Nondeterministic Finite Automata Research Articles

Once more about the state-minimization of the nondeterministic finite automata

In this paper we consider the corrected version of the algorithm of the state-minimization for the nondeterministic finite automata: we correct here a mistake of the previous paper, where the same problem was considered.

Computingϵ-Free NFA from Regular Expressions inO(nlog2(n)) Time

The standard procedure to transform a regular expression of size n to an ϵ -free nondeterministic finite automaton yields automata with O(n) states and O ( n 2 ) transitions. For a long time this was supposed to be also the lower bound, but a result by Hromkovic et al. showed how to build an ϵ -free NFA with only O ( n log 2 ( n )) transitions. The current lower bound on the number of transitions is Ω( n log( n )). A rough running time estimation for the common follow sets (CFS) construction proposed by Hromkovic et al. yields a cubic algorithm. In this paper we present a sequential algorithm for the CFS construction which works in time O ( n log( n ) + size of the output). On a CREW PRAM the CFS construction can be performed in time O (log( n )) using O ( n + (size of the output)/log( n )) processors. We also present a simpler proof of the lower bound on the number of transitions.

A NOTE ON THREE-WAY TWO-DIMENSIONAL PROBABILISTIC TURING MACHINES

This paper introduces a three-way two-dimensional probabilistic Turing machine (tr2-ptm), and investigates several properties of the machine. The tr2-ptm is a two-dimensional probabilistic Turing machine (2-ptm) whose input head can only move left, right, or down, but not up. Let 2-ptms (resp. tr2-ptms) denote a 2-ptm (resp. tr2-ptm) whose input tape is restricted to square ones, and let 2-PTMs(S(n)) (resp. TR2-PTMs(S(n))) denote the class of sets recognized by S(n) space-bounded 2-ptms's (resp. tr2-ptms's) with error probability less than ½, where S(n): N→N is a function of one variable n (= the side-length of input tapes). Let TR2-PTM(L(m,n)) denote the class of sets recognized by L(m,n) space-bounded tr2-ptm's with error probability less than ½, where L(m,n): N2→N is a function of two variables m (= the number of rows of input tapes) and n (= the number of columns of input tapes). The main results of this paper are: (1) 2-NFAs - TR2-PTMs(S(n))≠ϕ for any S(n)=o(log n), where 2-NFAs denotes the class of sets of square tapes accepted by two-dimensional nondeterministic finite automata, (2) TR2-PTMsS(n)[Formula: see text]2-PTMs(S(n)) for any S(n)=o(log n), and (3) for any function g(n)=o(log n) (resp. g(n)=o(log n/log log n)) and any monotonic nondecreasing function f(m) which can be constructed by some one-dimensional deterministic Turing machine, TR2-PTM(f(m)+g(n)) (resp. TR2-PTM(f(m)×g(n))) is not closed under column catenation, column closure, and projection. Additionally, we show that two-dimensional nondeterministic finite automata are equivalent to two-dimensional probabilistic finite automata with one-sided error in accepting power.

Tight bounds on the number of states of DFAs that are equivalent to [formula omitted]-state NFAs

It is shown that if α is an integer which can be expressed as 2 k or 2 k+1 for some integer 0⩽k⩽n/2−2, then there exist nondeterministic finite automata with n states whose equivalent deterministic finite automata need exactly 2 n−α states.

Increasing visualization and interaction in the automata theory course

In this paper we describe how to increase the visualization and interaction in the automata theory course through the use of the tools JFLAP and Pâté. We also describe new features in these tools that allow additional visualization and interaction. New features in JFLAP include the addition of regular expressions and exploring their conversion from and to nondeterministic finite automata (NFA), and increasing the interaction in the conversion of automata to grammars. New features in Pâté include the display of a parse tree while parsing unrestricted grammars, and improved interaction with parsing and the transformation of grammars.

New upper bounds to the limitedness of distance automata

A distance automaton is a finite nondeterministic automaton with a distance function which assigns zero or one to each atomic transition and assigns a nonnegative integer to each accepted word by the plus-min principle. In this paper, we prove that the distances of all accepted words of a distance automaton is bounded by some constant if and only if they are bounded by 2 4m 3+m log(m+2)+m , where m is the number of states of the automaton.

