A succinct lnguge for Dynmic epistemic logic Tristn Chrrier IRISA 263 Avenue du Generl Leclerc - CS 74205 35042 Rennes Cedex, Frnce tristn.chrrier@iris.fr Frncois Schwrzentruber ENS Rennes Avenue Robert Schumn 35170 Bruz, Frnce frncois.schwrzentruber@ens-rennes.fr ABSTRACT Dynmic epistemic logic (DEL) is n extension of modl multi-gent epistemic logic with dynmic opertors. We propose succinct version of DEL where Kripke models nd event models re described succinctly. Our proposl relies on Dynmic logic of propositionl ssignments (DLPA): epistemic reltions re described with so-clled mentl progrms written in DLPA. We give exmples of models tht re exponentilly more succinct in our frmework. Interestingly, the model checking of DEL is PSPACE-complete nd we show tht it remins in PSPACE for the succinct version. Keywords Dynmic epistemic logic, propositionl ssignment, model checking, succinctness, complexity. 1. INTRODUCTION Agents tke decisions ccording to their knowledge bout the world nd their knowledge bout other gents knowledge (higher-order knowledge). Dynmic epistemic logic ([4], [15]) is frmework designed for expressing higherorder knowledge properties nd dynmics. Actions in Dynmic epistemic logic re described by mens of grphs clled event models. Specific kinds of ctions hve been considered in the literture For instnce, public nnouncements re represented by single-node event models. Attention-bsed nnouncements [7] where some gents my listen to the source or not cn be represented too by event models, but their sizes re exponentil in the number of gents in the system. However, we clim tht this exponentil blow-up in the representtion is rtificil nd tht is why we ddress succinctness in dynmic epistemic logic. Usully, if the description lnguge is (exponentilly) more succinct, lgorithmic problems become (exponentilly) hrder. For instnce, deciding the existence of n Hmiltonin cycle is NP-complete but it becomes NEXPTIME-complete [12] when the input grph is described succinctly. A succinct representtion of grph with 2 b nodes is Boolen circuit C such tht there is n edge (i, j) { 0,..., 2 b 1 } 2 iff C ccepts the binry Appers in: Proc. of the 16th Interntionl Conference on Autonomous Agents nd Multigent Systems (AAMAS 2017), S. Ds, E. Durfee, K. Lrson, M. Winikoff (eds.), My 8 12, 2017, São Pulo, Brzil. Copyright c 2017, Interntionl Foundtion for Autonomous Agents nd Multigent Systems (www.ifms.org). All rights reserved. representtions of the b-bit integers i, j s inputs. In Dynmic epistemic logic, the results re surprising. Wheres the model checking of DEL is PSPACE-complete [1], we would expect tht the succinct version of the model checking is EXPSPACE-complete. Actully, we provide frmework where the representtion of ctions such s ttention-bsed nnouncements is exponentilly more succinct wheres the model checking remins in PSPACE. In our frmework, we do not use Boolen circuits trditionlly used for representing instnces for succinct decision problems (see [12], Chpter 20.1) but mentl progrms bsed on Dynmic Logic of Propositionl Assignments (DLPA) ([3], [2]) s developed in [11]. The reson of tht choice is tht our model checking lgorithm directly relies on DLPA. After hving reclled some bckground on Dynmic epistemic logic in Section 2, the min contribution is to provide succinct version of DEL in Section 3: we provide succinct models for DEL, prove tht DEL cn be embedded in succinct DEL (nd vice vers), nd prove tht succinct DEL is exponentilly more succinct thn DEL. The succinctness of succinct Kripke models is rther esy to prove but the proof of the succinctness of succinct event models relies on the definition of ction emultions [17]. In Section 4, we prove tht the model checking problem in succinct DEL is (still) in PSPACE. 2. BACKGROUND ON DEL We first strt by some nottions bout vlutions, then present Dynmic epistemic logic (DEL). Let AP be finite set of tomic propositions nd Ag be finite set of gents. 2.1 Vlutions The set of vlutions over AP is denoted by V(AP). Vlutions re denoted w, u,... nd they re represented by the set of true tomic propositions. The restriction of w to subset of propositions AP is w AP nd is noted w AP. For ny vlution w over AP, we note desc(w) the formul p w p p AP\w p tht describes the vlution w (AP is mde implicit for simplifying the nottion). For set of propositions AP, we note!(ap) the formul p AP p q AP q p q. The size of vlution defined over AP is the crdinl of AP, noted AP (we implement such vlution by bit rry of size AP). 2.2 Epistemic models Kripke models represent sttic epistemic situtions nd re defined s follows.
