A data word is a sequence of pairs of a letter from a finite alphabet and an element from an infinite set, where the latter can only be compared for equality. Safety one-way alternating automata with one register on infinite data words are considered, their nonemptiness is shown EXPSPACE-complete, and their inclusion decidable but not primitive recursive. The same complexity bounds are obtained for satisfiability and refinement, respectively, for the safety fragment of linear temporal logic with freeze quantification. Dropping the safety restriction, adding past temporal operators, or adding one more register, each causes undecidability.
Alternating timed automata on infinite words are considered. The main result is a characterization of acceptance conditions for which the emptiness problem for these automata is decidable. This result implies new decidability results for fragments of timed temporal logics. It is also shown that, unlike for MITL, the characterisation remains the same even if no punctual constraints are allowed.
Coalgebra is a currently quite active field, which aims to look at generic state-based systems (most prominently automata) from a very abstract point of view, mainly using tools from category theory. One of its achievements is to give a completely generic approach of determinization, unifying in an elegant manner non-deterministic automata, probabilistic automata or non-deterministic pushdown automata in one and the same model. However, the case of alternating automata fails to easily fit in this model. The aim of this internship was therefore to tackle this problem: can alternating automata a…
A notion of alternating timed automata is proposed. It is shown that such automata with only one clock have decidable emptiness problem over finite words. This gives a new class of timed languages which is closed under boolean operations and which has an effective presentation. We prove that the complexity of the emptiness problem for alternating timed automata with one clock is non-primitive recursive. The proof gives also the same lower bound for the universality problem for nondeterministic timed automata with one clock. We investigate extension of the model with epsilon-transitions and pro…
We introduce a new class of automata on infinite trees called \emph{alternating nonzero automata}, which extends the class of non-deterministic nonzero automata. We reduce the emptiness problem for alternating nonzero automata to the same problem for non-deterministic ones, which implies decidability. We obtain as a corollary algorithms for the satisfiability of a probabilistic temporal logic extending both CTL* and the qualitative fragment of pCTL*.
$μ$-Calculus and automata on infinite trees are complementary ways of describing infinite tree languages. The correspondence between $μ$-Calculus and alternating tree automaton is used to solve the satisfiability and model checking problems by compiling the modal $μ$-Calculus formula into an alternating tree automata. Thus advocating an automaton model specially tailored for working with modal $μ$-Calculus. The advantage of the automaton model is its ability to deal with arbitrary branching in a much simpler way as compare to the one proposed by Janin and Walukiewicz. Both problems (i.e., mode…
Algorithms for model checking and satisfiability of the modal $μ$-calculus start by converting formulas to alternating parity tree automata. Thus, model checking is reduced to checking acceptance by tree automata and satisfiability to checking their emptiness. The first reduces directly to the solution of parity games but the second is more complicated. We review the non-emptiness checking of alternating tree automata by a reduction to solving parity games of a certain structure, so-called emptiness games. Since the emptiness problem for alternating tree automata is EXPTIME-complete, the size …
We study alternating automata with qualitative semantics over infinite binary trees: alternation means that two opposing players construct a decoration of the input tree called a run, and the qualitative semantics says that a run of the automaton is accepting if almost all branches of the run are accepting. In this paper we prove a positive and a negative result for the emptiness problem of alternating automata with qualitative semantics. The positive result is the decidability of the emptiness problem for the case of Büchi acceptance condition. An interesting aspect of our approach is that we…
In this paper we deal with verification of safety properties of term-rewriting systems. The verification problem is translated to a purely logical problem of finding a finite countermodel for a first-order formula, which further resolved by a generic finite model finding procedure. A finite countermodel produced during successful verification provides with a concise description of the system invariant sufficient to demonstrate a specific safety property. We show the relative completeness of this approach with respect to the tree automata completion technique. On a set of examples taken from th…
One clock alternating timed automata (OCATA) have been introduced as natural extension of (one clock) timed automata to express the semantics of MTL. In this paper, we consider the application of OCATA to the problems of model-checking and satisfiability for MITL (a syntactic fragment of MTL), interpreted over infinite words. Our approach is based on the interval semantics (recently introduced in [BEG13] in the case of finite words) extended to infinite words. We propose region-based and zone-based algorithms, based on this semantics, for MITL model-checking and satisfiability. We report on th…
