Showing papers on "ω-automaton published in 2013"

Generalizing determinization from automata to coalgebras

Abstract: The powerset construction is a standard method for converting a nondeter- ministic automaton into a deterministic one recognizing the same language. In this paper, we lift the powerset construction from automata to the more general framework of coal- gebras with structured state spaces. Coalgebra is an abstract framework for the uniform study of different kinds of dynamical systems. An endofunctor F determines both the type of systems (F-coalgebras) and a notion of behavioural equivalence (�F) amongst them. Many types of transition systems and their equivalences can be captured by a functor F. For example, for deterministic automata the derived equivalence is language equivalence, while for non-deterministic automata it is ordinary bisimilarity. We give several examples of applications of our generalized determinization construc- tion, including partial Mealy machines, (structured) Moore automata, Rabin probabilistic automata, and, somewhat surprisingly, even pushdown automata. To further witness the generality of the approach we show how to characterize coalgebraically several equivalences which have been object of interest in the concurrency community, such as failure or ready semantics.

Advanced automata minimization

Abstract: We present an efficient algorithm to reduce the size of nondeterministic Buchi word automata, while retaining their language. Additionally, we describe methods to solve PSPACE-complete automata problems like universality, equivalence and inclusion for much larger instances (1-3 orders of magnitude) than before. This can be used to scale up applications of automata in formal verification tools and decision procedures for logical theories.The algorithm is based on new transition pruning techniques. These use criteria based on combinations of backward and forward trace inclusions. Since these relations are themselves PSPACE-complete, we describe methods to compute good approximations of them in polynomial time.Extensive experiments show that the average-case complexity of our algorithm scales quadratically. The size reduction of the automata depends very much on the class of instances, but our algorithm consistently outperforms all previous techniques by a wide margin. We tested our algorithm on Buchi automata derived from LTL-formulae, many classes of random automata and automata derived from mutual exclusion protocols, and compared its performance to the well-known automata tool GOAL.

Applications of symbolic finite automata

Abstract: Symbolic automata theory lifts classical automata theory to rich alphabet theories. It does so by replacing an explicit alphabet with an alphabet described implicitly by a Boolean algebra. How does this lifting affect the basic algorithms that lay the foundation for modern automata theory and what is the incentive for doing this? We investigate these questions here. In our approach we use state-of-the-art constraint solving techniques for automata analysis that are both expressive and efficient, even for very large and infinite alphabets. We show how symbolic finite automata enable applications ranging from modern regex analysis to advanced web security analysis, that were out of reach with prior methods.

A Fresh Approach to Learning Register Automata

Abstract: This paper provides an Angluin-style learning algorithm for a class of register automata supporting the notion of fresh data values. More specifically, we introduce session automata which are well suited for modeling protocols in which sessions using fresh values are of major interest, like in security protocols or ad-hoc networks. We show that session automata (i) have an expressiveness partly extending, partly reducing that of register automata, (ii) admit a symbolic regular representation, and (iii) have a decidable equivalence problem. We use these results to establish a learning algorithm that can infer inherently non-deterministic session automata. Moreover, in the case of deterministic automata, it has a better complexity wrt. membership queries than existing learning algorithms for register automata. We strengthen the importance of our automaton model by its characterization in monadic second-order logic.

Nondeterminism in the presence of a diverse or unknown future

Abstract: Choices made by nondeterministic word automata depend on both the past (the prefix of the word read so far) and the future (the suffix yet to be read). In several applications, most notably synthesis, the future is diverse or unknown, leading to algorithms that are based on deterministic automata. Hoping to retain some of the advantages of nondeterministic automata, researchers have studied restricted classes of nondeterministic automata. Three such classes are nondeterministic automata that are good for trees (GFT; i.e., ones that can be expanded to tree automata accepting the derived tree languages, thus whose choices should satisfy diverse futures), good for games (GFG; i.e., ones whose choices depend only on the past), and determinizable by pruning (DBP; i.e., ones that embody equivalent deterministic automata). The theoretical properties and relative merits of the different classes are still open, having vagueness on whether they really differ from deterministic automata. In particular, while DBP ⊆ GFG ⊆ GFT, it is not known whether every GFT automaton is GFG and whether every GFG automaton is DBP. Also open is the possible succinctness of GFG and GFT automata compared to deterministic automata. We study these problems for ω-regular automata with all common acceptance conditions. We show that GFT=GFG⊃DBP, and describe a determinization construction for GFG automata.

