Optimal Tableaux for Right Propositional Neighborhood Logic over Linear Orders

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1 Optimal Tableaux for Right Propositional Neighborhood Logic over Linear Orders Davide Bresolin 1, Angelo Montanari 2, Pietro Sala 2, and Guido Sciavicco 3 1 Department of Computer Science, University of Verona, Verona, Italy davide.bresolin@univr.it 2 Department of Mathematics and Computer Science, University of Udine, Udine, Italy {angelo.montanari pietro.sala}@dimi.uniud.it 3 Department of Information, Engineering and Communications, University of Murcia, Murcia, Spain guido@um.es Abstract. The study of interval temporal logics on linear orders is a meaningful research area in computer science and artificial intelligence. Unfortunately, even when restricted to propositional languages, most interval logics turn out to be undecidable. Decidability has been usually recovered by imposing severe syntactic and/or semantic restrictions. In the last years, tableau-based decision procedures have been obtained for logics of the temporal neighborhood and logics of the subinterval relation over specific classes of temporal structures. In this paper, we develop an optimal NEXPTIME tableau-based decision procedure for the future fragment of Propositional Neighborhood Logic over the whole class of linearly ordered domains. 1 Introduction Propositional interval temporal logics play a significant role in computer science and artificial intelligence, as they provide a natural framework for representing and reasoning about temporal properties. Unfortunately, the computational complexity of most of them (the two prominent interval temporal logics, namely, Halpern and Shoham s HS [8] and Venema s CDT [11], are highly undecidable) constitutes a barrier to their extensive use in practical applications. Not surprisingly, recent research in the area focused on the development of implementable deduction systems for them. Early work in this direction includes Bowman and Thompson s decision procedure for the propositional fragment of Moszkowski s ITL [9], interpreted over finite linearly ordered domains [1], and a non-terminating tableau system for CDT, interpreted over partially ordered domains [7]. In the former case, decidability is achieved by introducing a simplifying hypothesis, called locality principle, that constrains the relation between the truth value of a formula over an interval and its truth values over initial subintervals of that interval.

2 2 Tableau-based decision procedures have been recently obtained for logics of the temporal neighborhood and logics of the subinterval relation over specific classes of temporal structures, without resorting to any simplifying assumption. The logic D of the subinterval relation is a fragment of HS which features a single unary modality corresponding to the strict subinterval relation, where a subinterval has no endpoints in common with the current one. In [2], Bresolin et al. devise a sound and complete PSPACE-complete tableau system for D interpreted in the class of all dense linearly ordered sets. Moreover, they extended such a result to the logic D, where a subinterval may have (at most) one endpoint in common with the current one. The decision problem for D over other classes of temporal structures, including the whole class of linearly ordered domains, the class of discrete linearly ordered domains, N, and Z, is still open. The logic PNL of temporal neighborhood is the propositional fragment of Neighborhood Logic [6]. It can be viewed as a fragment of HS that features two modal operators A and A, that respectively correspond to the met-by and the meets relations. The logical properties of PNL have been systematically investigated in [3]. In particular, decidability (in fact, NEXPTIME-completeness) of PNL when interpreted over various classes of temporal structures, including all linearly ordered domains, all well-ordered domains, all finite linearly ordered domains, and N [3], has been shown via a reduction to the satisfiability problem for the two-variable fragment of first-order logic for binary relational structures over ordered domains [10]. Despite these significant achievements, the problem of devising decision procedures for PNL of practical interest has been only partially solved. In [5], a tableau system for its single-modality future fragment RPNL, interpreted over N, has been developed; such a result has been later extended to full PNL over Z [4]. In this paper, we focus our attention on RPNL interpreted in the whole class of linearly ordered domains, and we develop a NEXPTIME tableau system for it. Since NEXPTIME-completeness holds for PNL and its single-modality fragments, no matters which are the properties of the underlying linearly ordered temporal structure [3], the proposed solution turns out to be optimal. From a technical point of view, the proposed tableau system is quite different from the one for N [5]. While models for RPNL formulas over N can be generated by simply adding future points (possibly infinitely many) to a given partial model, the construction of a model for an RPNL formula over an arbitrary linearly ordered domain may require the addition of points (possibly infinitely many) in between existing ones. Such a difference is illustrated in Section 2 by means of a simple example. The paper is organized as follows. In Section 2, we introduce syntax and semantics of RPNL and we analyse its expressiveness. In Section 3, we describe a terminating tableau system for RPNL interpreted in the class of all linearly ordered domains. An example of the procedure at work concludes the section. In the next section, we prove its soundness, completeness, and optimality. Conclusions provide an assessment of the work and outline future research directions.

