Towards Symbolic Factual Change in Dynamic Epistemic Logic

Transcription

1 Towards Symbolic Factual Change in Dynamic Epistemic Logic Malvin Gattinger ILLC, Amsterdam July 18th 2017 ESSLLI Student Session Toulouse

2 Are there more red or more blue points?

3 Are there more red or more blue points?

4 6 4 Are there more red or more blue points?

5 Representation matters!

6 1. Dynamic Epistemic Logic 2. Symbolic Model Checking 3. Factual Change

7 Epistemic Logic Syntax ϕ ::= p ϕ ϕ ϕ K i ϕ Kripke Models M = (W, R i, Val) where W set of worlds R i W W indistinguishability relation Val : W P(P) valuation function Semantics M, w = K i ϕ iff wr i v implies M, v = ϕ

8 Dynamic Epistemic Logic: Action Models Action Models A = (A, R, pre, post) where A R i A A pre : A L post : A P L Product Update set of atomic events indistinguishability relation precondition function postcondition function M A := (W new, R new i, Val new ) where W new := {(w, a) W A M, w pre(a)} R new i := {((w, a), (v, b)) R i wv and R i ab} Val new ((w, a)) := {p V M, w post a (p)} (Baltag, Moss, and Solecki 1998,Benthem, Eijck, and Kooi (2006))

9 DEL Example: Coin Flip hidden from a a,b w a,b p a 1? p := a,b a = (w, a 1 ) p a a,b a 2? p := a,b (w, a 2 ) p

10 1. Dynamic Epistemic Logic 2. Symbolic Model Checking 3. Factual Change

11 Model Checking The Task Given a model and a formula, does it hold in the model? M, w = ϕ or M, w = ϕ???

12 Limits of Explicit Model Checking Set of possible worlds has to fit in memory. For large models (~ 1000 worlds) this gets slow. Runtime in seconds for n Muddy Children: n DEMO-S

13 1. Dynamic Epistemic Logic 2. Symbolic Model Checking 3. Factual Change

14 Symbolic Model Checking 2 3 Can we represent models in a more compact way?... such that we can still interpret all formulas?

15 Symbolic Model Checking 2 3 Yes! Can we represent models in a more compact way?... such that we can still interpret all formulas? 1. Represent M = (W, R i, Val) symbolically: F = (V, θ, O i ). 2. Translate DEL to equivalent boolean formulas. 3. Use Binary Decision Diagrams to speed it up.

16 Knowledge Structures Knowledge Structures F = (V, θ, O 1,, O n ) where V Vocabulary set of propositional variables θ State Law boolean formula over V O i V Observables propositions observable by i The set of states is {s V s θ}. Call (F, s) a scenario.

17 Knowledge Structures Knowledge Structures F = (V, θ, O 1,, O n ) where V Vocabulary set of propositional variables θ State Law boolean formula over V O i V Observables propositions observable by i The set of states is {s V s θ}. Call (F, s) a scenario. Symbolic Semantics F, s Kϕ iff s O i = s O i implies F, s ϕ

18 From Knowledge Structures to Kripke Models Theorem For every knowledge structure there is an equivalent S5 Kripke Model and vice versa. 2 3 Example The knowledge structure F = (V = {p, q}, θ = p q, O Alice = {p}, O Bob = {q}) is equivalent to this Kripke model: p Alice q Bob p, q

19 From Kripke Models to Knowledge Structures (This is the tricky direction.) Example 0 p Alice 1 p Alice Alice Bob 2 is equivalent to this knowledge structure: ( V = {p, p 2 }, θ = p 2 p, O Alice =, O Bob = {p 2 } ) with actual state: {p, p 2 }

20 Everything is boolean! Definition Fix a knowledge structure F = (V, θ, O 1,, O n ). We define a local boolean translation F : p F := p ϕ F := ϕ F ϕ 1 ϕ 2 F := ϕ 1 F ϕ 2 F K i ϕ F := (V \ O i )(θ ϕ F ) [!ϕ]ψ F := ϕ F ψ F ϕ

21 Everything is boolean! Definition Fix a knowledge structure F = (V, θ, O 1,, O n ). We define a local boolean translation F : p F := p ϕ F := ϕ F ϕ 1 ϕ 2 F := ϕ 1 F ϕ 2 F K i ϕ F := (V \ O i )(θ ϕ F ) [!ϕ]ψ F := ϕ F ψ F ϕ Theorem For all scenarios (F, s) and all formulas ϕ: F, s ϕ s ϕ F

