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 do not extend the classical solution for solving the emptiness problem of alternating automata, which first constructs an equivalent non-deterministic automaton. Instead, we directly construct an emptiness game making use of imperfect information.
The negative result is the undecidability of the emptiness problem for the case of co-Büchi acceptance condition. This result has two direct consequences: the undecidability of monadic second-order logic extended with the qualitative path-measure quantifier, and the undecidability of the emptiness problem for alternating tree automata with non-zero semantics, a recently introduced probabilistic model of alternating tree 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…
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.…
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…
We present a direct transformation of weak alternating $ω$-automata into equivalent backward deterministic $ω$-automata and show (1) how it can be used to obtain a transformation of non-deterministic Büchi automata into equivalent backward deterministic automata and (2) that it yields optimal equivalent backward deterministic automata when applied to linear-time temporal logic formulas. (1) uses the alternation-free fragment of the linear-time $μ$-calculus as an intermediate step; (2) is based on the straightforward translation of linear-time temporal logic into weak alternating $ω$-automata.
In this paper, we give a Nivat-like characterization for weighted alternating automata over commutative semirings (WAFA). To this purpose we prove that weighted alternating can be characterized as the concatenation of weighted finite tree automata (WFTA) and a specific class of tree homomorphism. We show that the class of series recognized by weighted alternating automata is closed under inverses of homomorphisms, but not under homomorphisms. We give a logical characterization of weighted alternating automata, which uses weighted MSO logic for trees. Finally we investigate the strong connectio…
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…
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 good-for-games (GFG) automata, i.e., alternating automata where both conjunctive and disjunctive choices can be resolved in an online manner, without knowledge of the suffix of the input word still to be read. We show that they can be exponentially more succinct than both their nondeterministic and universal counterparts. Furthermore, we lift many results from nondeterministic parity GFG automata to alternating ones: a single exponential determinisation procedure, an Exptime upper bound to the GFGness problem, a PTime algorithm for the GFGness problem of weak automata, and…
Self-loop alternating automata (SLAA) with Büchi or co-Büchi acceptance are popular intermediate formalisms in translations of LTL to deterministic or nondeterministic automata. This paper considers SLAA with generic transition-based Emerson-Lei acceptance and presents translations of LTL to these automata and back. Importantly, the translation of LTL to SLAA with generic acceptance produces considerably smaller automata than previous translations of LTL to Büchi or co-Büchi SLAA. Our translation is already implemented in the tool LTL3TELA, where it helps to produce small deterministic or nond…
To model real-world software systems, modelling paradigms should support a form of compositionality. In interface theory and model-based testing with inputs and outputs, conjunctive operators have been introduced: the behaviour allowed by composed specification s1 $\wedge$ s2 is the behaviour allowed by both partial models s1 and s2. The models at hand are non-deterministic interface automata, but the interaction between non-determinism and conjunction is not yet well understood. On the other hand, in the theory of alternating automata, conjunction and non-determinism are core aspects. Alterna…
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…
We study alternating parity good-for-games (GFG) automata, i.e., alternating parity automata where both conjunctive and disjunctive choices can be resolved in an online manner, without knowledge of the suffix of the input word still to be read. We show that they can be exponentially more succinct than both their nondeterministic and universal counterparts. Furthermore, we present a single exponential determinisation procedure and an Exptime upper bound to the problem of recognising whether an alternating automaton is GFG. We also study the complexity of deciding "half-GFGness", a property spec…
This paper proposes a new algorithm for the generation of unambiguous Büchi automata (UBA) from LTL formulas. Unlike existing tableau-based LTL-to-UBA translations, our algorithm deals with very weak alternating automata (VWAA) as an intermediate representation. It relies on a new notion of unambiguity for VWAA and a disambiguation procedure for VWAA. We introduce optimizations on the VWAA level and new LTL simplifications targeted at generating small UBA. We report on an implementation of the construction in our tool duggi and discuss experimental results that compare the automata sizes and c…
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…
We give a new version of fuzzy alternating $\mathrm{B\ddot{u}chi}$ automata over distributive lattices: weights are putting in every leaf node of run trees rather than along with edges from every node to its children. Such settings are great benefit to obtain complement just by taking dual operation and replacing each final weight with its complement. We prove that $L$-fuzzy nondeterministic $\mathrm{B\ddot{u}chi}$ automata have the same expressive power as $L$-fuzzy alternating $\mathrm{B\ddot{u}chi}$ ones. A direct construction (without related knowledge about $L$-fuzzy nondeterministic $\ma…
In this paper we give automata-based representation of LTL-FO$^+$ 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…
A word automaton recognizing a language $L$ is good for games (GFG) if its composition with any game with winning condition $L$ preserves the game's winner. While all deterministic automata are GFG, some nondeterministic automata are not. There are various other properties that are used in the literature for defining that a nondeterministic automaton is GFG, including "history-deterministic", "compliant with some letter game", "good for trees", and "good for composition with other automata". The equivalence of these properties has not been formally shown. We generalize all of these definitions…
Linear Temporal Logic (LTL) is a widely used specification framework for linear time properties of systems. The standard approach for verifying such properties is by transforming LTL formulae to suitable $ω$-automata and then applying model checking. We revisit Vardi's transformation of an LTL formula to an alternating $ω$-automaton and Wolper's LTL tableau method for satisfiability checking. We observe that both constructions effectively rely on a decomposition of formulae into linear factors. Linear factors have been introduced previously by Antimirov in the context of regular expressions. W…
In this article we consider two-way two-tape (alternating) automata accepting pairs of words and we study some closure properties of this model. Our main result is that such alternating automata are not closed under complementation for non-unary alphabets. This improves a similar result of Kari and Moore for picture languages. We also show that these deterministic, non-deterministic and alternating automata are not closed under composition.