3 3 2 RPNL over linearly ordered domains In this section, we first provide syntax and semantics of Right Propositional Neighborhood Logic (RPNL, for short); then, we show that RPNL is expressive enough to distinguish between satisfiability over N and over the class of all linearly ordered domains. The language of RPNL consists of a set AP of atomic propositions, the propositional connectives,, and the modal operator A (you can read A as adjacent). The other propositional connectives, as well as the logical constants (true) and (false) and the dual modal operator [A], are defined as usual. The formulas of RPNL, denoted by ϕ, ψ,..., are generated by the following abstract syntax: ϕ ::= p ϕ ϕ ϕ A ϕ. Given a linearly ordered domain D = D, <, an interval over D is an ordered pair [d i, d j ] such that d i < d j. An interval structure is a pair D, I(D), where I(D) is the set of all intervals over D. Logics of temporal neighborhood have been studied in different flavors, either including or excluding point intervals, that is, intervals of the form [d i, d i ], and a modal constant to capture them. In this paper, we assume the so-called strict semantics (point intervals are not admitted); however, similar results can be obtained for the non-strict case. The semantics of RPNL is given in terms of interpretations of the form M = D, I(D), V, where D, I(D) is an interval structure and V : AP 2 I(D) is a valuation function assigning a set of intervals to every atomic proposition. It is recursively defined by the satisfiability relation = as follows: for every p AP, M, [d i, d j ] = p iff [d i, d j ] V(p); M, [d i, d j ] = ψ iff M, [d i, d j ] = ψ; M, [d i, d j ] = ψ 1 ψ 2 iff M, [d i, d j ] = ψ 1 or M, [d i, d j ] = ψ 2 ; M, [d i, d j ] = A ψ iff [d j, d k ] I(D) such that M, [d j, d k ] = ψ. We denote by [A]ψ the formula A ψ. Note that [A]ψ means that every adjacent future interval must make p true, while [A][A]ψ means that ψ is true over every non-adjacent future interval. Given an RPNL-formula ϕ, we denote by (A)ϕ a formula of the form A ϕ or [A]ϕ. We define the closure of ϕ (denoted by CL(ϕ)) as the set of all subformulas of ϕ (including ϕ itself) and of their negations, and the temporal closure of ϕ (denoted by TF(ϕ)) as the set {(A)ψ (A)ψ CL(ϕ)}. To show that RPNL is expressive enough to distinguish between satisfiability over N and over the class of all linearly ordered domains, we exhibit a formula that is unsatisfiable over the former and satisfiable over the latter. Let [G] be the universally-in-the-future operator defined as follows: [G]ψ = ψ [A]ψ [A][A]ψ and let seq p be a shorthand for p A p. Consider the formula AccP oints = A p [G]seq p A [G] p. We will show that AccP oints is unsatisfiable over N, while it is satisfiable whenever the temporal structure in which it is interpreted has at least one accumulation point, that is, a point which is the right bound of an infinite (ascending) chain of points.

4 4 Proposition 1. The RPNL-formula AccP oints is satisfiable over the class of all linearly ordered domains, while it is not satisfiable over N. Proof. We first show that the formula AccP oints is not satisfiable over N. Suppose, by contradiction, that there exists an interpretation M, based on N, such that M, [d 0, d 1 ] = AccP oints. From M, [d 0, d 1 ] = A p [G]seq p, it follows that there exists a sequence of points d 1 < d j1 < d j2... such that M, [d 1, d j1 ] = p and M, [d ji, d ji+1 ] = p, for all i 1. Moreover, from M, [d 0, d 1 ] = A [G] p, it follows that there exists a point d i such that M, [d 1, d i ] = [G] p. Two cases may arise. Case (1). Suppose d i < d j1. From M, [d 1, d i ] = [A][A] p, it follows that M, [d i, d j1 ] = [A] p and thus M, [d j1, d j2 ] = p. As shown in Figure 1, this implies that both p and p hold over [d j1, d j2 ] (a contradiction). p, [A] p, d 0 d 1 d i d j1 AccP oints [A][A] p [A] p p, p d j2 p, A p, [A]seq p, [A][A]seq p p, Fig. 1. Unsatisfiability of AccP oints over N: case (1). Case (2). Suppose d j1 < d i. From M, [d 1, d i ] = [A][A] p, it follows that, for any point d k > d i, M, [d i, d k ] = [A] p and, for any point d m > d k, M, [d k, d m ] = p. Since AccP oints is interpreted over N, there exists a point d jh > d i such that p holds over [d jh, d jh+1 ]. Hence, as shown in Figure 2, both p and p hold over [d jh, d jh+1 ] (a contradiction). p, [A] p, d jh d 0 AccP oints d 1 [A][A] p d i [A] p d j1 d j2 d jh 1... p, p d jh+1 p, A p, p, p, A p [A]seq p, A p [A][A]seq p p, Fig. 2. Unsatisfiability of AccP oints over N: case (2). Let us consider now the class of all linearly ordered domains. A model satisfying AccP oints can be built as follows: we take an infinite sequence of points d j1 < d j2 < d j3 <... such that M, [d ji, d ji+1 ] = p, for every i 1, and then we add an accumulation point d ω greater than d ji, for every i 1, such that M, [d 1, d ω ] = [G] p. The definition of the valuation function can be easily completed without introducing any contradiction, thus showing that AccP oints is satisfiable (see Figure 3). RPNL interpreted over N thus differs from RPNL interpreted in the class of all linearly ordered domains. This prevents us from exploiting the tableau-based decision procedure for RPNL over N developed in [5] to check the satisfiability

5 5 d 0 AccP oints d 1 p, [A] p, [A][A] p d ω p, A p, [A]seq p, [A][A]seq p p, A p p, A p... p p p Fig. 3. A model for AccP oints over the class of linearly ordered domains. of RPNL formulas over the class of all linearly ordered domains. If applied to the formula AccP oints, such a procedure would correctly answer unsatisfiable, as there are no models satisfying it based on N. In the next section, we devise an original tableau system provided with the ability of dealing with accumulation points. 3 A tableau for RPNL over linearly ordered domains In the following, we first define the structure of a tableau for an RPNL-formula and then we show how to construct it. A tableau for an RPNL formula is a suitable labeled tree T. Every node n of T is labeled by a tuple ν(n) = [d i, d j ], Γ n, D n, where D n is a finite linear order, [d i, d j ] I(D n ), and Γ n CL(ϕ). Expansion rule. The expansion rule adds new nodes at the end of the branch to which it is applied. Given a branch B, B n 1 denotes the result of expanding B with the node n 1, while B n 1... n k denotes the result of adding k immediate successors nodes n 1,..., n k to B. A node n in a branch B such that the interval component [d i, d j ] of its labeling does not belong to the labeling of any other node in B is called an active node. In general, the same interval [d i, d j ] may belong to (the labeling of) different nodes in a branch B. In such a case, the farthest-from-the-root node in B labeled with [d i, d j ] is the active one, while the others are non-active. The expansion rule can be applied to active nodes only. With a little abuse of notation, given a branch B, we write ([d i, d j ], ψ) B if there exists a node n in B, labeled with [d i, d j ], Γ n, D n, such that ψ Γ n. Definition 1. Let B be a branch, D B be the linearly ordered set belonging to the label of the leaf of B, and n be an active node in B with label ν(n) = [d i, d j ], Γ n, D n. The expansion rule for n B is defined case-by-case as follows: OR: if ψ 1 ψ 2 Γ n, ψ 1 Γ n, and ψ 2 Γ n, expand B to B n 1 n 2, with ν(n 1 ) = [d i, d j ], Γ n {ψ 1 }, D B and ν(n 2 ) = [d i, d j ], Γ n {ψ 2 }, D B ; AND: if (ψ 1 ψ 2 ) Γ n and ψ 1 Γ n or ψ 2 Γ n, expand B to B n 1, with ν(n 1 ) = [d i, d j ], Γ n { ψ 1, ψ 2 }, D B ; NOT: if ψ Γ n and ψ Γ n, expand B to B n 1, with ν(n 1 ) = [d i, d j ], Γ n {ψ}, D B ;

6 6 DIAMOND: if A ψ Γ n and there exists no d k D B such that d k > d j and ([d j, d k ], ψ) B, then proceed as follows. Let d j+1,..., d j+k be the points in D B greater than d j. Expand B to B n 1... n k m 0... m k+1, where 1. for every 1 h k, if there exists an active node n B labeled with [d j, d j+h ], Γ, D, then ν(n h ) = [d j, d j+h ], Γ {ψ}, D B, otherwise ν(n h ) = [d j, d j+h ], {ψ}, D B ; 2. for every 0 h k, ν(m h ) = [d j, d h ], {ψ}, D h B, where Dh B is obtained from D B by adding a new point d h strictly in between d j+h and d j+h+1 (if h = k, the second condition is obviously missing); BOX: if A ψ Γ n