On the Complexity of Model Checking for Syntactically Maximal Fragments of the Interval Temporal Logic HS with Regular Expressions

Abstract

In this paper, we investigate the model checking (MC) problem for Halpern and Shoham's interval temporal logic HS. In the last years, interval temporal logic MC has received an increasing attention as a viable alternative to the traditional (point-based) temporal logic MC, which can be recovered as a special case. Most results have been obtained under the homogeneity assumption, that constrains a proposition letter to hold over an interval if and only if it holds over each component state. Recently, Lomuscio and Michaliszyn proposed a way to relax such an assumption by exploiting regular expressions to define the behaviour of proposition letters over intervals in terms of their component states. When homogeneity is assumed, the exact complexity of MC is a difficult open question for full HS and for its two syntactically maximal fragments AABBE and AAEBE. In this paper, we provide an asymptotically optimal bound to the complexity of these two fragments under the more expressive semantic variant based on regular expressions by showing that their MC problem is AEXP pol -complete, where AEXP pol denotes the complexity class of problems decided by exponential-time bounded alternating Turing Machines making a polynomially bounded number of alternations.

Why Propositional Quantification Makes Modal Logics on Trees Robustly Hard?

“…We have shown that the satisfiability problems for QCTL t X , QCTL t F and QCTL t XF are Tower-complete, and for N ≥ 2, SAT(QCTL t X,≤N ) is AExp pol -complete. Whereas AExp pol -hardness is established by reducing the alternating multi-tiling problem recently introduced in [31], the Tower-hardness of SAT(QCTL t X ) required to be able to express that a node has a number of children equal to some tower of exponentials of height k. Section V deals with the Tower-completeness of SAT(QCTL f t X ) and SAT(QCTL f t XF ), as well as Tower-completeness for the modal logics K, KD, GL, K4, D4 and S4 with propositional quantification but with adequate classes of tree-like structures. All our Tower-hardness results significantly improve what was known so far for fragments of QCTL t and for the above-mentioned well-known modal logics.…”

“…The complexity of several logical problems has been captured by the class AExp pol , see e.g. [45], [20], [31]. For proving AExp pol -hardness, we use an elegant modification of Tiling 1 , introduced in [31], [32].…”

“…[45], [20], [31]. For proving AExp pol -hardness, we use an elegant modification of Tiling 1 , introduced in [31], [32]. The extension amounts to consider a stack of n tilings, with a matching relation between two consecutive tile types on the same position of the grid, and quantifications over the tile types on the first row (initial conditions).…”

“…The matching lower bound is inherited from the problem QBF. In order to show that the problem is AExp pol -hard, we define below a reduction from AMTP presented in Section II and introduced in [31]. To define a grid [0, 2 n −1]×[0, 2 n − 1], a major part in the solution of an instance of AMTP, we specify a tree such that every node at distance less than 2n from the root ε has exactly two children (so there are exactly 2 2n nodes at distance 2n from ε) and each node at distance 2n encodes a position (H,…”

“…AExp pol is the class of problems decidable with an exponential-time alternating TM with a polynomial number of alternations. For AExp pol -hardness, our reduction is from the alternating multi-tiling problem introduced in [31], [32] (used to prove that the model-checking problem for the interval temporal logic BĒ with regular expressions is AExp pol -hard). ( ) More generally and despite the modest scope of EX, the satisfiability problem for QCTL t X is shown Tower-hard (Theorem 20), by a uniform reduction from k-NExpTime-complete tiling problems (uniformity is with respect to k).…”