Dynamic epistemic logic and lying
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1 Dynamic epistemic logic and lying Hans van Ditmarsch LORIA CNRS / Univ. de Lorraine & (associate) IMSc, Chennai hans.van-ditmarsch@loria.fr
2
3 The Ditmarsch Tale of Wonders I will tell you something. I saw two roasted fowls flying; they flew quickly and had their breasts turned to Heaven and their backs to Hell; and an anvil and a mill-stone swam across the Rhine prettily, slowly, and gently; and a frog sat on the ice at Whitsuntide and ate a ploughshare.... Open the window that the lies may fly out. Jacob Ludwig Grimm and Wilhelm Carl Grimm, Fairy Tales The Ditmarsch Tale of Wonders is called in German: Das Dietmarsische Lügenmärchen.
4 Lying and truth telling In the Grimm Brothers fairy tale it is clear that the speaker lies. Can you lie without the listener noticing? What are the informative consequences of lying? What are the informative consequences of telling the truth? Let us recall public announcement logic.
5 Consecutive numbers Anne and Bill are each going to be told a natural number. Their numbers will be one apart. The numbers are now being whispered in their respective ears. They are aware of this scenario. Suppose Anne is told 2 and Bill is told 3. The following truthful conversation between Anne and Bill now takes place: Bill: I do not know your number. Anne: I know your number. Bill: I know your number. Explain why is this possible. Oldest known source: Littlewood, A Mathematician s Miscellany, 1953
6 Consecutive numbers representing uncertainties (1,0) a (1,2) b (3,2) a (3,4)... (0,1) b (2,1) a (2,3) b (4,3)...
7 Consecutive numbers successive announcements (1,0) a (1,2) b (3,2) a (3,4)... (0,1) b (2,1) a (2,3) b (4,3)...??
8 Consecutive numbers successive announcements (1,0) a (1,2) b (3,2) a (3,4)... (0,1) b (2,1) a (2,3) b (4,3)... eliminated states
9 Consecutive numbers successive announcements (1,0) a (1,2) b (3,2) a (3,4)... (2,1) a (2,3) b (4,3)...
10 Consecutive numbers successive announcements (1,0) a (1,2) b (3,2) a (3,4)... (2,1) a (2,3) b (4,3)... Bill: I do not know your number.??
11 Consecutive numbers successive announcements (1,0) a (1,2) b (3,2) a (3,4)... (2,1) a (2,3) b (4,3)... Bill: I do not know your number. eliminated states
12 Consecutive numbers successive announcements (1,2) b (3,2) a (3,4)... (2,3) b (4,3)... Bill: I do not know your number.
13 Consecutive numbers successive announcements (1,2) b (3,2) a (3,4)... (2,3) b (4,3)... Bill: I do not know your number. Anne: I know your number.??
14 Consecutive numbers successive announcements (1,2) b (3,2) a (3,4)... (2,3) b (4,3)... Bill: I do not know your number. Anne: I know your number. eliminated states
15 Consecutive numbers successive announcements (1,2) (2,3) Bill: I do not know your number. Anne: I know your number.
16 Consecutive numbers successive announcements (1,2) (2,3) Bill: I do not know your number. Anne: I know your number. Bill: I know your number.??
17 Consecutive numbers successive announcements (1,2) (2,3) Bill: I do not know your number. Anne: I know your number. Bill: I know your number. already common knowledge
18 Consecutive numbers successive announcements (1,2) (2,3) Bill: I do not know your number. Anne: I know your number. Bill: I know your number.
19 Truthful (true) public announcement logic Jan Plaza, Logics of public communications, 1989 & 2007 Structures: pointed Kripke models E.g., (0,1) b (2,1) a (2,3)... Language: p ϕ ϕ ϕ B a ϕ [!ϕ]ϕ B a ϕ for agent a believes/knows ϕ ; [!ϕ]ψ for after truthful announcement of ϕ, ψ is true. Semantics: B a ϕ is true in a state iff: ϕ is true in all a-accessible states. In state (0,1) formula B b 1 b (b knows his number is 1 ) is true. [!ϕ]ψ is true in a state iff: whenever ϕ is true, in the restriction of the model to the ϕ-states, ψ is true. In state (0,1) formula B b 0 a [!0 a ]B b 0 a is true. Lying in public announcement logics Let us start with an example...