and there exists d k D B such that d j < d k and ([d j, d k ], ψ) B, expand B to B n 1. If there are no nodes in B labeled with [d j, d k ], then ν(n 1 ) = [d j, d k ], { ψ}, D B ; otherwise, ν(n 1 ) = [d j, d k ], Γ n { ψ}, D B, where n is the (unique) active node in B labeled with [d j, d k ]. Blocking condition. Given a branch B and a point d D B, the set REQ(d ) of the temporal requests of d is defined as follows: REQ(d ) = {ψ TF(ϕ) : [d, d ], ([d, d ], ψ) B}. Moreover, we define the set of past temporal requests of d as the following set of sets: PAST(d) = {REQ(d ) : d < d}. Definition 2. We say that a point d i D B is blocked if there exists a point d j D B, with d j < d i, such that (i) REQ(d i ) = REQ(d j ) and (ii) PAST(d i ) = PAST(d j ). Notice that for any pair d i, d j, PAST(d i ) = PAST(d j ) if and only if for every d < d i there exists d < d j such that REQ(d ) = REQ(d ). Expansion strategy. We say that a branch B is closed if there exist a formula ψ and an interval [d i, d j ] such that both ([d i, d j ], ψ) B and ([d i, d j ], ψ) B, otherwise we say that B is open. Moreover, we say that an expansion rule is applicable to a node n if n is active and its application generates at least one node with a new labeling. Definition 3. The expansion strategy for a branch B is defined as follows: i) apply the expansion rule to B only if B is open; ii) if B is open, apply the AND, OR, NOT, and BOX rules to the closest-tothe-root active node to which the expansion rule is applicable; iii) if B is open, apply the DIAMOND rule to the closest-to-the-root active node to which the expansion rule is applicable, provided that the right endpoint of the interval [d i, d j ] in its labeling is not blocked. Definition 4. A tableau for an RPNL-formula ϕ is any finite labeled tree obtained by expanding the one-node labeled tree [d 0, d 1 ], {ϕ}, {d 0 < d 1 } through successive application of the branch-expansion strategy to existing branches, until it cannot be applied anymore.

7 7 Open and closed tableau. Given a formula ϕ and a tableau T for it, we say that T is closed if all its branches are closed, otherwise it is open. Example. We illustrate the behaviour of the proposed tableau system by applying it to the formula ϕ = A p A [A] p [A](p A p). A portion of the resulting tableau is depicted in Figure 4. For the sake of readability, we will describe sequences of expansion steps that do not split the branch (applications of the AND, NOT, and BOX rules) as single expansion steps. Moreover, instead of explicitly representing the linear orders associated with the nodes, we will simply display the extensions to the linear order when they are introduced. Finally, in the textual explanation we will identify a branch with its leaf node. The root n 0 of the tableau contains the A -formulas A p and A [A] p. We n 0 = [d 0, d 1], A p, A [A] p, [A](p A p), {d 0 < d 1} n 1 = [d 1, d 2], p, A p, p A p, {d 1 < d 2} n 2 = [d 1, d 2], p, A p, n 3 = [d 1, d 3], p A p, p A p, [A] p [A] p, {d 1 < d 3 < d 2} closed n 4 = [d 1, d 3], n 5 = [d 1, d 3], p, p A p, [A] p n 7 = [d 3, d 2], p n 6 = [d 1, d 3], p A p, A p, [A] p closed p A p, [A] p, {d 2 < d 3}... n 8 = [d 2, d 4], p, {d 2 < d 4} n 14 = [d 3, d 5], p n 9 = [d 3, d 4], p n 13 = [d 4, d 5], p, {d 4 < d 5} n 10 = [d 1, d 4], p A p n 15 = [d 1, d 5], p A p n 11 = [d 1, d 4], p, p A p open n 12 = [d 1, d 4], p A p, A p n 16 = [d 1, d 5], p open n 17 = [d 1, d 5], A p, d 5 blocked Fig. 4. Part of the tableau for the formula ϕ = A p A [A] p [A](p A p) first apply the DIAMOND rule to A p. Since d 1 is the greatest point of the current linear order, we can only add a point d 2 to the right of d 1 and satisfy p over the interval [d 1, d 2 ] (node n 1 ). (In fact, node n 1 is obtained by an application of the DIAMOND rule followed by an application of the OR rule and the removal of the inconsistent node including both p and p.) Next, we apply the DIAMOND rule to the formula A [A] p in n 0 and generate the nodes n 2, n 3, and n 4. The node n 2 is closed, because it contains both A p and [A] p. The expansion proceeds by the application of the OR rule to n 3, that generates the nodes n 5 and n 6. The application of the BOX rule to n 5 generates the node n 7, which is further expanded by applying the DIAMOND rule to the formula A p in n 1. With two applications of the BOX rule (to nodes n 5 and n 0, respectively), we generate the node n 10. Then, the application of the OR rule to n 10 generates

