Agent Communication and Belief Change
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1 Agent Communication and Belief Change Satoshi Tojo JAIST November 30, 2014 Satoshi Tojo (JAIST) Agent Communication and Belief Change November 30, / 34
2 .1 Introduction.2 Introspective Agent.3 I know that you don t know.4 Opaqueness.5 Commitment and Permission.6 Now and Future Satoshi Tojo (JAIST) Agent Communication and Belief Change November 30, / 34
3 .1 Introduction.2 Introspective Agent.3 I know that you don t know.4 Opaqueness.5 Commitment and Permission.6 Now and Future Satoshi Tojo (JAIST) Agent Communication and Belief Change November 30, / 34
4 My Specialité Natural Language Grammar Theory Formal Semantics Logic of Knowledge and Belief Communicative Agent Legal Reasoning Language Acquisition/Evolution Grammar of Music Satoshi Tojo (JAIST) Agent Communication and Belief Change November 30, / 34
5 Classical Logic and Natural Language Transitivity A B and B C implies A C If it rained hard, it rained. If it rained, it didn t rain hard. Therefore it rained hard, it didn t rain hard? Contraposition A B implies B A If she wrote a letter to Santa Claus, she didn t get an answer. Therefore, if she got an answer from Santa Claus, she didn t write a letter to him? Strengthening (Weakening) A C implies A B C If Betty had been at the party, Bill would have a good time. Therefore, if Betty had been at the party and Bill had broken his leg, he would have had a good? Satoshi Tojo (JAIST) Agent Communication and Belief Change November 30, / 34
6 .1 Introduction.2 Introspective Agent.3 I know that you don t know.4 Opaqueness.5 Commitment and Permission.6 Now and Future Satoshi Tojo (JAIST) Agent Communication and Belief Change November 30, / 34
7 Satoshi Tojo (JAIST) Agent Communication and Belief Change November 30, / 34
8 Satoshi Tojo (JAIST) Agent Communication and Belief Change November 30, / 34
9 Satoshi Tojo (JAIST) Agent Communication and Belief Change November 30, / 34
10 Who knows what at which time? Satoshi Tojo (JAIST) Agent Communication and Belief Change November 30, / 34
11 Who knows what at which time? Satoshi Tojo (JAIST) Agent Communication and Belief Change November 30, / 34
12 History Modal Logic [Aristotle?] Kripke semantics [Kripke 1950s] Prolog Closed World Assumption [Colmerauer, Kowalski ca.1980] Truth-maintenance system [Doyle 1979] Default Logic [Reiter 1980] Autoepistemic Logic [Moore, Konolidge 1985] Circumscription [McCarthy 1980, Lifschitz 1985] Belief Revision AGM axioms [Gördenfors, Makinson 1988] Defeasible Logic [Nute 1994] Theory of Argumentation [Dung 1995] Dynamic Epistemic Logic [ca.2005 ] Satoshi Tojo (JAIST) Agent Communication and Belief Change November 30, / 34
13 Autoepistemic Logic Are the Rolling Stones giving a concert next week? No, because otherwise I would kave heard about it. Very strong introspection I know everything what is true. ϕ E implies Lϕ E. ϕ E implies Lϕ E. E: expansion of knowledge T of an introspective agent. Let LE be {Lϕ ϕ E}, LE c be { Lϕ ϕ E}, and Ω T (E) ={ϕ T LE LE c = ϕ}. E is an expansion of T iff E =Ω T (E). Satoshi Tojo (JAIST) Agent Communication and Belief Change November 30, / 34
14 Possible World Semantics w 0 w 1 w 2 p, q p, q p, q w 0 = Bp w 0 = Bq w 1 = B q w 2 = B p Satoshi Tojo (JAIST) Agent Communication and Belief Change November 30, / 34
15 Knowledge and Belief K a ϕ B a ϕ agent a knows ϕ agent a believes ϕ Logic of K.1 K a ϕ ϕ (T).2 K a ϕ K a K a ϕ (4).3 K a ϕ K a K a ϕ (5) Logic of B.1 B a ϕ B a ϕ (D).2 B a ϕ B a B a ϕ (4).3 B a ϕ B a B a ϕ (5) Satoshi Tojo (JAIST) Agent Communication and Belief Change November 30, / 34
16 .1 Introduction.2 Introspective Agent.3 I know that you don t know.4 Opaqueness.5 Commitment and Permission.6 Now and Future Satoshi Tojo (JAIST) Agent Communication and Belief Change November 30, / 34
17 I know that you don t know Now, I have a card, that is either 4, 5, 8, 4, 7, 1, 5, 1, 2, 8. Guess what I have. S only knows the suit and N only knows the number..1 N says: I don t know the answer..2 S: I have known that you don t know it..3 N: Now,Iknowtheanswer..4 S: If you know it, then I come to know it too. What is this card? Satoshi Tojo (JAIST) Agent Communication and Belief Change November 30, / 34
18 N doesn t know the suit (, 4) (, 5) (, 8) (, 7) (, 4) (, 1) (, 5) (, 1) (, 2) (, 8) Satoshi Tojo (JAIST) Agent Communication and Belief Change November 30, / 34
