Undecidability in Epistemic Planning

Transcription

1 Undecidability in Eistemic Planning Guillaume Aucher, Thomas Bolander To cite this version: Guillaume Aucher, Thomas Bolander. Undecidability in Eistemic Planning. [Research Reort] RR- 830, INRIA <hal > HAL Id: hal htts://hal.inria.fr/hal Submitted on 22 May 203 HAL is a multi-discilinary oen access archive for the deosit and dissemination of scientific research documents, whether they are ublished or not. The documents may come from teaching and research institutions in France or abroad, or from ublic or rivate research centers. L archive ouverte luridiscilinaire HAL, est destinée au déôt et à la diffusion de documents scientifiques de niveau recherche, ubliés ou non, émanant des établissements d enseignement et de recherche français ou étrangers, des laboratoires ublics ou rivés.

2 Undecidability in Eistemic Planning (extended version) Guillaume Aucher, Thomas Bolander RESEARCH REPORT N 830 Aril 203 Project-Team S4 ISSN ISRN INRIA/RR FR+ENG

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4 Undecidability in Eistemic Planning (extended version) Guillaume Aucher, Thomas Bolander Project-Team S4 Research Reort n 830 Aril ages Abstract: Dynamic eistemic logic (DEL) rovides a very exressive framework for multi-agent lanning that can deal with nondeterminism, artial observability, sensing actions, and arbitrary nesting of beliefs about other agents beliefs. However, as we show in this aer, this exressiveness comes at a rice. The lanning framework is undecidable, even if we allow only urely eistemic actions (actions that change only beliefs, not ontic facts). Undecidability holds already in the S5 setting with at least 2 agents, and even with agent in S4. It shows that multi-agent lanning is robustly undecidable if we assume that agents can reason with an arbitrary nesting of beliefs about beliefs. We also rove a corollary showing undecidability of the DEL model checking roblem with the star oerator on actions (iteration). Key-words: Automated lanning, dynamic eistemic logic, multi-agent systems, undecidability. This research reort is an extended version of the article ublished in the roceedings of the International Joint Conference on Artificial Intelligence (IJCAI), Beijing, August 203. Université de Rennes - INRIA, France DTU Coenhagen, Denmark RESEARCH CENTRE RENNES BRETAGNE ATLANTIQUE Camus universitaire de Beaulieu Rennes Cedex

5 Indécidabilité en lanification éistémique (version étendue) Résumé : La logique éistémique dynamique (DEL) est un formalisme logique très exressif our la lanification éistémique qui ermet de rendre comte du non-déterminisme, de l observation artielle, des actions de ercetion, et de croyances d ordre suérieur ortant sur les croyances d autres agents. Ceendant, comme nous le montrons dans ce raort, cette exressivité a un coût. Ce formalisme our la lanification est indécidable, et cela même si nous nous restreignons aux actions urement éistémiques (les actions qui changent seulement les croyances, as les faits du monde). L indécidabilité est déjà résente dans le cadre de la logique S5 avec au moins 2 agents, et même dans le cas d un seul agent avec la logique S4. Cela montre que la lanification multi-agent est indécidable de façon robuste si on suose que les agents euvent raisonner avec un enchevêtrement arbitraire de croyances d ordre suérieur sur les croyances d autres agents. Nous rouvons aussi un corollaire montrant que le roblème du model checking de la logique éistémique dynamique (DEL) est indécidable avec l oérateur étoile sur les actions (itération). Mots-clés : Planification automatique, logique éistémique dynamique, systèmes multi-agents, indécidabilité.

6 Undecidability in Eistemic Planning 3 Contents Introduction 4 2 Dynamic Eistemic Logic 5 2. Eistemic Models Event Models Product Udate Classes of Eistemic States and Actions Classical and Eistemic Planning 8 3. Classical Planning Eistemic Planning Two-counter Machines 9 5 Single-agent Eistemic Planning 0 5. The General Case Encoding of Configurations Encoding of the Comutation Function Encoding of the Halting Problem Eistemic Planning for K4, KT, K45, S4 and S Encoding of Configurations Encoding of the Comutation Function Encoding of the Halting Problem Multi-agent Eistemic Planning 6 6. Encoding of Configurations Encoding of the Comutation Function Encoding of the Halting Problem DEL Model Checking 8 8 Conclusion 9 8. Related Work Concluding Remarks RR n 830