w : {p} u : Figure 1: Exmple of n epistemic model Definition 1. A Kripke model M = (W, ( ) Ag, V ) is defined by non-empty set W of epistemic sttes/worlds, epistemic reltions ( ) Ag W W nd vlution function V : W 2 AP. Typiclly, W represent ll possible configurtions. V is lbeling for the sttes for describing the configurtion. The intuitive mening of w u is tht gent considers stte u s possible when the ctul stte is w. We do not require to be n equivlence reltion. The size of Kripke model M is implemented by lbeled grph nd ech reltion is represented by n djcency list. The size of M is thus O( Ag W 2 + W AP ). Exmple 1. Figure 1 depicts the model M = (W, ( ) Ag, V ) given by W = {w, u}, = b = W W, V (w) = {p} nd V (u) =. Agents nd b do not distinguish w from u, therefore they do not know the truth vlue of p (in both w nd u). Exmple 2. We consider the extension of the clssicl muddy children puzzle [14] where children my py ttention to the fther or not (s in [7]). We introduce tomic proposition m mening tht is muddy nd tomic proposition h mening tht gent hers (i.e. pys ttention to) the nnouncements of the fther. We consider the model M = (W, ( ) Ag, V ) where W = V({m, h Ag}), w u if (w = h iff u = h ) nd for ll b (w = m b iff u = m b ), nd V (w) = w for ll w W. In M, n gent will not distinguish two worlds w from u s long he sees the sme forehed sttes for the other gents nd his py ttention sttus is the sme in both worlds. A Kripke model represents the stte of mind of gents. A pointed Kripke model is pir (M, w) with w W, modeling the fct tht the current epistemic stte is w. 2.3 Syntx of epistemic lnguge The lnguge L EL extends the propositionl lnguge L P rop with modl opertors K nd is defined by the following BNF: ϕ ::= p ϕ (ϕ ϕ) K ϕ, with p AP, Ag. The formul K ϕ is red gent knows tht ϕ holds. We define the usul bbrevitions (ϕ 1 ϕ 2) for ( ϕ 1 ϕ 2) nd ˆK ϕ for K ϕ. The size of ϕ is the number of opertors needed to write ϕ. 2.4 Event models The dynmics of the system is modeled by event models. Definition 2. An event model E = (E, ( E ) Ag, pre, post) is defined by non-empty set of events E, epistemic reltions ( E ) Ag E E, precondition function pre : E L EL nd postcondition function post : E AP L P rop. The precondition function defines whether given event is executble or not. The postcondition updtes the vlues of tomic propositions fter executing n event e: the truth of p is ssigned to the vlue post(e, p). e : pre : p post : p b f : pre : Figure 2: Exmple of n event model Remrk 1. Some uthors ([16]) consider epistemic postcondition functions post : E AP L EL. Actully, it is lwys possible to compute n equivlent event model with propositionl postcondition function from n event model with n epistemic postcondition function. Such trnsformtion is given in [16] but is exponentil. Fortuntely, there exists polynomil lterntive trnsformtion where n event model with epistemic postcondition functions is trnsformed into sequence of event models with propositionl postcondition functions 1. A event model is without postconditions (i.e. it does not chnge physiclly the current stte) when post(e, p) = p for ll e E nd ll tomic propositions p, tht is when post is trivil. In tht cse, we usully omit the post function nd we write E = (E, ( E ) Ag, pre). A pointed event model is pir (E, e) with e E. A multi-pointed event model is pir (E, E 0) with E 0 E. The size of n event model is similrly defined thn the size of Kripke model. Exmple 3. Figure 2 shows the event model E = (E, ( E ) Ag, pre, post) where E = {e, f}, E = {(e, e), (f, f)}, E b = {(f, f), (e, f)}, pre(e) = p, pre(f) =, post(e, p) = nd post(f, p) = p. Exmple 4. We focus on the notion of ttention-bsed nnouncement of p s shown in [7]. In ddition to clssic tomic propositions, we dd propositions h for gent is listening to the nnouncement. The ttention-bsed nnouncement of p cn be then represented by the event model E = (E, ( E ) Ag, pre) where: E = V({p} {h Ag}) {idle}; for ll, e E f if e idle, f idle, e = h nd f = p; for ll, e E idle if e idle nd e = h ; for ll, idle E idle ; { desc(e) if e idle pre(e) = otherwise. Figure 4 shows the event model of the ttention-bsed nnouncement of p for two gents 1 nd 2. Event idle is the event where nothing hppens. The reltion E is defined s follows: if is listening (e = h ) then believes tht p hs been nnounced (f = p). If is not listening (e = h ) then believes nothing hppens. The precondition is defined to mtch the fct tht ttentive gents listen to the nnouncement of p (thus leding to events where p holds) nd tht other gents believe tht nothing hppened (thus the precondition on idle). The nnouncement is purely epistemic so the postcondition function is trivil. 1 The proof of this clim is in the long version of the pper.
