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We introduce deterministic suffix-reading automata (DSA), a new automaton model over finite words. Transitions in a DSA are labeled with words. From a state, a DSA triggers an outgoing transition on seeing a word ending with the transition's label. Therefore, rather than moving along an input word letter by letter, a DSA can jump along blocks of letters, with each block ending in a suitable suffix. This feature allows DSAs to recognize regular languages more concisely, compared to DFAs. In this work, we focus on questions around finding a "minimal" DSA for a regular language. The number of states is not a faithful measure of the size of a DSA, since the transition-labels contain strings of arbitrary length. Hence, we consider total-size (number of states + number of edges + total length of transition-labels) as the size measure of DSAs. We start by formally defining the model and providing a DSA-to-DFA conversion that allows to compare the expressiveness and succinctness of DSA with related automata models. Our main technical contribution is a method to derive DSAs from a given DFA: a DFA-to-DSA conversion. We make a surprising observation that the smallest DSA derived from the canonical DFA of a regular language L need not be a minimal DSA for L. This observation leads to a fundamental bottleneck in deriving a minimal DSA for a regular language. In fact, we prove that given a DFA and a number k, the problem of deciding if there exists an equivalent DSA of total-size at most k is NP-complete., Comment: In Proceedings GandALF 2024, arXiv:2410.21884

The well-known Nerode equivalence for finite words plays a fundamental role in our understanding of the class of regular languages. The equivalence leads to the Myhill-Nerode theorem and a canonical automaton, which in turn, is the basis of several automata learning algorithms. A Nerode-like equivalence has been studied for various classes of timed languages. In this work, we focus on timed automata with integer resets. This class is known to have good automata-theoretic properties and is also useful for practical modeling. Our main contribution is a Nerode-style equivalence for this class that depends on a constant K. We show that the equivalence leads to a Myhill-Nerode theorem and a canonical one-clock integer-reset timed automaton with maximum constant K. Based on the canonical form, we develop an Angluin-style active learning algorithm whose query complexity is polynomial in the size of the canonical form., Comment: 25 pages, 6 figures, to be published at the 44th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science

The translation of Metric Interval Temporal Logic (MITL) to timed automata is a topic that has been extensively studied. A key challenge here is the conversion of future modalities into equivalent automata. Typical conversions equip the automata with a guess-and-check mechanism to ascertain the truth of future modalities. Guess-and-check can be naturally implemented via alternation. However, since timed automata tools do not handle alternation, existing methods perform an additional step of converting the alternating timed automata into timed automata. This de-alternation step proceeds by an intricate finite abstraction of the space of configurations of the alternating automaton. Recently, a model of generalized timed automata (GTA) has been proposed. The model comes with several powerful additional features, and yet, the best known zone-based reachability algorithms for timed automata have been extended to the GTA model, with the same complexity for all the zone operations. We provide a new concise translation from MITL to GTA. In particular, for the timed until modality, our translation offers an exponential improvement w.r.t. the state-of-the-art. Thanks to this conversion, MITL model checking reduces to checking liveness for GTAs. However, no liveness algorithm is known for GTAs. Due to the presence of future clocks, there is no finite time-abstract bisimulation (region equivalence) for GTAs, whereas liveness algorithms for timed automata crucially rely on the presence of the finite region equivalence. As our second contribution, we provide a new zone-based algorithm for checking Buchi non-emptiness in GTAs, which circumvents this fundamental challenge.

Negotiations, introduced by Esparza et al., are a model for concurrent systems where computations involving a set of agents are described in terms of their interactions. In many situations, it is natural to impose timing constraints between interactions -- for instance, to limit the time available to enter the PIN after inserting a card into an ATM. To model this, we introduce a real-time aspect to negotiations. In our model of local-timed negotiations, agents have local reference times that evolve independently. Inspired by the model of networks of timed automata, each agent is equipped with a set of local clocks. Similar to timed automata, the outcomes of a negotiation contain guards and resets over the local clocks. As a new feature, we allow some interactions to force the reference clocks of the participating agents to synchronize. This synchronization constraint allows us to model interesting scenarios. Surprisingly, it also gives unlimited computing power. We show that reachability is undecidable for local-timed negotiations with a mixture of synchronized and unsynchronized interactions. We study restrictions on the use of synchronized interactions that make the problem decidable., Comment: A shorter version appears in FORMATS 2023