7 4 G. Aucher & T. Bolander Introduction Recently a number of authors have indeendently started develoing new and very exressive frameworks for automated lanning based on dynamic eistemic logic [Bolander and Andersen, 20, Löwe et al., 20, Aucher, 202, Pardo and Sadrzadeh, 202]. Dynamic eistemic logic (DEL) extends ordinary modal eistemic logic [Hintikka, 962] by the inclusion of event models to describe actions, and a roduct udate oerator that defines how eistemic models are udated as the consequence of executing actions described through event models [Baltag et al., 998]. Using eistemic models as states, event models as actions, and the roduct oerator as state transition function, one immediately gets a lanning formalism based on DEL. One of the main advantages of this formalism is exressiveness. Letting states of lanning tasks be eistemic models imlies that we have something that generalizes belief states, the classical aroach to lanning with nondeterminism and artial observability [Ghallab et al., 2004]. Comared to standard lanning formalisms using belief states, the DEL-based aroach has the advantage that actions (event models) encode both nondeterminism and artial observability [Andersen et al., 202], and hence that observability can be action deendent, and we do not need observation functions on to of action descritions. Active sensing actions are also exressible in the DEL-based framework. Another advantage of the DEL-based framework is that it generalizes immediately to the multi-agent case. Both eistemic logic and DEL are by default multi-agent formalisms, and the single-agent situation is simly a secial case. Hence the formalism rovides a lanning framework for multi-agent lanning integrating nondeterminism and artial observability. It can be used both for adversarial and cooerative multi-agent lanning. Finally, the underlying eistemic logic also allows agents to reresent their beliefs about the beliefs of other agents, hence allowing them to do Theory of Mind modeling. Theory of Mind (ToM) is a concet from cognitive sychology referring to the ability of attributing mental states (beliefs, intentions, etc.) to other agents [Premack and Woodruff, 978]. Having a ToM is essential for successful social interaction in human agents [Baron-Cohen, 997], hence can be exected to lay an equally imortant role in the construction of socially intelligent artificial agents. The fli side of the exressivity advantages of the DEL-based lanning framework is that the lan existence roblem is undecidable in the unrestricted framework. This was roven in [Bolander and Andersen, 20] by an encoding of Turing machines as 3-agent lanning tasks (leading to a reduction of the Turing machine halting roblem to the 3-agent lan existence roblem). The roof made essential use of actions with ostconditions, that is, ontic actions that make factual changes to the world (e.g. writing a symbol to a tae cell of a Turing machine). One could seculate that undecidability relied essentially on the inclusion of ontic actions, but in the resent aer we rove this not to be the case. We rove that lan existence is undecidable even when only allowing urely eistemic (non-ontic) actions, and already for 2 agents. This is by an encoding of two-counter machines as lanning tasks. We also rove that even single-agent lanning is undecidable on S4 frames. Given that we deal with multi-agent situations, it is imortant to secify our modeling aroach, and in articular whether the modeler/lanner is one of the agents. A classification of the different modeling aroaches and their resective formalisms can be found in [Aucher, 200]. For ease of resentation, we follow in this article the erfect external aroach of (dynamic) eistemic logic and model the situation from an external and omniscient oint of view. This said, all our results in this article transfer to the other modeling aroaches if we relace eistemic models with internal models or imerfect external models (i.e. multi-ointed models), which, as we said, generalize to a multi-agent setting the belief states of classical lanning [Bolander and Andersen, 20]. The article is structured as follows. In Section 2, we recall the core of the DEL framework. In Section 3, we relate our DEL-based aroach to the classical lanning aroach and rovide an examle of an A belief state can be modeled as a set of roositional valuations, which again can be modeled as a connected S5 model of eistemic logic. Inria

8 Undecidability in Eistemic Planning 5 eistemic multi-agent lanning task. In Section 4, we introduce two-counter machines which are used in Sections 5 and 6 to rove our undecidability results. In Section 7, we derive from our results the undecidability of the DEL model checking roblem (for the language with the star oerator on actions). Finally, we discuss related work and end with some concluding remarks in Section 8. 2 Dynamic Eistemic Logic In this section, we resent the basic notions from DEL required for the rest of the article (see [Baltag et al., 998, van Ditmarsch et al., 2007, van Benthem, 20] for more details). Following the DEL methodology, we slit our exosition into three subsections. In Section 2., we recall the syntax and semantics of the eistemic language. In Section 2.2, we define event models, and in Section 2.3, we define the roduct udate. Finally, in Section 2.4, we define secific classes of eistemic and event models that will be studied in the sequel. 2. Eistemic Models Throughout this article, P is a countable set of atomic roositions (roositional symbols) of cardinality at least two, and A is a non-emty finite set of agents. We will use symbols, q, r,... for atomic roositions and numbers 0,,... for agents. The eistemic language L(P, A) is generated by the following BNF: φ ::= φ φ φ i φ where P and i A. As usual, the intended interretation of a formula i φ is agent i believes φ or agent i knows φ. The formulas i φ, φ ψ and φ ψ are abbreviations of i φ, ( φ ψ), and φ ψ resectively. We define as an abbreviation for and as an abbreviation for for some arbitrarily chosen P. The semantics of L(P, A) is defined as usual through Krike models, here called eistemic models. Definition (Eistemic models and states). An eistemic model of L(P, A) is a trile M = (W, R, V ), where W, the domain, is a finite set of worlds; R : A 2 W W assigns an accessibility relation R(i) to each agent i A; V : P 2 W assigns a set of worlds to each atomic roosition; this is the valuation of that variable. The relation R(i) is usually abbreviated R i, and we write v R i (w) or wr i v when (w, v) R(i). For w W, the air (M, w) is called an eistemic state of L(P, A). Definition 2 (Truth conditions). Let an eistemic model M = (W, R, V ) be given. Let i A, w W and φ, ψ L(P, A). Then M, w = iff w V () M, w = φ iff M, w = φ M, w = φ ψ iff M, w = φ and M, w = ψ M, w = i φ iff for all v R i (w), M, v = φ Examle. Consider the following eistemic state of L({}, {0, }). RR n 830