(w, e) : b (w, f) : {p} (u, f) : Figure 3: Exmple of product b 2.5 Product The updte of Kripke model M with n event model E is defined by the synchronous product of both models, noted M E, nd defined s follows. Definition 3. Let M = (W, ( ) Ag, V ) be Kripke model. Let E = (E, ( E ) Ag, pre, post) be n event model. The product of M nd E is M E = (W, ( ), V ) where: W = {(w, e) W E M, w = pre(e)}; (w, e) (w, e ) iff w w nd e E e ; V ((w, e)) = {p AP M, w = post(e, p)}. Exmple 5. Figure 3 shows the product of the Kripke model of Figure 1 nd the event model of Figure 2. 2.6 Syntx The lnguge L DEL extends L EL nd is defined by the following BNF. ϕ ::= p ϕ (ϕ ϕ) K ϕ E, E 0 ϕ with p AP, Ag. Formul E, E 0 ϕ is red There exists n execution of (E, E 0) such tht ϕ holds. 2.7 Semntics We now define the semntics of formul ϕ of L DEL. Definition 4. We define M, w = ϕ (ϕ is true in the pointed Kripke model M, w) by induction on ϕ: M, w = p iff p V (w); M, w = ϕ iff M, w = ϕ; M, w = (ϕ ψ) iff M, w = ϕ or M, w = ψ; M, w = K ϕ iff for ll u W, w u implies M, u = ϕ; M, w = E, E 0 ϕ iff there exists e E 0 s.th. M, w = pre(e) nd M E, (w, e) = ϕ. Exmple 6. We consider the Kripke model M of Figure 1 nd the event model of Figure 2. We hve: As p V (u) nd w u, we hve M, w = K p. As p holds in ll -successors of (w, e) (Figure 3), M E, (w, e) = K p. We hve M, w = E, {e} K p. 2.8 Bisimultion nd equivlence We define notions of equivlence for models. 2.8.1 Bisimultions for Kripke models The usul notion of equivlence for Kripke models is bisimultion. Definition 5. Let AP be finite set of tomic propositions, M = (W, ( ) Ag, V ) nd M = (W, ( ) Ag, V ) be two Kripke models, w W nd w W. A reltion B W W is AP-bisimultion between M nd M iff for ll w W, w W such tht wbw : Invrince: V (w) AP = V (w ) AP ; Zig: for ll u W such tht w u there exists u W such tht w u nd ubu ; Zg: for ll u W such tht w u there exists u W such tht w u nd ubu. Two pointed Kripke models (M, w) nd (M, w ) re APbisimilr if there is AP-bisimultion B between M nd M with wbw. (M, w) nd (M, w ) re AP-bisimilr iff ((M, w) = ϕ iff (M, w ) = ϕ) for ll formuls ϕ of L DEL, whose propositions re in AP. When AP is cler from the context, we sy bisimilr insted of AP-bisimilr. Two Kripke models M nd M re bisimilr if there exists w W nd w W such tht (M, w) nd (M, w ) re bisimilr. 2.8.2 Equivlence of event models For event models, the equivlence is defined s follows. Definition 6. Let (E, e) nd (E, e ) be two pointed event models. They re equivlent if for ny pointed Kripke model (M, w), (M E, (w, e)) nd (M E, (w, e )) re bisimilr. Equivlence of event models without postconditions is chrcterized by ction emultions ([17]) defined s follows. Definition 7. Let E = (E, ( E ) Ag, pre) nd E = (E, ( E ) Ag, pre ) be two event models without postconditions Let Σ be the set of preconditions ppering in E nd E. Let ˆΣ be the set of formuls contining Σ nd closed under sub-formuls nd negtion (see [17] for more detils). Let CS(ˆΣ) be the set of mximl consistent subsets of ˆΣ. An ction emultion AE is set of reltions {AE Γ} Γ CS( ˆΣ) E E such tht whenever eae Γe : Invrince: pre(e) Γ nd pre(e ) Γ; Zig: For ll f E nd Γ ( CS(ˆΣ), if e E f, pre(f) Γ nd the formul ψ Γ ψ ˆK ) ψ Γ ψ is consistent then there exists f E such tht e E f nd fae Γ f ; Zg: symmetric of Zig for E. Action emultion is similr to bisimultion in the sense tht the types of rules re the sme, except tht insted of imposing equivlence for preconditions, we just sk here tht they re in sme mximl consistent subset of ˆΣ. Action emultion chrcterizes equivlence: (E, e) nd (E, e ) re equivlent if there is n ction emultion AE between E nd E such tht for ll nd Γ CS(ˆΣ) such tht pre(e) Γ, we hve eae Γe.
