Minimizing nfa's and regular expressions

AbstractWe show inapproximability results concerning minimization of nondeterministic finite automata (nfa's) as well as of regular expressions relative to given nfa's, regular expressions or deterministic finite automata (dfa's).We show that it is impossible to efficiently minimize a given nfa or regular expression with n states, transitions, respectively symbols within the factor o(n), unless P=PSPACE. For the unary case, we show that for any δ>0 it is impossible to efficiently construct an approximately minimal nfa or regular expression within the factor n1−δ, unless P=NP.Our inapproximability results for a given dfa with n states are based on cryptographic assumptions and we show that any efficient algorithm will have an approximation factor of at least npoly(logn). Our setup also allows us to analyze the minimum consistent dfa problem

The complexity of two finite-state models, optimizing transducers and counting automata

An optimizing finite-state transducer is a nondeterministic finite-state transducer in which states are either maximizing or minimizing. In a given state, the optimal output is the maximum or minimum--over all possible transitions--of the transition output concatenated with the optimal output of the resulting state. The ranges of optimizing finite-state transducers form a class in NL which includes a hierarchy based on the number of alternations of maximizing and minimizing states in a computation. The inequivalence problem--whether or not two transducers compute different functions, and the range inequivalence problem are shown to be undecidable. Some other problems associated with this model are shown to be complete for NL and NP;A counting finite-state automaton is a nondeterministic finite-state automaton which, on an input over its input alphabet, (magically) writes in binary the number of accepting computations on the input. We examine the complexity of computing the counting function of an NFA, and the complexity of recognizing its range as a set of binary strings. We also consider the pumping behavior of counting finite-state automata. The class of functions computed by counting NFAs (1) includes a class of functions computed by deterministic finite-state transducers; (2) is contained in the class of functions computed by polynomially time- and linearly space-bounded Turing transducers; (3) includes a function whose range is the composite numbers

Reduction of the Nondeterministic Finite Automata

Nedeterministický konečný automat je důležitým nástrojem, který se používá pro zpracování řetězců v mnoha různých oblastech programování. V rámci zvýšení efektivity programů je důležité snažit se o zmenšování jeho velikosti. Tento problém je však velmi výpočetně náročný, proto je potřeba hledat nové postupy. V této práci jsou uvedeny základy konečných automatů a poté jsou představeny různé metody zabývající se jejich redukcí. Použitelné redukční algoritmy jsou v práci podrobněji popsány, dále implementovány a otestovány. Nakonec jsou výsledky zhodnoceny.Nondeterministic finite automaton is an important tool, which is used to process strings in many different areas of programming. It is important to try to reduce its size for increasing programs' effectiveness. However, this problem is computationally hard, so we need to search for new techniques. Basics of finite automata are described in this work. Some methods for their reduction are then introduced. Usable reduction algorithms are described in greater detail. Then they are implemented and tested. The test results are finally evaluated.