9 6 G. Aucher & T. Bolander (M, w ) = w : 0, 0, 0, w 2 Each world is marked by its name followed by a list of the roositional symbols being true at the world (which is ossibly emty if none holds true). Edges are labelled with the name of the relevant accessibility relations (agents). We have e.g. (M, w ) = i i for i = 0, : neither agent knows the truth-value of. For eistemic states (M, w) we use the symbol to mark the designated world w. 2.2 Event Models Dynamic Eistemic Logic (DEL) introduces the concet of event model (or action model) for modeling the changes to eistemic states brought about by the execution of actions [Baltag et al., 998, Baltag and Moss, 2004]. Intuitively, in Definition 3 below, eq i e means that while the ossible event reresented by e is occurring, agent i considers ossible that the event reresented by e is in fact occurring. Definition 3 (Event models and eistemic actions). An event model of L(P, A) is a trile E = (E, Q, re), where E, the domain, is a finite non-emty set of events; Q : A 2 E E assigns an accessibility relation Q(i) to each agent i A; re : E L(P, A) assigns to each event a recondition. The relation Q(i) is generally abbreviated Q i, and we write v Q i (w) or wq i v when (w, v) Q(i). For e E, (E, e) is called an eistemic action of L(P, A). The event e in (E, e) is intended to denote the actual event that takes lace when the action is executed. Note that we assume that events do not cause factual changes in the world. Hence, we only consider so-called eistemic events and not ontic events with ostconditions, as in [van Ditmarsch et al., 2005, van Benthem et al., 2006]. Our assumtions for dealing with eistemic lanning will therefore also differ from the assumtions used in [Bolander and Andersen, 20]. 2.3 Product Udate Definition 4 (Alicability). An eistemic action (E, e) is alicable in an eistemic state (M, w) if (M, w) = re(e). The roduct udate yields a new eistemic state (M, w) (E, e) reresenting how the new situation which was reviously reresented by (M, w) is erceived by the agents after the occurrence of the event reresented by (E, e). Definition 5 (Product udate). Given an eistemic action (E, e) alicable in an eistemic state (M, w), where M = (W, R, V ) and E = (E, Q, re). The roduct udate of (M, w) with (E, e) is defined as the eistemic state (M, w) (E, e) = ((W, R, V ), (w, e)), where W ={(w, e) W E M, w = re(e)} R i ={((w, e), (v, f)) W W wr i v and eq i f} V () ={(w, e) W M, w = }. Inria

10 Undecidability in Eistemic Planning 7 L transitive Euclidean reflexive K KT K4 K45 S4 S5 Fig. : L-eistemic states and actions Examle 2. Continuing Examle, the following is an examle of an alicable eistemic action of L({}, {0, }) in (M, w ): (E, e ) = e : 0 0, e 2 : It corresonds to a rivate announcement of to agent 0, that is, agent 0 is told that holds (event e ), but agent thinks that nothing has haened (event e 2 ). The roduct udate is calculated as follows: (M, w ) (E, e ) = 0, 0 (w, e 2): (w, e ): 0, (w 2, e 2) 0, In the udated state, agent 0 knows (since 0 holds at (w, e )), but agent didn t learn anything (doesn t know and believes that 0 doesn t either). 2.4 Classes of Eistemic States and Actions In this article, we consider eistemic states and actions where the accessibility relations satisfy secific roerties, namely transitivity (for all w, v, u, wr i v and vr i u imly wr i u, defined by the axiom 4: i φ i i φ), Euclidicity (for all w, v, u, wr i v and wr i u imly vr i u, defined by the axiom 5: i φ i i φ) and reflexivity (for all w, wr i w, defined by the axiom T: i φ φ). Different conditions on the accessibility relations corresond to different assumtions on the notions of knowledge or belief [Fagin et al., 995, Meyer and van der Hoek, 995]. Modal logics of belief are usually considered to be at least as strong as K45, i.e., they should validate at least modus onens, necessitation (from φ, infer i φ) and ositive and negative introsection (Axioms 4 and 5 resectively). Modal logics of knowledge are usually considered to be at least as strong as S4, i.e., they should validate modus onens, necessitation, Truth (Axiom T) and ositive introsection (Axiom 4). In the literature, it is often assumed that the logic of knowledge is S5, i.e., it validates moreover negative introsection (Axiom 5). In the sequel we will refer to L-eistemic states and actions, where the conditions on these models is given in Figure. These classes of eistemic and event models are stable for the roduct udate, i.e., for all L {K, KT, K4, K45, S4, S5}, if (M, w) is a ointed L-eistemic model and (E, e) is a ointed L-event model alicable in (M, w) then (M, w) (E, e) is a ointed L-eistemic model. RR n 830