8 8 the nodes n 11 and n 12. The branch ending in n 11 can be easily shown to be open, because all the A formulas in it are fulfilled and no more expansion rules are applicable to it. Such a condition allows us to conclude that the formula ϕ is satisfiable. To give an example of the application of the blocking condition, we expand the tableau a bit more. By applying the DIAMOND rule to n 12, we obtain the node n 13. Two applications of the BOX rule to n 13 generates the nodes n 14 and n 15. The OR rule is then applied to n 15. The branch ending in n 16 is open, because all A formulas are fulfilled and no expansion rules can be applied to it. The branch ending in n 17 is not expanded anymore, because point d 5 is blocked (REQ(d 5 ) = REQ(d 4 ) and P AST (d 5 ) = P AST (d 4 )). 4 Soundness, completeness, and complexity In this section, we show that the proposed tableau method is sound, complete, and terminating. In addition, we prove that it is complexity optimal. As a preliminary step, we recall some basic notions (details can be found in [5]). Definition 5. A ϕ-atom is a set A CL(ϕ) such that, for every ψ CL(ϕ), ψ A iff ψ A, and, for every ψ 1 ψ 2 CL(ϕ), ψ 1 ψ 2 A iff ψ 1 A or ψ 2 A. We denote the set of all ϕ-atoms by A ϕ. Atoms are connected by the following binary relation. Definition 6. Let R ϕ be a binary relation over A ϕ such that, for every pair of atoms A, A A ϕ, A R ϕ A if and only if, for every [A]ψ CL(ϕ), if [A]ψ A, then ψ A. We now introduce a suitable labeling of interval structures based on ϕ-atoms. Definition 7. A ϕ-labeled interval structure ( LIS for short) is a pair L = D, I(D), L, where D, I(D) is an interval structure and L : I(D) A ϕ is a labeling function such that, for every pair of neighboring intervals [d i, d j ], [d j, d k ], L([d i, d j ]) R ϕ L([d j, d k ]). If we interpret the labeling function as a valuation function, LISs represent candidate models for ϕ. The truth of formulas devoid of temporal operators and that of [A]-formulas indeed follow from the definition of ϕ-atom and the definition of R ϕ, respectively. However, to obtain a model for ϕ we must also guarantee the truth of A -formulas. Definition 8. A ϕ-labeled interval structure L = D, I(D), L is fulfilling if and only if, for every temporal formula A ψ TF(ϕ) and every interval [d i, d j ] I(D), if A ψ L([d i, d j ]), then there exists d k > d j such that ψ L([d j, d k ]). Theorem 1 (Fulfilling LISs and satisfiability [5]). A formula ϕ is satisfiable if and only if there exists a fulfilling LIS L = D, I(D), L with ϕ L([d 0, d 1 ]).

9 9 Theorem 2 (Soundness). If ϕ is satisfiable, then the tableau for it is open. Proof. Let ϕ be a satisfiable formula and let L = D, I(D), L be a fulfilling LIS for ϕ. We show how an open tableau T for ϕ can be obtained from L. Since L is a fulfilling LIS for ϕ, there exists an interval [d 0, d 1 ] such that ϕ L([d 0, d 1 ]). We start the construction of T with the one-node initial tableau [d 0, d 1 ], {ϕ}, {d 0 < d 1 } and then we proceed in accordance with the expansion strategy. We prove by induction on the number of steps of the tableau construction that the current tableau T includes a branch B which satisfies the following invariant: for every n = [d i, d j ], Γ n, D n in B and every ψ Γ n, ψ L([d i, d j ]). By construction, the initial tableau satisfies the invariant. As for the inductive step, let T be the current tableau and let B be the branch of T that satisfies the invariant. Moreover, let n = [d i, d j ], Γ n, D n be the node in B taken into consideration by the expansion strategy. The following cases may arise: the OR rule is applied to n. We have that ψ 1 ψ 2 Γ n, ψ 1 Γ n, and ψ 2 Γ n. By the inductive hypothesis, ψ 1 ψ 2 L([d i, d j ]). By definition of L, there exists ψ k, with k {1, 