Showing papers on "ω-automaton published in 2011"
TL;DR: This paper tours a fragment of a vast literature documenting the importance of deterministic, nondeterministic, and alternating finite automata as an enormously valuable concept, and discusses developments relevant to infinite automata related problems like simulation of and by several types of finite Automata.
Abstract: Finite automata are probably best known for being equivalent to right-linear context-free grammars and, thus, for capturing the lowest level of the Chomsky-hierarchy, the family of regular languages. Over the last half century, a vast literature documenting the importance of deterministic, nondeterministic, and alternating finite automata as an enormously valuable concept has been developed. In the present paper, we tour a fragment of this literature. Mostly, we discuss developments relevant to finite automata related problems like, for example, (i) simulation of and by several types of finite automata, (ii) standard automata problems such as fixed and general membership, emptiness, universality, equivalence, and related problems, and (iii) minimization and approximation. We thus come across descriptional and computational complexity issues of finite automata. We do not prove these results but we merely draw attention to the big picture and some of the main ideas involved.
142 citations
31 Aug 2011
TL;DR: This work extends energy games to a multiweighted and parameterized setting, allowing them to model systems with multiple quantitative aspects and investigates the tractability of an extension of multi Weighted energy games in the setting of timed automata.
Abstract: Energy games have recently attracted a lot of attention. These are games played on finite weighted automata and concern the existence of infinite runs subject to boundary constraints on the accumulated weight, allowing e.g. only for behaviours where a resource is always available (nonnegative accumulated weight), yet does not exceed a given maximum capacity. We extend energy games to a multiweighted and parameterized setting, allowing us to model systems with multiple quantitative aspects. We present reductions between Petri nets and multiweighted automata and among different types of multiweighted automata and identify new complexity and (un)decidability results for both one- and two-player games. We also investigate the tractability of an extension of multiweighted energy games in the setting of timed automata.
93 citations
TL;DR: A tour of a vast literature on computational and descriptional complexity issues on multi-head finite automata documenting the importance of these devices and the borderline between decidable and undecidable problems is toured.
57 citations
TL;DR: The role of the distributive law for the truth valued domain of finite automata is analyzed and it is proved that the Kleene theorem holds in the frame of lattice-setting.
55 citations
TL;DR: This work introduces a new class of weight structures subsuming a range of these models as well as semirings and shows that such weighted automata and Kleene-type regular expressions are expressively equivalent both for finite and infinite words.
Abstract: Quantitative aspects of systems like consumption of resources, output of goods, or reliability can be modeled by weighted automata. Recently, objectives like the average cost or the longtime peak power consumption of a system have been modeled by weighted automata which are not semiring weighted anymore. Instead, operations like limit superior, limit average, or discounting are used to determine the behavior of these automata. Here, we introduce a new class of weight structures subsuming a range of these models as well as semirings. Our main result shows that such weighted automata and Kleene-type regular expressions are expressively equivalent both for finite and infinite words.
39 citations
26 May 2011
TL;DR: The objective of this survey is to present the ideal theory of monoids, the so-called Green's relations, and to illustrate the usefulness of this tool for solving automata related questions.
Abstract: The objective of this survey is to present the ideal theory of monoids, the so-called Green's relations, and to illustrate the usefulness of this tool for solving automata related questions.
We use Green's relations for proving four classical results related to automata theory: The result of Schutzenberger characterizing star-free languages, the theorem of factorization forests of Simon, the characterization of infinite words of decidable monadic theory due to Semenov, and the result of determinization of automata over infinite words of McNaughton.
37 citations
TL;DR: It is shown that various non-regular context-free languages, and non-context-free context-sensitive languages are accepted by RMFAs with a few heads, and an RMFA with three heads that accepts the language consisting of all words whose length is a prime number.
Abstract: A two-way reversible multi-head finite automaton (RMFA) is introduced as a simple model of reversible computing, and its language accepting capability is studied. We show that various non-regular context-free languages, and non-context-free context-sensitive languages are accepted by RMFAs with a few heads. For example, we give an RMFA with three heads that accepts the language consisting of all words whose length is a prime number. A construction method of a garbage-less RMFA from a given RFMA is also shown.
36 citations
TL;DR: Results are presented on the decidability and precise complexity of the emptiness, universality and language containment problems for such machines, thus answering questions central to the use of these models in formal verification.
Abstract: Probabilistic B\"uchi Automata (PBA) are randomized, finite state automata
that process input strings of infinite length. Based on the threshold chosen
for the acceptance probability, different classes of languages can be defined.
