Showing papers on "ω-automaton published in 2009"

The Theory of Stabilisation Monoids and Regular Cost Functions

Abstract: We introduce the notion of regular cost functions: a quantitative extension to the standard theory of regular languages. We provide equivalent characterisations of this notion by means of automata (extending the nested distance desert automata of Kirsten), of history-deterministic automata (history-determinism is a weakening of the standard notion of determinism, that replaces it in this context), and a suitable notion of recognisability by stabilisation monoids. We also provide closure and decidability results.

Tighter Bounds for the Determinisation of Büchi Automata

Abstract: The introduction of an efficient determinisation technique for Buchi automata by Safra has been a milestone in automata theory. To name only a few applications, efficient determinisation techniques for *** -word automata are the basis for several manipulations of *** -tree automata (most prominently the nondeterminisation of alternating tree automata) as well as for satisfiability checking and model synthesis for branching- and alternating-time logics. This paper proposes a determinisation technique that is simpler than the constructions of Safra, Piterman, and Muller and Schupp, because it separates the principle acceptance mechanism from the concrete acceptance condition. The principle mechanism intuitively uses a Rabin condition on the transitions; we show how to obtain an equivalent Rabin transition automaton with approximately (1.65 n ) n states from a nondeterministic Buchi automaton with n states. Having established this mechanism, it is simple to develop translations to automata with standard acceptance conditions. We can construct standard Rabin automata whose state-space is bilinear in the size of the input alphabet and the state-space of the Rabin transition automaton, or, for large input alphabets, contains approximately (2.66 n ) n states, respectively. We also provide a flexible translation to parity automata with O (n !2) states and 2n priorities based on a later introduction record, and hence connect the transformation of the acceptance condition to other record based transformations known from the literature.

Nondeterministic finite automata — recent results on the descriptional and computational complexity

Abstract: Nondeterministic finite automata (NFAs) were introduced in [68], where their equivalence to deterministic finite automata was shown. Over the last 50 years, a vast literature documenting the importance of finite automata as an enormously valuable concept has been developed. In the present paper, we tour a fragment of this literature. Mostly, we discuss recent developments relevant to NFAs related problems like, for example, (i) simulation of and by several types of finite automata, (ii) minimization and approximation, (iii) size estimation of minimal NFAs, and (iv) state complexity of language operations. We thus come across descriptional and computational complexity issues of nondeterministic finite automata. We do not prove these results but we merely draw attention to the big picture and some of the main ideas involved.

Alternating weighted automata

Abstract: Weighted automata are finite automata with numerical weights on transitions. Nondeterministic weighted automata define quantitative languages L that assign to each word w a real number L(w) computed as the maximal value of all runs over w, and the value of a run r is a function of the sequence of weights that appear along r. There are several natural functions to consider such as Sup, LimSup, LimInf, limit average, and discounted sum of transition weights. We introduce alternating weighted automata in which the transitions of the runs are chosen by two players in a turn-based fashion. Each word is assigned the maximal value of a run that the first player can enforce regardless of the choices made by the second player. We survey the results about closure properties, expressiveness, and decision problems for nondeterministic weighted automata, and we extend these results to alternating weighted automata. For quantitative languages L1 and L2, we consider the pointwise operations max(L1, L2), min(L1, L2), 1 - L1, and the sum L1 + L2. We establish the closure properties of all classes of alternating weighted automata with respect to these four operations. We next compare the expressive power of the various classes of alternating and nondeterministic weighted automata over infinite words. In particular, for limit average and discounted sum, we show that alternation brings more expressive power than nondeterminism. Finally, we present decidability results and open questions for the quantitative extension of the classical decision problems in automata theory: emptiness, universality, language inclusion, and language equivalence.

When Are Timed Automata Determinizable

Abstract: In this paper, we propose an abstract procedure which, given a timed automaton, produces a language-equivalent deterministic infinite timed tree. We prove that under a certain boundedness condition, the infinite timed tree can be reduced into a classical deterministic timed automaton. The boundedness condition is satisfied by several subclasses of timed automata, some of them were known to be determinizable (event-clock timed automata, automata with integer resets), but some others were not. We prove for instance that strongly non-Zeno timed automata can be determinized. As a corollary of those constructions, we get for those classes the decidability of the universality and of the inclusion problems, and compute their complexities (the inclusion problem is for instance EXPSPACE-complete for strongly non-Zeno timed automata).

Profinite Methods in Automata Theory

Abstract: This survey paper presents the success story of the topological approach to automata theory. It is based on profinite topologies, which are built from finite topogical spaces. The survey includes several concrete applications to automata theory.

