STIT is dangerously undecidable

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1 STIT is dangerously undecidable François Schwarzentruber and Caroline Semmling June 2, 2014 Abstract STIT is a potential logical framework to capture responsibility, counterfactual emotions and norms, which are main ingredients for specifying behaviors of virtual agents. We identify here a new fragment and its satisfiability problem is NPcomplete and in Σ 3 when the number of agents is unbounded. We also identify a slightly more expressive fragment which is undecidable. This fragment is used to prove that Broersen s XSTITwhen the semantics is standard is undecidable. We prove that with XSTIT-models, XSTITis NEXPTIME-hard when the number of agents is unbounded. Keywords: STIT theory, Satisfiability problem, Complexity 1 Introduction The STIT framework has its origin in philosophy. It comprises logics of agency where the types of action are not made explicit in the language. In fact, it is expressed that an individual or a group of agents sees to it that a certain state of affairs obtains, [J stit :ϕ]. The acronym STIT conveys this Seeing To It That. Developed by Belnap, Perloff, Horty [2, 3, 12, 13] the STIT framework is based on the idea to treat agency as a modality. It provides different semantics to interpret modal operators like [i stit : ϕ] and [J stit : ϕ] where the individual i or the group J sees to it that ϕ. We focus here on the version that is called Chellas STIT. It was defined for individuals and named so in [12] because of the similarity to the operator introduced by Chellas in [8]. 1 The individual STIT framework was extended to joint agency in [2, Section 10.C] and for the Chellas STIT to group agency in [13, Section 2.4]. Although developed as a formal framework in the philosophy of action, the STIT framework is increasingly taken into account as a potential framework for reasoning about concepts of artificial intelligence. At the moment the STIT framework is already the base for many extension. It has been suggested to combine these action operators with epistemic operators [5, 6], to describe counterfactual emotions within this STIT framework [16], to ascribe responsibility [9] and of course that it can be extended by deontic operators [7, 13]. 1 The original Chellas operator in [8] differs from the version used in [12] because it does not comply with the principle of independence of agents that plays a central role in the STIT framework. For discussion of achievement STIT and deliberative STIT semantics cf. [2, 12]. 1

2 In order to specify behaviors of artificial agents, it is essential to capture such concepts related to agency like knowledge, emotions, intentions and responsibilities. Hence STIT can be a promising candidate for the underlying necessary description of the agency. For instance, an agent a regrets that ϕ is true if a desires ϕ and he knows that ϕ is true and that other agents do not see to it that ϕ. Therefore we examine in this paper the STIT framework and focus on the satisfiability problem. It should be recalled that the satisfiability problem of atemporal group STIT is undecidable and that the atemporal group STIT is not finitely axiomatizable, when the number of agents is greater than three [11]. The individual atemporal STIT, 2 however, is finitely axiomatizable [24] and, if the number of agents is greater than two, it is NEXPTIME-complete [1]. Recently, a variant called XSTIT[6, 7] has been proven to be decidable [20] (the semantics of XSTITis not standard but given in terms of XSTITmodels). In this paper, we provide new complexity results concerning the STIT framework. More precisely: We identify a new fragment whose satisfiability problem is NP-complete (theorem 3); We prove that if the number of agents is unbounded, then its satisfiability problem is in Σ 3 (theorem 4); A slightly more expressive fragment is shown to be undecidable (theorem 5); XSTIT variant in the standard semantics of STIT is shown to be undecidable as well (corollary 1); XSTIT variant with the alternative non-equivalent semantics of XSTIT is NEXPTIMEhard (theorem 6). In section 2 we recall the semantics of STIT. In section 3 we discuss about fragments of STIT. In section 4, we consider the language of XSTITand we consider both the standard semantics of STIT and the alternative non-equivalent semantics of XSTIT, that was proposed by Broersen, interms of XSTITmodels. 2 The STIT framework In this section we define the language and the semantics of atemporal group STIT. 2.1 Syntax The language of group STIT are built from a denumerable infinite set of propositional variables ATM and a finite set of n agents, AGT = {1,..., n}. Well-formed formulas of the group STIT language are defined by: ϕ := p ϕ ϕ ϕ [J]ϕ, 2 Some authors [6, 23] use the term multi-agent STIT to designate the logic where operators are of the form [i cstit :]. We prefer here the use of the more explicit term individual STIT as in [11] to distinguish individual from group STIT. 2

