22

17 From Worlds to Sets, I M, w = ϕ ψ iff there is s, t such that M, s = ϕ and M, t = ψ and s t M, w = ϕ ψ iff there is s, t such that M, s = ϕ and M, t = ψ and s t M, w = ϕ ψ iff for all s there is a t such that M, s = ϕ implies M, t = ψ, and s t Logic and Artificial Intelligence 9/22

18 From Worlds to Sets, I M, w = ϕ ψ iff there is s, t such that M, s = ϕ and M, t = ψ and s t M, w = ϕ ψ iff there is s, t such that M, s = ϕ and M, t = ψ and s t M, w = ϕ ψ iff for all s there is a t such that M, s = ϕ implies M, t = ψ, and s t M, w = ϕ ψ iff for all s there is a t such that M, s = ϕ implies M, t = ψ, and s t Logic and Artificial Intelligence 9/22

19 From Worlds to Sets, II ϕ ψ := E(ϕ ψ) Logic and Artificial Intelligence 10/22

20 From Worlds to Sets, II ϕ ψ := E(ϕ ψ) ϕ ψ := E(ϕ ψ) Logic and Artificial Intelligence 10/22

21 From Worlds to Sets, II ϕ ψ := E(ϕ ψ) ϕ ψ := E(ϕ ψ) ϕ ψ := A(ϕ ψ) Logic and Artificial Intelligence 10/22

22 From Worlds to Sets, II ϕ ψ := E(ϕ ψ) ϕ ψ := E(ϕ ψ) ϕ ψ := A(ϕ ψ) ϕ ψ := A(ϕ ψ) Logic and Artificial Intelligence 10/22

23 From Worlds to Sets, III M, w = ϕ ψ iff for all s, for all t, M, s = ϕ and M, t = ψ implies s t Logic and Artificial Intelligence 11/22

24 From Worlds to Sets, III M, w = ϕ ψ iff for all s, for all t, M, s = ϕ and M, t = ψ implies s t M, w = ϕ ψ iff for all s, for all t, M, s = ϕ and M, t = ψ implies s t Logic and Artificial Intelligence 11/22

25 From Worlds to Sets,IV ϕ ψ := A(ψ ϕ) Logic and Artificial Intelligence 12/22

26 From Worlds to Sets,IV ϕ ψ := A(ψ ϕ) ϕ ψ := A(ψ ϕ) Logic and Artificial Intelligence 12/22

27 From Worlds to Sets,IV ϕ ψ := A(ψ ϕ) ϕ ψ := A(ψ ϕ) We must assume the ordering is total Logic and Artificial Intelligence 12/22

28 From Sets to Worlds P 1 >> P 2 >> P 3 >> >> P n x > y iff x and y differ in at least one P i and the first P i where this happens is one with P i x and P i y F. Liu and D. De Jongh. Optimality, belief and preference Logic and Artificial Intelligence 13/22

29 General Issues General Issues Once a semantics and language are fixed, then standard questions can be asked: eg. develop a proof theory, completeness, decidability, model checking. Logic and Artificial Intelligence 14/22

30 General Issues General Issues How should we compare the different logical systems? Embedding one logic in another: Logic and Artificial Intelligence 15/22

31 General Issues General Issues How should we compare the different logical systems? Embedding one logic in another: coalition logic is a fragment of ATL (tr([c]ϕ) = C ϕ) Logic and Artificial Intelligence 15/22

32 General Issues General Issues How should we compare the different logical systems? Embedding one logic in another: coalition logic is a fragment of ATL (tr([c]ϕ) = C ϕ) Compare different models for a fixed language: Logic and Artificial Intelligence 15/22

33 General Issues General Issues How should we compare the different logical systems? Embedding one logic in another: coalition logic is a fragment of ATL (tr([c]ϕ) = C ϕ) Compare different models for a fixed language: Alternating-Time Temporal Logics: Three different semantics for the ATL language. V. Goranko and W. Jamroga. Comparing Semantics of Logics for Multiagent Systems. KRA, Logic and Artificial Intelligence 15/22

34 General Issues General Issues How should we compare the different logical systems? Embedding one logic in another: coalition logic is a fragment of ATL (tr([c]ϕ) = C ϕ) Compare different models for a fixed language: Alternating-Time Temporal Logics: Three different semantics for the ATL language. V. Goranko and W. Jamroga. Comparing Semantics of Logics for Multiagent Systems. KRA, Comparing different frameworks: Logic and Artificial Intelligence 15/22