pre : p h 1 h 2 pre : p h 1 h 2 pre : p h 1 h 2 pre : p h 1 h 2 1, 2 1, 2 2 2 1 1, 2 2 pre : p h 1 h 2 pre : p h 1 h 2 pre : p h 1 h 2 pre : p h 1 h 2 1, 2 1 pre : 1 1, 2 Figure 4: The event model A p corresponding to n ttention-bsed nnouncement p for two gents 1 nd 2. The six edges pointing to the dshed box point to ll four events in the box. 2.9 Model checking The model checking problem tkes s input pointed Kripke model M, w, formul ϕ of L DEL nd returns yes if M, w = ϕ; no otherwise. Sets AP nd Ag re implicitly prt of the input: the number of tomic propositions nd the number of gents is unbounded nd specified by M, w, ϕ. Theorem 1. ([1]) The model checking problem is PSPACE-complete 2 3. SUCCINCT DEL In clssicl DEL Kripke models nd event models re represented explicitly. It fces some prcticl limits when sizes of models re exponentil in the number of tomic propositions or the number of gents (see Exmples 2 nd 4). In this section, we provide symbolic succinct representtion for both Kripke nd event models. It relies on mentl progrms, presented in Subsection 3.1, written in PDL-dilect clled Dynmic logic of propositionl ssignments (DLPA). In Subsections 3.2 nd 3.3, we define respectively succinct Kripke models nd succinct event models. For both cses, we give the semntics (how Kripke/event models re computed from their succinct representtions), the expressiveness (ll stndrd DEL models cn be represented in our succinct version) nd the succinctness (some succinct models re strictly more succinct thn stndrd DEL). In next Subsections, we present succinct representtion of the product updte nd the lnguge of succinct DEL. 3.1 Lnguge of mentl progrms A mentl progrm, simply clled progrm, succinctly describes reltion between vlutions. We write u v for v is successor of u by. The syntx for mentl progrms is defined by the following BNF. 2 1 ::= p β β? (; ) ( ) ( ) 1 where p AP, β is Boolen formul. Progrm p β is red ssign tomic proposition p to the truth vlue of 2 Actully, hrdness ws proven in [1] for lnguge with non-deterministic choice in the lnguge. For proof without such constrint, see [6]. β. Progrm β? is red test β. Progrm 1; 2 is red execute 1 then 2. Progrm ( 1 2) is red either execute 1 or 2. Progrm ( 1 2) is the intersection of 1 nd 2. Progrm 1 is the inverse of. We write set(p 1,..., p n) = (p 1 p 1 );... ; (p n p n ) for the progrm setting rbitrry vlues to p 1,..., p n. The semntics of mentl progrms is defined by induction s follows: w p β u iff (u = w\{p} nd w = β) w β? u iff w = u nd w = β?; or (u = w {p} nd w = β); w 1; 2 u iff there exists v s. th. w 1 v nd v 2 u; w 1 2 u iff w 1 u or w 2 u; w 1 2 u iff w 1 u nd w 2 u; w 1 u iff u w. The size of mentl progrm corresponds to the number of opertors needed to write the progrm. For instnce, the mentl progrm (p ) (q?; p ) hs size 10. The models re succinctly described by mens of mentl progrms. We re interested here in describing Kripke models, event models nd the product updte. 3.2 Succinct Kripke models We first define the succinct representtion of Kripke models, then show how to extrct Kripke model from succinct one nd vice vers nd finlly we give n exmple where the succinct representtion is indeed strictly more succinct. 3.2.1 Definition Definition 8. A succinct Kripke model is tuple M = AP M, β M, ( ) Ag where AP M is finite set of tomic propositions, β M is Boolen formul over AP M, nd is progrm over AP M for ech gent. The Boolen formul β M succinctly describes the set of epistemic sttes. Intuitively, ech succinctly describes the ccessibility reltion for n gent. A pointed succinct Kripke model is pir M, w where M = AP M, β M, ( ) Ag is succinct Kripke model nd w is vlution stisfying β M. 3.2.2 From succinct Kripke models to Kripke models We define the explicit Kripke model ˆM(M) ssocited to the succinct Kripke model M: the set of worlds is the set of vlutions stisfying β M nd the epistemic reltion is the reltion described by. Definition 9. Given succinct Kripke model M = AP M, β M, ( ) Ag, the Kripke model represented by M, noted ˆM(M) is the model M = (W, ( ) Ag, V ) where: W = {w V(AP M ) w = β M }; { } = (w, u) W 2 w u ; V (w) = w.
