MULTI-AGENT ONLY-KNOWING

Transcription

1 MULTI-AGENT ONLY-KNOWING Gerhard Lakemeyer Computer Science, RWTH Aachen University Germany AI, Logic, and Epistemic Planning, Copenhagen October 3, 2013 Joint work with Vaishak Belle

2 Contents of this talk Only-knowing The single-agent case The multi-agent case AILEP 2 / 18

3 Only-Knowing Levesque (1990) introduced the logic of only-knowing to capture the beliefs of a knowledge-base. (Other variants such as Halpern & Moses, Ben-David & Gafni, Waaler not discussed here.) EXAMPLE If all I know is that the father of George is a teacher, then I know that someone is a teacher; but not who the teacher is. Note: Does not work if all I know is replaced by I know Need to express that nothing else is known. AILEP 3 / 18

4 The Single-Agent Case Levesque considered only the single-agent case. Compelling (i.e. simple) model-theoretic account A possible-worlds framework for a first-order language An epistemic state is simply a set of worlds AILEP 4 / 18

5 The Single-Agent Case Levesque considered only the single-agent case. Compelling (i.e. simple) model-theoretic account A possible-worlds framework for a first-order language An epistemic state is simply a set of worlds Compelling proof-theoretic account (for the propositional fragment) K extra axioms AILEP 4 / 18

6 Connection with Default Reasoning Levesque showed only-knowing captures Moore s Autoepistemic Logic: beliefs that follow from only-knowing facts and defaults are precisely those contained in all autoepistemic expansions EXAMPLE If all I know is that Tweety is a bird and that birds fly unless known otherwise, then I believe that Tweety flies. AILEP 5 / 18

7 The Language ON L ON L is a first-order language with = infinitely many standard names n1, n 2,... syntactically like constants, serve as the fixed domain of discourse (rigid designators); variables x, y, z,... predicate symbols of every arity; the usual logical connectives and quantifiers:,, ; modal operators: Kα "at least" α is believed. Nα "at most" α is believed to be false; Only-knowing: Oα =. Kα N α AILEP 6 / 18

8 Levesque s Semantics Primitive Formula = predicate with standard names as args. A world w is set of primitive formulas An epistemic state e is a set of worlds. w e, w = P( n) iff P( n) w; e, w = x.α iff e, w = α x n for all n e, w = Kα iff for all w e, e, w = α; e, w = Nα iff for all w e, e, w = α AILEP 7 / 18

9 Levesque s Semantics Primitive Formula = predicate with standard names as args. A world w is set of primitive formulas An epistemic state e is a set of worlds. w e, w = P( n) iff P( n) w; e, w = x.α iff e, w = α x n for all n e, w = Kα iff for all w e, e, w = α; e, w = Nα iff for all w e, e, w = α (e, w = Oα iff for all w, w e iff e, w = α) AILEP 7 / 18

10 Axioms (propositional) objective: non-modal formulas subjective: all predicates within a modal Let L stand for both K and N: AILEP 8 / 18

11 Axioms (propositional) objective: non-modal formulas subjective: all predicates within a modal Let L stand for both K and N: 1. Axioms of propositional logic. 2. L(α β) Lα Lβ. 3. σ Lσ, where σ is subjective. 4. The N vs. K axiom: (Nφ Kφ), where φ is consistent and objective; 5. Oα (Kα N α). 6. Inference rules: Modus ponens and Necessitation (for K and N) AILEP 8 / 18

12 Many Agents Intuitively, only-knowing for many agents seems easy: EXAMPLE If Alice believes that all that Bob knows is that birds normally fly and that Tweety is a bird, then Alice believes that Bob believes that Tweety flies. AILEP 9 / 18

13 Many Agents Intuitively, only-knowing for many agents seems easy: EXAMPLE If Alice believes that all that Bob knows is that birds normally fly and that Tweety is a bird, then Alice believes that Bob believes that Tweety flies. But technically things were surprisingly cumbersome! The problem lies in the complexity in what agents consider possible: For a single agent possibilities are just worlds. For many agents possibilities include other agents beliefs. The problem is that is not so clear how to come up with models that contain all possibilities. AILEP 9 / 18

14 Some Previous Attempts (Lakemeyer 1993) uses the K45n -canonical model Yet, certain types of epistemic states cannot be constructed O a O b p is valid (Halpern 1993) proposes a tree approach Modalities do not interact in an intuitive manner In (Halpern and Lakemeyer 2001), a solution is proposed. But again uses canonical models proof theory needs to axiomatize validity AILEP 10 / 18