3 where p ATM and J AGT. The other Boolean connectives are defined as usual. The expression [J]ϕ is read as the group of agents J sees to it that ϕ. The expression J ϕ is introduced as abbreviation for [J] ϕ. It is read as the group of agents J does not prevent ϕ or the particular choices of all agents in J do not rule out ϕ. It is conventional to write [i]ϕ in place of [{i}]ϕ. The formula [ ]ϕ is read as it is historically necessary that ϕ and the formula ϕ expresses that it is historically possible that ϕ. The construction [J]ϕ says it is possible that J sees to it that ϕ. 2.2 Kripke Semantics In spite of the philosophically well-founded background of BT+AC structures [2, 3, 12], an alternative semantics is proposed in [14] that is closer to standard model semantics. Definition 1 A STIT-Kripke model is a tuple M = (W, R, V) where: W is a non-empty set, R is a function which assigns to each agent i AGT an equivalence relation R i on W such that for all w W, for all w i R (w) for i AGT it holds that R i (w i ), i AGT V is a valuation function V : ATM 2 W. The function R represents the choice-function. The constraint on R constitutes the independence condition of agents. The worlds in R i (w) to a given world w and agent i represents the choice-equivalent worlds for i in w. The collective choices of agents are also similarly defined as the collective choices on a Branching-Time structure with choice-functions, cf. [13]. For all non-empty J AGT, the sets of choice-equivalent worlds of a group J are defined by R J = i J R i, cf. [11]. The truth conditions of the Boolean connectives are standard and for the group STIT operator it is given for any J AGT by: M, w = [J]ϕ iff M, = ϕ for all u R J. A formula ϕ is satisfiable if there exists a Kripke model M and a world w such that M, w = ϕ. A formula ϕ is valid if for all Kripke models M and all worlds w in M, we have M, w = ϕ. Actually, the semantics described gives what we call here the non-deterministic STIT. Another alternative semantics is to restrict ourselves to deterministic Kripke model where we assume R i (w i ) is a singleton. This defines deterministic STIT. i AGT In deterministic STIT, [AGT]ϕ ϕ is a validity because it holds in all M, w where M is a deterministic Kripke model and all worlds w of M. 2.3 Product semantics It is given in terms of STIT-product models [17]. These models are defined in analogy to the normal form of games. 3

4 Definition 2 A STIT-product model is a tuple M = (W, V) defined as follows: W = Π i W i where W i are non-empty sets; V : W 2 ATM ; Accordingly, the truth condition of the group STIT operator is given by: M, w = [J]ϕ iff for all u W such that u j = w j for all j J we have M, u = ϕ. With STIT-product models, we obtain the same set of satisfiable formulas that for deterministic STIT. Actually, considering deterministic STIT is not in itself a restriction. Indeed, we obtain an algorithm for the satisfiability problem of non-deterministic STIT built from an algorithm for deterministic STIT because of the following embedding that is similar to the one presented in [10]: Theorem 1 Let AGT = AGT { } where is a fresh new agent. Let ϕ a STITformula in which the set of agents is AGT. Then ϕ is satisfiable in a non-deterministic STIT model iff tr 1 (ϕ) is satisfiable in a deterministic STIT-model where tr is defined by: tr 1 (p) = p; tr 1 ([J]ψ) = [J { }]tr 1 (ψ). Thus, in section 3, we only consider deterministic STIT. 3 Fragments In this section, we investigate fragments given by the following grammar: ϕ ::= χ ψ ϕ ϕ ϕ (1) ψ ( can formulas) ψ ::= [J]χ ψ ψ ψ (2) χ ::= p χ χ χ ( see-to-it formulas) (propositional formulas). where p ranges over atomic propositions and J ranges over subsets of agents. χ- formulas are propositional formulas. ψ-formulas are called see-to-it formulas and are Boolean formulas over formulas of the form [J]χ where J is a group of agents and χ is a Boolean formula. ϕ are Boolean formulas of χ-formulas and ψ-formulas but also constructions of the formula ψ. They are called can formulas. The fragment d f S T IT where negation constructions of type (1) are allowed but negation constructions of type (2) are not allowed has already been investigated in [16]. We recall the NP-completeness of it in subsection 3.1. The fragment d f S T IT2 where negation constructions of type (2) are allowed but negation constructions of type (1) is considered in subsection 3.2. Finally we consider bools T IT where negation constructions of both type (1) and type (2) are allowed. One starts down the slippery slope. In this section, we show that a slightly difference in the fragment can lead from decidability to undecidability. Remark that in this section, we deal with deterministic STIT. As the fragment presented here are stable by the translation given in Theorem 1, the upper-bound results also hold for non-deterministic STIT. 4

