Size Of Automaton Research Articles

Two-Way Non-Uniform Finite Automata

We consider two-tape automata where one tape contains the input word [Formula: see text] and the other an advice string [Formula: see text]. Such an automaton recognizes a language [Formula: see text] if there is an advice function for which every word on the input tape is correctly classified. This model has been introduced by Küçük et al. with the aim of modeling non-uniform computations on finite automata. So far, most results concerned automata with 1-way tapes, both for the input and the advice. First, we show that making just one of the tapes 2-way already increases the model’s power. Then we consider the case where both tapes are 2-way, which can also be viewed as a restricted version of the non-uniform families of automata used by Ibarra and Ravikumar to define the class NUDSPACE . We show this restriction to be not too significant since, e.g., [Formula: see text]DFA/poly), i.e., the languages recognized by automata with 2-way input and advice tape with polynomial advice, equals [Formula: see text]. Hence, we can show that many interesting problems concerning the state complexity of families of automata carry over to the problems concerning the advice size of non-uniform automata. In particular, the question whether there can be a more than polynomial gap in advice between determinism and nondeterminism is of great interest: The existence of a language that is recognized by a 2-way NFA with a constant number of advice heads and polynomial (resp. logarithmic) advice, while a corresponding 2-head DFA needs exponential (resp. polynomial) advice, would imply [Formula: see text] (resp. [Formula: see text]). We show that for advice of size [Formula: see text] there is no gap between determinism and nondeterminism, and that the gap is at most exponential.

On the Exploration of the Natural Sequence of Primes With Cellular Automata Targeting Enhanced Data Security and Privacy

Enhanced data security and privacy are one of the major concerns in today’s digital society. The role of Primes towards the enhancements of data security and privacy is undeniable. Though several prime generations were presented, yet a cost effective and an easy to implement generation of Prime sequence should always have an advantage targeting real life applications. Hence, prime sequence generation using Cellular Automata (CA) is presented in this article as CA based modelling are easy to implement at the cost of flip-flops. The main contribution of this research is to explore the natural sequence of primes (i.e., primes A000040) with a special class of group CA, at fixed boundary environment; which may potentially be used as a Prime source towards the enhancements of data security and privacy. Experimental results confirm that the first 50 members of A000040 series may be explored at automata size 8 only. Detailed investigations towards the CA configuration and its dynamics in view of the generation of prime A000040 sequence, are also presented in this article.

On the Exploration of the Natural Sequence of Primes With Cellular Automata Targeting Enhanced Data Security and Privacy

Enhanced data security and privacy are one of the major concerns in today’s digital society. The role of Primes towards the enhancements of data security and privacy is undeniable. Though several prime generations were presented, yet a cost effective and an easy to implement generation of Prime sequence should always have an advantage targeting real life applications. Hence, prime sequence generation using Cellular Automata (CA) is presented in this article as CA based modelling are easy to implement at the cost of flip-flops. The main contribution of this research is to explore the natural sequence of primes (i.e., primes A000040) with a special class of group CA, at fixed boundary environment; which may potentially be used as a Prime source towards the enhancements of data security and privacy. Experimental results confirm that the first 50 members of A000040 series may be explored at automata size 8 only. Detailed investigations towards the CA configuration and its dynamics in view of the generation of prime A000040 sequence, are also presented in this article.

Roots and Powers in Regular Languages: Recognizing Nonregular Properties by Finite Automata

It is well known that the set of powers of any given order, for example squares, in a regular language need not be regular. Nevertheless, finite automata can identify them via their roots. More precisely, we recall that, given a regular language L, the set of square roots of L is regular. The same holds true for the nth roots for any n and for the set of all nontrivial roots; we give a concrete construction for all of them. Using the above result, we obtain decision algorithms for many natural problems on powers. For example, it is decidable, given two regular languages, whether they contain the same number of squares at each length. Finally, we give an exponential lower bound on the size of automata identifying powers in regular languages. Moreover, we highlight interesting behavior differences between taking fractional powers of regular languages and taking prefixes of a fractional length. Indeed, fractional roots in a regular language can typically not be identified by finite automata.