In this paper, we consider a model of generalized timed automata (GTA) with two kinds of clocks, history and future, that can express many timed features succinctly, including timed automata, event-clock automata with and without diagonal constraints, and automata with timers. Our main contribution is a new simulation-based zone algorithm for checking reachability in this unified model. While such algorithms are known to exist for timed automata, and have recently been shown for event-clock automata without diagonal constraints, this is the first result that can handle event-clock automata with diagonal constraints and automata with timers. We also provide a prototype implementation for our model and show experimental results on several benchmarks. To the best of our knowledge, this is the first effective implementation not just for our unified model, but even just for automata with timers or for event-clock automata (with predicting clocks) without going through a costly translation via timed automata. Last but not least, beyond being interesting in their own right, generalized timed automata can be used for model-checking event-clock specifications over timed automata models., Comment: arXiv admin note: text overlap with arXiv:2207.02633

Timed automata have been introduced by Rajeev Alur and David Dill in the early 90's. In the last decades, timed automata have become the de facto model for the verification of real-time systems. Algorithms for timed automata are based on the traversal of their state-space using zones as a symbolic representation. Since the state-space is infinite, termination relies on finite abstractions that yield a finite representation of the reachable states. The first solution to get finite abstractions was based on extrapolations of zones, and has been implemented in the industry-strength tool Uppaal. A different approach based on simulations between zones has emerged in the last ten years, and has been implemented in the fully open source tool TChecker. The simulation-based approach has led to new efficient algorithms for reachability and liveness in timed automata, and has also been extended to richer models like weighted timed automata, and timed automata with diagonal constraints and updates. In this article, we survey the extrapolation and simulation techniques, and discuss some open challenges for the future., Comment: Invited contribution at FORMATS'22

Event-clock automata (ECA) are a well-known semantic subclass of timed automata (TA) which enjoy admirable theoretical properties, e.g., determinizability, and are practically useful to capture timed specifications. However, unlike for timed automata, there exist no implementations for checking non-emptiness of event-clock automata. As ECAs contain special prophecy clocks that guess and maintain the time to the next occurrence of specific events, they cannot be seen as a syntactic subclass of TA. Therefore, implementations for TA cannot be directly used for ECAs, and moreover the translation of an ECA to a semantically equivalent TA is expensive. Another reason for the lack of ECA implementations is the difficulty in adapting zone-based algorithms, critical in the timed automata setting, to the event-clock automata setting. This difficulty was studied by Geeraerts et al. in 2011, where the authors proposed a zone enumeration procedure that uses zone extrapolations for finiteness. In this article, we propose a different zone-based algorithm to solve the reachability problem for event-clock automata, using simulations for finiteness. A surprising consequence of our result is that for event-predicting automata, the subclass of event-clock automata that only use prophecy clocks, we obtain finiteness even without any simulations. For general event-clock automata, our new algorithm exploits the G-simulation framework, which is the coarsest known simulation relation in timed automata literature, and has been recently used for advances in other extensions of timed automata.

Updatable timed automata (UTA) are extensions of classic timed automata that allow special updates to clock variables, like x:= x - 1, x := y + 2, etc., on transitions. Reachability for UTA is undecidable in general. Various subclasses with decidable reachability have been studied. A generic approach to UTA reachability consists of two phases: first, a static analysis of the automaton is performed to compute a set of clock constraints at each state; in the second phase, reachable sets of configurations, called zones, are enumerated. In this work, we improve the algorithm for the static analysis. Compared to the existing algorithm, our method computes smaller sets of constraints and guarantees termination for more UTA, making reachability faster and more effective. As the main application, we get an alternate proof of decidability and a more efficient algorithm for timed automata with bounded subtraction, a class of UTA widely used for modelling scheduling problems. We have implemented our procedure in the tool TChecker and conducted experiments that validate the benefits of our approach., Comment: Shorter version of this article is accepted at FSTTCS 2020

We provide several positive and negative complexity results for solving games with imperfect recall. Using a one-to-one correspondence between these games on one side and multivariate polynomials on the other side, we show that solving games with imperfect recall is as hard as solving certain problems of the first order theory of reals. We establish square root sum hardness even for the specific class of A-loss games. On the positive side, we find restrictions on games and strategies motivated by Bridge bidding that give polynomial-time complexity.