In this paper, we present a number of results that clarify the power of such
machines and properties of the languages they define. The broad themes we focus
on are as follows. We present results on the decidability and precise
complexity of the emptiness, universality and language containment problems for
such machines, thus answering questions central to the use of these models in
formal verification. Next, we characterize the languages recognized by PBAs
topologically, demonstrating that though general PBAs can recognize languages
that are not regular, topologically the languages are as simple as
\omega-regular languages. Finally, we introduce Hierarchical PBAs, which are
syntactically restricted forms of PBAs that are tractable and capture exactly
the class of \omega-regular languages.
32 citations
TL;DR: This paper investigates functional weighted automata for four different measures: the sum, the mean, the discounted sum of weights along edges and the ratio between rewards and costs, showing that functionality is decidable for the four measures and whether the language associated with a given functional automaton can be defined with a deterministic one.
Abstract: A weighted automaton is functional if any two accepting runs on the same finite word have the same value. In this paper, we investigate functional weighted automata for four different measures: the sum, the mean, the discounted sum of weights along edges and the ratio between rewards and costs. On the positive side, we show that functionality is decidable for the four measures. Furthermore, the existential and universal threshold problems, the language inclusion problem and the equivalence problem are all decidable when the weighted automata are functional. On the negative side, we also study the quantitative extension of the realizability problem and show that it is undecidable for sum, mean and ratio. We finally show how to decide whether the language associated with a given functional automaton can be defined with a deterministic one, for sum, mean and discounted sum. The results on functionality and determinizability are expressed for the more general class of functional group automata. This allows one to formulate within the same framework new results related to discounted sum automata and known results on sum and mean automata. Ratio automata do not fit within this general scheme and different techniques are required to decide functionality.
30 citations
19 Jul 2011
TL;DR: This characterization is then used to show that transforming an n-state 2D FA over a one-letter alphabet to an equivalent sweeping 2DFA requires exactly n+1 states, and transforming it to aOne-way automaton requires exactly max0≤l≤n G(n - l) + l + 1 states, where G(k) is the maximum order of a permutation of k elements.
Abstract: A framework for the study of periodic behaviour of two-way deterministic finite automata (2DFA) is developed. Computations of 2DFAs are represented by a two-way analogue of transformation semigroups, every element of which describes the behaviour of a 2DFA on a certain string x. A subsemigroup generated by this element represents the behaviour on strings in x+. The main contribution of this paper is a description of all such monogenic subsemigroups up to isomorphism. This characterization is then used to show that transforming an n-state 2DFA over a one-letter alphabet to an equivalent sweeping 2DFA requires exactly n+1 states, and transforming it to a one-way automaton requires exactly max0≤l≤n G(n - l) + l + 1 states, where G(k) is the maximum order of a permutation of k elements.
27 citations
06 Sep 2011
TL;DR: A general framework for the specification and implementation of systems whose executions are words, or partial orders, over an infinite alphabet is developed, and class register automata, a one-way automata model over words with multiple data values is introduced.
Abstract: We develop a general framework for the specification and implementation of systems whose executions are words, or partial orders, over an infinite alphabet. As a model of an implementation, we introduce class register automata, a one-way automata model over words with multiple data values. Our model combines register automata and class memory automata. It has natural interpretations. In particular, it captures communicating automata with an unbounded number of processes, whose semantics can be described as a set of (dynamic) message sequence charts. On the specification side, we provide a local existential monadic second-order logic that does not impose any restriction on the number of variables. We study the realizability problem and show that every formula from that logic can be effectively, and in elementary time, translated into an equivalent class register automaton.
01 Jan 2011
TL;DR: Survey results that compare the recognition power of different variants, consider their basic closure properties and study decidability questions are surveyed.
Abstract: Picture walking automata were introduced by M. Blum and C. Hewitt in 1967 as a generalization of one-dimensional two-way finite automata to recognize pictures, or two-dimensional words. Several variants have been investigated since then, including deterministic, non-deterministic and alternating transition rules; four-, three- and two-way movements; single- and multi-headed variants; automata that must stay inside the input picture, or that may move outside. We survey results that compare the recognition power of different variants, consider their basic closure properties and study decidability questions.
01 Jan 2011
TL;DR: It is shown that emptiness for such automata is decidable, both over finite and infinite words, under reasonable computability assumptions on the linear order.
Abstract: In this paper we work over linearly ordered data domains equipped with finitely many unary predicates and constants. We consider nondeterministic automata processing words and storing finitely many variables ranging over the domain. During a transition, these automata can compare the data values of the current configuration with those of the previous configuration using the linear order, the unary predicates and the constants.