We consider systems composed of an unbounded number of uniformly designed linear hybrid automata, whose dynamic behavior is determined by their relation to neighboring systems. We present a class of such systems and a class of safety properties whose verification can be reduced to the verification of (small) families of neighbouring systems of bounded size, and identify situations in which such verification problems are decidable, resp. fixed parameter tractable. We illustrate the approach with an example from coordinated vehicle guidance, and describe an implementation which allows us to perf…
One clock alternating timed automata OCATA have been recently introduced as natural extension of (one clock) timed automata to express the semantics of MTL (Ouaknine, Worrell 2005). We consider the application of OCATA to problem of model-checking MITL formulas (a syntactic fragment of MTL) against timed automata. We introduce a new semantics for OCATA where, intuitively, clock valuations are intervals instead of single real values. Thanks to this new semantics, we show that we can bound the number of clock copies that are necessary to allow an OCATA to recognise the models of an MITL formula.…
We consider pushdown timed automata (PTAs) that are timed automata (with dense clocks) augmented with a pushdown stack. A configuration of a PTA includes a control state, dense clock values and a stack word. By using the pattern technique, we give a decidable characterization of the binary reachability (i.e., the set of all pairs of configurations such that one can reach the other) of a PTA. Since a timed automaton can be treated as a PTA without the pushdown stack, we can show that the binary reachability of a timed automaton is definable in the additive theory of reals and integers. The resu…
University of Twente（特文特大学）；University of Colorado Boulder（科罗拉多大学博尔德分校）；University of Liverpool（利物浦大学）
When omega-regular objectives were first proposed in model-free reinforcement learning (RL) for controlling MDPs, deterministic Rabin automata were used in an attempt to provide a direct translation from their transitions to scalar values. While these translations failed, it has turned out that it is possible to repair them by using good-for-MDPs (GFM) Büchi automata instead. These are nondeterministic Büchi automata with a restricted type of nondeterminism, albeit not as restricted as in good-for-games automata. Indeed, deterministic Rabin automata have a pretty straightforward translation to…
In this paper, we study the extension of 1-clock Alternating Timed Automata (1-ATA) with the ability to read in both forward and backward direction, the 2-Way 1-clock Alternating Timed Automata (2-Way 1-ATA). We show that subclass of 2-Way 1-ATA with reset free loops (2-Way 1-ATA-rfl) is expressively equivalent to MSO[<] extended with Guarded Metric Quantifiers (GQMSO). Emptiness Checking problem for 2-Way 1-ATA-rfl (and hence GQMSO) is undecidable, in general. We propose a "non-punctuality" like restriction, called non-adjacency, for 2-Way 1-ATA-rfl, and also for GQMSO, for which the emptines…
A data tree is an unranked ordered tree whose every node is labelled by a letter from a finite alphabet and an element ("datum") from an infinite set, where the latter can only be compared for equality. The article considers alternating automata on data trees that can move downward and rightward, and have one register for storing data. The main results are that nonemptiness over finite data trees is decidable but not primitive recursive, and that nonemptiness of safety automata is decidable but not elementary. The proofs use nondeterministic tree automata with faulty counters. Allowing upward …
We study alternating register automata on data words and data trees in relation to logics. A data word (resp. data tree) is a word (resp. tree) whose every position carries a label from a finite alphabet and a data value from an infinite domain. We investigate one-way automata with alternating control over data words or trees, with one register for storing data and comparing them for equality. This is a continuation of the study started by Demri, Lazic and Jurdzinski. From the standpoint of register automata models, this work aims at two objectives: (1) simplifying the existent decidability pr…
The coalgebraic modelling of alternating automata and of probabilistic automata has long been obstructed by the absence of distributive laws of the powerset monad over itself, respectively of the powerset monad over the finite distribution monad. This can be fixed using the framework of weak distributive laws. We extend this framework to the case when one of the monads is only a functor. We provide abstract compositionality results, a generalized determinization procedure, and systematic soundness of up-to techniques. Along the way, we apply these results to alternating automata as a motivatin…
Quantitative languages are an extension of boolean languages that assign to each word a real number. Mean-payoff automata are finite automata with numerical weights on transitions that assign to each infinite path the long-run average of the transition weights. When the mode of branching of the automaton is deterministic, nondeterministic, or alternating, the corresponding class of quantitative languages is not robust as it is not closed under the pointwise operations of max, min, sum, and numerical complement. Nondeterministic and alternating mean-payoff automata are not decidable either, as …