22 Example: Symbolic Muddy Children I F 0 = V = {p 1, p 2, p 3 }, θ 0 =, At least one of you is muddy. O 1 = {p 2, p 3 } O 2 = {p 1, p 3 } O 3 = {p 1, p 2 }

23 Example: Symbolic Muddy Children I F 0 = V = {p 1, p 2, p 3 }, θ 0 =, At least one of you is muddy. O 1 = {p 2, p 3 } O 2 = {p 1, p 3 } O 3 = {p 1, p 2 } F 1 = V = {p 1, p 2, p 3 }, θ 1 = (p 1 p 2 p 3 ), O 1 = {p 2, p 3 } O 2 = {p 1, p 3 } O 3 = {p 1, p 2 }

24 Example: Symbolic Muddy Children I F 0 = V = {p 1, p 2, p 3 }, θ 0 =, At least one of you is muddy. O 1 = {p 2, p 3 } O 2 = {p 1, p 3 } O 3 = {p 1, p 2 } F 1 = V = {p 1, p 2, p 3 }, θ 1 = (p 1 p 2 p 3 ), Do you know if you are muddy?... Nobody reacts. This announcement is equivalent to: O 1 = {p 2, p 3 } O 2 = {p 1, p 3 } O 3 = {p 1, p 2 } i I( (K i p i K i p i )) F1 = (p 2 p 3 ) (p 1 p 3 ) (p 1 p 2 )

25 Magic from 1986: Binary Decision Diagrams (Read the classic Bryant 1986 for more details.)

26 BDD Magic I How long do you need to compare these two formulas? p 3 (p 1 p 2 ) (p 1 p 2 ) p 3

27 BDD Magic I How long do you need to compare these two formulas? p 3 (p 1 p 2 ) (p 1 p 2 ) p 3 Here are is their BDDs:

28 BDD Magic II This was not an accident, BDDs are canonical. Theorem: ϕ ψ BDD(ϕ) = BDD(ψ) Equivalence checks are free and we have fast algorithms to compute BDD( ϕ), BDD(ϕ ψ), BDD(ϕ ψ) etc.

29 Results Why translate DEL to boolean formulas? Because computers are incredibly good at dealing with them! n DEMO-S5 SMCDEL

30 So far, so good... This was S5 PAL, what about: 1. Non-S5 relations for belief? 2. Action Models with factual change?

31 Motivating Example: Sally-Anne

32 Motivating Example: Sally-Anne To model this we need non-s5 action models with factual change!

33 Extension to non-s5 O i P always defines equivalence relations! How can we describe other relations?

34 Extension to non-s5 O i P always defines equivalence relations! How can we describe other relations? Copy P to describe reachability (Gorogiannis and Ryan 2002) Fresh variables V := {p p V } and formulas L(V V ) Actually: use BDD of boolean formula describing the relation

35 Extension to non-s5: From relations to BDDs p 1 p 2 p 1, p 2 Relation R ( p 1 p 2 p 1 p 2 ) ( p 1 p 2 p 1 p 2 ) ( p 1 p 2 p 1 p 2 ) ( p 1 p 2 p 1 p 2 ) ( p 1 p 2 p 1 p 2 ) (p 1 p 2 p 1 p 2 ) (p 1 p 2 p 1 p 2 ) (p 1 p 2 p 1 p 2 ) Formula Φ(R). p 1 p 1 p 2 p 2 BDD of Φ(R)

36 Extension to non-s5: Belief Structures F = (V, θ, Ω 1,, Ω n ) where V Vocabulary propositional variables θ State Law boolean formula over V Ω i L(V V ) Observables encoded relation for i The equivalent Kripke model is given by: R i xy : (x y ) Ω i New translation for modalities: i ψ F := V (θ (Ω i ( ϕ F ) ))

37 The return of the postcondition 1. Dynamic Epistemic Logic 2. Symbolic Model Checking 3. Factual Change

38 Transformers Definition A transformer for V is a tuple X = (V +, θ +, V, θ, Ω + ) where V + is a set of fresh atomic propositions s.t. V V + = θ + is a possibly epistemic formula from L(V V + ) V V is the modified subset of the original vocabulary θ : V L B (V V + ) encodes postconditions Ω + i L B (V + V + ) describe observations for each i to transform F = (V, θ, Ω i ), let F X := (V new, θ new, Ω new i ) where 1. V new := V V + V 2. θ new := [ V /V ] (θ θ + F ) ( [ q V q V /V ] (θ (q)) ) 3. Ω new i := ([ V /V ] [ (V ) /(V ) ] Ω i ) Ω + i