5 3.1 A decidable fragment of the state of the art In [16], the authors have introduced a decidable fragment called d f S T IT. It is defined with the help of three grammar rules that are: ϕ ::= χ ψ ϕ ϕ ϕ ψ ψ ::= [J]χ ψ ψ χ ::= p χ χ χ ( can formulas) ( see-to-it formulas) (propositional formulas). The d f S T IT fragment contains formulas as ([J]p [K]q) ([J]q p) [K]p. It however does not allow formulas like ( [J]p [K]q). Although it is possible to negate can formulas, it is not possible to negate a see-to-it formula in the scope of an -operator. The fragment d f S T IT is then defined as the set of all formulas that can be generated from the ϕ rule in the previous BNF. The satisfiability problem of a formula of d f S T IT is NP-complete. More precisely, the proof in [16] gives the following theorem: Theorem 2 If a formula ϕ of d f S T IT is satisfiable, then it is satisfiable in a model where each agent has at most k choices where k is the number of modal operators appearing in ϕ. A challenging open question concerning d f S T IT would be to provide a tight complexity upper bound for the satisfiability problem when the number of agents is unbounded. 3.2 Results about another decidable fragment If we disallow negation of can formulas ϕ, but allow negation of arbitrary see-toit formulas in the scope of a -operator, we obtain the following fragment, called d f S T IT2: ϕ ::= χ ψ ϕ ϕ ψ ψ ::= [J]χ ψ ψ ψ χ ::= p χ χ χ ( can formulas) ( see-to-it formulas) (propositional formulas). For instance, this fragment is interesting for writing formulas as [{Ann}]letter [{Ann, Bob}](carry [{Ann}]letter) saying that it is possible for Ann to write a letter but actually Ann and Bob see to it that the box is carried and she is not seeing to it that the letter is written. This fragment is decidable and also NP-complete because of the following result: Theorem 3 If a formula ϕ of d f S T IT2 is satisfiable, then it is satisfiable in a model where each agent has at most k choices where k is the number of modal operators appearing in ϕ. 5