35 General Issues General Issues How should we compare the different logical systems? Embedding one logic in another: coalition logic is a fragment of ATL (tr([c]ϕ) = C ϕ) Compare different models for a fixed language: Alternating-Time Temporal Logics: Three different semantics for the ATL language. V. Goranko and W. Jamroga. Comparing Semantics of Logics for Multiagent Systems. KRA, Comparing different frameworks: eg. PDL vs. Temporal Logic, PDL vs. STIT, STIT vs. ATL, etc. Logic and Artificial Intelligence 15/22

36 General Issues General Issues How should we merge the different logical systems? Logic and Artificial Intelligence 16/22

37 General Issues General Issues How should we merge the different logical systems? Combining logics is hard! Explain D. Gabbay, A. Kurucz, F. Wolter and M. Zakharyaschev. Many Dimensional Modal Logics: Theory and Applications Logic and Artificial Intelligence 16/22

38 General Issues General Issues How should we merge the different logical systems? Combining logics is hard! Explain D. Gabbay, A. Kurucz, F. Wolter and M. Zakharyaschev. Many Dimensional Modal Logics: Theory and Applications Theorem ϕ ϕ is provable in combinations of Epistemic Logics and PDL with certain cross axioms ( [a]ϕ [a] ϕ) (and full substitution). R. Schmidt and D. Tishkovsky. On combinations of propositional dynamic logic and doxastic modal logics. JOLLI, Logic and Artificial Intelligence 16/22

39 General Issues Merging Logics of Rational Agency Entangling Knowledge/Beliefs and Preferences Epistemizing Logics of Action and Ability BDI (Belief + Desires + Intentions) Logics Logic and Artificial Intelligence 17/22

40 General Issues Logics of Knowledge and Preference K(ϕ ψ): Ann knows that ϕ is at least as good as ψ Kϕ Kψ: knowing ϕ is at least as good as knowing ψ Logic and Artificial Intelligence 18/22

41 General Issues Logics of Knowledge and Preference K(ϕ ψ): Ann knows that ϕ is at least as good as ψ Kϕ Kψ: knowing ϕ is at least as good as knowing ψ M = W,,, V Logic and Artificial Intelligence 18/22

42 General Issues Logics of Knowledge and Preference K(ϕ ψ): Ann knows that ϕ is at least as good as ψ Kϕ Kψ: knowing ϕ is at least as good as knowing ψ M = W,,, V J. van Eijck. Yet more modal logics of preference change and belief revision. manuscript, F. Liu. Changing for the Better: Preference Dynamics and Agent Diversity. PhD thesis, ILLC, Logic and Artificial Intelligence 18/22

43 General Issues A(ψ ϕ) vs. K(ψ ϕ) Logic and Artificial Intelligence 19/22

44 General Issues A(ψ ϕ) vs. K(ψ ϕ) Should preferences be restricted to information sets? Logic and Artificial Intelligence 19/22

45 General Issues A(ψ ϕ) vs. K(ψ ϕ) Should preferences be restricted to information sets? M, w = ϕ iff there is a v with w v and w v such that M, v = ϕ K(ψ ϕ) Logic and Artificial Intelligence 19/22

46 General Issues Merging Logics of Rational Agency Entangling Knowledge/Beliefs and Preferences Epistemizing Logics of Action and Ability BDI (Belief + Desires + Intentions) Logics Logic and Artificial Intelligence 20/22

47 General Issues Knowing how to win Consider the following game: Two cards, Ace and Joker, lie face down and the agent i must choose one. The Ace wins, the Joker loses. Logic and Artificial Intelligence 21/22

48 General Issues Knowing how to win Consider the following game: Two cards, Ace and Joker, lie face down and the agent i must choose one. The Ace wins, the Joker loses. Does the agent i have a strategy to win the game? Logic and Artificial Intelligence 21/22

49 General Issues Knowing how to win Consider the following game: Two cards, Ace and Joker, lie face down and the agent i must choose one. The Ace wins, the Joker loses. Does the agent i have a strategy to win the game? Does the agent i know that she has a strategy to win the game? Logic and Artificial Intelligence 21/22

50 General Issues Knowing how to win Consider the following game: Two cards, Ace and Joker, lie face down and the agent i must choose one. The Ace wins, the Joker loses. Does the agent i have a strategy to win the game? Does the agent i know that she has a strategy to win the game? Does the agent i know a strategy to win the game? Logic and Artificial Intelligence 21/22

51 General Issues J. Fantl. Knowing-how and knowing-that. Philosophy Compass, 3 (2008), M.P. Singh. Know-how. In Foundations of Rational Agency (1999), M. Wooldridge and A. Rao, Eds., pp Logic and Artificial Intelligence 22/22