In the mid 80s, Lichtenstein, Pnueli, and Zuck proved a classical theorem stating that every formula of Past LTL (the extension of LTL with past operators) is equivalent to a formula of the form $\bigwedge_{i=1}^n \mathbf{G}\mathbf{F} φ_i \vee \mathbf{F}\mathbf{G} ψ_i$, where $φ_i$ and $ψ_i$ contain only past operators. Some years later, Chang, Manna, and Pnueli built on this result to derive a similar normal form for LTL. Both normalisation procedures have a non-elementary worst-case blow-up, and follow an involved path from formulas to counter-free automata to star-free regular expressions a…
In this paper we give automata-based representation of LTL-FO$^+$ properties. LTL-FO$^+$ is an extension of LTL that includes first-order quantification over bounded variable, thus greatly increasing the expressivity of the language. An automata representation of this formalism allows greater ease in writing and understanding properties, as well as in performing manipulations, such as negation or emptiness checking. The automata representation of an LTL-FO$^+$ formula has finite size regardless of the domain of quantified variables, and the number of states that is linear in the size of the pr…
The aim of this work is to thoroughly investigate Buchi automata augmented with spatial constraints. The input trees of such an automaton are infinite k-ary Sigma-trees, with the nodes standing for time points, and Sigma including, additionally to its uses in classical k-ary Sigma-trees, the description of the snapshot of an n-object spatial scene of interest. The constraints, from an RCC8-like spatial Relation Algebra (RA) x, are used to impose spatial constraints on objects of the spatial scene, eventually at different nodes of the input trees. We show that a Buchi alternating automaton augm…
Parikh automata extend finite automata by counters that can be tested for membership in a semilinear set, but only at the end of a run, thereby preserving many of the desirable algorithmic properties of finite automata. Here, we study the extension of the classical framework onto infinite inputs: We introduce reachability, safety, Büchi, and co-Büchi Parikh automata on infinite words and study expressiveness, closure properties, and the complexity of verification problems. We show that almost all classes of automata have pairwise incomparable expressiveness, both in the deterministic and the n…
Hyperproperties elevate the traditional view of trace properties form sets of traces to sets of sets of traces and provide a formalism for expressing information-flow policies. For trace properties, algorithms for verification, monitoring, and synthesis are typically based on a representation of the properties as omega-automata. For hyperproperties, a similar, canonical automata-theoretic representation is, so far, missing. This is a serious obstacle for the development of algorithms, because basic constructions, such as learning algorithms, cannot be applied. In this paper, we present a canon…
A timed network consists of an arbitrary number of initially identical 1-clock timed automata, interacting via hand-shake communication. In this setting there is no unique central controller, since all automata are initially identical. We consider the universal safety problem for such controller-less timed networks, i.e., verifying that a bad event (enabling some given transition) is impossible regardless of the size of the network. This universal safety problem is dual to the existential coverability problem for timed-arc Petri nets, i.e., does there exist a number $m$ of tokens, such that st…
We generalize some of the central results in automata theory to the abstraction level of coalgebras and thus lay out the foundations of a universal theory of automata operating on infinite objects. Let F be any set functor that preserves weak pullbacks. We show that the class of recognizable languages of F-coalgebras is closed under taking unions, intersections, and projections. We also prove that if a nondeterministic F-automaton accepts some coalgebra it accepts a finite one of the size of the automaton. Our main technical result concerns an explicit construction which transforms a given alt…
This article introduces ATAB, a tool that automatically generates pairwise reachability checks for action trees. Action trees can be used to study the behaviour of real-world concurrent programs. ATAB encodes pairwise reachability checks into alternating tree automata that determine whether an action tree has a schedule where any pair of given points in the program are simultaneously reachable. Because the pairwise reachability problem is undecidable in general ATAB operates under a restricted form of lock-based concurrency. ATAB produces alternating tree automata that are more compact and mor…
Loop under-approximation is a technique that enriches C programs with additional branches that represent the effect of a (limited) range of loop iterations. While this technique can speed up the detection of bugs significantly, it introduces redundant execution traces which may complicate the verification of the program. This holds particularly true for verification tools based on Bounded Model Checking, which incorporate simplistic heuristics to determine whether all feasible iterations of a loop have been considered. We present a technique that uses \emph{trace automata} to eliminate redunda…
We consider probabilistic automata on infinite words with acceptance defined by safety, reachability, Büchi, coBüchi, and limit-average conditions. We consider quantitative and qualitative decision problems. We present extensions and adaptations of proofs for probabilistic finite automata and present a complete characterization of the decidability and undecidability frontier of the quantitative and qualitative decision problems for probabilistic automata on infinite words.