6 Proof. A formula ϕ of d f S T IT2 is a conjunction of formulas ψ 1,..., ψ n and ψ where ψ 1,..., ψ n and ψ are Boolean formulas over propositions and constructions of the form [J]χ or [J]χ where χ is purely propositional. Let M, w a pointed model satisfying ϕ. The idea is to then select points for each ψ i and for each subformulas [J]χ of ψ i or ψ that is false. Then we construct a model M made up of those selected points. If the number of agents is unbounded, that is, AGT is infinite the previous theorem says that each formula ϕ is satisfiable in a model of exponential size k card(agt ϕ) where AGT ϕ is the (finite) set of agents that appears in ϕ. Therefore, the satisfiability problem is in NEXPTIME. Nevertheless, the result can be improved. We will show that the satisfiability problem is in Σ 3 = NP co NPNP. The class Σ 3 includes problems that can been solved by an alternating algorithm running in polynomial time where the alternation of quantifiers is. For more information to Σ 3, one may refer to [19]. Theorem 4 If the number of agents is unbounded, d f S T IT2 is in Σ 3, where Σ 3 = NP co NPNP. Proof. We want to check if ϕ is satisfiable. Let suppose that AGT is finite and contains all agents that appear in ϕ. Without loss of generality (w.l.o.g.) we suppose that ϕ does not contain any [AGT]-operator, because for any STIT-product model M = (W, V), it holds that R AGT = id W and M, w = [AGT]ψ ψ for all w W. Then the algorithm is as follows: Step 1: w.l.o.g. we suppose that ϕ is a conjunction of formulas of the form ψ i for i = 1... n. If not, we add an -operator in front of the Boolean subformulas ψ. For instance, if ϕ = (p q) r, it is transformed to ϕ = (p q) r. Note that ϕ is satisfiable iff ϕ is satisfiable; Step 2: for each formula ψ i, we assume a valuation for ψ i so that we can rewrite each formula ψ i in the following form: (π i J χ i, j J [J]χ i J J AGT j J where J is a finite, but big enough set of integers and π i, χ i, j J, χ i J are propositional formulas for all i = 1,..., n and j J. Note that many of the subformulas χ i, j i J and χ J are not present. Such a subformula is considered as. Step 3: A i = π i i i {1,...,n} χ i J χ J for all i {1,..., n}; Bi, J j = χi, j J i i {1,...,n} χ i J J χ J for all i = 1..n, j J ; C J1,...,J n = ( i {1,...,n} J i J χ i i J i AGT and J k J l = for all k l. ), for all J 1,..., J n such that i {1,...,n} J i = 6

7 We are going to prove that ϕ is satisfiable iff each propositional formula A i, B i, j and C J1,...,J n above are satisfiable. Let M = (W, V) a STIT-product model, and w W such that M, w = ϕ. We show that the formulas A i, B i, j and C J1,...,J n are satisfiable. For all i {1,..., n}, there exists a world w i such that M, w i = ψ i, thus it holds that M, w i = A i for all i {1,..., n}. For all j J, because of M, w i = J χ i, j J, there exists a world wi, j R J (w i ) such that M, w i, j = χ i, j J. The world wi, j also satisfies B i, j. Let us consider J 1,..., J n such that i {1,...,n} J i = AGT and J k J l = for all k l. Let u be the point defined by for all k {1,..., n}, u k = w i k where i is the unique element such that k J i. It follows w i R Ji u for all i {1,..., n}. Hence, the formula C J1,...,J n is true at u. We construct a STIT-product model M = (W, R, V) such that there exists a world w W such that M, w = ϕ. For all i {1,..., n}, W i = {1,..., n} ({1,..., n} J 2 AGT ). The valuation V is defined as follows: V((i) AGT ) is a valuation that satisfies A i ; V((i) J + (i, j, J) J) is a valuation that satisfies B i, j (note that it is well defined because J AGT) for all others points u, V(u) is a valuation that satisfies C J 1,...,J n such that J i = {k {1,..., n} u k = i} (note that there exists C 1,..., C n such that i {1,...,n} J i = AGT and J k J l = for all k l such that = C J1,...,J n C J 1,...,J n ) Now let us prove that there exists w W such that M, w = ϕ. For that, we prove that for all i {1,..., n}, M, (i) AGT = ψ i. Note that M, (i) AGT = π i, since M, (i) AGT = A i. Let J AGT, j J. It holds that M, (i) AGT = J χ i, j J because there exists a world, namely u = (i) J + (i, j, J) J, such that (i) AGT R J u and M, u = χ i, j J. i We prove that M, (i) AGT = [J]χ J. Let u R J ((i) AGT ). If u is of the form (i) J +(i, j, J) J, as Bi, J j χ i J is valid, hence M, u = χ i J. Otherwise, M, u = C J 1 such that J J,...,J n i. By definition of C J 1,...,J n, we have = C J 1 χ,...,j n i J hence again M, u = χ i J. Thus, it i is stated that M, (i) AGT = [J]χ J. As a consequence, there exists an alternating algorithm that works as follows: first we execute Step 2. It consists in choosing subformulas for all ψ ( -choice). Then we execute Step 3. First we check that formulas 1. and 2. are satisfiable ( -choice). Then, concerning the point 3., we choose J 1,..., J n and the algorithm succeeds if for each choice of J 1,..., J n ( -choice), the formulas 3. is satisfiable ( -choice). The quantifications involved in the algorithm is. Therefore the problem of satisfiability is in Σ 3 = NP co NPNP. Proposition 1 If the number of agents is unbounded, the satisfiability problem in d f S T IT d f S T IT2 is NP-hard and co-np-hard. 7