On the state complexity of semi-quantum finite automata

Abstract: Some of the most interesting and important results concerning quantum finite automata are those showing that they can recognize certain languages with (much) less resources than corresponding classical finite automata \cite{Amb98,Amb09,AmYa11,Ber05,Fre09,Mer00,Mer01,Mer02,Yak10,ZhgQiu112,Zhg12}. This paper shows three results of such a type that are stronger in some sense than other ones because (a) they deal with models of quantum automata with very little quantumness (so-called semi-quantum one- and two-way automata with one qubit memory only); (b) differences, even comparing with probabilistic classical automata, are bigger than expected; (c) a trade-off between the number of classical and quantum basis states needed is demonstrated in one case and (d) languages (or the promise problem) used to show main results are very simple and often explored ones in automata theory or in communication complexity, with seemingly little structure that could be utilized.

Method and apparatus for compilation of finite automata

Abstract: A method and corresponding apparatus are provided implementing run time processing using Deterministic Finite Automata (DFA) and Non-Deterministic Finite Automata (NFA) to find the existence of a pattern in a payload. A subpattern may be selected from each pattern in a set of one or more regular expression patterns based on at least one heuristic and a unified deterministic finite automata (DFA) may be generated using the subpatterns selected from all patterns in the set, and at least one non-deterministic finite automata (NFA) may be generated for at least one pattern in the set, optimizing run time performance of the run time processing.

Generating small automata and the Černý conjecture

Abstract: We present a new efficient algorithm to generate all nonisomorphic automata with given numbers of states and input letters. The generation procedure may be restricted effectively to strongly connected automata. This is used to verify the Cerný conjecture for all binary automata with n≤11 states, which improves the results in the literature. We compute also the distributions of the length of the shortest reset word for binary automata with n≤10 states, which completes the results reported by other authors.

Two-Variable Logic on 2-Dimensional Structures.

Abstract: This paper continues the study of the two-variable fragment of first-order logic (FO^2) over two- dimensional structures, more precisely structures with two orders, their induced successor relations and arbitrarily many unary relations. Our main focus is on ordered data words which are finite sequences from the set \Sigma x D where \Sigma is a finite alphabet and D is an ordered domain. These are naturally represented as labelled finite sets with a linear order <=_l and a total preorder <=_p. We introduce ordered data automata, an automaton model for ordered data words. An ordered data automaton is a composition of a finite state transducer and a finite state automaton over the product Boolean algebra of finite and cofinite subsets of N. We show that ordered data automata are equivalent to the closure of FO^2(+1_l,<=_p,+1_p) under existential quantification of unary relations. Using this automaton model we prove that the finite satisfiability problem for this logic is decidable on structures where the <=_p-equivalence classes are of bounded size. As a corollary, we obtain that finite satisfiability of FO^2 is decidable (and it is equivalent to the reachability problem of vector addition systems) on structures with two linear order successors and a linear order corresponding to one of the successors. Further we prove undecidability of FO^2 on several other two-dimensional structures.

Deciding the weak definability of Büchi definable tree languages

Abstract: Weakly definable languages of infinite trees are an expressive subclass of regular tree languages definable in terms of weak monadic second-order logic, or equivalently weak alternating automata. Our main result is that given a Buchi automaton, it is decidable whether the language is weakly definable. We also show that given a parity automaton, it is decidable whether the language is recognizable by a nondeterministic co-Buchi automaton. The decidability proofs build on recent results about cost automata over infinite trees. These automata use counters to define functions from infinite trees to the natural numbers extended with infinity. We reduce to testing whether the functions defined by certain "quasi-weak" cost automata are bounded by a finite value.