2}, such that ψ k L([d i, d j ]). The expansion of B into B n k, with ν(n k ) = [d i, d j ], Γ n {ψ k }, D B, maintains the invariant true. the AND rule is applied to n. We have that (ψ 1 ψ 2 ) Γ n and ψ 1 Γ n or ψ 2 Γ n. By the inductive hypothesis, (ψ 1 ψ 2 ) L([d i, d j ]). By definition of L, both ψ 1 L([d i, d j ]) and ψ 2 L([d i, d j ]). It immediately follows that the expanded branch B n 1, with ν(n 1 ) = [d i, d j ], Γ n { ψ 1, ψ 2 }, D B, preserves the invariant. the NOT rule is applied to n. We have that ψ Γ n and ψ Γ n. By the inductive hypothesis, ψ L([d i, d j ]) and thus ψ L([d i, d j ]). The expanded branch B n 1, with ν(n 1 ) = [d i, d j ], Γ n {ψ}, D B, satisfies the invariant. the DIAMOND rule is applied to n. We have that A ψ Γ n and there exists no d k D B such that d k > d j and ([d j, d k ], ψ) B. Since A ψ L([d i, d j ]), by definition of fulfilling LIS, ψ L([d j, d ]) for some d > d j in D. Two cases may arise: d D B. If there are no nodes in B labeled with [d j, d ], we expand B with a node n, with ν(n) = [d j, d ], {ψ}, D B ; otherwise, we expand B with a node n, with ν(n) = [d j, d ], Γ n {ψ}, D B where n is the (unique) active node in B labeled with [d j, d ]. In both cases, the expansion of B with n preserves the invariant. d / D B. Let d j+1,..., d j+h, with h 0, be the points in D B greater than d j. We have that d j+k < d < d j+k+1 for some 0 k h. Let D be the linear order obtained from D B by putting the new point d in between d j+k and d j+k+1. The expansion of B with a node m k, with ν(m k ) = [d j, d ], {ψ}, D, preserves the property.

10 10 the BOX rule is applied to n. We have that A ψ Γ n and there exists d k D B such that d j < d k, and ([d j, d k ], ψ) B. Since A ψ = [A] ψ L([d i, d j ]), ψ L([d j, d h ]) for all d j < d h, and thus ψ L([d j, d k ]). It can be easily seen that the expansion of B with node n 1, whose labeling is defined according to the BOX rule, preserves the truth of the invariant. The above argument guarantees that the resulting tableau T features (at least) one branch B that satisfies the invariant. Suppose now, by contradiction, that B is closed. This implies that there exist an interval [d i, d j ] and a formula ψ CL(ϕ) such that both ([d i, d j ], ψ) B and ([d i, d j ], ψ) B. Given the truth of the invariant, it follows that both ψ L([d i, d j ]) and ψ L([d i, d j ]) (contradiction). Hence, B is open and thus, by definition, the tableau for ϕ is open. Completeness is proved by showing how to construct a fulfilling LIS satisfying ϕ from a fulfilling branch B in a tableau T for ϕ. From Theorem 1, it follows immediately that ϕ has a model. We will take advantage of the following lemma. Lemma 1. Let B be an open branch of a tableau. For every [d i, d j ] I(D B ), the following conditions hold: 1. for any ψ CL(ϕ), it never happens that both (ψ, [d i, d j ]) B and ( ψ, [d i, d j ]) B; 2. if (ψ 1 ψ 2, [d i, d j ]) B, then (ψ 1, [d i, d j ]) B or (ψ 2, [d i, d j ]) B; 3. if ( (ψ 1 ψ 2 ), [d i, d j ]) B, then ( ψ 1, [d i, d j ]) B and ( ψ 2, [d i, d j ]) B; 4. if ( ψ, [d i, d j ]) B, then (ψ, [d i, d j ]) B; 5. if ( A ψ, [d i, d j ]) B and d j is not a blocked point, then there exists d k > d j such that (ψ, [d j, d k ]) B; 6. if ( A ψ, [d i, d j ]) B, then, for every d k > d j, ( ψ, [d j, d k ]) B. Proof. The thesis follows from the tableau rules and the expansion strategy. Theorem 3 (Completeness). If the tableau for ϕ is open, then ϕ is satisfiable. Proof. Let T be the tableau for ϕ. By definition, if T is open, then there exists an open branch B in T. We distinguish two cases. 1. There are no blocked points in B. A fulfilling LIS L = D, I(D), L can be obtained as follows. As a first step, we execute the following operations: D = D B ; for every ψ CL(ϕ) such that ([d 0, d 1 ], ψ) B, we let ψ L([d 0, d 1 ]); for every [d i, d j ] I(D B ), with d i > d 0, and every ψ CL(ϕ) such that ([d i, d j ], ψ) B, we let ψ L([d i, d j ]); for every d j > d 1, we let L([d 0, d j ]) = L([d 1, d j ]) (notice that for every [d 0, d j ] I(D B ), with d j > d 1, there are no nodes in B with an interval [d 0, d j ] in their label.