3.2.3 From Kripke models to succinct models We define succinct Kripke model M M representing the Kripke model M with respect to set of propositions AP. Definition 10. Let M = (W, ( ) Ag, V ) be Kripke model. We define the succinct Kripke model M M = AP M, β M, ( ) Ag where: AP M = AP {p w w W }; β M =!({p w w W }) w W pw desc(v (w)) = w u pw?; set(ap M ); pu?. The intended mening of the fresh tomic propositions p w is to designte the world w (s nominls in hybrid logic [5]). Formul β M describes the set W nd the vlution V. Progrm performs non-deterministic choice over edges w u nd then simulte the trnsition w u. The following proposition sttes tht M M indeed represents M. ˆM(MM ), {p w} V (w) nd M, w re AP- Proposition 1. bisimilr. Proof. We note M = (W, ( ) Ag, V ) nd ˆM(M M ) = (W, ( ) Ag, V ). We define B := {(u, p u V (u)) u W }. Let us prove tht B is APbisimultion. Invrince of B is routine. For Zig, consider w W. For ll u W, if w u then we cn verify tht ˆM(M M ), u V (u) = β M nd tht p w V (w) p u V (u) so we hve w u. The Zg property is symmetricl. Exmple 7. The Kripke model M from Figure 1 is modeled by the succinct Kripke model M M = AP M, β M, ( ) Ag with AP M = {p, p w, p u}, β M =!({p u, p w}) (p w p) (p u p) nd = v 1,v 2 W pv 1?; set(ap M ); p v2?. 3.2.4 Succinctness of succinct Kripke models To show the succinctness of succinct Kripke models, we describe fmily of Kripke models hving exponentilly more compct representtions with succinct Kripke models. Let us consider the fmily of models M given in Exmple 2 for ll number of gents n. The Kripke model M is succinctly represented by the succinct Kripke model M = AP M, β M, ( ) Ag defined by β M =, AP M = {h, m Ag} nd = set({m } {h b b }). Formul β M = mens tht the set of possible worlds is the set of ll vlutions. Progrm chnges propositions gent is uncertin of (m nd h b for ll b ) while the truth vlues of other propositions remin unchnged. Pointed Kripke models M, w nd ˆM(M), w re bisimilr. The number of worlds in M is 2 2n while the size of M is O(n 2 ) (ech progrm is of size O(n)). Furthermore, there is no bisimilr Kripke model M with less worlds thn M since ll vlutions pper once in M. 3.3 Succinct event models We dopt similr method for succinct event models. 3.3.1 Definition Definition 11. A succinct event model is tuple E = AP M, AP E, χ E, (,E) Ag, post where: AP M nd AP E re two finite disjoint sets of tomic propositions; χ E is formul of L EL over AP M AP E where tomic propositions of AP E re not under the scope of modl opertor K ;,E is progrm over AP M AP E for ll Ag; post is progrm over AP M AP E. AP M is the set of propositions used to describe preconditions while AP E re new fresh propositions to potentilly encode distinct events with the sme precondition. The formul χ E describes the set of events nd their preconditions. Ech,E corresponds to the symbolic representtion of n ccessibility reltion E for n gent. post encodes the postcondition function. 3.3.2 From succinct event models to event models Let sub(χ E) be the union of AP M nd the set of subformuls of χ E of the form K ψ not under the scope of nother K opertor. Exmple 8. If AP M = {p, q, r} nd AP E = {p e, p f } then sub(( p f K K b p) q K r) = {p, q, r, K K b p, K r}. Definition 12. Given succinct event model E = AP M, AP E, χ E, (,E) Ag, post, the event model represented by E, noted Ê(E) is the model (E, ( E ) Ag, pre, post) where E = {(v pre, v post) V(sub(χ E) AP E) V(AP M ) v pre = χ E nd v pre APM AP E v post (v pre APE )}; { } ((vpre, v post), (v E pre, v post )) E 2 =,E ;set(sub(χ E )) ; v pre v pre post pre((v pre, v post)) = desc(v pre sub(χe ) ); { if p vpost post((v pre, v post), p) = otherwise. In n event e = (v pre, v post), v pre represents the vlution before the execution of e (it encodes the precondition nd the truth vlue of propositions in AP E). v post represents the vlution fter the execution of e (it tkes into ccount the effect of the postconditions). The reltion E is defined with the progrm,e but we lso chnge rbitrrily propositions of sub(χ E). The precondition of n event e = (v pre, v post) is the description of v pre restricted to sub(χ E). For instnce, if v pre = {p, p e, K r} then pre((v pre, v post)) = p K r. The postcondition is inferred from v post. 3.3.3 From event models to succinct event models We define succinct event model E E representing the event model E. Definition 13. Let E = (E, ( E ) Ag, pre, post) be n event model. We define the succinct event model E E = AP M, AP E, χ E, (,E) Ag, post where AP M is superset of propositions ppering in pre; AP E = {p e e E}; χ E =!(AP E) e E (pe pre(e));,e = e E f pe?; set(ap M AP E); p f?; post = ( ) e E pe?; p AP M p post(e, p).