Rabin-Mostowski Index Problem: A Step beyond Deterministic Automata

Abstract: For a given regular language of infinite trees, one can ask about the minimal number of priorities needed to recognise this language with a non-deterministic or alternating parity automaton. These questions are known as, respectively, the non-deterministic and the alternating Rabin-Mostowski index problems. Whether they can be answered effectively is a long-standing open problem, solved so far only for languages recognisable by deterministic automata (the alternating variant trivialises). We investigate a wider class of regular languages, recognisable by so-called game automata, which can be seen as the closure of deterministic ones under complementation and composition. Game automata are known to recognise languages arbitrarily high in the alternating Rabin-Mostowski index hierarchy, i.e., the alternating index problem does not trivialise any more. Our main contribution is that both index problems are decidable for languages recognisable by game automata. Additionally, we show that it is decidable whether a given regular language can be recognised by a game automaton.

On the Complexity of Equivalence and Minimisation for Q-weighted Automata

Abstract: This paper is concerned with the computational complexity of equivalence and minimisation for automata with transition weights in the field Q of rational numbers. We use polynomial identity testing and the Isolation Lemma to obtain complexity bounds, focussing on the class NC of problems within P solvable in polylogarithmic parallel time. For finite Q-weighted automata, we give a randomised NC procedure that either outputs that two automata are equivalent or returns a word on which they differ. We also give an NC procedure for deciding whether a given automaton is minimal, as well as a randomised NC procedure that minimises an automaton. We consider probabilistic automata with rewards, similar to Markov Decision Processes. For these automata we consider two notions of equivalence: expectation equivalence and distribution equivalence. The former requires that two automata have the same expected reward on each input word, while the latter requires that each input word induce the same distribution on rewards in each automaton. For both notions we give algorithms for deciding equivalence by reduction to equivalence of Q-weighted automata. Finally we show that the equivalence problem for Q-weighted visibly pushdown automata is logspace equivalent to the polynomial identity testing problem.

Regular Real Analysis

Abstract: We initiate the study of regular real analysis, or the analysis of real functions that can be encoded by automata on infinite words. It is known that ω-automata can be used to represent {relations} between real vectors, reals being represented in exact precision as infinite streams. The regular functions studied here constitute the functional subset of such relations. We show that some classic questions in function analysis can become elegantly computable in the context of regular real analysis. Specifically, we present an automata-theoretic technique for reasoning about limit behaviors of regular functions, and obtain, using this method, a decision procedure to verify the continuity of a regular function. Several other decision procedures for regular functions-for finding roots, fix points, minima, etc.-are also presented. At the same time, we show that the class of regular functions is quite rich, and includes functions that are highly challenging to encode using traditional symbolic notation.

Two-Way Finite Automata: Old and Recent Results

Abstract: The notion of two-way automata was introduced at the very beginning of automata theory. In 1959, Rabin and Scott and, independently, Shepherdson, proved that these models, both in the deterministic and in the nondeterministic versions, have the same power of one-way automata, namely, they characterize the class of regular languages. In 1978, Sakoda and Sipser posed the question of the costs, in the number of the states, of the simulations of one-way and two-way non-deterministic automata by two-way deterministic automata. They conjectured that these costs are exponential. In spite of all attempts to solve it, this question is still open. In the last ten years the problem of Sakoda and Sipser was widely reconsidered and many new results related to it have been obtained. In this work we discuss some of them. In particular, we focus on the restriction to the unary case, namely the case of automata defined over the one letter input alphabet, and on the connections with open questions in space complexity.

Simultaneous Finite Automata: An Efficient Data-Parallel Model for Regular Expression Matching

Abstract: Automata play important roles in wide area of computing and the growth of multicores calls for their efficient parallel implementation. Though it is known in theory that we can perform the computation of a finite automaton in parallel by simulating transitions, its implementation has a large overhead due to the simulation. In this paper we propose a new automaton called simultaneous finite automaton (SFA) for efficient parallel computation of an automaton. The key idea is to extend an automaton so that it involves the simulation of transitions. Since an SFA itself has a good property of parallelism, we can develop easily a parallel implementation without overheads. We have implemented a regular expression matcher based on SFA, and it has achieved over 10-times speedups on an environment with dual hexa-core CPUs in a typical case.