We show that emptiness for such automata is decidable, both over finite and infinite words, under reasonable computability assumptions on the linear order.
Finally, we show how our automata model can be used for verifying properties of workflow specifications in the presence of an underlying database.
13 Jul 2011
TL;DR: An approach to finding the shortest reset word of a finite synchronizing automaton by using a SAT solver and the results allow us to formulate a hypothesis that the length of the shortest Reset Word of a random finite automaton with n states and 2 input letters with high probability is sublinear with respect to n and can be estimated as 1.95n0.55.
Abstract: In this paper we describe an approach to finding the shortest reset word of a finite synchronizing automaton by using a SAT solver. We use this approach to perform an experimental study of the length of the shortest reset word of a finite synchronizing automaton. The largest automata we considered had 100 states. The results of the experiments allow us to formulate a hypothesis that the length of the shortest reset word of a random finite automaton with n states and 2 input letters with high probability is sublinear with respect to n and can be estimated as 1.95n0.55.
26 Mar 2011
TL;DR: In this article, a game-based deterministic timed automata determinization algorithm is proposed, which is at once more general than the determinization procedure of [4] and more precise than the approximation algorithm of [11].
Abstract: Timed automata are frequently used to model real-time systems. Their determinization is a key issue for several validation problems. However, not all timed automata can be determinized, and determinizability itself is undecidable. In this paper, we propose a game-based algorithm which, given a timed automaton with e-transitions and invariants, tries to produce a language-equivalent deterministic timed automaton, otherwise a deterministic over-approximation. Our method subsumes two recent contributions: it is at once more general than the determinization procedure of [4] and more precise than the approximation algorithm of [11].
20 Jun 2011
TL;DR: This paper discusses APAs over dissimilar alphabets, a determinisation operator, conjunction of non-deterministic APAs, and an APA-embedding of Interface Automata, and concludes introducing a tool for automatic manipulation of APAs.
Abstract: Probabilistic Automata (PAs) are a recognized framework for modeling and analysis of nondeterministic systems with stochastic behavior. Recently, we proposed Abstract Probabilistic Automata (APAs) -- an abstraction framework for PAs. In this paper, we discuss APAs over dissimilar alphabets, a determinisation operator, conjunction of non-deterministic APAs, and an APA-embedding of Interface Automata. We conclude introducing a tool for automatic manipulation of APAs.
TL;DR: By estimating the number of regular expressions that have e as a partial derivative, a lower bound of the average number of mergings of states in and its asymptotic behaviour is computed.
Abstract: The partial derivative automaton () is usually smaller than other nondeterministic finite automata constructed from a regular expression, and it can be seen as a quotient of the Glushkov automaton (). By estimating the number of regular expressions that have e as a partial derivative, we compute a lower bound of the average number of mergings of states in and describe its asymptotic behaviour. This depends on the alphabet size, k, and for growing k's its limit approaches half the number of states in . The lower bound corresponds to consider the automaton for the marked version of the regular expression, i.e. where all its letters are made different. Experimental results suggest that the average number of states of this automaton, and of the automaton for the unmarked regular expression, are very close to each other.
26 Mar 2011
TL;DR: The Buchi Store is introduced, an open repository of Buchi automata formodel-checking practice, research, and education that contains Buchi Automata and their complements for common specification patterns and numerous temporal formulae.
Abstract: We introduce the Buchi Store, an open repository of Buchi automata formodel-checking practice, research, and education.The repository contains Buchi automata and their complements for common specification patterns and numerous temporal formulae. These automata are made as small as possible by various construction techniques, in view that smaller automata are easier to understand and often help in speeding up the model-checking process. The repository is open, allowing the user to add new automata or smaller ones that are equivalent to some existing automaton. Such a collection of Buchi automata is also useful as a benchmark for evaluating complementation or translation algorithms and as examples for studying Buchi automata and temporal logic.
TL;DR: For a given weighted finite automaton over a strong bimonoid, the reduced Nerode automaton is constructed, which is crisp-deterministic and equivalent to the original weighted automaton with respect to the initial algebra semantics and gives smaller crisp-Deterministic automata than any other known determinization algorithm.
TL;DR: This work studies the conversion of self-verifying automata to deterministic automata from a descriptional complexity point of view and the main result is the exact cost of such a simulation.