The fresh tomic proposition p e designtes event e. Formul χ E describes the set E nd the precondition pre. Progrm,E performs non-deterministic choice in the sme spirit of in Definition 10. Progrm post non-deterministiclly chooses the current event e nd pplies the postcondition ssignments in prllel. The following proposition sttes tht E E indeed represents E. Proposition 2. Ê(E E) nd E re equivlent. Proof. Let E = (E, ( E ) Ag, pre, post) nd Ê(EE) = (E, ( E ) Ag, pre, post ). We prove tht for ny Kripke model M = (W, ( M ) Ag, V ), M E nd M Ê(EE) re bisimilr. Let B be the set of tuples ((w, e), (w, e )) with w W, e E nd e = (v e pre, v e post ) E such tht v e pre = {p e} v e with v e sub(χ E), V (w) v e, v e = pre(e) nd v e post = V (w, e). Such n e is well defined by Definitions 12 nd 13. We prove tht B is bisimultion. We hve V ((w, e )) = V ((w, e)) by Definition 12 so invrince holds. For Zig, if (w, e) M E (u, f) then M, u = pre(f). Therefore there exists v f sub(χ E) such tht V (u) v f, v f = pre(f) nd {p f } v f = χ E. By definition of post, we hve V (u) {p f } post V ((u, f)) {p f }. We deduce tht f = ({p f } v f, V ((u, f))) E. We hve w M u so,e ;set(sub(χ E )) {p e} v e {p f } v f. We conclude tht (u, f) E (u, f ). The Zg property is proven similrly. Exmple 9. The event model E of Figure 2 is modeled by the succinct event model E E = AP M, AP E, χ E, (,E) Ag, post with AP M = {p, p w, p u}, AP E = {p e, p f }, χ E =!(AP E) (p e p) (p f ),,E = e 1 E e 2 pe 1?; set(ap M AP E); p e2? nd post = (p e?; p ) (p f?) (trivil postcondition counterprt in post hs been omitted). 3.3.4 Succinctness of succinct event models As for succinct Kripke models, we provide fmily of event models hving exponentilly more compct representtions with succinct event models. Let us consider the fmily of models (E n) n N given in Exmple 4 for ll number of gents n. The event model E is succinctly represented by the succinct event model E = AP M, AP E, χ E, (,E) Ag, post defined by χ E = ;,E = ( p idle?; (h?; p ; set({h b, b Ag})) ( h?; set(ap M ); p idle )) (p idle?; set(ap M )); post =?. Atomic proposition p idle intuitively mens tht the event idle is occurring (t the bottom in Figure 4). Formul χ E = mens tht the set of possible events is unconstrined. Progrm,E works s follows: if p idle is flse, if h is true, ssign to p nd rbitrrily chnge h b for ll b ; if h is flse, chnge vlutions of propositions in AP M nd set p idle to true; otherwise if p idle is true, chnge ll truth vlues of propositions in AP M. The number of worlds in E is 2 n+1 + 1 while the size of E is O(n 2 ) (ech progrm,e is of size O(n)). Now we prove tht stndrd event models cnnot represent E n s succinctly s succinct event models. Theorem 2. There is no propositionl event model E n equivlent to E n with less tht 2 n events. Proof. We use the chrcteriztion of equivlent models with ction emultions (Definition 7). We suppose tht there is n ction emultion AE between (E n, e) nd (E n, e ) for some e E n nd e E n. Let Σ be the set of preconditions of E n nd E n. Note tht ˆΣ (defined s in Definition 7) is set of propositionl formuls. Let e 1 nd e 2 be two vlutions in V({p} {h Ag}) such tht e 1 = p nd e 2 = p nd e 1 e 2. E n hs less thn 2 n events, so there is n event e 0 of E n, Γ 1, Γ 2 such tht e 1AE Γ1 e 0 nd e 2AE Γ2 e 0. As e 1 e 2, there is n gent such tht e 1 = h nd e 2 = h (we swp e 1 nd e 2 if e 1 = h nd e 2 = h ). Then e 1 En idle. We consider the mximl consistent subset Γ = {ϕ ˆΣ {h, ( Ag} = ϕ}. We hve pre(idle) Γ nd the formul ψ ˆK ) ψ Γ1 ψ Γ ψ is consistent (becuse Γ 1 nd Γ re propositionl). By Zig, there exists f E such tht e E n f ( with idleae Γ f. By Zg, s e 2AE Γ2 e nd the formul ψ ˆK ) ψ Γ2 ψ Γ ψ is consistent, there exists f E such tht e En f nd fae Γ f. By invrince we obtin pre(f) Γ. However pre(f) = desc(f) nd p f so {h, Ag} = pre(f). We derive contrdiction, so there re t lest 2 n events. 3.4 Succinct product updtes Now, we generlize Definition 9 to obtin succinct representtion for updtes. Definition 14. Let M = AP M, β M, ( ) Ag be succinct Kripke model. Let E 1,..., E n be sequence of succinct event models with E i = AP M, AP Ei, χ Ei, (,Ei ) Ag, post i with AP E1,..., AP En being disjoint sets. The succinct product updte of M nd E 1,..., E n, noted M E 1 E n, is M n = AP n, L n, ( n ) Ag defined by induction on n: M 0 = AP M, [β M ], ( ) Ag ; For n 1, M n = AP n 1 AP En, L n 1 :: [χ En ; post n ], ( n ) Ag with n = post 1 n ; ((n 1 ; set(ap En )),En ); post n nd :: is the conctention opertor. Contrry to Definition 9, we now represent the set of worlds by list L n. For n = 0, Definition 14 nd Definition 9 coincides in the sense tht [β M ] is considered the sme s β M. For n 1, we push χ En ; post n t the end of L n 1. The intuition behind the definition of mentl progrm n is s follows: we undo the effect of the postconditions of E n; we then simulte the conjunction w w nd e E e of Definition 3 by executing the intersection of progrm n 1 ; set(ap En ) nd,en ; finlly we repply the postconditions of E n. Next definition explins how to build the Kripke model tht corresponds to the succinct product updte. Definition 15. Given succinct Kripke model M = AP M, β M, ( ) Ag nd succinct event models E 1,..., E n where E i = AP M, AP Ei, χ Ei, (,Ei ) Ag, post i, the Kripke model represented by the product updte of M nd E 1,..., E n, noted ˆM(M E 1 E n) is the model M n = (W n, (,n) Ag, V n), defined by induction on n: M 0 = ˆM(M);
