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Only-Knowing G Lakemeyer M ULTI -AGENT O NLY -K NOWING Gerhard Lakemeyer Computer Science, RWTH Aachen University Germany AI, Logic, and Epistemic Planning, Copenhagen October 3, 2013 Joint work with Vaishak Belle Only-Knowing G Lakemeyer Contents of this talk I Only-knowing I The single-agent case I The multi-agent case AILEP 2 / 18 Only-Knowing G Lakemeyer 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.) E XAMPLE If all I know is that the father of George is a teacher, then I I know that someone is a teacher; I but not who the teacher is. Note: Does not work if all I know is replaced by I know I Need to express that nothing else is known. AILEP 3 / 18 Only-Knowing G Lakemeyer The Single-Agent Case Levesque considered only the single-agent case. I Compelling (i.e. simple) model-theoretic account I I AILEP A possible-worlds framework for a first-order language An epistemic state is simply a set of worlds 4 / 18 Only-Knowing G Lakemeyer The Single-Agent Case Levesque considered only the single-agent case. I Compelling (i.e. simple) model-theoretic account I I A possible-worlds framework for a first-order language An epistemic state is simply a set of worlds I Compelling proof-theoretic account (for the propositional fragment) I AILEP K45 + 2 extra axioms 4 / 18 Only-Knowing G Lakemeyer Connection with Default Reasoning Levesque showed only-knowing captures Moore’s Autoepistemic Logic: I beliefs that follow from only-knowing facts and defaults are precisely those contained in all autoepistemic expansions E XAMPLE 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 Only-Knowing G Lakemeyer The Language ON L ON L is a first-order language with = I infinitely many standard names n1 , n2 , . . . syntactically like constants, serve as the fixed domain of discourse (rigid designators); I I I I variables x, y , z, . . . predicate symbols of every arity; the usual logical connectives and quantifiers: ∧, ¬, ∀; modal operators: I I I AILEP Kα "at least" α is believed. N α "at most" α is believed to be false; Only-knowing: . Oα = Kα ∧ N ¬α 6 / 18 Only-Knowing G Lakemeyer 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 I e, w |= P(~n) iff P(~n) ∈ w; I e, w |= ∀x.α iff e, w |= αnx for all n I e, w |= Kα iff for all w 0 ∈ e, e, w 0 |= α; I e, w |= N α iff for all w 0 6∈ e, e, w 0 |= α AILEP 7 / 18 Only-Knowing G Lakemeyer 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 I e, w |= P(~n) iff P(~n) ∈ w; I e, w |= ∀x.α iff e, w |= αnx for all n I e, w |= Kα iff for all w 0 ∈ e, e, w 0 |= α; I e, w |= N α iff for all w 0 6∈ e, e, w 0 |= α I (e, w |= Oα iff for all w 0 , w 0 ∈ e iff e, w 0 |= α) AILEP 7 / 18 Only-Knowing G Lakemeyer Axioms (propositional) objective: non-modal formulas subjective: all predicates within a modal Let L stand for both K and N : AILEP 8 / 18 Only-Knowing G Lakemeyer Axioms (propositional) objective: non-modal formulas subjective: all predicates within a modal Let L stand for both K and N : 1. 2. 3. 4. Axioms of propositional logic. L(α ⊃ β) ⊃ Lα ⊃ Lβ. σ ⊃ Lσ, where σ is subjective. The N vs. K axiom: (N φ ⊃ ¬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 Only-Knowing G Lakemeyer Many Agents Intuitively, only-knowing for many agents seems easy: E XAMPLE 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 Only-Knowing G Lakemeyer Many Agents Intuitively, only-knowing for many agents seems easy: E XAMPLE 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: I For a single agent possibilities are just worlds. I For many agents possibilities include other agents beliefs. I The problem is that is not so clear how to come up with models that contain all possibilities. AILEP 9 / 18 Only-Knowing G Lakemeyer Some Previous Attempts I (Lakemeyer 1993) uses the K45n -canonical model I Yet, certain types of epistemic states cannot be constructed ¬Oa ¬Ob p is valid I (Halpern 1993) proposes a tree approach I