15 The Logic ON L n. ON Ln = multi-agent version of ON L Here, only for a and b (K a, K b, N a, N b ) depth: alternating nesting of modalities a notion of a-depth and b-depth EXAMPLE (a-depth) p: 1 Ka p: 1 Kb p: 2 Ka K b p: 2 a-objective: formulas not in scope of Ka or N a : p K b p is a-objective, p K a p is not. AILEP 11 / 18

16 Beyond Sets of Worlds Alice s epistemic state is again a set of states of affairs, but where a state of affairs consists of a world and Bob s epistemic state Similarly, Bob s epistemic state is again a set of affairs where a state of affairs consists of a world and Alice s epistemic state (that determines her beliefs at this state) To be well-defined, we can do this only to some finite depth For formulas of a-depth k and b-depth j, it is sufficient to look at an epistemic state for Alice of depth k and an epistemic state for Bob of depth j. AILEP 12 / 18

17 Formal Semantics Define an epistemic state for Alice as a set of pairs e 1 a = { w, {}, w, {}...} (for formulas of a-depth 1) e k a = { w, e k 1 b,...} Similarly, an epistemic state for Bob eb 1 = { w, {}...} e j b = { w, ej 1 a,...}. (k, j)-model = e k a, e j b, w. Given a formula of a-depth k and b-depth j e k a, e j b, w = K aα iff for all w, e k 1 b ea, k ea, k e k 1 b, w = α ea, k e j b, w = N aα iff for all w, e k 1 b ea, k ea, k e k 1 b, w = α Oa α K a α N a α. AILEP 13 / 18

18 Some Properties A formula of a-depth k and b-depth j. is valid if it is true at all (k, j )-models for k k, j j. K45n (for K i and N i ) Ka (α β) K a α K a β K a α K a K a α K a α K a K a α Mutual introspection: Ka α N a K a α Barcan formula x Ka α K a x α AILEP 14 / 18

19 Examples of Only-Knowing Oa (p O b p) entails Ka p Ka K b p K a K b K a p but not K a K b K a p AILEP 15 / 18

20 Examples of Only-Knowing Oa (p O b p) entails Ka p Ka K b p K a K b K a p but not K a K b K a p Ka O b x T (x) entails Ka K b ( x T (x) K b T (x)) AILEP 15 / 18

21 Examples of Only-Knowing Oa (p O b p) entails Ka p Ka K b p K a K b K a p but not K a K b K a p Ka O b x T (x) entails Ka K b ( x T (x) K b T (x)) Let δb = x.k b Bird(x) K b Fly(x) Fly(x) K a O b (δ b Bird(tweety)) entails Ka K b Fly(tweety). AILEP 15 / 18

22 Proof Theory for Many Agents K45n + Axiom defining O i + new version of the N vs K axiom. Recall, in the single-agent case: (Nφ Kφ), where φ is consistent and objective. Two things to generalize here: Objective formulas to i-objective formulas But generalizing the notion of consistency of i-objective formulas is circular! AILEP 16 / 18

23 N i vs. K i Axioms Idea: break the circularity by considering a hierarchy of sub-languages based on the nesting of N i. Define a family of languages Let ON L 1. n = no N j in the scope of K i, N i (i j) Let ON L t+1 n formed from ON L t n, K i α and N i α for all α ON L t n AILEP 17 / 18

24 N i vs. K i Axioms Idea: break the circularity by considering a hierarchy of sub-languages based on the nesting of N i. Define a family of languages Let ON L 1. n = no N j in the scope of K i, N i (i j) Let ON L t+1 n formed from ON L t n, K i α and N i α for all α ON L t n Proof theory is K45n + Def. of O i + A 1 n N iα K i α if α is a K45 n -consistent i-objective formula A t+1 n N i α K i α, if α ON L t n, is i-objective and consistent wrt. K45 n, A 1 n A t n THEOREM For all α ON L t n, = α iff Axioms t α AILEP 17 / 18

25 Conclusions First-order modal logic for multi-agent only-knowing Faithfully generalizes intuitions of Levesque s logic Semantics not based on Kripke structures or canonical models, and thus avoids some problems of previous approaches In other work, we incorporated this notion of only-knowing into a mult-agent variant of the situation calculus for reasoning about knowledge and action. AILEP 18 / 18