Probabilistic Weighted Automata

Abstract: Nondeterministic weighted automata are finite automata with numerical weights on transitions. They define quantitative languages L that assign to each word w a real number L (w ). The value of an infinite word w is computed as the maximal value of all runs over w , and the value of a run as the supremum, limsup, liminf, limit average, or discounted sum of the transition weights. We introduce probabilistic weighted automata, in which the transitions are chosen in a randomized (rather than nondeterministic) fashion. Under almost-sure semantics (resp. positive semantics), the value of a word w is the largest real v such that the runs over w have value at least v with probability 1 (resp. positive probability). We study the classical questions of automata theory for probabilistic weighted automata: emptiness and universality, expressiveness, and closure under various operations on languages. For quantitative languages, emptiness and universality are defined as whether the value of some (resp. every) word exceeds a given threshold. We prove some of these questions to be decidable, and others undecidable. Regarding expressive power, we show that probabilities allow us to define a wide variety of new classes of quantitative languages, except for discounted-sum automata, where probabilistic choice is no more expressive than nondeterminism. Finally, we give an almost complete picture of the closure of various classes of probabilistic weighted automata for the following pointwise operations on quantitative languages: max, min, sum, and numerical complement.

Size Complexity of Two-Way Finite Automata

Abstract: This is a talk on the size complexity of two-way finite automata. We present the central open problem in the area, explain a motivation behind it, recall its early history, and introduce some of the concepts used in its study. We then sketch a possible future, describe a natural systematic way of pursuing it, and record some of the progress that has been achieved. We add little to what is already known--only exposition, terminology, and questions.

Ranking functions for size-change termination

Abstract: This article explains how to construct a ranking function for any program that is proved terminating by size-change analysis.The “principle of size-change termination” for a first-order functional language with well-ordered data is intuitive: A program terminates on all inputs, if every infinite call sequence (following program control flow) would imply an infinite descent in some data values. Size-change analysis is based on information associated with the subject program's call-sites. This information indicates, for each call-site, strict or weak data decreases observed as a computation traverses the call-site. The set DESC of call-site sequences for which the size-changes imply infinite descent is ω-regular, as is the set FLOW of infinite call-site sequences following the program flowchart. If FLOW ⊆ DESC (a decidable problem), every infinite call sequence would imply infinite descent in a well-ordering—an impossibility—so the program must terminate.This analysis accounts for termination arguments applicable to different call-site sequences, without indicating a ranking function for the program's termination. In this article, it is explained how one can be constructed whenever size-change analysis succeeds. The constructed function has an unexpectedly simple form; it is expressed using only min, max, and lexicographic tuples of parameters and constants. In principle, such functions can be tested to determine whether size-change analysis will be successful. As a corollary, if a program verified as terminating performs only multiply recursive operations, the function that it computes is multiply recursive.The ranking function construction is connected with the determinization of the Buchi automaton for DESC. While the result is not practical, it is of value in addressing the scope of size-change reasoning. This reasoning has been applied broadly, in the analysis of functional and logic programs, as well as term rewrite systems.

Concavely-Priced Probabilistic Timed Automata

Abstract: Concavely-priced probabilistic timed automata, an extension of probabilistic timed automata, are introduced. In this paper we consider expected reachability, discounted, and average price problems for concavely-priced probabilistic timed automata for arbitrary initial states. We prove that these problems are EXPTIME-complete for probabilistic timed automata with two or more clocks and PTIME-complete for automata with one clock. Previous work on expected price problems for probabilistic timed automata was restricted to expected reachability for linearly-priced automata and integer valued initial states. This work uses the boundary region graph introduced by Jurdzinski and Trivedi to analyse properties of concavely-priced (non-probabilistic) timed automata.

Descriptional and Computational Complexity of Finite Automata

Abstract: 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, e.g., 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.

On Parallel Implementations of Deterministic Finite Automata

Abstract: We present implementations of parallel DFA run methods and find whether and under what conditions is worthy to use the parallel methods of simulation of run of finite automata. First, we introduce the parallel DFA run methods for general DFA, which are universal, but due to the dependency of simulation time on the number of states |Q | of automaton being run, they are suitable only for run of automata with the smaller number of states. Then we show that if we apply some restrictions to properties of automata being run, we can reach the linear speedup compared to the sequential simulation method. We designed methods benefiting from k -locality that allows optimum parallel run of exact and approximate pattern matching automata. Finally, we show the results of experiments conducted on two types of parallel computers (Cluster of workstations and Symmetric shared-memory multiprocessors).

Power of Randomization in Automata on Infinite Strings

Abstract: Probabilistic Buchi 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 precisely characterize the complexity of the emptiness, universality, and language containment problems for such machines, answering canonical 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 *** -regular languages. Finally, we introduce Hierarchical PBAs, which are syntactically restricted forms of PBAs that are tractable and capture exactly the class of *** -regular languages.