8 Proof. We reduce the satisfiability problem of Classical Boolean logic hence the satisfiability problem of d f S T IT d f S T IT2 if the number of agents is unbounded is NP-hard (trivial). We reduce the validity problem of Classical Boolean logic hence the satisfiability problem of d f S T IT d f S T IT2 if the number of agents is unbounded is co-np-hard in the following way. To check whether the Boolean formula χ(p 1,..., p n ) is satisfiable we check the following formula in d f S T IT d f S T IT 2 is STIT-satisfiable: ( [i]p i [i] p i ) [ ]χ. i {1..n} So the satisfaibility problem of d f S T IT d f S T IT2 if the number of agents is unbounded is conp-hard. 3.3 Undecidability results Now we extend the previous fragments slightly to obtain all formulas where each modal operator is followed by an arbitrary Boolean formula except the -operator which can be followed by an arbitrary formula of modal depth one.the difference to the former fragments is not big. We just allow negations in both the ψ and ϕ rule. The BNF is read as follows: ϕ ::= χ ψ ϕ ϕ ϕ ψ ψ ::= [J]χ ψ ψ ψ χ ::= p χ χ χ ( can formulas) ( see-to-it formulas) (propositional formulas). The new fragment we consider is generated by the ϕ rule of the grammar and is called bools T IT. Unfortunately, we have the following result: Theorem 5 The satisfiability problem of bools T IT is undecidable. Proof. We reduce the satisfiability problem of group STIT to the satisfiability problem of bools T IT. We use here the idea of [18] consisting in replacing all subformulas by atoms. Let ϕ 0 a group STIT formula. Let S F(ϕ 0 ) the set of all subformulas of ϕ 0. For all ψ S F(ϕ 0 ) we introduce a new proposition x ψ and we construct the formula S IM ψ depending on the form of ψ: if ψ is a proposition p, S IM ψ = [ ](p x p ); if ψ = χ, S IM ψ = [ ](x χ x χ ); if ψ = χ 1 χ 2, S IM ψ = [ ](x χ1 χ 2 (x χ1 x χ2 )); if ψ = [J]χ, S IM ψ = [ ](x [J]χ [J]x χ ). We leave the reader to prove by induction that ϕ is STIT-satisfiable iff ψ S F(ϕ 0 ) S IM ψ x ϕ is satisfiable. In practice, we may be interesting in the following decision problem: 8

9 input: a STIT formula ϕ, an integer k; output: yes whether there exists a model M where each agents has at most k choices and a world w in M such that M, w = ϕ. A naïve procedure is to guess a pointed model M, w of size k card(agt) and to do model checking of ϕ in M, w. The latter problem is in NEXPTIME (it is exponential in the number of digits to represent k). 4 Discussion about XSTIT Since group STIT is undecidable, Broersen in [6, 7] proposes the XSTIT logic to consider non-instantaneous STIT operators that are evaluated at a determined set of next - states. Broersen s XSTIT logic is decidable and corresponds to the standard view in formal models of computation where an action at the actual state only takes effect in some next states [6, 7]. He also mentions a philosophical concern. An action is associated with a period of time that it takes in order to perform the action, so the view is defensible that action takes place in time, cf. [6]. The XSTIT language is the following: ϕ ::= p ϕ ϕ ϕ [ ]ϕ [J]Xϕ 3 where p ATM and J AGT. The operators [J]X are fused operators of the STIT operators [J] and the next moment operator X. 4.1 Standard semantics For the moment we neglect the semantics given by Broersen [7]. As shown in [21], the original semantics for a group STIT with next operator is also equivalent to Kripke models M = (W, R, F X, V), where (W, R, V) is a STIT-Kripke model, cf. Definition 1, and F X is a total function F X : W W such that R J F X F X R (no choice between undivided histories). The function F X represents the next relation on the set of worlds. Then the truth condition for the next operator is given by: M, w = Xϕ iff M, F X (w) = ϕ. To address the satisfiability problem of the XSTIT language in the original semantics, we define the following translation tr 2 from bools T IT in the XSTIT as follows: tr 2 (p) = Xp; tr 2 ([ ]ψ) = [ ]tr 2 (ψ); tr 2 (ϕ ψ) = (tr 2 (ϕ) tr 2 (ψ)) ; tr 2 ([J]ϕ) = [J]Xϕ. Example 1 tr 2 ( ([J]p [K]q) [L]r s) = ([J]Xp [K]Xq) [L]Xr Xs). 3 Broersen in [7] denotes the operator [ ] by and the operators [J]X by [J xstit]. We change the BNF to show some similarities to the bools T IT fragment of group STIT with next operator. 9