20 Consecutive numbers, with lying (0,1) (2,1) (2,3) (4,3)... b a b Bill: I do not know your number. Anne: I know your number. Bill: I know your number.
21 Consecutive numbers, with lying (0,1) (2,1) (2,3) (4,3)... b a b Bill: I do not know your number. Anne: I know your number. Bill: I know your number. Anne: I know your number. Anne is lying
22 Consecutive numbers, with lying (0,1) (2,1) (2,3) (4,3)... b a b Bill: I do not know your number. Anne: I know your number. Bill: I know your number. Anne: I know your number. Anne is lying Bill: You re lying.
23 Consecutive numbers, with lying (0,1) (2,1) (2,3) (4,3)... b a b Bill: I do not know your number. Anne: I know your number. Bill: I know your number.
24 Consecutive numbers, with lying (0,1) (2,1) (2,3) (4,3)... b a b Bill: I do not know your number. Anne: I know your number. Bill: I know your number.
25 Consecutive numbers, with lying (0,1) (2,1) (2,3) (4,3)... b a b Bill: I do not know your number. Anne: I know your number. Bill: I know your number. Bill: I know your number.
26 Consecutive numbers, with lying (0,1) (2,1) (2,3) (4,3)... b a b Bill: I do not know your number. Anne: I know your number. Bill: I know your number. Bill: I know your number. Bill is lying
27 Consecutive numbers, with lying (0,1) (2,1) (2,3) (4,3)... b a b Bill: I do not know your number. Anne: I know your number. Bill: I know your number. Bill: I know your number. Bill is lying Anne: I know your number.
28 Consecutive numbers, with lying (0,1) (2,1) (2,3) (4,3)... b a b Bill: I do not know your number. Anne: I know your number. Bill: I know your number. Bill: I know your number. Bill is lying Anne: I know your number. Anne is mistaken. Anne thinks to know that Bill has 1.
29 Consecutive numbers, with lying (0,1) (2,1) (2,3) (4,3)... b a b Bill: I do not know your number. Anne: I know your number. Bill: I know your number. Bill: I know your number. Bill is lying Anne: I know your number. Anne is mistaken. Anne thinks to know that Bill has 1. (Bill: I know your number. By now, this is true!)
30 What is a lie? You are lying if you say to me that ϕ (is true), but believe that ϕ (is true). (With the intention for me to believe ϕ.) The lie was effective (the intention has been successfully realized) if I now believe that ϕ was true. ( Was, not is, for technical reasons.) For me to believe your lie that ϕ, I must consider it possible that ϕ. (Otherwise I will believe that you re lying!) Lying by an outside observer Lying public announcement The agents are the listeners whose beliefs are modelled. Lies are announcements made by an outsider (not modelled). The announcements are always believed.