For a given regular language of infinite trees, one can ask about the minimal number of priorities needed to recognize this language with a non-deterministic, alternating, or weak alternating parity automaton. These questions are known as, respectively, the non-deterministic, alternating, and weak Rabin-Mostowski index problems. Whether they can be answered effectively is a long-standing open problem, solved so far only for languages recognizable by deterministic automata (the alternating variant trivializes). We investigate a wider class of regular languages, recognizable by so-called game au…
An improved translation from alternating parity automata on infinite words to alternating weak automata is given. The blow-up of the number of states is related to the size of the smallest universal ordered trees and hence it is quasi-polynomial, and only polynomial if the asymptotic number of priorities is logarithmic in the number of states. This is an exponential improvement on the translation of Kupferman and Vardi (2001) and a quasi-polynomial improvement on the translation of Boker and Lehtinen (2018). Any slightly better such translation would (if---like all presently known such transla…
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…
We introduce Symbolic Alternating Finite Automata (s-AFA) as an expressive, succinct, and decidable model for describing sets of finite sequences over arbitrary alphabets. Boolean operations over s-AFAs have linear complexity, which is in sharp contrast with the quadratic cost of intersection and union for non-alternating symbolic automata. Due to this succinctness, emptiness and equivalence checking are PSpace-hard. We introduce an algorithm for checking the equivalence of two s-AFAs based on bisimulation up to congruence. This algorithm allows us to exploit the power of SAT and SMT solvers t…
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.
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…
Promise problems were mainly studied in quantum automata theory. Here we focus on state complexity of classical automata for promise problems. First, it was known that there is a family of unary promise problems solvable by quantum automata by using a single qubit, but the number of states required by corresponding one-way deterministic automata cannot be bounded by a constant. For this family, we show that even two-way nondeterminism does not help to save a single state. By comparing this with the corresponding state complexity of alternating machines, we then get a tight exponential gap betw…
This thesis is focused on techniques for finite automata and their use in practice, with the main emphasis on nondeterministic tree automata. This concerns namely techniques for size reduction and language inclusion testing, which are two problems that are crucial for many applications of tree automata. For size reduction of tree automata, we adapt the simulation quotient technique that is well established for finite word automata. We give efficient algorithms for computing tree automata simulations and we also introduce a new type of relation that arises from a combination of tree automata do…
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 …
We present here Imitator II, a new version of Imitator, a tool implementing the "inverse method" for parametric timed automata: given a reference valuation of the parameters, it synthesizes a constraint such that, for any valuation satisfying this constraint, the system behaves the same as under the reference valuation in terms of traces, i.e., alternating sequences of locations and actions. Imitator II also implements the "behavioral cartography algorithm", allowing us to solve the following good parameters problem: find a set of valuations within a given bounded parametric domain for which t…
We introduce first order alternating automata, a generalization of boolean alternating automata, in which transition rules are described by multisorted first order formulae, with states and internal variables given by uninterpreted predicate terms. The model is closed under union, intersection and complement, and its emptiness problem is undecidable, even for the simplest data theory of equality. To cope with this limitation, we develop an abstraction refinement semi-algorithm based on lazy annotation of the symbolic execution paths with interpolants, obtained by applying (i) quantifier elimin…
University of Sciences and Technology Houari Boumediène（乌阿里·布迈丁科学技术大学）
The aim of this work is to provide a family of qualitative theories for spatial change in general, and for motion of spatial scenes in particular. To achieve this, we consider a spatio-temporalisation MTALC(Dx), of the well-known ALC(D) family of Description Logics (DLs) with a concrete domain: the MTALC(Dx) concepts are interpreted over 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 roles split into m+n immediate-succ…
We consider probabilistic automata on infinite words with acceptance defined by parity conditions. We consider three qualitative decision problems: (i) the positive decision problem asks whether there is a word that is accepted with positive probability; (ii) the almost decision problem asks whether there is a word that is accepted with probability 1; and (iii) the limit decision problem asks whether for every epsilon > 0 there is a word that is accepted with probability at least 1 - epsilon. We unify and generalize several decidability results for probabilistic automata over infinite words, a…