10 Proposition 2 A bools T IT-formula ϕ is STIT-satisfiable iff tr 2 (ϕ) is STIT-satisfiable. Proof. Let ϕ be a bools T IT-formula. Suppose there exists a model M = (W, R, V) and world w W such that M, w = ϕ 0. We define M = (W {0, 1}, R, V, F X ) where: R J = {((w, 0), (u, 0)) (w, u) R J} {((w, 1), (u, 1)) w W}; V (p) = {(w, 1) w V(p)}; F X ((w, 0)) = (w, 1) and F X ((w, 1)) = (w, 1) for all w W. Now we have to prove that M, (w, 0) = tr 2 (ϕ 0 ). To this aim, we prove by induction over Boolean formulas χ, M, (w, 1) = tr 2 (χ) iff M, w = χ. Then we prove by induction over formulas ϕ in bools T IT that M, (w, 0) = tr 2 (ϕ) iff M, w = ϕ. Suppose there exists M = (W, R, V, F X ) and a world w W such that M, w = tr 2 (ϕ 0 ). We define M = (W, R, V) as: W = W ; R = R ; V(p) = {w W F X (w) V (p)}. We have to prove that M, w = ϕ 0. We prove by induction that over Boolean formulas χ, M, F X (w) = tr 2 (χ) iff M, w = χ. Then we prove by induction over formulas ϕ in bools T IT that M, w = tr 2 (ϕ) iff M, w = ϕ. Corollary 1 Deciding whether a XSTIT-formula is STIT-satisfiable is undecidable. Proof. We reduce the STIT-satisfiability problem of a bools T IT-formula to the STITsatisfiability problem of a XSTIT-formula. 4.2 Broersen s semantics Broersen gave an alternative non-equivalent semantics to XSTIT. Recently it has been shown that the corresponding satisfiability problem is decidable [20]. Here, we prove that the satisfiability problem is NEXPTIME-hard. Before that, we recall the Broersen s semantics as defined in [7] is defined as follows. We first define the class of models called XSTIT models. Then we define truth conditions. Definition 3 (XSTIT-models) A XSTIT model M X = (S, H, R [ ], {R J J AGT}) consists of: S is a non-empty set of so called states and H is a non-empty sets of so called histories, where a history is a set of states. If s h for s S and h H, the pair (h, s) is named a world. The set of all worlds is denoted by W. R [ ] W W is the so called necessitiy-relation over worlds for all (h, s), (h, s ) W: (h, s)r [ ] (h, s ) iff s = s. For all J AGT, R J W W such that: 10

