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 and back to formulas. We improve on both points. We present a direct and purely syntactic normalisation procedure for LTL yielding a normal form, comparable to the one by Chang, Manna, and Pnueli, that has only a single exponential blow-up. As an application, we derive a simple algorithm to translate LTL into deterministic Rabin automata. The algorithm normalises the formula, translates it into a special very weak alternating automaton, and applies a simple determinisation procedure, valid only for these special automata.
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…
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 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…
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…
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…
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…
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…
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 …
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…
A key result in the theory of the modal mu-calculus is the disjunctive normal form theorem by Janin & Walukiewicz, stating that every mu-calculus formula is semantically equivalent to a so-called disjunctive formula. These disjunctive formulas have good computational properties and play a pivotal role in the theory of the modal mu-calculus. It is therefore an interesting question what the best normalisation procedure is for rewriting a formula into an equivalent disjunctive formula of minimal size. The best constructions that are known from the literature are automata-theoretic in nature and c…
Combining ideas from distributed algorithms and alternating automata, we introduce a new class of finite graph automata that recognize precisely the languages of finite graphs definable in monadic second-order logic. By restricting transitions to be nondeterministic or deterministic, we also obtain two strictly weaker variants of our automata for which the emptiness problem is decidable. As an application, we suggest how suitable graph automata might be useful in formal verification of distributed algorithms, using Floyd-Hoare logic.
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…
Stone-type dualities provide a powerful mathematical framework for studying properties of logical systems. They have recently been fruitfully explored in understanding minimisation of various types of automata. In Bezhanishvili et al. (2012), a dual equivalence between a category of coalgebras and a category of algebras was used to explain minimisation. The algebraic semantics is dual to a coalgebraic semantics in which logical equivalence coincides with trace equivalence. It follows that maximal quotients of coalgebras correspond to minimal subobjects of algebras. Examples include partially o…
We study the expressive power of Alternating Parity Krivine Automata (APKA), which provide operational semantics to Higher-Order Modal Fixpoint Logic (HFL). APKA consist of ordinary parity automata extended by a variation of the Krivine Abstract Machine. We show that the number and parity of priorities available to an APKA form a proper hierarchy of expressive power as in the modal mu-calculus. This also induces a strict alternation hierarchy on HFL. The proof follows Arnold's (1999) encoding of runs into trees and subsequent use of the Banach Fixpoint Theorem.
Parikh's game logic is a PDL-like fixpoint logic interpreted on monotone neighbourhood frames that represent the strategic power of players in determined two-player games. Game logic translates into a fragment of the monotone $μ$-calculus, which in turn is expressively equivalent to monotone modal automata. Parity games and automata are important tools for dealing with the combinatorial complexity of nested fixpoints in modal fixpoint logics, such as the modal $μ$-calculus. In this paper, we (1) discuss the semantics a of game logic over neighbourhood structures in terms of parity games, and (…
It is well known that the verification of resource-constrained multiagent systems is undecidable in general. In many such settings, resources are private to agents. In this paper, we investigate the model checking problem for a resource logic based on Alternating-Time Temporal Logic (ATL) with shared resources. Resources can be consumed and produced up to any amount. We show that the model checking problem is undecidable if two or more of such unbounded resources are available. Our main technical result is that in the case of a single shared resource, the problem becomes decidable. Although in…
Spot is a C ++ 17 library for LTL and $ω$-automata manipulation, with command-line utilities, and Python bindings. This paper summarizes its evolution over the past six years, since the release of Spot 2.0, which was the first version to support $ω$-automata with arbitrary acceptance conditions, and the last version presented at a conference. Since then, Spot has been extended with several features such as acceptance transformations, alternating automata, games, LTL synthesis, and more. We also shed some lights on the data-structure used to store automata.