11 8 G. Aucher & T. Bolander 3 Classical and Eistemic Planning In this section, we briefly relate the eistemic lanning aroach of DEL as roounded in [Bolander and Andersen, 20, Löwe et al., 20] with the classical lanning aroach [Ghallab et al., 2004]. For more detailed connections, we refer the reader to [Bolander and Andersen, 20]. 3. Classical Planning Following [Ghallab et al., 2004], any classical lanning domain can be reresented as a restricted statetransition system. Definition 6 (Restricted state-transition system). A restricted state-transition system is a tule Σ = (S, A, γ), where S is a finite or recursively enumerable set of states; A is a finite set of actions; γ : S A S is a artial and comutable state-transition function. The state-transition function is artial, i.e., for any (s, a) S A, either γ(s, a) is undefined or γ(s, a) S. Definition 7 (Classical lanning task). A classical lanning task is reresented as a trile (Σ, s 0, S g ), where Σ is a restricted state-transition system; s 0 is the initial state, a member of S; S g is the set of goal states, a subset of S. Definition 8 (Solution to a classical lanning task). A solution to a classical lanning task (Σ, s 0, S g ) is a finite sequence of actions (a lan) a, a 2,..., a n such that:. For all i n, γ(γ(... γ(γ(s 0, a ), a 2 ),..., a i ), a i ) is defined; 2. γ(γ(... γ(γ(s 0, a ), a 2 ),..., a n ), a n ) S g. Note that finding solutions to classical lanning tasks is always at least semi-decidable: given a lanning roblem, we can comute its state sace (the sace of states reachable by a sequence of actions alied to the initial state) in a breadth-first manner, and if one of the goal states is reachable, we will eventually find it. 3.2 Eistemic Planning Here is the definition of eistemic lanning tasks, which are secial cases of classical lanning tasks: Definition 9 (Eistemic lanning tasks). An eistemic lanning task is a trile (s 0, A, φ g ) where s 0 is a finite eistemic state, the initial state; A is a finite set of finite eistemic actions; φ g is a formula in L(P, A), the goal formula. Any eistemic lanning task (s 0, A, φ g ) canonically induces a classical lanning task ((S, A, γ), s 0, S g ) given by: S = {s 0 a a n n 0, a i A}. Inria

12 Undecidability in Eistemic Planning 9 S g = {s S s = φ g }. { s a if a is alicable in s γ(s, a) = undefined otherwise. Hence, eistemic lanning tasks are secial cases of classical lanning tasks. A solution to an eistemic lanning task (s 0, A, φ g ) is a solution to the induced classical lanning task. Examle 3. Let a denote the eistemic action (E, e ) of Examle 2 and let a 2 denote the result of relacing 0 by and by 0 everywhere in a. The eistemic action a 2 is a rivate announcement of to agent. Now consider an eistemic lanning task (s 0, A, φ g ), where s 0 = (M, w ) is the eistemic state from Examle, and A {a, a 2 }. Let the goal be that both 0 and know, but don t know that each other knows: φ g = It is easy to check that a solution to this eistemic lanning task is the action sequence a, a 2, since we have s 0 a a 2 = φ g. Hence a solution to the task of both agents knowing without susecting that each other does, is to first announce rivately to 0 then rivately to. The following definition is adated from [Erol et al., 995]. Definition 0 (Plan existence roblem). Let n and L {K, KT, K4, K45, S4, S5}. PlanEx(L, n) is the following decision roblem: Given an eistemic lanning task T = (s 0, A, φ g ) where s 0 is an L-eistemic state, A is a set of L-eistemic actions and A = n, does T have a solution? 4 Two-counter Machines We will rove undecidability of the lan existence roblem, PLANEX(L, n), for various classes of eistemic lanning tasks. Each roof is by a reduction of the halting roblem for two-counter machines to the lan existence roblem for the relevant class of lanning tasks. So, we first introduce two-counter machines [Minsky, 967, Hamson and Kurucz, 202]. Definition (Two-counter machines). A two-counter machine M is a finite sequence of instructions (I 0,..., I T ), where each instruction I t, for t < T, is from the set {inc(i), jum(j), jzdec(i, j) i = 0,, j T } and I T = halt. A configuration of M is a trile (k, l, m) N 3 with k being the index of the current instruction, and l, m being the current contents of counters 0 and, resectively. The comutation function f M : N N 3 of M mas time stes into configurations, and is given by f M (0) = (0, 0, 0) and if f M (n) = (k, l, m) then (k +, l +, m) if I k = inc(0) (k +, l, m + ) if I k = inc() (j, l, m) if I k = jum(j) (j, l, m) if I k = jzdec(0, j) and l = 0 f M (n + ) = (j, l, m) if I k = jzdec(, j) and m = 0 (k +, l, m) if I k = jzdec(0, j) and l > 0 (k +, l, m ) if I k = jzdec(, j) and m > 0 (k, l, m) if I k = halt. We say that M halts if f M (n) = (T, l, m) for some n, l, m N. Theorem. [Minsky, 967] The halting roblem for two-counter machines is undecidable. RR n 830