39 Simple Example: Coin Flip hidden from a a,b w a,b p a 1? p := a,b a = (w, a 1 ) p a a,b a 2? p := a,b (w, a 2 ) p (V = {p}, θ = p, Ω a =, Ω b = ) (V + = {q} θ + =, Ω + a =, Ω + b = q q ) V = {p}, θ (p) := q, = (V = {p, q, p } θ = p (p q), Ω a = Ω b = q q )

40 Monster Example: Symbolic Sally-Anne I Sally is in the room: p, Marble is in the box: t Question after the actions: Does Sally believe that the marble is in the box: s t?

41 Monster Example: Symbolic Sally-Anne I Sally is in the room: p, Marble is in the box: t Question after the actions: Does Sally believe that the marble is in the box: s t? Initial structure: (({p, t}, (p t),, ), p) X 1 : Sally puts the marble in the basket: ((,, {t}, θ (t) =,, ), ). X 2 : Sally leaves: ((,, {p}, θ (p) =,, ), ). X 3 : Anne puts the marble in the box, not observed by Sally: (({q},, {t}, θ (t) = ( q t) (q ), q, q q ), {q}). X 4 : Sally comes back: ((,, {p}, θ (p) =,, ), ).

42 The slide we will skip (({p, t}, (p t),, ), p) ((,, {t}, θ (t) =,, ), ) = (({p, t, t }, (p t ) t,, ), {p, t}) ((,, {p}, θ (p) =,, ), ) = (({p, t, t, p }, (p t ) t p,, ), {t, p }) V (({p, t}, t p,, ), {t}) (({q},, {t}, θ (t) = ( q t) (q ), q, q q ), {q}) = (({p, t, q, t }, t p (t (( q t ) (q ))), q, q q ), {q} = (({p, t, q, t }, t p (t q), q, q q ), {q}) V (({p, t, q}, p (t q), q, q q ), {q}) ((,, {p}, θ (p) =,, ), ) = (({p, t, q, p }, p (t q) p, q, q q ), {p, q}) V (({p, t, q}, (t q) p, q, q q ), {p, q})

43 Everything on the previous scary slide are boolean operations, so we blindly trust a computer to deal with it.

44 Monster Example: Symbolic Sally-Anne IIII In the last scene Sally believes the marble is in the basket: {p, q} S t {p, q} V (θ (Ω S t )) {p, q} {p, t, q }((t q ) p ( q t )) {p, q} (François Schwarzentruber: Hintikka s world)

45 Summary representation matters! symbolic model checking DEL gives a big speed-up knowledge/belief structures encode Kripke models transformers provide a modular approach for ontic and epistemic actions Future Work Comparison with (Charrier and Schwarzentruber 2017) Find big examples and benchmark! Limit postconditions to and? github.com/jrclogic/smcdel malvin@w4eg.eu

46 Bonus Slide: Comparison with Mental Programs / Succinct DEL Similar approach in (Charrier and Schwarzentruber 2017). Mental programs describe which change is allowed: q, p q,... Observational BDDs and mental programs are dual in memory consumption: Relation BDD Mental Program Identity 2 V 1 Total 1 2 V

47 References Baltag, Alexandru, Lawrence S. Moss, and Slawomir Solecki The logic of public announcements, common knowledge, and private suspicions. In TARK 98, edited by I. Bilboa, Benthem, Johan van, Jan van Eijck, and Barteld Kooi Logics of communication and change. Information and Computation 204 (11). Elsevier: Bryant, Randal E Graph-Based Algorithms for Boolean Function Manipulation. IEEE Transaction on Computers C-35 (8): Charrier, Tristan, and François Schwarzentruber A Succinct Language for Dynamic Epistemic Logic. In Proceedings of the 16th Conference on Autonomous Agents and Multiagent Systems, AAMAS 17. São Paulo, Brazil: International Foundation for Autonomous Agents; Multiagent Systems. Gorogiannis, Nikos, and Mark D. Ryan Implementation of Belief Change Operators Using BDDs. Studia Logica 70 (1). Kluwer Academic Publishers: doi: /a: