Abstract. We study linear-time temporal logics interpreted over data words with multiple attributes. We restrict the atomic formulas to equalities of attribute values in successive positions and to repetitions of attribute values in the future or past. We demonstrate correspondences between satisfiability problems for logics and reachability-like decision problems for counter systems. We show that allowing/disallowing atomic formulas expressing repetitions of values in the past corresponds to the reachability/coverability problem in Petri nets. This gives us 2expspace upper bounds for several satisfiability problems. We prove matching lower bounds by reduction from a reachability problem for a newly introduced class of counter systems. This new class is a succinct version of vector addition systems with states in which counters are accessed via pointers, a potentially useful feature in other contexts. We strengthen further the correspondences between data logics and counter systems by characterizing the complexity of fragments, extensions and variants of the logic. For instance, we precisely characterize the relationship between the number of attributes allowed in the logic and the number of counters needed in the counter system.
This paper studies the boundedness and termination problems for vector addition systems equipped with one stack. We introduce an algorithm, inspired by the Karp & Miller algorithm, that solves both problems for the larger class of well-structured pushdown systems. We show that the worstcase running time of this algorithm is hyper-Ackermannian for pushdown vector addition systems. For the upper bound, we introduce the notion of bad nested words over a wellquasi-ordered set, and we provide a general scheme of induction for bounding their lengths. We derive from this scheme a hyper-Ackermannian upper bound for the length of bad nested words over vectors of natural numbers. For the lower bound, we exhibit a family of pushdown vector addition systems with finite but large reachability sets (hyperAckermannian).
Abstract. We study linear-time temporal logics interpreted over data words with multiple attributes. We restrict the atomic formulas to equalities of attribute values in successive positions and to repetitions of attribute values in the future or past. We demonstrate correspondences between satisfiability problems for logics and reachability-like decision problems for counter systems. We show that allowing/disallowing atomic formulas expressing repetitions of values in the past corresponds to the reachability/coverability problem in Petri nets. This gives us 2expspace upper bounds for several satisfiability problems. We prove matching lower bounds by reduction from a reachability problem for a newly introduced class of counter systems. This new class is a succinct version of vector addition systems with states in which counters are accessed via pointers, a potentially useful feature in other contexts. We strengthen further the correspondences between data logics and counter systems by characterizing the complexity of fragments, extensions and variants of the logic. For instance, we precisely characterize the relationship between the number of attributes allowed in the logic and the number of counters needed in the counter system.
We introduce two-player games which build words over infinite alphabets, and we study the problem of checking the existence of winning strategies. These games are played by two players, who take turns in choosing valuations for variables ranging over an infinite data domain, thus generating multi-attributed data words. The winner of the game is specified by formulas in the Logic of Repeating Values, which can reason about repetitions of data values in infinite data words. We prove that it is undecidable to check if one of the players has a winning strategy, even in very restrictive settings. However, we prove that if one of the players is restricted to choose valuations ranging over the Boolean domain, the games are effectively equivalent to single-sided games on vector addition systems with states (in which one of the players can change control states but cannot change counter values), known to be decidable and effectively equivalent to energy games.Previous works have shown that the satisfiability problem for various variants of the logic of repeating values is equivalent to the reachability and coverability problems in vector addition systems. Our results raise this connection to the level of games, augmenting further the associations between logics on data words and counter systems.We denote by Z the set of integers and by N the set of non-negative integers. We let N + denote the set of integers that are strictly greater than 0. For any set S, we denote by S * (resp. S ω ) the set of all finite (resp. countably infinite) sequences of elements in S. For a sequence σ ∈ S * , we denote its length by |σ|. We denote by P(S) (resp. P + (S)) the set of all subsets (resp. non-empty subsets) of S. Logic of repeating valuesWe recall the syntax and semantics of the logic of repeating values from [10,11]. This logic extends the usual propositional linear temporal logic with the ability to reason about repetitions of data values from an infinite domain. We let this logic use both Boolean variables (i.e., propositions) and data variables ranging over an infinite data domain D. The Boolean variables can be simulated by data variables. However, we need to consider fragments of the logic, for which explicitly having Boolean variables is convenient. Let BVARS = {q, t, . . .} be a countably infinite set of Boolean variables ranging over { , ⊥}, and let DVARS = {x, y, . . .} be a countably infinite set of 'data' variables ranging over D. We denote by LRV the logic whose formulas are defined as follows: 1A valuation is the union of a mapping from BVARS to { , ⊥} and a mapping from DVARS to D. A model is a finite or infinite sequence of valuations. We use σ to denote models and σ(i) denotes the i th valuation in σ, where i ∈ N + . For any model σ and position i ∈ N + , the satisfaction relation |= is defined inductively as follows. The semantics of temporal operators next (X), previous (X −1 ), until (U), since (S) and the Boolean connectives are defined in the usual way, but for the sake of completeness we provide their 1 In a pr...
We introduce ω-Petri nets (ωPN), an extension of plain Petri nets with ωlabeled input and output arcs, that is well-suited to analyse parametric concurrent systems with dynamic thread creation. Most techniques (such as the Karp and Miller tree or the Rackoff technique) that have been proposed in the setting of plain Petri nets do not apply directly to ωPN because ωPN define transition systems that have infinite branching. This motivates a thorough analysis of the computational aspects of ωPN. We show that an ωPN can be turned into an plain Petri net that allows to recover the reachability set of the ωPN, but that does not preserve termination. This yields complexity bounds for the reachability, (place) boundedness and coverability problems on ωPN. We provide a practical algorithm to compute a coverability set of the ωPN and to decide termination by adapting the classical Karp and Miller tree construction. We also adapt the Rackoff technique to ωPN, to obtain the exact complexity of the termination problem. Finally, we consider the extension of ωPN with reset and transfer arcs, and show how this extension impacts the decidability and complexity of the aforementioned problems. ReachabilityDecidable and EX- PSPACE-hard (4) Undecidable (6) Undecidable (6) Place-boundedness EXPSPACE-c (4) Boundedness Decidable (6) Coverability Decidable and Ackerman-hard (6) Problem ωPN ωOPN+T, ωOPN+R ωIPN+T, ωIPN+R Termination EXPSPACE-c (5) Undecidable (6) Decidable and Ackerman-hard (6) complexity of (plain) PN problems apply to ωPN too. However, it does not preserve termination. Thus, we study, in Section 5, as a third contribution, an extension of the self-covering path technique due to Rackoff [19]. This technique allows to provide a direct proof of EXPSPACE upper bounds for several classical decision problems, and in particular, this allows to prove EXPSPACE completeness of the termination problem.Finally, in Section 6, as a additional contribution, and to get a complete picture, we consider extensions of ωPN with reset and transfer arcs [7]. For those extensions, the decidability results for reset and transfer nets (without ω arcs) also apply to our extension with the notable exception of the termination problem that becomes, as we show here, undecidable. The summary of our results are given in Table 1.Related works ωPN are well-structured transition systems [10]. The set saturation technique [1] and so symbolic backward analysis can be applied to them while the finite tree unfolding is not applicable because of the infinite branching property of ωPN. For the same reason, ωPN are not well-structured nets [11].In [3], Bradzil et al. extends the Rackoff technique to VASS games with ω output arcs. While this extension of the Rackoff technique is technically close to ours, we cannot directly use their results to solve the termination problem of ωPN.Several works (see for instance [4,5] rely on Petri nets to model parametric systems and perform parametrised verification. However, in all these works, the dynamic creation of thread...
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