13 0 G. Aucher & T. Bolander n + worlds CHAIN(,k) CHAIN(2,l) CHAIN(3,m) Fig. 2: CHAIN(, n) Fig. 3: s (k,l,m) γ 0 m + events γ 0 γ n γ 0 γ n γ γ 0 γ γ 0 γ n γ 0 γ 0 γ γ n γ n Fig. 4: INC() Fig. 5: DEC() Fig. 6: REPL(, n, m) 5 Single-agent Eistemic Planning In this section, we assume that the set A is a singleton. 5. The General Case We encode the halting roblem of a two-counter machine M as an eistemic lanning task in three stes:. We define eistemic models CHAIN(, n) for encoding natural numbers, and eistemic states s (k,l,m) for encoding configurations; 2. We define a finite set of eistemic actions F M for encoding the comutation function f M ; 3. We encode the halting roblem as an eistemic lanning task using these models. 5.. Encoding of Configurations For each roositional symbol P and each n N, we define an eistemic model CHAIN(, n) as in Figure 2. For each (k, l, m) N 3, we define the eistemic state s (k,l,m) as in Figure 3. It encodes the configurations (k, l, m) of two-counter machines Encoding of the Comutation Function First, we need some formal reliminaries: Definition 2 (Path formulas). For every n N, we define the n-ath formula as follows: γ n := n. Lemma. Let n N and let (M, w) be an eistemic state. Then (M, w) = γ n iff there is a ath of length n starting in w and ending in a world with no successor (a sink). Inria

14 Undecidability in Eistemic Planning For each roositional symbol P and each m, n N we define three event models INC(), DEC() and REPL(, n, m) as in Figures 4 6. We have omitted edge labels, as we are in the single-agent case. Lemma 2. For all m, n N, for all P,. CHAIN(, n) INC() = CHAIN(, n + ) 2. if n > 0, CHAIN(, n) DEC() = CHAIN(, n ) 3. CHAIN(, n) REPL(, n, m) = CHAIN(, m). Proof. We only rove items and 3. We first rove item. Introducing names for the nodes and events, we can calculate as follows: CHAIN(, n) INC() = CHAIN(, n + ). n + w : w 2: w n+: w n+2: e : γ 0 e 2 : γ = e 3 : γ 0 n + 2 (w,e ): (w n+,e ): (w n+,e 2): (w n+2,e 3): = Now, we rove item 3. Introducing names for the nodes and events, we can calculate as follows: w : e : γ 0 γ n CHAIN(, n) REPL(, n, m) = n + w 2: w n+: m + e 2: γ 0 γ n e m+: γ 0 γ n = w n+2: e m+2: γ n γ n (w,e ): m + (w,e 2): (w,e m+): = CHAIN(, m). (w 2,e m+2): Definition 3. For all k N, we define the formulas φ k as follows: φ k := ( γ k γ k+ ). () Using Lemma and the definition of s (k,l,m), we immediately get the following result: Fact. For all k, l, m, k N, s (k,l,m) = φ k iff k = k. (2) Let M = (I 0,..., I T ) be a two-counter machine. For all k < T and all l, m N, we define an eistemic action a M (k, l, m) as in Figures 7 0 deending on the values of I k, l and m. If k, l, m, k, l, m N, we write (k, l, m) (k, l, m ) when the following holds: { k = k l = 0 iff l = 0 if I k = jzdec(0, j) and m = 0 iff m = 0 if I k = jzdec(, j) RR n 830

15 2 G. Aucher & T. Bolander φ k 3 φ k INC() INC(2) Fig. 7: The action a M (k, l, m) when I k = inc(0). The case I k = inc() is by relacing 2 with 3 everywhere. REPL(,k, j) 2 3 Fig. 8: The action a M (k, l, m) when I k = jum(j). φ k ( 2 γ ) φ k ( 2 γ ) REPL(,k, j) 2 3 INC() DEC(2) 3 Fig. 9: The action a M (k, l, m) when I k = jzdec(0, j), l = 0. Case I k = jzdec(, j), m = 0 is by relacing 2 with 3. Fig. 0: The action a M (k, l, m) when I k = jzdec(0, j), l > 0. Case I k = jzdec(, j), m > 0 is by relacing 2 with 3. Note that when (k, l, m) (k, l, m ) then a M (k, l, m) = a M (k, l, m ), hence the following set is finite: F M := {a M (k, l, m) k {0,..., T }, l, m N}. Lemma 3 below shows that F M really encodes the comutation stes of the comutation function. Lemma 3. Let M = (I 0,..., I T ) be a two-counter machine, l, m, n N and k < T. Then, the following holds:. a M (k, l, m) is alicable in s fm (n) iff (k, l, m) f M (n); 2. s fm (n) a M (f M (n)) = s fm (n+). Proof sketch. Assume that f M (n) = (k, l, m ). Item is by case of I k. We only consider the cases I k = inc(0) and I k = jum(j). In these cases, a M (k, l, m) is an eistemic action of the form (E, e) with re(e) = φ k. Hence using Fact we get: a M (k, l, m) is alicable in s fm (n) s (k,l,m ) = φ k k = k (k, l, m) (k, l, m ), because I k = inc(0) or I k = jum(j). Item 2 is by case of I k. We only consider the case I k = inc(0): s fm (n) a M (f M (n)) = s (k,l,m ) a M (k, l, m ) = s (k +,l +,m) = s fm (n+), using Lemma 2 and that a M (k, l, m ) is the eistemic action of Fig. 7. Inria