Synchronizing heuristics: Speeding up the fastest

Testing is the most expensive and time consuming phase in the development of complex systems. Model–based testing is an approach that can be used to automate the generation of high quality test suites, which is the most challenging part of testing. Formal models, such as finite state machines or automata, have been used as specifications from which the test suites can be automatically generated. The tests are applied after the system is synchronized to a particular state, which can be accomplished by using a synchronizing word. Computing a shortest synchronizing word is of interest for practical purposes, e.g. for a shorter testing time. However, computing a shortest synchronizing word is an NP–hard problem. Therefore, heuristics are used to compute short synchronizing words. Greedy and Cycle are the fastest synchronizing heuristics currently known. In this paper, without sacrificing the quality of the synchronizing words, we improve the time performance of Greedy and Cycle, by restructuring the algorithms and applying some optimizations. Our experimental results show that depending on the automata size, Cycle and Greedy can be made 25 × and 95 × faster, respectively. The suggested improvements become more effective as the size of the automaton increases.

Общий подход к решению задач на графах коллективом автоматов

We propose a general method to solve graph problems by a set of automata (computational agents) located in vertices of undirected ordered connected rooted graph and communicating by passing messages along graph edges. The automata are semi-robots, i.e., their internal memory size is sufficient to store values depending on the number of vertices and edges of the graph (n and m, correspondingly) but is too small to store the complete graph description. Section 2 presents classification of graph-based distributed computational models depending on internal memory size of vertex automaton, vertex automaton operation time, and edge capacity (the number of messages that are passing along an edge simultaneously). We choose for further use the model of maximum parallelism, having zero automaton operation time and unbounded edge capacity. It allows to obtain lower complexity bounds on distributed graph problems. Section 3 describes algorithm complexity assessment rules. Section 4 presents basic message processing procedures and message classification according to paths passed by them and methods of message processing by vertex automaton. We propose to construct algorithms solving graph problems on the base of the procedures considered, and present some examples in further sections. Sections 5-9 describe distributed algorithms for spanning tree construction, for task termination detection (based on detection of absence of messages used by the task), for shortest paths tree construction, for graph vertices enumeration, for collecting graph structure information in the root automaton memory (if it is unbounded). The algorithms proposed has linear time complexity in n and m, and use linear in n and m number of messages. Section 10 considers construction of maximum weight path in a weighted graph, which is the NP problem. We propose the algorithm that for the sake of using unbounded number of messages can solve this problem in linear time in n and m. The conclusion summarizes the paper and draws directions of further research: constructing algorithms for other graph problems and generalization of the approach to directed, non-deterministic and dynamic graphs.

The number of DFAs for a given spanning tree

In the last few decades, several techniques to randomly generate a deterministic finite automaton have been developed. These techniques have implications in the enumeration and random generation of automata of size n. One of the ways to generate a finite automaton is to generate a random tree and to complete it to a deterministic finite automaton, assuming that the tree will be the automaton's breadth-first spanning tree. In this paper we explore further this method, and the string representation of a tree, and use it to counting the number of automata having a tree as a breadth-first spanning subtrees. We introduce the notions of characteristic and difference sequence of a tree, and define the weight of a tree. We also present a recursive formula for this quantity in terms of the "derivative" of a tree. Finally, we analyze the implications of this formula in terms of exploring trees with the largest and smallest number of automata in the span of the tree and present simple applications for finding densities of subsets of DFAs.

FROM REGULAR TO STRICTLY LOCALLY TESTABLE LANGUAGES

A classical result (often credited to Y. Medvedev) states that every language recognized by a finite automaton is the homomorphic image of a local language, over a much larger so-called local alphabet, namely the alphabet of the edges of the transition graph. Local languages are characterized by the value k = 2 of the sliding window width in the McNaughton and Papert's infinite hierarchy of strictly locally testable languages (k-slt). We generalize Medvedev's result in a new direction, studying the relationship between the width and the alphabetic ratio telling how much larger the local alphabet is. We prove that every regular language is the image of a k-slt language on an alphabet of doubled size, where the width logarithmically depends on the automaton size, and we exhibit regular languages for which any smaller alphabetic ratio is insufficient. More generally, we express the trade-off between alphabetic ratio and width as a mathematical relation derived from a careful encoding of the states. At last we mention directions for theoretical development and application.

