Reversed Squares of Opposition in PAL and DEL

Transcription

1 Center for Logic and Analytical Philosophy, University of Leuven SQUARE 2010, Corsica

2 Goal of the talk public announcement logic (PAL), and dynamic epistemic logic (DEL) in general, are recent field of epistemic logic that study how knowledge changes under the influence of various actions (e.g. communication between agents) many applications in computer science (protocol security), AI (multi-agent systems), game theory (backward induction paradox), linguistics (formal semantics) etc... the goal of this talk is to connect PAL with the philosophical tradition of the square of opposition: squares/hexagons arise in a nontrivial way in PAL squares/hexagons are compact representations of the subtle relations between knowledge and dynamics studied in PAL

3 Outline of the talk 1 Public Announcement Logic and Dynamic Epistemic Logic Intuitive idea Syntax and semantics Some key validities 2 3

4 Disclaimer Intuitive idea Syntax and semantics Some key validities dynamic epistemic logic studies any type of epistemic actions (whispering, cheating, private communication etc... ) public announcement logic is a small fragment: studies only public announcements for expository reasons, we will focus on PAL everything that follows can be generalized to full DEL

5 Intuitive idea Intuitive idea Syntax and semantics Some key validities a community Ag of agents an outside source S announces that ϕ publicly and truthfully to all agents i Ag explanation: outside source: S / Ag ( S is God ) truthful: S can only announce truths, falsehoods cannot be announced (true = true before the announcement) public: each agent i Ag hears the announcement, but i also sees that the others have heard the announcement, and i sees that the others have seen that i has heard the announcement, etc...

6 Models Intuitive idea Syntax and semantics Some key validities Multi-agent Kripke model M = W, {R i } i Ag, V W : nonempty set of possible worlds R i : equivalence relation on W (for each agent i Ag) V : Prop (W ): valuation (Equivalence relations R i, so we will get S5-knowledge operators.)

7 Updating a model Intuitive idea Syntax and semantics Some key validities Take three ingredients: a multi-agent Kripke model M = W, {R i } i Ag, V a world w W a formula ϕ L If (and only if!) M, w = ϕ, we define the updated model and world M ϕ, w ϕ := W ϕ, {R ϕ i } i Ag, V ϕ, w ϕ : W ϕ := {v W M, v = ϕ} R ϕ i := R i ( W ϕ W ϕ) V ϕ (p) := V (p) W ϕ for any p Prop w ϕ := w

8 Formal language Intuitive idea Syntax and semantics Some key validities L is defined as follows ϕ ::= p ϕ ϕ ϕ K i ϕ [!ϕ]ϕ K i ϕ: agent i knows that ϕ [!ϕ]ψ: ψ is true after any public announcement of ϕ any might be an empty quantifier here: ϕ might be false and thus not announcable dually:!ϕ ψ := [!ϕ] ψ ϕ can be announced, and afterwards ψ is true

9 Semantics Intuitive idea Syntax and semantics Some key validities M, w = p iff w V (p) M, w = ϕ iff M, w = ϕ M, w = ϕ ψ iff M, w = ϕ and M, w = ψ M, w = K i ϕ iff M, v = ϕ for all v W such that wr i v M, w = [!ϕ]ψ iff if M, w = ϕ then M ϕ, w = ψ M, w =!ϕ ψ iff M, w = ϕ and M ϕ, w = ψ We say that PAL = ϕ iff M, w = ϕ for all M, w.

10 Some key validities Intuitive idea Syntax and semantics Some key validities truthfulness: PAL = ϕ!ϕ partiality: PAL =!ϕ PAL = [!ϕ]ψ!ϕ ψ (consider M, w = ϕ) functionality: PAL =!ϕ ψ [!ϕ]ψ distributivity: PAL = [!ϕ](ψ χ) ([!ϕ]ψ [!ϕ]χ) necessitation: if PAL = ψ then also PAL = [!ϕ]ψ these are all structural validities interaction of knowledge and public announcement: later

11 A failed first attempt we now start looking for squares of opposition in PAL [!ϕ] satisfies distr. and necess. normal modal operator given the modal analysis of knowledge in classical (static) epistemic logic, it is tempting to construct a square like this: but since PAL = [!ϕ]ψ!ϕ ψ, this doesn t work!

12 Building the right square but recall that we have functionality: PAL =!ϕ ψ [!ϕ]ψ, to get subalternation working in reversed order :

13 Building the right square since!ϕ ψ just abbreviates [!ϕ] ψ, we get the contradictories:

14 Building the right square!ϕ ψ and!ϕ ψ can both be false (if ϕ is false), but they cannot both be true (check the semantics!), so they are contraries:

15 Building the right square [!ϕ]ψ and [!ϕ] ψ can both be true (if ϕ is false), but they cannot both be false, so they are subcontraries:

16 Building the right square using well-known techniques from Sesmat and Blanché, we turn the square into a hexagon:

17 Building the right square but!ϕ ψ!ϕ ψ just says that ϕ is announcable at all, and thus true; similarly [!ϕ]ψ [!ϕ] ψ says that ϕ is not announcable, and thus false:

18 Summary we have constructed a square (hexagon) for public announcements in most well-known squares, implications go from the universal notion (,, K, O) to the existential notion (,, ˆK, P) in our square the implications are reversed! other relations (contradiction, contr., subcontr.) still ok the reversed square is really natural!

19 Some more validities We will now extend the hexagon constructed above with knowledge First some validities about the interaction between knowledge and public announcement: 1 PAL =!ϕ K i ψ K i [!ϕ]ψ 2 PAL = K i [!ϕ]ψ [!ϕ]k i ψ 3 PAL =!ϕ K i ψ K i [!ϕ]ψ (contrapositive of 2) 4 PAL = K i [!ϕ]ψ [!ϕ] K i ψ (contrapositive of 1)

20 Building the second square take the square/hexagon constructed above, but replace ψ with Kψ (we drop agent indices):

21 Building the second square using the four validities above, we extend the hexagon:

22 Building the second square obviously K[!ϕ]ψ and K[!ϕ]ψ are contradictory:

23 Building the second square two more contrariety relations (can both be false if ϕ false):

24 Building the second square two more subcontrariety relations (can both be true if ϕ false):

25 Summary we have extended the first (reversed) hexagon with knowledge by adding two more formulas, we obtained: four more implication (altern/subaltern) relations two more contradiction relations two more contrariety relations two more subcontrariety relations PAL provides a detailed account of the subtle interactions between knowledge and public announcement the (extended) hexagon is a compact representation of this account

26 and future work aim: connect PAL (DEL) with the rich philosophical tradition of the square of opposition our squares/hexagons are: interesting: they arise in a nontrivial way useful: compact representation of a lot of information future work: study this at higher level of abstraction partial functionality is essential for the reversed squares (also in generalization to DEL) drop the epistemic perspective altogether reversed squares for dynamic logic of (deterministic) computer programs

27 Thank you! Handout available at demey