11 R AGT is a next -time relation, such that if (h, s)r AGT (h, s ) then h = h and R AGT is serial and deterministic. Thus, the relation R AGT defines a linear ordering on the states of every history. R J R K, if K J, R [ ] R AGT R ; R J R [ ] R AGT for all J AGT; R AGT R [ ] R J for all J AGT (no choice between undivided histories). for all w, w, w W, J, K AGT with J K = it holds: if wr [ ] w and wr [ ] w, then there is a w W such that wr [ ] w and for all w iv, w v W if w R J w iv, then w R J w iv and if w R K w v, then w R K w v (independency of agents). The truth conditions are defined as follows. The Boolean connectives are defined as usual. M X, w = [ ]ϕ iff for all w W with wr [ ] w it holds that M, w = ϕ. M X, w = [J]Xϕ iff for all w W with wr J w it holds that M, w = ϕ. The property of independence of agents in Broersen s XSTIT semantics makes the formula [1]Xp 1... [n]xp n ([1]Xp 1... [n]xp n ) valid. As a consequence, this enables us to prove the following result on the complexity of the satisfiability problem over Broersen s two dimensional Kripke models: Theorem 6 If the number of agents is unbounded, the satisfiability problem of a XSTITformula in Broersen s XSTIT-model is NEXPTIME-hard. Proof. We will reduce a NEXPTIME-complete tiling problem [4] to the satisfiability problem of a XSTIT-formula in Broersen s XSTIT-model. Let k be a natural number. A tile type t is a 4-tuple of colors t = (le f t(t), right(t), up(t), down(t)). The tiling problem is defined as follows. Input: a finite set T of tile types, a t 0 T and a natural number k written in its binary form. Output: yes iff we can tile a k k grid with the tile types of T and t 0 being placed onto (0, 0). In other words, the problem is to decide whether there exists a function τ from {1,... k} 2 to T satisfying the following constraints: 1. τ(0, 0) = t 0 ; 2. up(τ(x, y)) = down(τ(x, y + 1)) for all x {1,..., k}, y {1,..., k 1}; 3. right(τ(x, y)) = le f t(τ(x + 1, y)) for all x {1,..., k 1}, y {1,..., k}. 11

12 W.l.o.g. we may assume that k = 2 n. Let us consider an instance (T, t 0, k) of the tiling problem. The propositions p i encodes the coordinates. The truth values of the propositions p i, where i < n, correspond to the binary representation of the abscissa x and the truth values of the propositions p i, where n i < 2n, correspond to the binary representation of the ordinate y. For n = 4 e.g., the location where x = 4 and y = 3 is represented by the following valuation: p 0, p 1, p 2, p } {{ } 3 p 4, p 5, p 6, p } {{ } We introduce also proposition p i for a second tiling in the same way. The global idea is now to design a XSTIT formula that impose: the existence of two tilings (formulas 1, 2, 3, 7, 8); the fact that the two tilings are equal (formula 4); the fact that the horizontal color constraints are satisfied (we impose it between a tile t of the first tiling and the tile on the right of t but in the second tiling) (formula 5); the fact that the vertical color constraints are satisfied (formula 6). t 0 in (0, 0) in the first tiling (formula 9). Now we assign agents to these propositions. For every proposition p i and p i we add a STIT operator [p i ] resp. [p i ]. We then encode the problem in the XSTIT language: 1. [ ]([p]xp [p] Xp) where p ranges over p i or p i ; 2. [p]xp [p]x p where p ranges over p i or p i. Formulas 1. and 2. enforces that all valuations over p 1,..., p n, p 1,..., p n are available in next moments. The worlds in the next moments are considered as points representing couples (a, b) where a is a tile location in the first tiling (where coordinates are described by p i s) and b is a tile location in the second tiling (where coordinates are described by p i s). All couples can be found in the model because of the independence of agents constraints on XSTIT models. It validates the formula ( [p 0 ]p 0... [p k ]p k ) ([p 0 ]p 0... [p k ]p k ). We also introduce atomic propositions for each tile of the first tiling, t 1, t 2,... and prime versions likewise for the second tiling t 1, t 2,.... Instead of writing down cryptic propositional formulas, we prefer to write the intuitive meaning and we leave to the reader the formalization in classical propositional logic of the following formulas: 3. [ ] uniqueness and existence of a tile for the first and second tiling ; ( 4. [ ]( abscissa from pi s = abscissa from p i s ordinate from p i s = ordinate from p i s ) same tile in the first and second tiling ) ; 12