From Regular to Strictly Locally Testable Languages

A classical result (often credited to Y. Medvedev) states that every language recognized by a finite automaton is the homomorphic image of a local language, over a much larger so-called local alphabet, namely the alphabet of the edges of the transition graph. Local languages are characterized by the value k=2 of the sliding window width in the McNaughton and Papert's infinite hierarchy of strictly locally testable languages (k-slt). We generalize Medvedev's result in a new direction, studying the relationship between the width and the alphabetic ratio telling how much larger the local alphabet is. We prove that every regular language is the image of a k-slt language on an alphabet of doubled size, where the width logarithmically depends on the automaton size, and we exhibit regular languages for which any smaller alphabetic ratio is insufficient. More generally, we express the trade-off between alphabetic ratio and width as a mathematical relation derived from a careful encoding of the states. At last we mention some directions for theoretical development and application.

DETERMINISTIC PUSHDOWN AUTOMATA AND UNARY LANGUAGES

The simulation of deterministic pushdown automata defined over a one-letter alphabet by finite state automata is investigated from a descriptional complexity point of view. We show that each unary deterministic pushdown automaton of size s can be simulated by a deterministic finite automaton with a number of states that is exponential in s. We prove that this simulation is tight. Furthermore, its cost cannot be reduced even if it is performed by a two-way nondeterministic automaton. We also prove that there are unary languages for which deterministic pushdown automata cannot be exponentially more succinct than finite automata. In order to state this result, we investigate the conversion of deterministic pushdown automata into context-free grammars. We prove that in the unary case the number of variables in the resulting grammar is strictly smaller than the number of variables needed in the case of nonunary alphabets.

Memory Efficient String Matching Algorithm for Network Intrusion Management System

As the core algorithm and the most time consuming part of almost every modern network intrusion management system (NIMS), string matching is essential for the inspection of network flows at the line speed. This paper presents a memory and time efficient string matching algorithm specifically designed for NIMS on commodity processors. Modifications of the Aho-Corasick (AC) algorithm based on the distribution characteristics of NIMS patterns drastically reduce the memory usage without sacrificing speed in software implementations. In tests on the Snort pattern set and traces that represent typical NIMS workloads, the Snort performance was enhanced 1.48%-20% compared to other well-known alternatives with an automaton size reduction of 4.86-6.11 compared to the standard AC implementation. The results show that special characteristics of the NIMS can be used into a very effective method to optimize the algorithm design.

A complete characterization of deterministic regular liveness properties

Many systems can be modeled formally by nondeterministic Büchi-automata. The complexity of model checking then essentially depends on deciding subset conditions on languages that are recognizable by these automata and that represent the system behavior and the desired properties of the system. The involved complementation process may lead to an exponential blow-up in the size of the automata. We investigate a rich subclass of properties, called deterministic regular liveness properties, for which complementation at most doubles the automaton size if the properties are represented by deterministic Büchi-automata. In this paper, we will present a decomposition theorem for this language class that entails a complete characterization of the deterministic regular liveness properties, and extend an existing incomplete result which then, too, characterizes the deterministic regular liveness properties completely.

EVENTUALLY NUMBER-CONSERVING CELLULAR AUTOMATA

Although it is undecidable whether a one-dimensional cellular automaton obeys a given conservation law over its limit set, it is however possible to obtain sufficient conditions to be satisfied by a one-dimensional cellular automaton to be eventually number-conserving. We present a preliminary study of two-input one-dimensional cellular automaton rules called eventually number-conserving cellular automaton rules whose limit sets, reached after a number of time steps of the order of the cellular automaton size, consist of states having a constant number of active sites. In particular, we show how to find rules having given limit sets satisfying a conservation rule. Viewed as models of systems of interacting particles, these rules obey a kind of Darwinian principle by either annihilating unnecessary particles or creating necessary ones.

The size of subsequence automaton

Given a set of strings, the subsequence automaton accepts all subsequences of these strings. We derive a lower bound for the maximum number of states of this automaton. We prove that the size of the subsequence automaton for a set of k strings of length n is Ω ( n k / ( k + 1 ) k k ! ) for any k ⩾ 1 . It solves an open problem because only the case k ⩽ 2 was shown before.