We introduce Visibly Linear Dynamic Logic (VLDL), which extends Linear Temporal Logic (LTL) by temporal operators that are guarded by visibly pushdown languages over finite words. In VLDL one can, e.g., express that a function resets a variable to its original value after its execution, even in the presence of an unbounded number of intermediate recursive calls. We prove that VLDL describes exactly the $ω$-visibly pushdown languages. Thus it is strictly more expressive than LTL and able to express recursive properties of programs with unbounded call stacks. The main technical contribution of t…
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…
We propose a novel approach for coping with alternating quantification as the main source of nonelementary complexity of deciding WS1S formulae. Our approach is applicable within the state-of-the-art automata-based WS1S decision procedure implemented, e.g. in MONA. The way in which the standard decision procedure processes quantifiers involves determinization, with its worst case exponential complexity, for every quantifier alternation in the prefix of a formula. Our algorithm avoids building the deterministic automata---instead, it constructs only those of their states needed for (dis)proving…
We introduce Parametric Linear Dynamic Logic (PLDL), which extends Linear Dynamic Logic (LDL) by temporal operators equipped with parameters that bound their scope. LDL itself was proposed as an extension of Linear Temporal Logic (LTL) that is able to express all omega-regular specifications while still maintaining many of LTL's desirable properties like intuitive syntax and semantics and a translation into non-deterministic Büchi automata of exponential size. But LDL lacks capabilities to express timing constraints. By adding parameterized operators to LDL, we obtain a logic that is able to e…
We introduce Parametric Linear Dynamic Logic (PLDL), which extends Linear Dynamic Logic (LDL) by temporal operators equipped with parameters that bound their scope. LDL was proposed as an extension of Linear Temporal Logic (LTL) that is able to express all $ω$-regular specifications while still maintaining many of LTL's desirable properties like an intuitive syntax and a translation into non-deterministic Büchi automata of exponential size. But LDL lacks capabilities to express timing constraints. By adding parameterized operators to LDL, we obtain a logic that is able to express all $ω$-regul…
In this article, we develop a new and somewhat unexpected connection between higher-order model-checking and linear logic. Our starting point is the observation that once embedded in the relational semantics of linear logic, the Church encoding of any higher-order recursion scheme (HORS) comes together with a dual Church encoding of an alternating tree automata (ATA) of the same signature. Moreover, the interaction between the relational interpretations of the HORS and of the ATA identifies the set of accepting states of the tree automaton against the infinite tree generated by the recursion s…
Runtime monitoring is one of the central tasks in the area of operational decision support for business process management. In particular, it helps process executors to check on-the-fly whether a running process instance satisfies business constraints of interest, providing an immediate feedback when deviations occur. We study runtime monitoring of properties expressed in LTL on finite traces (LTLf), and in its extension LDLf. LDLf is a powerful logic that captures all monadic second order logic on finite traces, and that is obtained by combining regular expressions with LTLf, adopting the syn…
We use modal logic as a framework for coalgebraic trace semantics, and show the flexibility of the approach with concrete examples such as the language semantics of weighted, alternating and tree automata, and the trace semantics of generative probabilistic systems. We provide a sufficient condition under which a logical semantics coincides with the trace semantics obtained via a given determinization construction. Finally, we consider a condition that guarantees the existence of a canonical determinization procedure that is correct with respect to a given logical semantics. That procedure is …
We study an extension of $\mtl$ in pointwise time with rational expression guarded modality $\reg_I(\re)$ where $\re$ is a rational expression over subformulae. We study the decidability and expressiveness of this extension ($\mtl$+$φ\ureg_{I, \re} φ$+$\reg_{I,\re}φ$), called $\regmtl$, as well as its fragment $\sfmtl$ where only star-free rational expressions are allowed. Using the technique of temporal projections, we show that $\regmtl$ has decidable satisfiability by giving an equisatisfiable reduction to $\mtl$. We also identify a subclass $\mitl+\ureg$ of $\regmtl$ for which our equi-sat…
Hyperproperties are commonly used in computer security to define information-flow policies and other requirements that reason about the relationship between multiple computations. In this paper, we study a novel class of hyperproperties where the individual computation paths are chosen by the strategic choices of a coalition of agents in a multi-agent system. We introduce HyperATL*, an extension of computation tree logic with path variables and strategy quantifiers. HyperATL* can express strategic hyperproperties, such as that the scheduler in a concurrent system has a strategy to avoid inform…