Abstract: Self-verifying automata are a special variant of finite automata with a symmetric kind of nondeterminism. We study the conversion of self-verifying automata to deterministic automata from a descriptional complexity point of view. The main result is the exact cost, in terms of the number of states, of such a simulation.
TL;DR: This work transfers Antimirov's partial derivatives from regular word expressions to regular tree expressions for an analysis of the size of the resulting automaton as well as for algorithmic improvements.
Abstract: Since recognizable tree languages are closed under the rational
operations, every regular tree expression denotes a recognizable
tree language. We provide an alternative proof to this fact that
results in smaller tree automata. To this aim, we transfer
Antimirov's partial derivatives from regular word expressions to
regular tree expressions. For an analysis of the size of the
resulting automaton as well as for algorithmic improvements, we also
transfer the methods of Champarnaud and Ziadi from words to trees.
22 Aug 2011
TL;DR: This work considers the emptiness problem, which asks whether some word is accepted by a given probabilistic automaton, and the universality problem, and provides reductions to establish the PSPACE-completeness of the two problems.
Abstract: Probabilistic automata are finite-state automata where the transitions are chosen according to fixed probability distributions. We consider a semantics where on an input word the automaton produces a sequence of probability distributions over states. An infinite word is accepted if the produced sequence is synchronizing, i.e. the sequence of the highest probability in the distributions tends to 1. We show that this semantics generalizes the classical notion of synchronizing words for deterministic automata. We consider the emptiness problem, which asks whether some word is accepted by a given probabilistic automaton, and the universality problem, which asks whether all words are accepted. We provide reductions to establish the PSPACE-completeness of the two problems.
01 Jan 2011
TL;DR: The main result is a Rabin-style characterization theorem: a function is quasi-weak definable if and only if it is definable using two dual forms of nondeterministic Buchi cost automata, which yields a new decidability result for cost functions over infinite trees.
Abstract: Cost automata have a finite set of counters which can be manipulated on each transition but do not aect control flow. Based on the evolution of the counter values, these automata define functions from a domain like words or trees to N[{1}, modulo an equivalence relation which ignores exact values but preserves boundedness properties. These automata have been studied by Colcombet et al. as part of a “theory of regular cost functions”, an extension of the theory of regular languages which retains robust equivalences, closure properties, and decidability like the classical theory. We extend this theory by introducing quasi-weak cost automata. Unlike traditional weak automata which have a hard-coded bound on the number of alternations between accepting and rejecting states, quasi-weak automata bound the alternations using the counter values (which can vary across runs). We show that these automata are strictly more expressive than weak cost automata over infinite trees. The main result is a Rabin-style characterization theorem: a function is quasi-weak definable if and only if it is definable using two dual forms of nondeterministic Buchi cost automata. This yields a new decidability result for cost functions over infinite trees. 1998 ACM Subject Classification F.1.1 Models of Computation
04 Jul 2011
TL;DR: In contrast to the classical (unbounded) reachability problem, the timed-bounded version is decidable for rectangular hybrid automata provided only nonnegative rates are allowed.
Abstract: This paper investigates the time-bounded version of the reachability problem for hybrid automata. This problem asks whether a given hybrid automaton can reach a given target location within T time units, where T is a constant rational value. We show that, in contrast to the classical (unbounded) reachability problem, the timed-bounded version is decidable for rectangular hybrid automata provided only non-negative rates are allowed. This class of systems is of practical interest and subsumes, among others, the class of stopwatch automata. We also show that the problem becomes undecidable if either diagonal constraints or both negative and positive rates are allowed.
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TL;DR: A deterministic model for associating costs with strings that is parameterized by operations of interest, a notion of regularity that provides a yardstick to measure expressiveness, and study decision problems and theoretical properties of resulting classes of cost functions are proposed.
Abstract: Motivated by the successful application of the theory of regular languages to formal verification of finite-state systems, there is a renewed interest in developing a theory of analyzable functions from strings to numerical values that can provide a foundation for analyzing quantitative properties of finitestate systems. In this paper, we propose a deterministic model for associating costs with strings that is parameterized by operations of interest (such as addition, scaling, and min), a notion of regularity that provides a yardstick to measure expressiveness, and study decision problems and theoretical properties of resulting classes of cost functions. Our definition of regularity relies on the theory of string-to-tree transducers, and allows associating costs with events that are conditional upon regular properties of future events. Our model of cost register automata allows computation of regular functions using multiple “write-only” registers whose values can be combined using the allowed set of operations. We show that classical shortest-path algorithms as well as algorithms designed for computing discounted costs, can be adopted for solving the min-cost problems for the more general classes of functions specified in our model. Cost register automata with min and increment give a deterministic model that is equivalent to weighted automata, an extensively studied nondeterministic model, and this connection results in new insights and new open problems.