For n 1: M n = (W n, (,n) Ag, V n) where w V(AP M n i=1 AP E i ), s.t. there W n = exists v W n 1 nd e V(AP En ) s.t. M n 1, v e = χ En nd v e post n w; { },n= (w, u) Wn 2 w n u ; V n(w) = w. We sy tht w is stte of M E 1 E n if w W n where W n is given in Definition 15. Proposition 3. Let M be succinct Kripke model nd E 1,..., E n sequence of succinct event models. The models ˆM(M) Ê(E1)... Ê(En) nd ˆM(M E 1... E n) re AP M n i=1 AP E -bisimilr. i Proof. The result is proven by recurrence on n. Cse n = 0 is direct. For the cse n > 1, with the induction hypothesis it suffices to prove tht ˆM(M E1... E n) nd ˆM(M E 1... E n 1) Ê(En) re AP M n i=1 AP E - i bisimilr. The proof technique is similr to the proof of Proposition 2, the detils re left to the reder. When the context is cler, we write M E for M E 1... E n. 3.5 Lnguge of succinct DEL We define the lnguge L succdel: ϕ ::= p ϕ (ϕ ϕ) K ϕ E, β 0 ϕ where E is succinct event model over (AP M, AP E) nd β 0 is Boolen formul over AP M AP E. The syntx of succinct DEL is similr to the syntx of DEL itself except tht opertors E, E 0 (where E, E 0 is multi-pointed event model) re replced by E, β 0 where β 0 is Boolen formul tht succinctly represents set of events E 0. The semntics of E, β 0 ϕ is: M, w = E, β 0 ϕ iff there exists e Ê(E) such tht e = β 0, M, w = pre(e) nd M Ê(E), (w, e) = ϕ where pre(e) is the precondition of e in Ê(E). We write M E, w = ϕ for ˆM(M E), w = ϕ. 4. MODEL CHECKING The succinct model checking problem tkes s input pointed succinct Kripke model M, w nd formul ϕ of L succdel, nd returns yes if M, w = ϕ, no otherwise. As the model checking of DLPA is PSPACE-hrd [2][3], the succinct model checking problem is PSPACE-hrd. Now we provide the upper bound: Theorem 3. The succinct model checking problem is in PSPACE. To prove Theorem 3, s APTIME = PSPACE [8] (where APTIME is to the clss of problems decided by n lternting Turing mchine in polynomil time), we provide n lternting lgorithm deciding the succinct model checking problem in polynomil time. 4.1 Bckground on lternting lgorithms The internl nodes of the computtion tree of our lgorithm on n input M, w, ϕ re either existentil configurtions (s for non-deterministic lgorithms) or universl configurtions. Plyer ( ) (respectively ( )) chooses the next configurtion in existentil (universl) configurtions. Lefs re either ccepting or rejecting configurtions. Plyer ( ) wins if the gme reches n ccepting configurtion. The lgorithm ccepts its input M, w, ϕ if plyer ( ) hs winning strtegy. The running time of n lternting lgorithm is the height of the computtion tree. 4.2 Description of our lgorithm The full pseudo-code of our lgorithm is given in Figure 5 nd extends lgorithms for vrints of DLPA given in [9] nd [11] in order to hndle succinct event models. The min procedure min for model checking succinct DEL first clls mc yes(m, w, ϕ). If this cll is not rejecting the input (tht is if M, w = ϕ) then it ccepts its input M, w, ϕ. We implicitly define the dul versions mc no, stteof no, issucc no obtined from procedures mc yes, stteof yes, issucc yes by switching ( ) with ( ), nd with or, nd indices yes with indices no. The specifictions of ll procedures re given in the following Theorem: Theorem 4. For ny succinct product updte M E, ny vlutions w, u ny formul ϕ of L succdel nd ny mentl progrm : mc yes rejects M E, w, ϕ iff M E, w = ϕ; mc no rejects M E, w, ϕ iff M E, w = ϕ; stteof yes rejects w, M E iff w is not stte of M E; stteof no rejects w, M E iff w is stte of M E; issucc yes rejects w, u, iff w u; issucc no rejects w, u, iff w u. Procedures mc yes nd mc no. The cses ϕ = p nd ϕ = (ϕ 1 ϕ 2) in mc yes re strightforwrd. The cse ϕ = ψ consists in clling the dul procedure mc no with ψ. In the cse ϕ = K ψ, we compute the progrm n from the Kripke model (M E 1... E n), given in Definition 14, then universlly choose vlution u nd finlly check tht one of the three conditions holds: u is not in the set of sttes of (M E 1... E n) (the corresponding cll to stteof no does not reject its input), mening tht the chosen vlution u is irrelevnt. w u (the cll issucc no does not reject w, u, ), mening tht the chosen vlution u is irrelevnt. or tht (M E 1... E n), u = ψ (the cll mc yes does not reject M E 1... E n, u, ψ)). This condition is prticulrly relevnt when u is ctully in the set of sttes of (M E 1... E n) nd w u. In the cse ϕ = E, β 0 ψ, plyer ( ) chooses vlution e with e = β 0. Then the lgorithm checks tht w e is stte of M E 1... E n nd tht M E 1... E n, E, (w e) = ψ. Procedures stteof yes nd stteof no. The cll stteof yes(w, M, E 1,..., E n) rejects if nd only if the vlution u does not correspond to stte of M, E 1,..., E n. In order to check tht the vlution u is not stte of