Modalities do not interact in an intuitive manner I In (Halpern and Lakemeyer 2001), a solution is proposed. But I I AILEP again uses canonical models proof theory needs to axiomatize validity 10 / 18 Only-Knowing G Lakemeyer The Logic ON Ln I ON Ln =. multi-agent version of ON L I Here, only for a and b (Ka , Kb , Na , Nb ) I depth: alternating nesting of modalities I a notion of a-depth and b-depth E XAMPLE (a- DEPTH ) I I I I p: 1 Ka p: 1 Kb p: 2 Ka Kb p: 2 I a-objective: formulas not in scope of Ka or Na : p ∧ Kb p is a-objective, p ∧ Ka p is not. AILEP 11 / 18 Only-Knowing G Lakemeyer Beyond Sets of Worlds I 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 I 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) I To be well-defined, we can do this only to some finite depth I 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 Only-Knowing G Lakemeyer Formal Semantics I Define an epistemic state for Alice as a set of pairs I I ea1 = {hw, {}i, hw 0 , {}i . . .} (for formulas of a-depth 1) eak = {hw, ebk −1 i, . . .} I Similarly, an epistemic state for Bob I I eb1 = {hw, {}i . . .} ebj = {hw, eaj−1 i, . . .} I (k , j)-model =. heak , ebj , wi. I I I AILEP Given a formula of a-depth k and b-depth j eak , ebj , w |= Ka α iff for all hw 0 , ebk −1 i ∈ eak , eak , ebk −1 , w 0 |= α eak , ebj , w |= Na α iff for all hw 0 , ebk −1 i 6∈ eak , eak , ebk −1 , w 0 |= α Oa α ≡ Ka α ∧ Na ¬α. 13 / 18 Only-Knowing G Lakemeyer Some Properties A formula of a-depth k and b-depth j. is valid if it is true at all (k 0 , j 0 )-models for k 0 ≥ k , j 0 ≥ j. I K45n (for Ki and Ni ) I I I Ka (α ⊃ β) ⊃ Ka α ⊃ Ka β Ka α ⊃ Ka Ka α ¬Ka α ⊃ Ka ¬Ka α I Mutual introspection: I Ka α ⊃ Na Ka α I Barcan formula I AILEP ∀x Ka α ⊃ Ka ∀x α 14 / 18 Only-Knowing G Lakemeyer Examples of Only-Knowing I Oa (p ∧ Ob p) entails I I I I AILEP Ka p Ka Kb p Ka ¬Kb Ka p but not Ka Kb ¬Ka p 15 / 18 Only-Knowing G Lakemeyer Examples of Only-Knowing I Oa (p ∧ Ob p) entails I I I I Ka p Ka Kb p Ka ¬Kb Ka p but not Ka Kb ¬Ka p I Ka Ob ∃x T (x) entails I AILEP Ka Kb (∃x T (x) ∧ ¬Kb T (x)) 15 / 18 Only-Knowing G Lakemeyer Examples of Only-Knowing I Oa (p ∧ Ob p) entails I I I I Ka p Ka Kb p Ka ¬Kb Ka p but not Ka Kb ¬Ka p I Ka Ob ∃x T (x) entails I Ka Kb (∃x T (x) ∧ ¬Kb T (x)) I Let δb = ∀x.Kb Bird(x) ∧ ¬Kb ¬Fly (x) ⊃ Fly (x) Ka Ob (δb ∧ Bird(tweety )) entails I AILEP Ka Kb Fly (tweety ). 15 / 18 Only-Knowing G Lakemeyer Proof Theory for Many Agents I K45n + Axiom defining Oi + new version of the N vs K axiom. I Recall, in the single-agent case: (N φ ⊃ ¬Kφ), where ¬φ is consistent and objective. I Two things to generalize here: I I AILEP Objective formulas to i-objective formulas But generalizing the notion of consistency of i-objective formulas is circular! 16 / 18 Only-Knowing G Lakemeyer Ni vs. Ki Axioms Idea: break the circularity by considering a hierarchy of sub-languages based on the nesting of Ni . I Define a family of languages I I AILEP . Let ON L1n = no Nj in the scope of Ki , Ni (i 6= j) t+1 Let ON Ln formed from ON Ltn , Ki α and Ni α for all α ∈ ON Ltn 17 / 18 Only-Knowing G Lakemeyer Ni vs. Ki Axioms Idea: break the circularity by considering a hierarchy of sub-languages based on the nesting of Ni . I Define a family of languages I I . Let ON L1n = no Nj in the scope of Ki , Ni (i 6= j) t+1 Let ON Ln formed from ON Ltn , Ki α and Ni α for all α ∈ ON Ltn I Proof theory is K45n + Def. of Oi + A1n Ni α ⊃ ¬Ki α if ¬α is a K45n -consistent i-objective formula At+1 Ni α ⊃ ¬Ki α, if ¬α ∈ ON Ltn , is i-objective and n consistent wrt. K45n , A1n − Atn T HEOREM For all α ∈ ON Ltn , |= α iff Axiomst ` α AILEP 17 / 18 Only-Knowing G Lakemeyer Conclusions I First-order modal logic for multi-agent only-knowing I Faithfully generalizes intuitions of Levesque’s logic I Semantics not based on Kripke structures or canonical models, and thus avoids some problems of previous approaches I 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