11 11 The resulting structure is not necessarily a LIS: it could be the case that, for some interval [d i, d j ] and formula ψ CL(ϕ), neither ψ L([d i, d j ]) nor ψ L([d i, d j ]). However, it can be extended to a complete LIS as follows (we proceed by induction on the structure of ψ): if ψ = p or ψ = p, we let p L([d i, d j ]); if ψ = ψ 1 ψ 2, we let ψ L([d i, d j ]) if and only if ψ 1 L([d i, d j ]) or ψ 2 L([d i, d j ]); if ψ = ψ 1, we let ψ L([d i, d j ]) if and only if ψ 1 L([d i, d j ]); if ψ = A ψ 1, we let ψ L([d i, d j ]) if and only if there exists d k > d j such that ψ 1 L([d j, d k ]). Such a completion procedure produces a fulfilling LIS L: by Lemma 1, for each pair of neighboring intervals [d i, d j ], [d j, d k ], if [A]ψ L([d i, d j ]), then ψ L([d j, d k ]). Moreover, since there are no blocked points in B, L is fulfilling. By Theorem 1, we can conclude that ϕ is satisfiable. 2. There is at least one blocked point in B. We proceed as in the previous case. Obviously, the resulting structure is in general not fulfilling. We can turn it into a fulfilling LIS L as follows. Let (b 1,..., b k ) be the list of all blocked points in D B, arranged in an arbitrary order. For every blocked point b j (b 1,..., b k ), let m j be the corresponding maximal blocking point, that is, the greatest non-blocked point in D B such that m j < b j, REQ(b j ) = REQ(m j ), and PAST(b j ) = PAST(m j ). Consider now a specific point b i {b 1,..., b k } and assume that A ψ 1,..., A ψ n are the A -formulas in REQ(b i ) which are not fulfilled by the current structure. Since m i is non-blocked and REQ(b i ) = REQ(m i ), for every A ψ j REQ(b i ) there exists an interval [m i, d ψj ] such that ψ j L([m i, d ψj ]). Two cases may arise: d ψj > b i. In such a case, we fulfill A ψ j REQ(b i ) by replacing the current labeling of [b i, d ψj ] with L([b i, d ψj ]) = L([m i, d ψj ]); m i < d ψj b i. In such a case, we add a new point e ψj in between b i and its immediate successor (if any) and we let L([b i, e ψj ]) = L([m i, d ψj ]). Such a construction must be repeated for all A ψ j { A ψ 1,..., A ψ n }. At the end, there can be a number of intervals, generated by the new points it introduces, with an incomplete labeling. Let e ψj be one of the new points. We complete the labeling of the interval starting/ending at e ψj as follows: for every interval [e ψj, d], we let L([e ψj, d]) = L([d ψj, d]); for every interval [d, e ψj ], with d < d ψj, we let L([d, e ψj ]) = L([d, d ψj ]); for every interval [d, e ψj ], with d ψj d < e ψj, let d be a point such that d < d ψj and REQ(d ) = REQ(d). Since PAST(b i ) = PAST(m i ), such point is guaranteed to exists. We let L([d, e ψj ]) = L([d, d ψj ]). At the end of this completion process, we remove b i from the list of blocked points. For every added point e ψj, if there there exists a A -formula in REQ(e ψj ) whose request is not fulfilled by the current structure, we insert e ψj at the end of the current list of blocked points. By possibly repeating the above expansion step infinitely many times, we guarantee that every point added to the list of blocked point is eventually expanded. The resulting (limit) structure is thus a fulfilling labeled structure, and, by Theorem 1, we can conclude that ϕ is satisfiable.