Characterization of Glushkov automata

Glushkov's algorithm computes a nondeterministic finite automaton without ε-transitions and with n+1 states from a regular expression having n occurrences of letters. The aim of this paper is to give a set of necessary and sufficient conditions characterizing this automaton. Our characterization theorem is formulated in terms of directed graphs. Moreover these conditions allow us to produce an algorithm of conversion of a Glushkov automaton into a regular expression of small size.

Automata of asynchronous behaviors

We survey three applications that use finite automata to specify behaviors of concurrent processes in general, and asynchronous circuits in particular. The applications are: verification of concurrent processes, liveness properties, and delay insensitivity of asynchronous networks. In all three cases, we start with a common model of a nondeterministic finite automaton, and then add certain application-specific features. Typically, the added features involve separating the alphabet or the state set of the automaton into several disjoint subsets. For each application we provide the motivation, describe the type of automaton used, define the most important operations, and state some of the key results. For process verification, we describe a BDD-based tool that implements the respective automata and operations.

A new algorithm of the state-minimization for the nondeterministic finite automata

The problem of the state-minimization for the nondeterministic finite Rabin-Scott’s automata is considered. A new algorithm for this problem is obtained. The obtained algorithm has the exponential effectiveness, like the earlier-known algorithms for this problem. But each of previous algorithms amounts to the search of minimum generative system for local reaction of equal automaton of canonical form, and unlike them, we use in this paper two special functions, marking states of the given automaton.

Faster Approximate String Matching

We present a new algorithm for on-line approximate string matching. The algorithm is based on the simulation of a nondeterministic finite automaton built from the pattern and using the text as input. This simulation uses bit operations on a RAM machine with word length w = Ω (log n) bits, where n is the text size. This is essentially similar to the model used in Wu and Manber's work, although we improve the search time by packing the automaton states differently. The running time achieved is O(n) for small patterns (i.e., whenever mk = O(log n)) , where m is the pattern length and k<m is the number of allowed errors. This is in contrast with the result of Wu and Manber, which is O(kn) for m=O(log n) . Longer patterns can be processed by partitioning the automaton into many machine words, at O(mk/w n) search cost. We allow generalizations in the pattern, such as classes of characters, gaps, and others, at essentially the same search cost. We then explore other novel techniques to cope with longer patterns. We show how to partition the pattern into short subpatterns which can be searched with less errors using the simple automaton, to obtain an average cost close to $ O(\sqrt{mk/w} n) $ . Moreover, we allow the superimposition of many subpatterns in a single automaton, obtaining near $ O(\sqrt{mk/(\sigma w)} n) $ average complexity (σ is the alphabet size). We perform a complete analysis of all the techniques and show how to combine them in an optimal form, also obtaining new tighter bounds for the probability of an approximate occurrence in random text. Finally, we show experimental results comparing our algorithms against previous work. These experiments show that our algorithms are among the fastest for typical text searching, being the fastest in some cases. Although we aim mainly at text searching, we believe that our ideas can be successfully applied to other areas such as computational biology.

NondeterministicNC1Computation

We define the counting classes #NC1,GapNC1,PNC1, andC=NC1. We prove that boolean circuits, algebraic circuits, programs over nondeterministic finite automata, and programs over constant integer matrices yield equivalent definitions of the latter three classes. We investigate closure properties. We observe that #NC1⊆#L, thatPNC1⊆L, and thatC=NC1⊆L. Then we exploit our finite automaton model and extend the padding techniques used to investigate leaf languages. Finally, we draw some consequences from the resulting body of leaf language characterizations of complexity classes, including the unconditional separations ofACC0fromMOD-PHand that ofTC0from the counting hierarchy. Moreover, we obtain that if dlogtime-uniformity and logspace-uniformity forAC0coincide then the polynomial time hierarchy equalsPSPACE.

On finite automata with limited nondeterminism

We develop a new algorithm for determining if a given nondeterministic finite automaton is limited in nondeterminism. From this, we show that the number of nondeterministic moves of a finite automaton, if limited, is bounded by \(2^{n} - 2\) where \(n\) is the number of states. If the finite automaton is over a one-letter alphabet, using Gohon's result the number of nondeterministic moves, if limited, is less than \(n^{2}\). In both cases, we present families of finite automata demonstrating that the upper bounds obtained are almost tight. We also show that the limitedness problem of the number of nondeterministic moves of finite automata is PSPACE-hard. Since the problem is already known to be in PSPACE, it is therefore PSPACE-complete.