Security policies enforcement using finite and pushdown edit automata

Abstract: Edit automata have been introduced by J.Ligatti et al. as a model for security enforcement mechanisms which work at run time. In a distributed interacting system, they play a role of a monitor that runs in parallel with a target program and transforms its execution sequence into a sequence that obeys the security property. In this paper, we characterize security properties which are enforceable by finite edit automata (i.e. edit automata with a finite set of states) and deterministic context-free edit automata (i.e. finite edit automata extended with a stack). We prove that the properties enforceable by finite edit automata are a sub-class of regular sets. Moreover, given a regular set $$P$$ , one can decide in time $$O(n^2)$$ , whether $$P$$ is enforceable by a finite edit automaton (where $$n$$ is the number of states of the finite automaton recognizing $$P$$ ) and we give an algorithm to synthesize the controller. Moreover, we prove that safety policies are always enforced by a deterministic context-free edit automaton. We also prove that it is possible to check if a policy is a safety policy in $$O(n^4)$$ . Finally, we give a topological condition on the deterministic automaton expressing a regular policy enforceable by a deterministic context-free edit automaton.

On Infinite Words Determined by Stack Automata.

Abstract: We characterize the infinite words determined by one-way stack automata. An infinite language L determines an infinite word alpha if every string in L is a prefix of alpha. If L is regular or context-free, it is known that alpha must be ultimately periodic. We extend this result to the class of languages recognized by one-way nondeterministic checking stack automata (1-NCSA). We then consider stronger classes of stack automata and show that they determine a class of infinite words which we call multilinear. We show that every multilinear word can be written in a form which is amenable to parsing. Finally, we consider the class of one-way multihead deterministic finite automata (1:multi-DFA). We show that every multilinear word can be determined by some 1:multi-DFA, but that there exist infinite words determined by 1:multi-DFA which are not multilinear.

From equivalence to almost-equivalence, and beyond: minimizing automata with errors

Abstract: We introduce E-equivalence, which is a straightforward generalization of almost-equivalence. While almost-equivalence asks for ordinary equivalence up to a finite number of exceptions, in E-equival...

Matrix approach to stabilizability of deterministic finite automata

Abstract: This paper investigates the state feedback stabilizing problem of deterministic finite automata using a matrix approach. With the help of semi-tensor product, a matrix-based expression for finite automata is given, and the dynamics of automata are expressed in the form of a discrete-time bilinear equation. After providing the notions of equilibrium and cycle stability, we give necessary and sufficient algebraic conditions for the stabilizability of deterministic finite automata for the two respective cases. Then, based on the matrix expression, we focus on a special case where the controlled state trajectories to the target equilibrium is minimal. All the state feedback controllers can be obtained by solving a matrix inequality. Examples are also given for illustration.

Queue Automata of Constant Length

Abstract: We introduce and study the notion of constant length queue automata, as a formalism for representing regular languages. We show that their descriptional power outperforms that of traditional finite state automata, of constant height pushdown automata, and of straight line programs for regular expressions, by providing optimal exponential and double-exponential size gaps. Moreover, we prove that constant height pushdown automata can be simulated by constant length queue automata paying only by a linear size increase, and that removing nondeterminism in constant length queue automata requires an optimal exponential size blow-up, against the optimal double-exponential cost for determinizing constant height pushdown automata.

Strength-Based decomposition of the property Büchi automaton for faster model checking

Abstract: The automata-theoretic approach for model checking of linear-time temporal properties involves the emptiness check of a large Buchi automaton. Specialized emptiness-check algorithms have been proposed for the cases where the property is represented by a weak or terminal automaton. When the property automaton does not fall into these categories, a general emptiness check is required. This paper focuses on this class of properties. We refine previous approaches by classifying stronglyconnected components rather than automata, and suggest a decomposition of the property automaton into three smaller automata capturing the terminal, weak, and the remaining strong behaviors of the property. The three corresponding emptiness checks can be performed independently, using the most appropriate algorithm. Such a decomposition approach can be used with any automata-based model checker. We illustrate the interest of this new approach using explicit and symbolic LTL model checkers.