We introduce a tableau decision method for deciding realizability of specifications expressed in a safety fragment of LTL that includes bounded future temporal operators. Tableau decision procedures for temporal and modal logics have been thoroughly studied for satisfiability and for translating temporal formulae into equivalent Büchi automata, and also for model checking, where a specification and system are provided. However, to the best of our knowledge no tableau method has been studied for the reactive synthesis problem. Reactive synthesis starts from a specification where propositional v…
Graz University of Technology（格拉茨技术大学）；Silicon Austria Labs（奥地利硅实验室）；Lamarr Security Research（拉马尔安全研究院）；Aalborg University（奥尔堡大学）
Safety is still one of the major research challenges in reinforcement learning (RL). In this paper, we address the problem of how to avoid safety violations of RL agents during exploration in probabilistic and partially unknown environments. Our approach combines automata learning for Markov Decision Processes (MDPs) and shield synthesis in an iterative approach. Initially, the MDP representing the environment is unknown. The agent starts exploring the environment and collects traces. From the collected traces, we passively learn MDPs that abstractly represent the safety-relevant aspects of th…
We introduce improvements in the algorithm by Gastin and Oddoux translating LTL formulae into Büchi automata via very weak alternating co-Büchi automata and generalized Büchi automata. Several improvements are based on specific properties of any formula where each branch of its syntax tree contains at least one eventually operator and at least one always operator. These changes usually result in faster translations and smaller automata. Other improvements reduce non-determinism in the produced automata. In fact, we modified all the steps of the original algorithm and its implementation known a…
While the complexity of translating future linear temporal logic (LTL) into automata on infinite words is well-understood, the size increase involved in turning automata back to LTL is not. In particular, there is no known elementary bound on the complexity of translating deterministic $ω$-regular automata to LTL. Our first contribution consists of tight bounds for LTL over a unary alphabet: alternating, nondeterministic and deterministic automata can be exactly exponentially, quadratically and linearly more succinct, respectively, than any equivalent LTL formula. Our main contribution consist…
Motivated by the success of bounded model checking framework for finite state machines, Ouaknine and Worrell proposed a time-bounded theory of real-time verification by claiming that restriction to bounded-time recovers decidability for several key decision problem related to real-time verification. In support of this theory, the list of undecidable problems recently shown decidable under time-bounded restriction is rather impressive: language inclusion for timed automata, emptiness problem for alternating timed automata, and emptiness problem for rectangular hybrid automata. The objective of …
A data word is a sequence of pairs of a letter from a finite alphabet and an element from an infinite set, where the latter can only be compared for equality. To reason about data words, linear temporal logic is extended by the freeze quantifier, which stores the element at the current word position into a register, for equality comparisons deeper in the formula. By translations from the logic to alternating automata with registers and then to faulty counter automata whose counters may erroneously increase at any time, and from faulty and error-free counter automata to the logic, we obtain a c…
Timed automata and register automata are well-known models of computation over timed and data words respectively. The former has clocks that allow to test the lapse of time between two events, whilst the latter includes registers that can store data values for later comparison. Although these two models behave in appearance differently, several decision problems have the same (un)decidability and complexity results for both models. As a prominent example, emptiness is decidable for alternating automata with one clock or register, both with non-primitive recursive complexity. This is not by cha…
In this paper we address the synthesis problem for specifications given in linear temporal single-agent epistemic logic, KLTL (or $KL_1$), over single-agent systems having imperfect information of the environment state. Van der Meyden and Vardi have shown that this problem is 2Exptime complete. However, their procedure relies on complex automata constructions that are notoriously resistant to efficient implementations as they use Safra-like determinization. We propose a "Safraless" synthesis procedure for a large fragment of KLTL. The construction transforms first the synthesis problem into th…
The classic approaches to synthesize a reactive system from a linear temporal logic (LTL) specification first translate the given LTL formula to an equivalent omega-automaton and then compute a winning strategy for the corresponding omega-regular game. To this end, the obtained omega-automata have to be (pseudo)-determinized where typically a variant of Safra's determinization procedure is used. In this paper, we show that this determinization step can be significantly improved for tool implementations by replacing Safra's determinization by simpler determinization procedures. In particular, w…
University of Corunna（科鲁尼亚大学）；Université d’Angers（昂热大学）；University of Potsdam（波茨坦大学）
We explore different ways of implementing temporal constraints expressed in an extension of Answer Set Programming (ASP) with language constructs from dynamic logic. Foremost, we investigate how automata can be used for enforcing such constraints. The idea is to transform a dynamic constraint into an automaton expressed in terms of a logic program that enforces the satisfaction of the original constraint. What makes this approach attractive is its independence of time stamps and the potential to detect unsatisfiability. On the one hand, we elaborate upon a transformation of dynamic formulas in…
University of California at Los Angeles（加州大学洛杉矶分校）；University of Pennsylvania（宾夕法尼亚大学）
We propose a method to construct finite-state reactive controllers for systems whose interactions with their adversarial environment are modeled by infinite-duration two-player games over (possibly) infinite graphs. The proposed method targets safety games with infinitely many states or with such a large number of states that it would be impractical---if not impossible---for conventional synthesis techniques that work on the entire state space. We resort to constructing finite-state controllers for such systems through an automata learning approach, utilizing a symbolic representation of the u…
We introduce an automata-theoretic method for the verification of distributed algorithms running on ring networks. In a distributed algorithm, an arbitrary number of processes cooperate to achieve a common goal (e.g., elect a leader). Processes have unique identifiers (pids) from an infinite, totally ordered domain. An algorithm proceeds in synchronous rounds, each round allowing a process to perform a bounded sequence of actions such as send or receive a pid, store it in some register, and compare register contents wrt. the associated total order. An algorithm is supposed to be correct indepe…