12 12 To determine the computational complexity of the proposed method, we take advantage of the following lemma. Lemma 2. The length of every branch in a tableau T for ϕ is bounded by ϕ ϕ. Proof. Let B be a branch of T and D B = D B, < be the linear order associated with it. For every [d i, d j ] I(D B ), there exists at most one active node in B labeled with [d i, d j ]. Moreover, according to the expansion rule, a node n becomes inactive when a new active node n is added to B, with Γ n Γ n. Hence, for every [d i, d j ] I(D B ), there exist at most ϕ nodes in B, where ϕ denotes the number of elements (and thus of subformulas) of ϕ. The only rule that adds new points to D B is the DIAMOND rule, which is applicable only to nodes labeled with intervals [d i, d j ] whose right endpoint is not blocked (according to Definition 2, a point d j is blocked if there exists a point d i < d j such that REQ(d i ) = REQ(d j ) and PAST(d i ) = PAST(d j )). For every formula A ψ TF(ϕ), either A ψ REQ(d j ) or [A] ψ REQ(d j ) or neither A ψ nor [A] ψ belongs to REQ(d j ). Hence, the number of different sets of requests that can be associated with a point is less than or equal to 3 TF 2. Along the branch B, the pasts of points are obviously ordered by inclusion, that is, given d, d D B, with d < d, PAST(d) PAST(d ). This implies that the number of different PAST sets that may occur in B is bounded by the number of REQ sets (that is, 3 TF 2 ). Hence, the number of non-blocked points is less than or equal to 3 TF 2 3 TF 2 = 3 TF. Now, for every non-blocked point d i, the number of A -formulas in REQ(d i ) is less than or equal to TF 2. Hence, by the application of the DIAMOND rule, every non-blocked point can introduce at most TF 2 new points. In the worst case, the total number of points in D B is thus TF 2 3 TF. This implies that the TF 2 number of intervals in I(D B ) is bounded by TF, and that the number TF 2 of nodes in B is bounded by ϕ TF ϕ ϕ. Lemma 2 allows us to conclude that the satisfiability of an RPNL formula ϕ can be checked by a non-deterministic procedure that generates an open branch of the tableau, if any, in non-deterministic exponential time. From the NEXPTIME-hardness of the satisfiability problem for RPNL [5], the optimality of the proposed tableau method immediately follows. Theorem 4. The tableau method for RPNL interpreted over all linearly ordered domains is optimal. 5 Conclusions In this paper, we focused our attention on the logic RPNL. Its decidability over various classes of linear orders immediately follows from results in [3]. A limited research effort was devoted to the development of decision procedures for RPNL

13 13 to be used in practice: an optimal tableau method for RPNL over N is given in [5] (later extended to full PNL over Z in [4]). In this paper, we devise a computationally optimal tableau method for RPNL interpreted in the whole class of linearly ordered domains, which turned out to be substantially different from that for N. We are currently investigating the possibility of generalizing the proposed tableau method to cope with full PNL. Besides additional rules for the past-time modalities A and [A], a revision of the definition of blocked points is needed, to distinguish between right-blocked (points that do not require the addition of new points to their future) and left-blocked (points that do not require the addition of new points to their past) points. These modifications have a relevant impact on the soundness, completeness, and termination of the method. In parallel, we are exploring the possibility of adpating the tableau method to the case of RPNL (and PNL) over dense linearly ordered domains, whose decision problem is still open. References 1. H. Bowman and S. Thompson. A decision procedure and complete axiomatization of finite interval temporal logic with projection. Journal of Logic and Computation, 13(2): , D. Bresolin, V. Goranko, A. Montanari, and P. Sala. Tableau systems for logics of subinterval structures over dense orderings. In Proc. of TABLEAUX 2007, volume 4548 of LNAI, pages Springer, D. Bresolin, V. Goranko, A. Montanari, and G. Sciavicco. On Decidability and Expressiveness of Propositional Interval Neighborhood Logics. In Proc. of the International Symposium on Logical Foundations of Computer Science (LFCS), volume 4514 of LNCS, pages Springer, D. Bresolin, A. Montanari, and P. Sala. An optimal tableau-based decision algorithm for Propositional Neighborhood Logic. In Proc. of the 24th International Symposium on Theoretical Aspects of Computer Science (STACS), volume 4393 of LNCS, pages Springer, D. Bresolin, A. Montanari, and G. Sciavicco. An optimal decision procedure for Right Propositional Neighborhood Logic. Journal of Automated Reasoning, 38(1-3): , Z. Chaochen and M. R. Hansen. An adequate first order interval logic. In W.P. de Roever, H. Langmaak, and A. Pnueli, editors, Compositionality: the Significant Difference, number 1536 in LNCS, pages Springer, V. Goranko, A. Montanari, G. Sciavicco, and P. Sala. A general tableau method for propositional interval temporal logics: Theory and implementation. Journal of Applied Logic, 4(3): , Joseph Y. Halpern and Yoav Shoham. A propositional modal logic of time intervals. Journal of the ACM, 38(4): , B. Moszkowski. Reasoning about digital circuits. Tech. Rep , Dept. of Computer Science, Stanford University, Stanford, CA, M. Otto. Two variable first-order logic over ordered domains. Journal of Symbolic Logic, 66(2): , Y. Venema. A modal logic for chopping intervals. Journal of Logic and Computation, 1(4): , 1991.