M E 1... E n, the procedure mimics Definition 14 nd distinguishes two cses. In the cse n = 0, we check tht β M holds in w. In the cse n 1, we first check χ En. Then we simulte the progrm post n by universlly choosing vlution u such tht u post n w. Finlly, we check tht χ En holds in u (we check tht the precondition of E n holds). Procedures issucc yes nd issucc no. The cses over follow the semntics for mentl progrms (Subsection 3.1). Procedure min(m, w, ϕ) mc yes(m, w, ϕ) ccept Procedure mc yes(m E 1... E n, w, ϕ) Cse ϕ = p: if w = p then reject Cse ϕ = (ϕ 1 ϕ 2): ( ) choose i {1, 2} mc yes((m E 1... E n), w, ϕ i) Cse ϕ = ψ: mc no((m E 1... E n), w, ψ). Cse ϕ = K ψ: Get n from (M E 1... E n). ( ) choose u over AP M E1... E n ( )stteof no(u, (M E 1... E n)) or issucc no(w, u, ) or mc yes((m E 1... E n), u, ψ) Cse ϕ = E, β 0 ψ: ( ) choose e such tht e = β 0 ( )stteof yes(w e, (M E 1... E n E n)) nd mc yes(m E 1... E n E, (w e), ψ). Procedure stteof yes(w, M E 1... E n) Cse n = 0: mc yes(m, w, β M ) Cse n > 0: ( ) choose u V(AP M E1... E n ) ( )issucc yes(u, w, post n ) nd stteof yes(u APM E1... E n 1, M E 1,..., E n 1) nd mc yes(m, E 1,..., E n 1, u APM E1... E n 1,, χ En ) Procedure issucc yes(w, u, ) Cse = p β: if (w = β nd u w {p}) or (w = β nd u w\{p}) then reject Cse = β?: if w u or w = β then reject Cse = ( 1; 2): ( ) choose vlution v ( )issucc yes(w, v, 1) nd issucc yes(v, u, 2) Cse = ( 1 2): ( ) choose i {1, 2} issucc yes(w, u, i) Cse = ( 1 2): ( ) choose i {1, 2} issucc yes(w, u, i) Cse = 1 : issucc yes(u, w, ). 4.3 Sketch of proofs Figure 5: Pseudo-code Lemm 1. The procedure min runs in polynomil time in the size of (M E 1... E n, w, ϕ). Proof. The size of the rgument of the procedures is strictly decresing t ech step, so the execution is in liner time in respect to the size of the input. Thus, Mc runs in polynomil time in respect to the size of (M E 1... E n, w, ϕ). Therefore, the height of the computtion tree is polynomil in the size of the input. Theorem 4 is proved by mutul induction for mc no, stteof no, issucc no obtined from mc yes, stteof yes, issucc yes on the size of their inputs. 4.4 Impct The trnsltion, with strightforwrd bse cses, tr( E, E 0 ϕ) = E E, p e p e tr(ϕ) ( ) e E 0 e E 0 is such tht M, w = ϕ iff M M, p w desc(v (w)) = tr(ϕ). Thus, Theorem 3 gives n lterntive proof for the PSPACE upper bound of the model checking of DEL Interestingly, if for some E, E 0 whose size depends on the input (s the ttention-bsed nnouncement in Exmple 4) nd for which there exists succinct pointed event model E, β 0 of polynomil size in the input, then, insted of (*), we use: tr( E, E 0 ϕ) = E, β 0 tr(ϕ) nd the model checking remins in PSPACE. For instnce, the model checking of n epistemic formul tht contins n rbitrry finite number of gents nd ttention-bsed nnouncement modlities [p!] t is in PSPACE. 5. CONCLUSION AND PERSPECTIVES In this pper, we provide n exponentilly more succinct version of Dynmic epistemic logic (DEL) nd of the model checking problem of DEL tht remins in PSPACE. It opens long-term reserch trck for studying lgorithmic problems in DEL such s stisfibility problem nd epistemic plnning in their succinct versions. Vn Bethem et l. [13] studied symbolic version of DEL. Yet, it does not include postconditions nd is not bsed on lnguge for specifying events. The succinct stisfibility problem is the following decision problem: determine whether formul ϕ L succdel is stisfible. There exists tbleu method for determining whether formul ϕ L DEL is stisfible [1] tht yields non-deterministic lgorithm in exponentil time for the stisfibility problem. We conjecture tht the tbleu method cn be dpted for formul ϕ L succdel in order to prove tht the succinct stisfibility problem is in NEXPTIME. Chrrier et l. [10] provides succinct lnguge for itertions of event models: they provide modlities (E, e) i where i is n integer written in binry insted of the explicit sequence (E, e);... ; (E, e). The corresponding model checking of DEL is PSPACE-complete even with such succinct itertions, when preconditions re propositionl nd with trivil postconditions. It would be interesting to study the extension of our succinct DEL with succinct itertions. Also, interestingly, the stisfibility problem of ttentionbsed nnouncement logic is PSPACE-complete [7] even if corresponding event models re exponentil in the number of gents (see Figure 4). So the gp between PSPACE nd NEXPTIME is not due to the size of models but in the very structure of event models. We clim tht our succinct version DEL my help in chrcterizing frgments of DEL with lower complexity for model checking/stisfibility problem. Acknowledgements. We would like to thnk Mlvin Gttinger nd Sophie Pinchint for their useful insights nd the nonymous reviewers for their comments.
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