19 N knows the number, then S knows. (, 4) (, 5) (, 8) (, 1) (, 5) Satoshi Tojo (JAIST) Agent Communication and Belief Change November 30, / 34
20 Sum and Product A says to S and P: I have chosen two natural numbers x and y such that 1 < x < y and x + y 100. I am now going to announce their sum s = x + y to S only, and their product p = x y to P only. The content of these announcement remains a secret..1 P says: I don t know the answer..2 S: I have known that you don t know it..3 P: Now,Iknowtheanswer..4 S: If you know it, then I come to know it too. Determine the numbers x and y. Tips: If the division of p is one way (e.g., 2 4, 3 9,,orx and y are primes), P would immediately answer I know the numbers ; if there is no such pair for s, S can say I knew that P didn t know the numbers. cf. Goldbach conjecture. Satoshi Tojo (JAIST) Agent Communication and Belief Change November 30, / 34
21 Lupus and Tabula Satoshi Tojo (JAIST) Agent Communication and Belief Change November 30, / 34
22 .1 Introduction.2 Introspective Agent.3 I know that you don t know.4 Opaqueness.5 Commitment and Permission.6 Now and Future Satoshi Tojo (JAIST) Agent Communication and Belief Change November 30, / 34
23 Japanese Raccoon Dog Satoshi Tojo (JAIST) Agent Communication and Belief Change November 30, / 34
24 Possibilities in Raccoon or Badger Satoshi Tojo (JAIST) Agent Communication and Belief Change November 30, / 34
25 Belief in Possible Worlds Semantics w 0 w 1 w 2 p, q p, q p, q w 0 = Bp w 0 = Bq w 1 = B q w 2 = B p w 0 w 1 w 2 p, q p, q p, q w 0 = Bp w 0 = Bq w 1 = B q w 2 = B p Satoshi Tojo (JAIST) Agent Communication and Belief Change November 30, / 34
26 How many parallel worlds do we need? Given n atomic propositions, then the possibility is 2 n ; is it enough? Actually no. We need to provide those with different accessibility. Satoshi Tojo (JAIST) Agent Communication and Belief Change November 30, / 34
27 Accessibility Akripkeframe:M = W, A, (R k ) k A, V (A is a set of agents and R k is the accessibility of belief modal operator B k ). B k =(R k )= to w 1 w 2 w 3 w 4 from {}}{ w 1 w 2 w 3 w represents R k = {w 1 Rw 1 w 1 Rw 2, w 2 Rw 2, w 2 Rw 3, w 3 Rw 3, w 3 Rw 4, w 4 Rw 1, w 4 Rw 2 } in W = {w 1, w 2, w 3, w 4 }. We can easily investigate if T, D, 4 or 5. Satoshi Tojo (JAIST) Agent Communication and Belief Change November 30, / 34
28 Belief Revision by Public Announcement Let calculus be Boolean: = 1. Ex. Suppose W = {w 1, w 2, w 3, w 4 } and V(ϕ) ={w 1, w 3 },then[ϕ!] prohibits accesses to {w 2, w 4 }; i.e., the matrix is the unit matrix with 2nd and 4th 1 s are knocked out by zeros. Now, belief update is }{{} revised: B i = }{{} [ϕ!] } {{ } original: B i Satoshi Tojo (JAIST) Agent Communication and Belief Change November 30, / 34
29 Digression Do we need Modal Logic? Satoshi Tojo (JAIST) Agent Communication and Belief Change November 30, / 34
30 Programming Language factorial(0,1). factorial(n,f):- N1 is N-1, factorial(n1, F1), F is N*F1. Satoshi Tojo (JAIST) Agent Communication and Belief Change November 30, / 34
31 .1 Introduction.2 Introspective Agent.3 I know that you don t know.4 Opaqueness.5 Commitment and Permission.6 Now and Future Satoshi Tojo (JAIST) Agent Communication and Belief Change November 30, / 34
32 Commitment and Permission [ϕ i j ]: after i informs j of ϕ if there is a channel. M, w = [ϕ i j ]ψ M[ϕ i j ], w = ψ, where M [ϕ i j ] = W, (R k ) k G, (C ij ) i,j V and R j (x) := { R j (x) ϕ M R j (x) if x =@andx C ij otherwise. Satoshi Tojo (JAIST) Agent Communication and Belief Change November 30, / 34
33 Let M = W, (R i ) i V. commitment M, w = [com j (ϕ)]ψ M [com j (ϕ)], w = ψ, where M [com j (ϕ)] = W, (R i ) i G\{j}, S V and permission S j (x) := { R j (x) ϕ M R j (x) if x =@, otherwise. M, w = [per j (ϕ)]ψ M [per j (ϕ)], w = ψ, where M [per j (ϕ)] = W, (R i ) i G\{j}, S V and { S j R j (x) ϕ M if x =@, (x) := R j (x) otherwise. Satoshi Tojo (JAIST) Agent Communication and Belief Change November 30, / 34
34 .1 Introduction.2 Introspective Agent.3 I know that you don t know.4 Opaqueness.5 Commitment and Permission.6 Now and Future Satoshi Tojo (JAIST) Agent Communication and Belief Change November 30, / 34
35 Now and Future Reliability Detective Story as Logical Puzzle Channel Communication Linear Algebraic Representation Liar to represent Who Knows What at Which Time. Satoshi Tojo (JAIST) Agent Communication and Belief Change November 30, / 34
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