31 Example of truthful public announcement Let p be the proposition Oranges freeze in Sevilla. Agent a does not know whether this is true. This uncertainty can be modelled as follows: p p a After the announcement of p we get: p p a!p p To model lying, later on, we need a more explicit visualization: a p p a a!p p a
32 A different semantics: Believed announcements An alternative to the logic of truthful public announcements is the logic of believed (public) announcements. The effect of the announcement of ϕ, is that only states where ϕ is true remain accessible for the agents. The announcement may be false. a p p a a p p p a a After the announcement, a believes that oranges freeze in Sevilla. No matter what the truth is. Believed announcements are investigated in: Jelle Gerbrandy, Bisimulations on Planet Kripke, ILLC 1999 Barteld Kooi, Expressivity (...) via reduction axioms. Journal of Applied Non-Classical Logics 17(2): , 2007
33 Lie as Lying public announcement In case of a lie that ϕ: ϕ is false; it is announced that ϕ is true; after the announcement, agent a believes that ϕ (was true). Lying public announcement for which we write ϕ is one execution of believed announcement: a p p a a p p p a a Truthful public announcement!ϕ is another execution of believed announcement: a p p a a!p p p a a
34 Principles of lying public announcement Axioms for truthful public announcement: Dual axioms for lying: [!ϕ]p ϕ p [!ϕ] ψ ϕ [!ϕ]ψ [!ϕ](ψ 1 ψ 2 ) [!ϕ]ψ 1 [!ϕ]ψ 2 [!ϕ]b i ψ ϕ B i [!ϕ]ψ [ ϕ]p ϕ p [ ϕ] ψ ϕ [ ϕ]ψ [ ϕ](ψ 1 ψ 2 ) [ ϕ]ψ 1 [ ϕ]ψ 2 [ ϕ]b i ψ ϕ B i [!ϕ]ψ Combined, the principles deliver: (where [ϕ]ψ ([!ϕ]ψ [ ϕ]ψ)) [ϕ]b i ψ B i (ϕ [ϕ]ψ)
35 Principles of lying public announcement [ ϕ]b i ψ ϕ B i [!ϕ]ψ (After the lie that ϕ, agent i believes that ψ,) iff, (on condition that ϕ is false, agent i believes that ψ after truthful announcement that ϕ). In lying (and truthful) public announcement agents may go crazy (empty access / believe inconsistencies). This is a problem. p p a a! p p p Going crazy can be avoided elegantly by requiring that the listeners consider it possible that the lie is true: preconditions B a ϕ. Hans van Ditmarsch, Jan van Eijck, Yanjing Wang, Floor Sietsma. On the logic of lying, LNCS 7010, pp , 2012.
36 Lying public announcement to lying agent announcement In lying public announcement it is implicit that the speaker believes that the announcement is false. We can make this explicit. The result is the logic of (lying) agent announcement. Consider the information state where a does not know whether p, b knows whether p, and p is true. b a p p a a b Oranges freeze in Sevilla. Bill (b) knows whether this is true. Anne (a) is ignorant. (And this is common knowledge.)
37 Lying agent announcement example Clearly, a public announcement is not a lie from b to a. b a p a p a b p a b!p A public lie is also not a lie from b to a. b a p a p a b b a p a p p Instead, a lie from b to a should have the following effect: b a p a p a b b p b a p p a b
38 Lying agent announcement example b a p a p a b b p b a p p a b After agent b lies to a that p, we have that: b still believes that p; a believes that p; a believes that b believes p; (a believes that a and b have common belief of p.)
39 Semantics and principles of agent announcement The accessibility relation for speaker b does not change. The accessibility relation for listener a changes: only the states where (speaker) b believes ϕ remain accessible for a. Preconditions of agent announcements (by b) that ϕ Truthful agent announcement! b ϕ: B b ϕ Lying agent announcement b ϕ: B b ϕ Bluffing agent announcement! b ϕ: (B b ϕ B b ϕ) Principles for b lying to a that ϕ (abbreviation [ b ϕ]ψ ([! b ϕ]ψ [ b ϕ]ψ [! b ϕ]ψ)) [ b ϕ]b a ψ B b ϕ B a [! b ϕ]ψ [ b ϕ]b b ψ B b ϕ B b [ b ϕ]ψ [! b ϕ]b a ψ (B b ϕ B b ϕ) B a [! b ϕ]ψ...
40 When speaker b is caught as a liar This lie is believed: b a p a p a b b p b a p p a b This lie should not be believed: p a b b p p b Agent a now believes everything. (There are no arrows for a.) We can elegantly solve this by strengthening the precondition to B a B b ϕ (listener a considers possible that speaker b believes ϕ). Hans van Ditmarsch, Dynamics of lying, Synthese, 2013
41 The invention of lying
42 Consecutive numbers with lying (0,1) (2,1) (2,3) (4,3)... b a b
43 Consecutive numbers with lying ab ab ab ab (0,1) (2,1) (2,3) (4,3)... b a b
44 Consecutive numbers with lying ab ab ab ab (0,1) (2,1) (2,3) (4,3)... b a b Anne: I know your number. Anne is lying
45 Consecutive numbers with lying ab ab ab ab (0,1) (2,1) (2,3) (4,3)... b a b Anne: I know your number. Anne is lying ab a a a (0,1) b (2,1) a (2,3) (4,3)...