Limited Automata and Regular Languages

Abstract: Limited automata are one-tape Turing machines that are allowed to rewrite the content of any tape cell only in the first d visits, for a fixed constant d. In the case d = 1, namely, when a rewriting is possible only during the first visit to a cell, these models have the same power of finite state automata. We prove state upper and lower bounds for the conversion of 1-limited automata into finite state automata. In particular, we prove a double exponential state gap between nondeterministic 1-limited automata and one-way deterministic finite automata. The gap reduces to single exponential in the case of deterministic 1-limited automata. This also implies an exponential state gap between nondeterministic and deterministic 1-limited automata. Another consequence is that 1-limited automata can have less states than equivalent two-way nondeterministic finite automata. We show that this is true even if we restrict to the case of the one-letter input alphabet. For each d ≥ 2, d-limited automata are known to characterize the class of context-free languages. Using the Chomsky-Schutzenberger representation for context-free languages, we present a new conversion from context-free languages into 2-limited automata.

Fresh-Variable Automata: Application to Service Composition

Abstract: We introduce fresh-variable automata, a natural extension of finite-state automata over infinite alphabet. In this model the transitions are labeled with constants or variables that can be refreshed in some specified states. We prove several closure properties for this class of automata and study their decision problems. We show the applicability of our model in modeling Web services handling data from an infinite domain. We introduce a notion of simulation that enables us to reduce the Web service composition problem to the construction of a simulation of a target service by the asynchronous product of existing services, and prove that this construction is computable.

Routing alogrithm over semi-regular tessellations

Abstract: Path discovery or routing algorithms are challenging when the nodes are distributed over not on just regular grid like rectangular type but on semiregular grids Investigations in the study of finite state automata that move about in a two dimensional space are suitable to tackle this context The model proposed by H Muller [1] is used here to construct new automaton which can explore the path through obstacles over the grid This model is to be applied for routing phase for data transmission The earlier results were shown for static obstacles distributed over integer grid and the automaton in this case was constructed to interact on the rectangular grid location endowed with four neighborhood directional states In this paper we allow higher degree of neighborhood and mixing the types cells It has been verified that the finite automaton with number of printing (output) symbols determined by the maximum out degree of a cell in the underlying semi-regular grid can find the target

Reachability Problem for Weak Multi-Pushdown Automata

Abstract: This paper is about reachability analysis in a restricted subclass of multi-pushdown automata. We assume that the control states of an automaton are partially ordered, and all transitions of an automaton go downwards with respect to the order. We prove decidability of the reachability problem, and computability of the backward reachability set. As the main contribution, we identify relevant subclasses where the reachability problem becomes NP-complete. This matches the complexity of the same problem for communication-free vector addition systems, a special case of stateless multi-pushdown automata.

Size of Unary One-Way Multi-head Finite Automata

Abstract: We investigate the descriptional complexity of deterministic one-way multi-head finite automata accepting unary languages It is known that in this case the languages accepted are regular Thus, we study the increase of the number of states when an n-state k-head finite automaton is simulated by a classical (one-head) deterministic or nondeterministic finite automaton In the former case an upper bound of O(n·F(t·n) k − 1) and a lower bound of n·F(n) k − 1 states is shown, where t is a constant and F denotes Landau’s function Since both bounds are of order \(e^{\Theta(\sqrt{n \cdot \ln(n)})}\), the trade-off for the simulation is tight in the order of magnitude For the latter case we obtain an upper bound of O(n 2k ) and a lower bound of Ω(n k ) states We investigate also the costs for the conversion of one-head nondeterministic finite automata to deterministic k-head finite automata, that is, we trade nondeterminism for heads Finally, as an application of the simulation results, we show that decidability problems for unary deterministic k-head finite automata such as emptiness or equivalence are LOGSPACE-complete

An Introduction to Timed Automata

Abstract: We introduce timed automata and show how they can be used for the specification of timed systems. We also present some syntactical extensions useful for modeling and in particular networks of timed automata. Then we present two techniques for the analysis of timed automata: the zone graph and the region graph. The former can be more efficient for reachability analysis and related problems but it is not always finite. In contrast, the latter is a finite graph and can be used for model-checking. We compare the properties fulfilled by finite automata and timed automata emphasizing their main differences: languages of timed automata are not closed under complementation and the inclusion of languages is undecidable for timed automata.