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.

A lower bound technique for the size of nondeterministic finite automata

In this note, we prove a simple theorem that provides a lower bound on the size of nondeterministic finite automata which accept a given regular language.

Learning two-tape automata from queries and counterexamples

We investigate the learning problem of two-tape deterministic finite automata (2-tape DFAs) from queries and counterexamples. Instead of accepting a subset of ∑*, a 2-tape DFA over an alphabet ∑ accepts a subset of ∑* × ∑*, and therefore, it can specify a binary relation on ∑*. In [3] Angluin showed that the class of deterministic finite automata (DFAs) is learnable in polynomial time from membership queries and equivalence queries, namely, from a minimally adequate teacher (MAT). In this article we show that the class of 2-tape DFAs is learnable in polynomial time from MAT. More specifically, we show an algorithm that, given any languageL accepted by an unknown 2-tape DFAM, learns from MAT a two-tape nonde-terministic finite automaton (2-tape NFA)M′ acceptingL in time polynomial inn andl, wheren is the size ofM andl is the maximum length of any counterexample provided during the learning process.

On path equivalence of nondeterministic finite automata

Two nondeterministic finite automata (NFAs) are said to be path equivalent if each string is accepted by the two automata via the same number of computation paths. In this paper we show the following. 1. (1) The path equivalence problem for NFAs without λ-cycles is solvable not only in polynomial sequential time but also in O( log 2( n)) parallel time using a polynomial number of processors, where n is the total number of states in two NFAs. 2. (2) The path equivalence problem for NFAs with λ-cycles is PSPACE-complete, hence decidable.

Exponential upper and lower bounds for the order of a regular language

The order of the language recognized by a nondeterministic finite automaton with n states is either infinite or at most 3 2 n ( n+1)−1. For each even n there is a deterministic finite automaton having n states and recognizing a language of order 2 (n−2) 2 −1 .

Parallel finite automata for modeling concurrent software systems

The class of parallel finite automata (PFAs) is described that naturally expresses the interleaving parallelism inherent in Petri net notation without admitting the possibility of an infinite state space. The equivalence of this class to deterministic finite automata (DFAs) is demonstrated via an algorithm for generating an equivalent nondeterministic finite automaton (NFA) from a PFA. A composition rule is given for constructing a PFA from a regular expression with the interleaving operator. Finally, the languages generated by this class are related to known classes of Petri net languages. Though class PFA is equivalent in recognition power to class DFA, the fact that DFAs are a structural subset of PFAs makes the PFA representation preferable for many applications requiring finite automata models. As an example, we discuss the usefulness of PFA notation as a structural model for hypertex and nonlinear interactive information networks in general.

A CHARACTERIZATION OF RECOGNIZABLE PICTURE LANGUAGES

This paper first shows that REC, the family of recognizable picture languages in Giammarresi and Restivo,3 is equal to the family of picture languages accepted by two-dimensional on-line tessellation acceptors in Inoue and Nakamura.5 By using this result, we then solve open problems in Giammarresi and Restivo,3 and show that (i) REC is not closed under complementation, and (ii) REC properly contains the family of picture languages accepted by two-dimensional nondeterministic finite automata even over a one letter alphabet.

Equations and regular-like expressions for afa

lternating finite automata are a generalization of non-deterministic finite automata and a mechanism that models parallel computation. It has been shown in [6] that alternating finite automata (AFA) have several theoretical properties and practical features. In this paper, we further study alternating finite automata. A new type of system of equations is introduced. The systems of equations to be considered involve Boolean expressions over a finite set X and the symbols of an alphabet A. Each AFA can be described as a system of equations that has a unique fixpoint, and whose solutions are precisely the regular languages. We present an algebraic interpretation of AFA, which parallels that of regular expressions and of linear equations. We also explore direct ways of solving such systems and generating equivalent AFA from regular expressions or even extended regular expressions.