The dynamics of lying. Hans van Ditmarsch, LORIA CNRS / Université de Lorraine
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1 The dynamics of lying Hans van Ditmarsch, LORIA CNRS / Université de Lorraine hvd@us.es
2
3 Das Dietmarsische Lügenmärchen Ich will euch etwas erzählen. Ich sah zwei gebratene Hühner fliegen, flogen schnell und hatten die Bäuche gen Himmel gekehrt, die Rücken nach der Hölle, und ein Amboß und ein Mühlstein schwammen ber den Rhein, fein langsam und leise, und ein Frosch saß und fraß eine Pflugschar zu Pfingsten auf dem Eis. Da waren drei Kerle, wollten einen Hasen fangen, gingen auf Krücken und Stelzen, der eine war taub, der zweite blind, der dritte stumm und der vierte konnte keinen Fuß rühren. Wollt ihr wissen, wie das geschah? Der Blinde, der sah zuerst den Hasen ber Feld traben, der Stumme rief dem Lahmen zu, und der Lahme faßte ihn beim Kragen. Etliche, die wollten zu Land segeln und spannten die Segel im Wind und schifften ber große Äcker hin: da segelten sie ber einen hohen Berg, da mußten sie elendig ersaufen. Ein Krebs jagte einen Hasen in die Flucht, und hoch auf dem Dach lag eine Kuh, die war hinaufgestiegen. In dem Lande sind die Fliegen so groß als hier die Ziegen. Mache das Fenster auf, damit die Lügen hinausfliegen. Gebrüder Grimm
4 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
5 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?
6 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.
7 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.
8 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)...
9 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)...??
10 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
11 Consecutive numbers successive announcements (1,0) a (1,2) b (3,2) a (3,4)... (2,1) a (2,3) b (4,3)...
12 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.??
13 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
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.
15 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.??
16 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
17 Consecutive numbers successive announcements (1,2) (2,3) Bill: I do not know your number. Anne: I know your number.
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 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
20 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.
21 Public announcement logic Jan Plaza, Logics of public communications, 1989 & 2007 We can model truth-telling in public announcement logic. We can model lying in versions and extensions of that logic. Let us start with an example...
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.
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. Anne: I know your number. Anne is lying
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. Anne: I know your number. Anne is lying Bill: You re lying.
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.
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.
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.
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
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.
30 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.
31 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!)
32 What is a lie? You are lying if you say to me that ϕ (is true), but believe that ϕ (is true). The lie is effective 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 The single agent is the listener whose beliefs are modelled. Lies are announcements made by an outsider (not modelled). First version: The announcements are always believed.
33 Ignorance 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
34 A different semantics for announcements An alternative to the logic of public announcements is the logic of believed announcements. The effect of the announcement of ϕ is that only states where ϕ is true are accessible for the agents. The announcement may be false. This has been called manipulative announcement (update). a p p a a!p p p a a After the announcement, the agent believes that Oranges freeze in Sevilla. It is unrelated to whether this is true. Alternative semantics for announcement: Jelle Gerbrandy, Bisimulations on Planet Kripke, ILLC 1999 Barteld Kooi, Expressivity (...) via reduction axioms. Journal of Applied Non-Classical Logics 17(2): , 2007
35 Lie In case of a lie that p: p is false; it is announced that p is true; after the announcement, agent a believes that p. There one execution of manipulative announcement: a p p a a p p p a a The other execution of manipulative announcement is: a p p a a!p p p a a
36 Principles of public lying 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 the familiar axiom: [ ϕ]b i ψ B i (ϕ [ ϕ]ψ)
37 Principles of public lying [ 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. A problem with lying (and with truthful announcements): agents going crazy (empty access / believing everything). p p a a!p p p Hans van Ditmarsch, Jan van Eijck, Yanjing Wang, Floor Sietsma. On the logic of lying, LNCS 7010, pp , 2012.
38 Agent-to-agent lying In public lying it is implicit that the speaker believes that the announcement is false. We can make this explicit in multi-agent epistemic logic. 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.)
39 More agents 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
40 More agents 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
41 Agent b lies to a that p b a p a p a b b p b a p p a b After this successful lie we have that: b still believes that p; a believes that p; a believes that a and b have common belief of p.
42 Agent b is lying or telling the truth to agent a that ϕ States where b believes ϕ remain accessible to b. States where b believes ϕ remain accessible to b. States where b believes ϕ remain accessible to a. States where b believes ϕ are no longer accessible to a. Preconditions of agent announcements (by b) that ϕ Truthful agent announcement: B b ϕ Lying agent announcement: B b ϕ Bluffing agent announcement: (B b ϕ B b ϕ) Principles for b lying to a that ϕ [ 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...
43 When agent b is caught out as a liar This lie is believed: b a p a p a b b p b a p p a b This lie is not believed: p a b b p p b Agent b now believes everything. (There are no arrows for a.) Hans van Ditmarsch, The Ditmarsch Tale of Wonders the dynamics of lying, manuscript, 2012
44 The invention of lying
45 Consecutive numbers with lying (0,1) (2,1) (2,3) (4,3)... b a b
46 Consecutive numbers with lying ab ab ab ab (0,1) (2,1) (2,3) (4,3)... b a b
47 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
48 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)...
49 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.
50 Consecutive numbers with lying ab ab ab ab (0,1) (2,1) (2,3) (4,3)... b a b
51 Consecutive numbers with lying ab ab ab ab (0,1) b (2,1) a (2,3) b (4,3)...
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) (2,1) (2,3) (4,3)... b a b
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
54 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
55 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.
56 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
57 Action models: agent perspectives on actions Truthful public announcement of ϕ: Same as: ϕ a ϕ ϕ a a A lie to a that ϕ: ϕ ϕ a a Action model for b truthtelling, lying and bluffing to a: a ab a b b (B a ϕ B a ϕ) B a ϕ B a ϕ
58 Skeptical agents and plausible information 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. A lie to a that ϕ: A lie to a skeptical agent: ϕ a ϕ a a a ϕ B a ϕ ϕ B a ϕ The truth that ϕ, is more plausible than the lie that ϕ: a a a ϕ ϕ a a
59 Oranges in Sevilla p = Oranges freeze in Sevilla a = Hans b = you Truthful announcement that p: B a p and! a p Lying announcement that p: B a p and a p Bluffing announcement that p: (B a p B a p) and! a p Honest mistake that p: p B a p en! a p The postcondition that holds: B b p If you are skeptical, precondition: B b p
60 The Lying Game Question-answer games with lies. Strongly negative payoffs for being caught as a liar.
61 Further issues with lying Incorporating common knowledge/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. Signal analysis: noise versus intentional noise. Liar s Paradox in dynamic epistemic logic!
62 Further issues with lying Incorporating common knowledge/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. Signal analysis: noise versus intentional noise. Liar s Paradox in dynamic epistemic logic! Thank you!
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