Complementation of coalgebra automata

Abstract: Coalgebra automata, introduced by the second author, generalize the well-known automata that operate on infinite words/streams, trees, graphs or transition systems. This coalgebraic perspective on automata lays foundation to a universal theory of automata operating on infinite models of computation. In this paper we prove a complementation lemma for coalgebra automata. More specifically, we provide a construction that transforms a given coalgebra automaton with parity acceptance condition into a device of similar type, which accepts exactly those pointed coalgebras that are rejected by the original automaton. Our construction works for automata operating on coalgebras for an arbitrary standard set functor which preserves weak pullbacks and restricts to finite sets. Our proof is coalgebraic in flavour in that we introduce and use a notion of game bisimilarity between certain kinds of parity games.

Multi-Head Finite Automata: Characterizations, Concepts and Open Problems

Abstract: Multi-head finite automata were introduced in (Rabin, 1964) and (Rosenberg, 1966). Since that time, a vast literature on computational and descriptional complexity issues on multi-head finite automata documenting the importance of these devices has been developed. Although multi-head finite automata are a simple concept, their computational behavior can be already very complex and leads to undecidable or even non-semi-decidable problems on these devices such as, for example, emptiness, finiteness, universality, equivalence, etc. These strong negative results trigger the study of subclasses and alternative characterizations of multi-head finite automata for a better understanding of the nature of non-recursive trade-offs and, thus, the borderline between decidable and undecidable problems. In the present paper, we tour a fragment of this literature.

Graph-Rewriting Automata as a Natural Extension of Cellular Automata

Abstract: We introduce a framework called graph-rewriting automata to model evolution processes of networks. It is a natural extension of cellular automata in the sense that a fixed lattice space of cellular automata is extended to a dynamic graph structure by introducing local graph-rewriting rules. We consider three different constructions of rule sets to show that various network evolution is possible: hand-coding, evolutionary generation, and exhaustive search. Graph-rewriting automata provide a new tool to describe various complex systems and to approach many scientific problems.

Genetic approaches to search for computing patterns in cellular automata

Abstract: The emergence of computation in complex systems "with simple components is a hot topic in the science of complexity. A uniform framework to study emergent computation in complex systems are cellular automata. They are discrete systems in which an array of cells evolves from generation to generation on the basis of local transition rules. The well-established problem of emergent computation and universality in cellular automata has been tackled by a number of people in the last thirty years and remains an area "where amazing phenomena at the edge of theoretical computer science and nonlinear science can be discovered. Future work could also evaluate all discovered cellular automata and calculate for each cellular automaton some rule-based parameters, e.g., Langtons lamda. All cellular automata simulating an AND gate may have similar values for these parameters that could lead to answer the question Where are the edges of computational universality? and may therefore lead to a better understanding of the emergence of computation in complex systems with local interactions.

Strongly transitive automata and the Černý conjecture

Abstract: The synchronization problem is investigated for a new class of deterministic automata called strongly transitive. An extension to unambiguous automata is also considered.

Regular Expressions with Numerical Constraints and Automata with Counters

Abstract: Regular expressions with numerical constraints are an extension of regular expressions, allowing to bound numerically the number of times that a subexpression should be matched. Expressions in this extension describe the same languages as the usual regular expressions, but are exponentially more succinct. We define a class of finite automata with counters and a deterministic subclass of these. Deterministic finite automata with counters can recognize words in linear time. Furthermore, we describe a subclass of the regular expressions with numerical constraints, a polynomial-time test for this subclass, and a polynomial-time construction of deterministic finite automata with counters from expressions in the subclass.

Picture Recognizability with Automata Based on Wang Tiles

Abstract: We introduce a model of automaton for picture language recognition which is based on tiles and is called Wang automaton, since its description relies on the notation of Wang systems. Wang automata combine features of both online tessellation acceptors and 4-ways automata: as in online tessellation acceptors, computation assigns states to each picture position; as in 4-way automata, the input head visits the picture moving from one pixel to an adjacent one, according to some scanning strategy. We prove that Wang automata recognize the class REC, i.e. they are equivalent to tiling systems or online tessellation acceptors, and hence strictly more powerful than 4-way automata. We also consider a very natural notion of determinism for Wang automata, and study the resulting class, comparing it with other deterministic classes considered in the literature, like DREC and Snake-DREC.

An nlogn Algorithm for Hyper-minimizing States in a (Minimized) Deterministic Automaton

Abstract: We improve a recent result [ A. Badr : Hyper-Minimization in O (n 2). In Proc. CIAA , LNCS 5148, 2008] for hyper-minimized finite automata. Namely, we present an O (n logn ) algorithm that computes for a given finite deterministic automaton (dfa) an almost equivalent dfa that is as small as possible--such an automaton is called hyper-minimal. Here two finite automata are almost equivalent if and only if the symmetric difference of their languages is finite. In other words, two almost-equivalent automata disagree on acceptance on finitely many inputs. In this way, we solve an open problem stated in [ A. Badr , V. Geffert , I. Shipman : Hyper-minimizing minimized deterministic finite state automata. RAIRO Theor. Inf. Appl. 43(1), 2009] and by Badr . Moreover, we show that minimization linearly reduces to hyper-minimization, which shows that the time-bound O (n logn ) is optimal for hyper-minimization.