Monitoring of a signal plays an essential role in the runtime verification of cyber-physical systems. Qualitative timed pattern matching is one of the mathematical formulations of monitoring, which gives a Boolean verdict for each sub-signal according to the satisfaction of the given specification. There are two orthogonal directions of extension of the qualitative timed pattern matching. One direction on the result is quantitative: what engineers want is often not a qualitative verdict but the quantitative measurement of the satisfaction of the specification. The other direction on the algori…
Sampled semantics of timed automata is a finite approximation of their dense time behavior. While the former is closer to the actual software or hardware systems with a fixed granularity of time, the abstract character of the latter makes it appealing for system modeling and verification. We study one aspect of the relation between these two semantics, namely checking whether the system exhibits some qualitative (untimed) behaviors in the dense time which cannot be reproduced by any implementation with a fixed sampling rate. More formally, the \emph{sampling problem} is to decide whether there…
Alternating automata have been widely used to model and verify systems that handle data from finite domains, such as communication protocols or hardware. The main advantage of the alternating model of computation is that complementation is possible in linear time, thus allowing to concisely encode trace inclusion problems that occur often in verification. In this paper we consider alternating automata over infinite alphabets, whose transition rules are formulae in a combined theory of booleans and some infinite data domain, that relate past and current values of the data variables. The data th…
This paper studies the complexity of languages of finite words using automata theory. To go beyond the class of regular languages, we consider infinite automata and the notion of state complexity defined by Karp. Motivated by the seminal paper of Rabin from 1963 introducing probabilistic automata, we study the (deterministic) state complexity of probabilistic languages and prove that probabilistic languages can have arbitrarily high deterministic state complexity. We then look at alternating automata as introduced by Chandra, Kozen and Stockmeyer: such machines run independent computations on …
In this paper we study a subclass of pebble automata (PA) for data languages for which the emptiness problem is decidable. Namely, we introduce the so-called top view weak PA. Roughly speaking, top view weak PA are weak PA where the equality test is performed only between the data values seen by the two most recently placed pebbles. The emptiness problem for this model is decidable. We also show that it is robust: alternating, nondeterministic and deterministic top view weak PA have the same recognition power. Moreover, this model is strong enough to accept all data languages expressible in Li…
We introduce the notion of adaptive synchronisation for pushdown automata, in which there is an external observer who has no knowledge about the current state of the pushdown automaton, but can observe the contents of the stack. The observer would then like to decide if it is possible to bring the automaton from any state into some predetermined state by giving inputs to it in an \emph{adaptive} manner, i.e., the next input letter to be given can depend on how the contents of the stack changed after the current input letter. We show that for non-deterministic pushdown automata, this problem is…
We study the qualitative and quantitative zero-reachability problem in probabilistic multi-counter systems. We identify the undecidable variants of the problems, and then we concentrate on the remaining two cases. In the first case, when we are interested in the probability of all runs that visit zero in some counter, we show that the qualitative zero-reachability is decidable in time which is polynomial in the size of a given pMC and doubly exponential in the number of counters. Further, we show that the probability of all zero-reaching runs can be effectively approximated up to an arbitraril…
In this document, we propose a description, via a Haskell implementation, of a generalization of the notion of regular expression allowing us to group the definitions and the methods of (tree or word) automata constructions over one generic structure, based on enriched category theory tools. We first recall several methods of conversion from expressions to automata, enlightening the similarities between the words and trees cases. We then produce an original study of the power of enriched category theory applied 1) to automata and expressions implementation, and 2) to the study of associated al…
Quantitative analysis of computing systems is an emerging area in automated formal analysis. Such properties address aspects such as costs and rewards, quality measures, resource consumption, distance metrics, etc. Existing solutions for problems in quantitative analysis face two challenges, namely lack of generalizability and separation-of-techniques. Lack of generalizability refers to the issue that solution approaches are specialized to cost model. Different cost models deploy such disparate algorithms that there is no transfer of knowledge from one cost model to another. Separation-of-tech…