13 ( 5. [ ] abscissa from p i s = abscissa from p i s + 1 the right color of the tile of the first tiling is equal to the left color of the tile of the second tiling ( 6. [ ] ordinate from p i s = ordinate from p i s + 1 ) the bottom color of the tile of the first tiling is equal to the top color of the tile of the second tiling ) ; ) ; 7. [ ](Xt [{p i 0 i < 2n}]Xt), for all tiles in the first tiling propositions t; (this means that if you have tile t for the first tiling, then if you do not change coordinates of the tiling location for the first tiling, you will still have t as a tile for the first tiling) 8. [ ](Xt [{p i 0 i < 2n}]Xt ) for all propositions t for the second tiling. 9. [ ](X( p 1... p n ) t 0 ) (t 0 is placed at coordinates (0, 0) in the first tiling) The reduction is comptable in polynomial time and is such that we can tile a 2 n 2 n squares with the tile types of T and t 0 being placed onto (0, 0) iff the conjunction of the formulas 1,..., 9 is satisfiable in a XSTIT-model. 5 Conclusion The main concern of this paper deals with the satisfiability problem of the formulas over standard semantics of atemporal STIT framework. As it is wellknown, the atemporal group STIT logic and the product logic S5 n are related: one can polynomial reduce the satisfiability problem of S5 n to the satisfiability problem of atemporal group STIT logic [11] and vice versa. So all results about S5 n can been transferred to atemporal STIT logic and vice versa, first of all the undecidability of the SAT problem. We have seen that there is a fine line between decidability and undecidability. Even though the modal depth is restricted to two, there are fragments of STIT which are undecidable. Agi Kurucz shows that the product S5 3 has not the finite model property [15] by exhibiting a formula with just four propositions. A model satisfying that formula has to have an infinite number of worlds. Consequently, atemporal group STIT also does not have the FMP. An open question, however, is the complexity of the satisfiability problem of atemporal group STIT when the number of proposition is restricted to three. At the end of the paper we discuss some complexity issues of Broersen XSTIT logic. Since it is decidable, as Broersen point out in [6] and as it has been proven [20], this XSTIT logic cannot be equal to a fragment of STIT with an additional next 13

14 operator where the group STIT operators are fused with the next operator. Nevertheless, we prove that the satisfiability problem of Broersen s XSTIT logic is NEXPTIME-hard. STIT can be a key feature of reasoners about counterfactual emotions, responsability, etc. As some fragments are NP-complete, a far good pratical approach would be to use SAT techniques. Another idea may be to develop efficient tableau method that can be integrated in already existing solvers that rely on this technique for modal and description logics. Handling XSTIT seems far more difficult in pratice as it is already NEXPTIMEhard. The upper bound of XSTIT is an open issue. References [1] Philippe Balbiani, Andreas Herzig, and Nicolas Troquard. Alternative axiomatics and complexity of deliberative STIT theories. Journal of Philosophical Logic, [2] Nuel Belnap, Michael Perloff, and Ming Xu. Facing the Future: Agents and Choices in Our Indeterminist World. Oxford University Press, Oxford, [3] Nuel Belnap and Micheal Perloff. Seeing to it that: a canonical form for agentives. Theoria, 54: , [4] P.V.E. Boas. The convenience of tilings. In Complexity, Logic, and Recursion Theory, pages Marcel Dekker Inc, [5] J. Broersen. A logical analysis of the interaction between obligation-to-do and knowingly doing. Deontic Logic in Computer Science, pages , [6] J. Broersen. A complete stit logic for knowledge and action, and some of its applications. Declarative Agent Languages and Technologies VI, pages 47 59, [7] J.M. Broersen. Deontic epistemic stit logic distinguishing modes of mens rea. Journal of Applied Logic, 9(2): , [8] Brian F. Chellas. Time and modality in the logic of agency. Studia Logica, 51(3/4): , [9] Roberto Ciuni and Rosja Mastop. Attributing distributed responsibility in stit logic. In LORI, pages 66 75, [10] V. Goranko and W. Jamroga. Comparing semantics of logics for multi-agent systems. Synthese, 139: , [11] A. Herzig and F. Schwarzentruber. Properties of logics of individual and group agency. In Advances in Modal Logic, pages , [12] John Horty and Nuel Belnap. The deliberative stit: a study of action, omission, ability and obligation. Journal of Philosophical Logic, 24(6): ,

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