We investigate a zone based approach for the reachability problem in timed automata. The challenge is to alleviate the size explosion of the search space when considering networks of timed automata working in parallel. In the timed setting this explosion is particularly visible as even different interleavings of local actions of processes may lead to different zones. Salah et al. in 2006 have shown that the union of all these different zones is also a zone. This observation was used in an algorithm which from time to time detects and aggregates these zones into a single zone. We show that such aggregated zones can be calculated more efficiently using the local time semantics and the related notion of local zones proposed by Bengtsson et al. in 1998. Next, we point out a flaw in the existing method to ensure termination of the local zone graph computation. We fix this with a new algorithm that builds the local zone graph and uses abstraction techniques over (standard) zones for termination. We evaluate our algorithm on standard examples. On various examples, we observe an order of magnitude decrease in the search space. On the other examples, the algorithm performs like the standard zone algorithm., Comment: A shorter version appears in proceedings of CONCUR 2019

A popular method for solving reachability in timed automata proceeds by enumerating reachable sets of valuations represented as zones. A na\"ive enumeration of zones does not terminate. Various termination mechanisms have been studied over the years. Coming up with efficient termination mechanisms has been remarkably more challenging when the automaton has diagonal constraints in guards. In this paper, we propose a new termination mechanism for timed automata with diagonal constraints based on a new simulation relation between zones. Experiments with an implementation of this simulation show significant gains over existing methods., Comment: Shorter version of this article to appear in CAV 2019

We consider the reachability problem for timed automata having diagonal constraints (like x - y < 5) as guards in transitions. The best algorithms for timed automata proceed by enumerating reachable sets of its configurations, stored in the form of a data structure called "zones". Simulation relations between zones are essential to ensure termination and efficiency. The algorithm employs a simulation test "is-Z-simulated-by-Z' ?" which ascertains that zone Z does not reach more states than zone Z', and hence further enumeration from Z is not necessary. No effective simulations are known for timed automata containing diagonal constraints as guards. In this paper, we propose a simulation relation LU-d for timed automata with diagonal constraints. On the negative side, we show that deciding Z-is-not-LU-d-simulated-by-Z' is NP-complete. On the positive side, we identify a witness for non-simulation and propose an algorithm to decide the existence of such a witness using an SMT solver. The shape of the witness reveals that the simulation test is likely to be efficient in practice., Comment: Shorter version of this article appears in CONCUR 2018

Designing query languages for graph structured data is an active field of research, where expressiveness and efficient algorithms for query evaluation are conflicting goals. To better handle dynamically changing data, recent work has been done on designing query languages that can compare values stored in the graph database, without hard coding the values in the query. The main idea is to allow variables in the query and bind the variables to values when evaluating the query. For query languages that bind variables only once, query evaluation is usually NP-complete. There are query languages that allow binding inside the scope of Kleene star operators, which can themselves be in the scope of bindings and so on. Uncontrolled nesting of binding and iteration within one another results in query evaluation being PSPACE-complete. We define a way to syntactically control the nesting depth of iterated bindings, and study how this affects expressiveness and efficiency of query evaluation. The result is an infinite, syntactically defined hierarchy of expressions. We prove that the corresponding language hierarchy is strict. Given an expression in the hierarchy, we prove that it is undecidable to check if there is a language equivalent expression at lower levels. We prove that evaluating a query based on an expression at level i can be done in $\Sigma_i$ in the polynomial time hierarchy. Satisfiability of quantified Boolean formulas can be reduced to query evaluation; we study the relationship between alternations in Boolean quantifiers and the depth of nesting of iterated bindings., Comment: Improvements from ICALP 2016 review comments

Designing query languages for graph structured data is an active field of research. Evaluating a query on a graph results in a relation on the set of its nodes. In other words, a query is a mechanism for defining relations on a graph. Some relations may not be definable by any query in a given language. This leads to the following question: given a graph, a query language and a relation on the graph, does there exist a query in the language that defines the relation? This is called the definability problem. When the given query language is standard regular expressions, the definability problem is known to be PSPACE-complete. The model of graphs can be extended by labeling nodes with values from an infinite domain. These labels induce a partition on the set of nodes: two nodes are equivalent if they are labeled by the same value. Query languages can also be extended to make use of this equivalence. Two such extensions are Regular Expressions with Memory (REM) and Regular Expressions with Equality (REE). In this paper, we study the complexity of the definability problem in this extended model when the query language is either REM or REE. We show that the definability problem is EXPSPACE-complete when the query language is REM, and it is PSPACE-complete when the query language is REE. In addition, when the query language is a union of conjunctive queries based on REM or REE, we show coNP-completeness., Comment: Small corrections based on reviews from PODS 2015, results unchanged