01 Jan 2011
TL;DR: In this article, the authors studied the polynomial state complexity of two-way alternating finite automata (2@S"k and 2@P"k) and showed that for any k = 2,3,4, the complexity of 2@Sk-k automata is infinite.
Abstract: We study the polynomial state complexity classes 2@S"k and 2@P"k, that is, the hierarchy of problems that can be solved with a polynomial number of states by two-way alternating finite automata (2Afas) making at most k-1 alternations between existential and universal states, starting in an existential or universal state, respectively. This hierarchy is infinite: for k=2,3,4,..., both 2@S"k"-"1 and 2@P"k"-"1 are proper subsets of 2@S"k and of 2@P"k, since the conversion of a one-way @S"k- or @P"k-alternating automaton with n states into a two-way automaton with a smaller number of alternations requires 2^n^/^4^-^O^(^k^) states. The same exponential blow-up is required for converting a @S"k-bounded 2Afa into a @P"k-bounded 2Afa and vice versa, that is, 2@S"k and 2@P"k are incomparable. In the case of @S"k-bounded 2Afas, the exponential gap applies also for intersection, while in the case of @P"k-bounded 2Afas for union. The same results are established for one-way alternating finite automata. This solves several open problems raised in [C. Kapoutsis, Size complexity of two-way finite automata, in: Proc. Develop. Lang. Theory, in: Lect. Notes Comput. Sci., vol. 5583, Springer-Verlag, 2009, pp. 47-66.]
11 Oct 2011
TL;DR: It is shown that universal weighted automata in the sum semantics can represent all polynomials, and it is argued that a summation semantics is of interest too, as it captures the intuition that one has to pay for the cost of all conjuncts.
Abstract: In the traditional Boolean setting of formal verification, alternating automata are the key to many algorithms and tools In this setting, the correspondence between disjunctions/conjunctions in the specification and nondeterministic/universal transitions in the automaton for the specification is straightforward A recent exciting research direction aims at adding a quality measure to the satisfaction of specifications of reactive systems The corresponding automata-theoretic framework is based on weighted automata, which map input words to numerical values In the weighted setting, nondeterminism has a minimum semantics - the weight that an automaton assigns to a word is the cost of the cheapest run on it For universal branches, researchers have studied a (dual) maximum semantics We argue that a summation semantics is of interest too, as it captures the intuition that one has to pay for the cost of all conjuncts
We introduce and study alternating weighted automata on finite words in both the max and sum semantics We study the duality between the min and max semantics, closure under max and sum, the added power of universality and alternation, and arithmetic operations on automata In particular, we show that universal weighted automata in the sum semantics can represent all polynomials
TL;DR: The relationships between the structure of the suffix and factor automata and classical combinatorial parameters related to the special factors of w are studied and formulae for the number of states of these automata are derived.
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TL;DR: In this article, the authors determined the asymptotic proportion of minimal automata, within n-state accessible deterministic complete automata over a k-letter alphabet, with the uniform distribution over the possible transition structures, and a binomial distribution over terminal states, with arbitrary parameter b.
Abstract: We determine the asymptotic proportion of minimal automata, within n-state accessible deterministic complete automata over a k-letter alphabet, with the uniform distribution over the possible transition structures, and a binomial distribution over terminal states, with arbitrary parameter b. It turns out that a fraction ~ 1-C(k,b) n^{-k+2} of automata is minimal, with C(k,b) a function, explicitly determined, involving the solution of a transcendental equation.
TL;DR: This is the first demonstration of the superiority of a quantum model to the corresponding classical one in the real-time case with an error bound less than 1, by demonstrating a non-context-free language that can be recognized with perfect soundness by a rtQ1CA.
Abstract: The question of whether quantum real-time one-counter automata (rtQ1CAs) can outperform their probabilistic counterparts has been open for more than a decade. We provide an affirmative answer to this question, by demonstrating a non-context-free language that can be recognized with perfect soundness by a rtQ1CA. This is the first demonstration of the superiority of a quantum model to the corresponding classical one in the real-time case with an error bound less than 1. We also introduce a generalization of the rtQ1CA, the quantum one-way one-counter automaton (1Q1CA), and show that they too are superior to the corresponding family of probabilistic machines. For this purpose, we provide general definitions of these models that reflect the modern approach to the definition of quantum finite automata, and point out some problems with previous results. We identify several remaining open problems.