16 Undecidability in Eistemic Planning Encoding of the Halting Problem From Lemma 3, we derive the following lemma: Lemma 4. Let M = (I 0,..., I T ) be a two-counter machine. Define T M as the following single-agent eistemic lanning task on K: T M = (s (0,0,0), F M, φ T ). Then T M has a solution iff M halts. Proof. Using Lemma 3 and induction on m, we get that for all m N, (a) a M (f M (m)) is alicable in the state s (0,0,0) a M (f M (0)) a M (f M (m )), and is the only alicable action from T M in this state. (b) s (0,0,0) a M (f M (0)) a M (f M (m)) = s fm (m+). For an action sequence a 0,..., a n to be a solution to T M it must by definition (Section 3) satisfy: (i) For all m n, a m is alicable in s (0,0,0) a 0 a a m. (ii) s (0,0,0) a 0 a a n = φ T. First we rove that if T M has a solution, M halts. Assume a 0,..., a n is solution to T M. Then (i) and (ii) holds. From (i) we get, using (a), that a m = a M (f M (m)) for all m n. This imlies s (0,0,0) a 0 a n = s fm (n+), using (b), and hence (ii) gives us s fm (n+) = φ T. Fact now imlies that f M (n + ) = (T, l, m). Since T is the index of the halting instruction of M, this shows that M halts. We have now roven that if T M has a solution then M halts. Assume conversely that M halts. Then f M (n+) = (T, l, m) for some n. Define a m = a M (f M (m)) for all m n. If we can rove (i) and (ii) we are done. (i) follows immediately from (a). From (b) we can conclude s (0,0,0) a 0 a n = s fm (n+). Since f M (n + ) = (T, l, m), we must have s fm (n+) = φ T, by Fact. Hence, we can conclude that (ii) holds. So, from Lemma 4 and Theorem, we obtain: Theorem 2. PLANEX(K,) is undecidable. 5.2 Eistemic Planning for K4, KT, K45, S4 and S5 We are going to rove an even stronger result than Theorem 2, namely that the lan existence roblem for S4 lanning tasks is undecidable. The roof of this undecidability result generalizes the revious roof, but we need an extra atomic roosition r. For better readability, we use four atomic roositions, 2, 3 and r, although we could use only two. The idea underlying the roof is to relace worlds with meta-worlds, which are in fact eistemic models Encoding of Configurations The chains as defined in Figure 2 are relaced by the meta-chains of Figure. A rectangle meta-world is the eistemic model delimited in Figure by a rectangle, and an ellise meta-world is the eistemic model delimited in Figure by the ellise. Note that the worlds in an ellise meta-world are related to each other in both directions, unlike the worlds in a rectangle meta-world where the arrow is in only one direction. Then, we define the eistemic model META-S (k,l,m) encoding the configuration (k, l, m) as in Figure 3. Note that the roots of the meta-chains and the root w 0 of META-S (k,l,m) are related to each other in both directions, unlike the revious roof for the logic K. RR n 830

17 4 G. Aucher & T. Bolander r, w 0 n + r, r, META-CHAIN(,k) META-CHAIN(2,l) META-CHAIN(3,m) Fig. 2: META-S (k,l,m) Fig. : META-CHAIN(, n) Encoding of the Comutation Function The sub-model generated by a rectangle meta-world in a meta-chain is called a meta-ath, and the length of a meta-ath is the number of rectangle meta-worlds in such a sequence minus one. Definition 4 (Meta-ath formulas). For all n N, we define the formulas χ n inductively as follows: We then obtain a counterart of Lemma : χ 0 := ( r r) χ n+ := ( r (r χ n)) Lemma 5. Let k, l, m N and let w META-S (k,l,m).. If w is in a rectangle meta-world and n N, then, META-S (k,l,m), w = χ n iff there is a meta-ath of length at least n starting at the rectangle meta-world containing w and ending in a rectangle meta-world with no successor (rectangle) meta-world. 2. w is in a rectangle meta-world of META-S (k,l,m) iff META-S (k,l,m), w = for some {, 2, 3 }. Proof sketch. The roof of the first item is by induction on n and by observing that, in a meta-ath, χ 0 holds only at the bottom last rectangle meta-world of the meta-ath. The roof of the second item relies on the fact that the worlds of an ellise meta-chain are all connected to the root of META-S (k,l,m), which satisfies neither, 2 nor 3. Then, we define the event models META-INC(), META-DEC() and META-REPL(, n, m) as in Figures 3 5. Moreover, we define the following formulas which allow us to count the length of metachains: Definition 5. For all k N and all P, we define the formulas ψ k () as follows: ψ k () := { ( χ 0 ) if k = 0; ( χ k ) ( χ k ) if k > 0. (3) By Lemma 5, we easily get the following result: Inria