We propose a new efficient algorithm for detecting if a cycle in a timed automaton can be iterated infinitely often. Existing methods for this problem have a complexity which is exponential in the number of clocks. Our method is polynomial: it essentially does a logarithmic number of zone canonicalizations. This method can be incorporated in algorithms for verifying B\"uchi properties on timed automata. We report on some experiments that show a significant reduction in search space when our iteratability test is used.

We consider the reachability problem for timed automata. A standard solution to this problem involves computing a search tree whose nodes are abstractions of zones. For efficiency reasons, they are parametrized by the maximal lower and upper bounds (LU-bounds) occurring in the guards of the automaton. We propose an algorithm that is updating LU-bounds during exploration of the search tree. In order to keep them as small as possible, the bounds are refined only when they enable a transition that is impossible in the unabstracted system. So our algorithm can be seen as a kind of lazy CEGAR algorithm for timed automata. We show that on several standard benchmarks, the algorithm is capable of keeping very small LU-bounds, and in consequence reduce the search space substantially.

The reachability problem for timed automata asks if there exists a path from an initial state to a target state. The standard solution to this problem involves computing the zone graph of the automaton, which in principle could be infinite. In order to make the graph finite, zones are approximated using an extrapolation operator. For reasons of efficiency in current algorithms extrapolation of a zone is always a zone and in particular it is convex. In this paper, we propose to solve the reachability problem without such extrapolation operators. To ensure termination, we provide an efficient algorithm to check if a zone is included in the so called region closure of another. Although theoretically better, closure cannot be used in the standard algorithm since a closure of a zone may not be convex. An additional benefit of the proposed approach is that it permits to calculate approximating parameters on-the-fly during exploration of the zone graph, as opposed to the current methods which do it by a static analysis of the automaton prior to the exploration. This allows for further improvements in the algorithm. Promising experimental results are presented., Comment: Extended version of FSTTCS 2011 paper

We consider the reachability problem for timed automata. A standard solution to this problem involves computing a search tree whose nodes are abstractions of zones. These abstractions preserve underlying simulation relations on the state space of the automaton. For both effectiveness and efficiency reasons, they are parametrized by the maximal lower and upper bounds (LU-bounds) occurring in the guards of the automaton. We consider the aLU abstraction defined by Behrmann et al. Since this abstraction can potentially yield non-convex sets, it has not been used in implementations. We prove that aLU abstraction is the biggest abstraction with respect to LU-bounds that is sound and complete for reachability. We also provide an efficient technique to use the aLU abstraction to solve the reachability problem., Comment: Extended version of LICS 2012 paper (conference paper till v6). in Information and Computation, available online 27 July 2016

An infinite run of a timed automaton is Zeno if it spans only a finite amount of time. Such runs are considered unfeasible and hence it is important to detect them, or dually, find runs that are non-Zeno. Over the years important improvements have been obtained in checking reachability properties for timed automata. We show that some of these very efficient optimizations make testing for Zeno runs costly. In particular we show NP-completeness for the LU-extrapolation of Behrmann et al. We analyze the source of this complexity in detail and give general conditions on extrapolation operators that guarantee a (low) polynomial complexity of Zenoness checking. We propose a slight weakening of the LU-extrapolation that satisfies these conditions.

The B\"uchi non-emptiness problem for timed automata refers to deciding if a given automaton has an infinite non-Zeno run satisfying the B\"uchi accepting condition. The standard solution to this problem involves adding an auxiliary clock to take care of the non-Zenoness. In this paper, it is shown that this simple transformation may sometimes result in an exponential blowup. A construction avoiding this blowup is proposed. It is also shown that in many cases, non-Zenoness can be ascertained without extra construction. An on-the-fly algorithm for the non-emptiness problem, using non-Zenoness construction only when required, is proposed. Experiments carried out with a prototype implementation of the algorithm are reported., Comment: Published in the Special Issue on Computer Aided Verification - CAV 2010; Formal Methods in System Design, 2011