46 Consecutive numbers with lying ab ab ab ab (0,1) (2,1) (2,3) (4,3)... b a b Anne: I know your number. Anne is lying ab a a a (0,1) b (2,1) a (2,3) (4,3)... Bill: That s a lie.
47 Consecutive numbers with lying ab ab ab ab (0,1) (2,1) (2,3) (4,3)... b a b
48 Consecutive numbers with lying ab ab ab ab (0,1) b (2,1) a (2,3) b (4,3)...
49 Consecutive numbers with lying ab ab ab ab (0,1) b (2,1) a (2,3) b (4,3)... a ab ab ab (0,1) (2,1) (2,3) (4,3)... b a b
50 Consecutive numbers with lying ab ab ab ab (0,1) b (2,1) a (2,3) b (4,3)... a ab ab ab (0,1) b (2,1) a (2,3) b (4,3)... Bill: I know your number. Bill is lying
51 Consecutive numbers with lying ab ab ab ab (0,1) b (2,1) a (2,3) b (4,3)... a ab ab ab (0,1) b (2,1) a (2,3) b (4,3)... Bill: I know your number. Bill is lying a ab b b (0,1) (2,1) (2,3) (4,3)... b a b
52 Consecutive numbers with lying ab ab ab ab (0,1) b (2,1) a (2,3) b (4,3)... a ab ab ab (0,1) b (2,1) a (2,3) b (4,3)... Bill: I know your number. Bill is lying a ab b b (0,1) b (2,1) a (2,3) b (4,3)... Anne: I know your number. Anne is mistaken.
53 Consecutive numbers with lying ab ab ab ab (0,1) b (2,1) a (2,3) b (4,3)... a ab ab ab (0,1) b (2,1) a (2,3) b (4,3)... Bill: I know your number. Bill is lying a ab b b (0,1) b (2,1) a (2,3) b (4,3)... Anne: I know your number. Anne is mistaken. a ab b (0,1) (2,1) (2,3) (4,3)... b a b
54 Results for lying announcement Public announcement and agent announcement can be represented by an action model, i.e., a general framework for epistemic dynamics. This translation provides axiomatizations (and upper bounds for complexity issues). A skeptical agent does not accept new information ϕ if it already believes ϕ: more complex preconditions. Agents may distinguish between more and less plausible states, and more and less plausible actions: a truthful announcement is more plausible than a lying announcement (and a bluffing announcement in between the two). Unless you receive information to the contrary, you will assume the announcement is truthful. Otherwise, that it was bluffing. Otherwise, that it was a lie. In fair games players can distinguish lies from mistakes. Let p stand for Oranges freeze in Sevilla...
55 Oranges in Sevilla p = Oranges freeze in Sevilla (as Hans claims) a = speaker (me) b = listener (you) Truthful announcement that p: Lying announcement that p: Bluffing announcement that p: Honest mistake that p: The postcondition that holds: If you are skeptical, precondition: if B a p and! a p B a p and a p (B a p B a p) and! a p p B a p en! a p B b p B b B a p
56 Further issues with lying Incorporating common knowledge / common belief: B a ϕ B b ϕ C ab ((B a ϕ B a ϕ) (B b ϕ B b ϕ)) Insincere or strategic voting in social choice is a form of lying. Protocols with few liars or few lies. (Ulam Games) Signal analysis: noise versus intentional noise. Modelling a Liar Paradox in dynamic epistemic logic. Computational complexity of lying versus truthfulness Ref: Hans van Ditmarsch, Dynamics of Lying, Synthese, Contains references to other recent publications on lying. Thank you!
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