18 Undecidability in Eistemic Planning 5 r r r r r χ 0 m + r r r r r Fig. 3: META-INC() ψ k ( ) Fig. 4: META-DEC() Fig. 5: META-REPL(, n, m) ψ k ( ) META-INC() META-INC(2) META-REPL(,k, j) Fig. 6: The action e M (k, l, m) when I k = inc(0). The case I k = inc() is by relacing 2 and 3 everywhere. Fig. 7: The action e M (k, l, m) when I k = jum(j). Fact 2. For all k, l, m, k N, META-S (k,l,m), w 0 = ψ k ( ) iff k = k META-S (k,l,m), w 0 = ψ l ( 2 ) iff l = l META-S (k,l,m), w 0 = ψ m ( 3 ) iff m = m. Let M = (I 0,..., I T ) be a two-counter machine. For all k < T and all l, m N, we define an eistemic action e M (k, l, m) as in Figures 6 9 by relacing in Figures 7 0 INC, DEC and REPL with META-INC, META-DEC and META-REPL resectively and by relacing φ k with ψ k. Note again that the roots of META-INC, META-DEC and META-REPL and the root w 0 of META-S (k,l,m) are related to each other in both directions Encoding of the Halting Problem If CHAIN, INC, DEC and the function a M are relaced in Lemmata 2 and 3 with META-CHAIN, META-INC, META-DEC and the function e M resectively, then these Lemmata still hold. Therefore, Lemma 4 and Theorem 2 also generalize to this S4 setting, and we finally obtain that: Theorem 3. PLANEX(S4, ) is undecidable. RR n 830

19 6 G. Aucher & T. Bolander ψ k ( ) ψ 0( 2) ψ k ( ) ψ 0( 2) META-REPL(,k, j) META-INC() META-DEC(2) Fig. 8: The action e M (k, l, m) when I k = jzdec(0, j), l = 0. Case I k = jzdec(, j), m = 0 is by relacing 2 with 3. Fig. 9: The action e M (k, l, m) when I k = jzdec(0, j), l > 0. Case I k = jzdec(, j), m > 0 is by relacing 2 with 3. As a direct corollary of Theorem 3, we have that single-agent eistemic lanning for K4 is undecidable as well. Desite all these negative results, there is still room for decidability in the single-agent case if we assume that knowledge or belief are negatively introsective: Theorem 4. PLANEX(K45, ) and PLANEX(S5, ) are decidable. Proof. Any formula of K45 (and hence also of S5) is rovably equivalent to a normal form formula of degree [Meyer and van der Hoek, 995], i.e., a conjunction of disjunctions of the form φ i φ 0 i φ... i φ n. Therefore, any eistemic lanning task can be reduced equivalently to a classical lanning task whose states are eistemic models of height at most. So, because there is a finite number of eistemic models of height at most (for a finite set of roositional atoms), the state sace of the classical lanning roblem is finite. Hence, we immediately get decidability. 6 Multi-agent Eistemic Planning In the multi-agent setting, we rove a strong result, namely that multi-agent eistemic lanning is undecidable for any logic between K and S5. The roof of this undecidability result generalizes the roof for single-agent K given in Section 5.. Like for the roof of undecidability of the revious section for S4, the idea underlying the roof is to relace worlds with meta-worlds. 6. Encoding of Configurations We encode configurations as eistemic states of two-agent S5. The worlds in CHAIN(, n) of Figure 2 are relaced with the eistemic models META-WORLD () of Figure 20. The way meta-worlds are connected to each other to form a META-CHAIN (, n) is shown in Figure 2. Then, for each configuration (k, l, m) N 3, we define an eistemic state META-S (k,l,m) by relacing in Figure 3 CHAIN with META-CHAIN and by labeling the accessibility relations originating from the designated world with agent. Note that as we are in S5, all relations are equivalence relations, but the reflexive, symmetric and transitive closure is left imlicit in figures. 6.2 Encoding of the Comutation Function Similarly to the case of single-agent K and S4, we define ath formulas. Inria

20 Undecidability in Eistemic Planning 7 =, r 0 Fig. 20: META-WORLD () n + META-WORLD () = w :, r w 3n 2:, r w 3n+:, r w 2: 0 w 3: w 3n : 0 w 3n: 0 w 3n+2: w 3n+3: Fig. 2: META-CHAIN (, n) τ n µ n 0 µ n λ 0 0 µ 0 3(m+) events τ n µ n 0 µ n µ 0 τ 0 0 µ 0 λ 0 µ 0 τ 0 τ n µ n 0 µ n Fig. 22: META-INC () Fig. 23: META-DEC () Fig. 24: META-REPL (, n, m) Definition 6 (Meta-ath formulas). For all P and n N, we define the formulas λ n (), µ n () and τ n () inductively by: We then obtain a counterart of Lemma : λ 0 () := r µ 0 () := 0 λ 0 () λ 0 () τ 0 () := r µ 0 () λ n+ () := µ n () µ n () r µ n+ () := 0 λ n+ () λ n+ () τ n+ () := r µ n+ (). Lemma 6. For all P, n N, 0 i n, j 3n + 3: META-CHAIN (, n), w j = λ i () iff j = 3n + 3 3i META-CHAIN (, n), w j = µ i () iff j = 3n + 2 3i META-CHAIN (, n), w j = τ i () iff j = 3n + 3i. In other words, λ i holds in the bottom world of the (i+)th to last meta-world of META-CHAIN (, n), µ i in the to right world of the same meta-world and τ i in the to left world of the same meta-world. Now we define META-INC (), META-DEC () and META-REPL (, n, m) as in Figures RR n 830