On the power of nondeterminism and Las Vegas randomization for two-dimensional finite automata

The goal of this work is to investigate the computational power of nondeterminism and Las Vegas randomization for two-dimensional finite automata. The following three results are the main contribution of this paper: (i) Las Vegas (three-way) two-dimensional finite automata are more powerful than (three-way) two-dimensional deterministic ones. (ii) Three-way two-dimensional nondeterministic finite automata are more powerful than three-way two-dimensional Las Vegas finite automata. (iii) There is a strong hierarchy based on the number of computations (as measure of the degree of nondeterminism) for three-way two-dimensional finite automata. These results contrast with the situation for one-way and two-way finite automata, where all these computation modes have the same acceptance power, and the differences may occur only in the sizes of automata. Results (i) and (ii) provide the first such simultaneous acceptance separation between nondeterminism, Las Vegas, and determinism for a computing model.

Efficient Symbolic Representations for Arithmetic Constraints in Verification

In this paper we discuss efficient symbolic representations for infinite-state systems specified using linear arithmetic constraints. We give algorithms for constructing finite automata which represent integer sets that satisfy linear constraints. These automata can represent either signed or unsigned integers and have a lower number of states compared to other similar approaches. We present efficient storage techniques for the transition function of the automata and extend the construction algorithms to formulas on both boolean and integer variables. We also derive conditions which guarantee that the pre-condition computations used in symbolic verification algorithms do not cause an exponential increase in the automata size. We experimentally compare different symbolic representations by using them to verify non-trivial concurrent systems. Experimental results show that the symbolic representations based on our construction algorithms outperform the polyhedral representation used in Omega Library, and the automata representation used in LASH.

Uniqueness and efficiency of Nash equilibrium in a family of randomly generated repeated games

This paper describes the results of an analysis of the Nash equilibrium in randomly generated repeated games. We study two families of games: symmetric bimatrix games G( A, B) with B = A ⊤ and nonsymmetric bimatrix games (the first includes the classical games of prisoner dilemma, battle of the sexes, and chickens). We use pure strategies, implemented by automata of size two, and different strategy domination criteria. We observe that, in this environment, the uniqueness and efficiency of equilibria outcomes is the typical result.

On the Power of Las Vegas for One-Way Communication Complexity, OBDDs, and Finite Automata

The study of the computational power of randomized computations is one of the central tasks of complexity theory. The main goal of this paper is the comparison of the power of Las Vegas computation and deterministic respectively nondeterministic computation. We investigate the power of Las Vegas computation for the complexity measures of one-way communication, ordered binary decision diagrams, and finite automata.(i) For the one-way communication complexity of two-party protocols we show that Las Vegas communication can save at most one half of the deterministic one-way communication complexity. We also present a language for which this gap is tight.(ii) The result (i) is applied to show an at most polynomial gap between determinism and Las Vegas for ordered binary decision diagrams.(iii) For the size (i.e., the number of states) of finite automata we show that the size of Las Vegas finite automata recognizing a language L is at least the square root of the size of the minimal deterministic finite automaton recognizing L. Using a specific language we verify the optimality of this lower bound.

Characteristic Sets for Polynomial Grammatical Inference

When concerned about efficient grammatical inference two issues are relevant: first one is to determine quality of result, and second is to try to use polynomial time and space. A typical idea to deal with first point is to say that an algorithm performs well if it infers {\it the limit} correct language. The second point has led to debate about how to define polynomial time: main definitions of polynomial inference have been proposed by Pitt and Angluin. We return in this paper to a definition proposed by Gold that requires a characteristic set of strings to exist for each grammar, and this set to be polynomial in size of grammar or automaton that is to be learned, where size of sample is sum of lengths of all strings it includes. The learning algorithm must also infer correctly as soon as characteristic set is included in data. We first show that this definition corresponds to a notion of teachability as defined by Goldman and Mathias. By adapting their teacher/learner model to grammatical inference we prove that languages given by context-free grammars, simple deterministic grammars, linear grammars and nondeterministic finite automata are not identifiable in limit from polynomial time and data.

Nonuniform complexity and the randomness of certain complete languages

This paper contains further study of the randomness properties of languages. The connection between time-/space-bounded Kolmogorov complexity and nonuniform complexity defined by grammar and automaton size is investigated. We show that certain languages that are complete under nonuniform one-way log-space reductions are weakly random with respect to other language classes contained in P. For example, it is shown that every context-free language that is complete under nonuniform one-way log-space reductions is weakly random with respect to the class of deterministic context-free languages, and every deterministic context-free language that is complete under nonuniform one-way log-space reductions is weakly random with respect to the class of linear context-free languages.