21 8 G. Aucher & T. Bolander Let M = (I 0,..., I T ) be a two-counter machine. For all k < T and all l, m N, we define similarly to the eistemic actions of Figures 6 9 an eistemic action e M (k, l, m) from the eistemic actions of Figures 7 0 by:. relacing INC, DEC and REPL with META-INC, META-DEC and META-REPL resectively; 2. labeling the accessibility relations originating from the designated worlds with agent ; 3. relacing φ k with µ k ( ); 4. relacing ( 2 γ ) with µ 0 ( 2 ). 6.3 Encoding of the Halting Problem If CHAIN, INC, DEC, a M and s (k,l,m) are relaced in Lemmata 2 and 3 with META-CHAIN, META-INC, META-DEC, e M and META-S (k,l,m) resectively, then these Lemmata still hold. Therefore, Lemma 4 and Theorem 2 also generalize to this two-agent S5 setting, and we finally obtain that: Theorem 5. PLANEX(S5, n) is undecidable for any n 2. 7 DEL Model Checking The DEL language L DEL is defined by the following BNF [van Ditmarsch et al., 2007]: φ ::= φ (φ φ) i φ [π]φ π ::= (E, e) (π π) (π; π) π where P, i A and (E, e) is any eistemic action. In DEL, one assumes that A >. The truth conditions for the rograms π are defined as follows: M, w = [E, e]φ iff M, w = re(e) imlies (M, w) (E, e) = φ M, w = [π γ]φ iff M, w = [π]φ and M, w = [γ]φ M, w = [π; γ]φ iff M, w = [π][γ]φ M, w = [π ]φ iff for all finite sequences π;... ; π, M, w = [π;... ; π]φ The formula [E, e]φ reads as after the execution of the eistemic action (E, e), it holds that φ. The model checking roblem is the following: Given an eistemic state (M, w), a formula φ L DEL, is it the case that M, w = φ?. As an immediate corollary of our results, we have the following theorem. It comlements the result of [Miller and Moss, 2005] stating that the satisfiability roblem of DEL is undecidable. Theorem 6. The model checking roblem of the language L DEL is undecidable. Proof. PLANEX(S5, n) is reducible to the model checking roblem of the language L DEL : an eistemic lanning task T = (s 0, A, φ g ) has a solution iff s 0 = [A ] φ g holds. Inria

22 Undecidability in Eistemic Planning 9 Single-agent Multi-agent lanning lanning K UD UD KT UD UD K4 UD UD K45 D UD S4 UD UD S5 D UD Fig. 25: Summary of results (D=Decidable, UD=UnDecidable) 8 Conclusion 8. Related Work Alternatives to the DEL-based aroach to multi-agent lanning with ToM abilities can be found both in the literature on temoral eistemic logics [van der Hoek and Wooldridge, 2002] and in the literature on POMDP-based lanning [Gmytrasiewicz and Doshi, 2005]. However, these alternative formalisms exress lanning tasks in terms of an exlicitly given state sace, and hence do not address how to exress actions in a comact and convenient formalism (and how to ossibly avoid building the entire state sace when solving lanning tasks). In the DEL-based formalism the state sace is induced by the action descritions as in classical lanning. Note that our assumtions in DEL-based lanning corresond to the infinite horizon case of lanning based on POMDPs, in which already ordinary, single-agent lanning is undecidable [Madani et al., 999]. 8.2 Concluding Remarks Our results are summarized in the table of Figure 25 (we recall that they hold only for P 2). From this table, we notice that in the single-agent setting, the roerty of Euclidicity (defined by Axiom 5: i φ i i φ) draws the borderline between decidability and undecidability: if 5 is added to K4 or S4, we immediately obtain decidability. Given these results, an imortant quest of course becomes to find fragments of the formalism in which interesting roblems can still be formulated, but where the comlexity is comarable to the comlexity of other standard lanning formalisms (varying from PSPACE-comleteness for classical lanning [Bylander, 994] u to 2-EXP-comleteness for lanning under nondeterminism and artial observability [Rintanen, 2004]). We leave the quest for decidable fragments to future work. Initial results in this direction can be found in [Löwe et al., 20]. RR n 830

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25 RESEARCH CENTRE RENNES BRETAGNE ATLANTIQUE Camus universitaire de Beaulieu Rennes Cedex Publisher Inria Domaine de Voluceau - Rocquencourt BP Le Chesnay Cedex inria.fr ISSN