Equivalent Minimal Deterministic Finite Automaton Research Articles

Descriptional Complexity of Non-Unary Self-Verifying Symmetric Difference Automata

Unary self-verifying symmetric difference automata have a known tight bound of [Formula: see text] for their state complexity. We now consider the non-unary case and show that, for every [Formula: see text], there is a regular language [Formula: see text] accepted by a non-unary self-verifying symmetric difference nondeterministic automaton with [Formula: see text] states, such that its equivalent minimal deterministic finite automaton has [Formula: see text] states. Furthermore, given any SV-XNFA with [Formula: see text] states, it is possible, up to isomorphism, to find at most another [Formula: see text] equivalent SV-XNFA. Finally, we show that for a certain set of non-unary SV-XNFA, [Formula: see text] is a tight bound on the state complexity.

Descriptional Complexity of Non-Unary Self-Verifying Symmetric Difference Automata

Previously, self-verifying symmetric difference automata were defined and a tight bound of 2^n-1-1 was shown for state complexity in the unary case. We now consider the non-unary case and show that, for every n at least 2, there is a regular language L_n accepted by a non-unary self-verifying symmetric difference nondeterministic automaton with n states, such that its equivalent minimal deterministic finite automaton has 2^n-1 states. Also, given any SV-XNFA with n states, it is possible, up to isomorphism, to find at most another |GL(n,Z_2)|-1 equivalent SV-XNFA.

THE MAGIC NUMBER PROBLEM FOR SUBREGULAR LANGUAGE FAMILIES

We investigate the magic number problem, that is, the question whether there exists a minimal n-state nondeterministic finite automaton (NFA) whose equivalent minimal deterministic finite automaton (DFA) has α states, for all n and α satisfying n ≤ α ≤ 2n. A number α not satisfying this condition is called a magic number (for n). It was shown that no magic numbers exist for general regular languages, whereas trivial and non-trivial magic numbers for unary regular languages were identified. We obtain similar results for automata accepting subregular languages like, for example, star-free languages, prefix-, suffix-, and infix-closed languages, and prefix-, suffix-, and infix-free languages, showing that there are only trivial magic numbers, when they exist. For finite languages we obtain some partial results showing that certain numbers are non-magic.

MAGIC NUMBERS AND TERNARY ALPHABET

A number α, in the range from n to 2n, is magic for n with respect to a given alphabet size s, if there is no minimal nondeterministic finite automaton of n states and s input symbols whose equivalent minimal deterministic finite automaton has α states. We show that in the case of a ternary alphabet, there are no magic numbers. For all n and α satisfying n ⩽ α ⩽ 2n, we define an n-state nondeterministic finite automaton with a three-letter input alphabet that requires exactly α deterministic states.

The Magic Number Problem for Subregular Language Families

We investigate the magic number problem, that is, the question whether there exists a minimal n-state nondeterministic finite automaton (NFA) whose equivalent minimal deterministic finite automaton (DFA) has alpha states, for all n and alpha satisfying n less or equal to alpha less or equal to exp(2,n). A number alpha not satisfying this condition is called a magic number (for n). It was shown in [11] that no magic numbers exist for general regular languages, while in [5] trivial and non-trivial magic numbers for unary regular languages were identified. We obtain similar results for automata accepting subregular languages like, for example, combinational languages, star-free, prefix-, suffix-, and infix-closed languages, and prefix-, suffix-, and infix-free languages, showing that there are only trivial magic numbers, when they exist. For finite languages we obtain some partial results showing that certain numbers are non-magic.

DETERMINISTIC BLOW-UPS OF MINIMAL NONDETERMINISTIC FINITE AUTOMATA OVER A FIXED ALPHABET

We show that for all integers n and α such that n ⩽ α ⩽ 2n, there exists a minimal nondeterministic finite automaton of n states with a four-letter input alphabet whose equivalent minimal deterministic finite automaton has exactly α states. It follows that in the case of a four-letter alphabet, there are no "magic numbers", i.e., the holes in the hierarchy. This improves a similar result obtained by Geffert for a growing alphabet of size n + 2.

Deterministic blow-ups of minimal NFA's

The paper treats the question whether there always exists a minimal nondeterministic finite automaton of n states whose equivalent minimal deterministic finite automaton has α states for any integers n and α with n ≤ α ≤ 2 n . Partial answers to this question were given by Iwama, Kambayashi, and Takaki (2000) and by Iwama, Matsuura, and Paterson (2003). In the present paper, the question is completely solved by presenting appropriate automata for all values of n and α . However, in order to give an explicit construction of the automata, we increase the input alphabet to exponential sizes. Then we prove that 2n letters would be sufficient but we describe the related automata only implicitly. In the last section, we investigate the above question for automata over binary and unary alphabets.

MAGIC NUMBERS FOR SYMMETRIC DIFFERENCE NFAS

Iwama et al. showed that there exists an n-state binary nondeterministic finite automaton such that its equivalent minimal deterministic finite automaton has exactly 2n - α states, for all n ≥ 7 and 5 ≤ α ≤ 2n-2, subject to certain coprimality conditions. We investigate the same question for both unary and binary symmetric difference nondeterministic finite automata. In the binary case, we show that for any n ≥ 4, there is an n-state symmetric difference nondeterministic finite automaton for which the equivalent minimal deterministic finite automaton has 2n - 1 + 2k - 1 - 1 states, for 2 < k ≤ n - 1. In the unary case, we consider a large practical subclass of unary symmetric difference nondeterministic finite automata: for all n ≥ 2, we argue that there are many values of α such that there is no n-state unary symmetric difference nondeterministic finite automaton with an equivalent minimal deterministic finite automaton with 2n - α states, where 0 < α < 2n - 1. For each n ≥ 2, we quantify such values of α precisely.

Generalized acceptance, succinctness and supernondeterministic finite automata

We define generalized acceptance in nondeterministic finite automata, in order to achieve more instances of succinct descriptions of regular languages in practical applications. We show that generalized acceptance enable the construction of a hierarchy of succinct nondeterministic descriptions for finite automata. The hierarchy corresponds to deterministic finite automata on level 0 and nondeterministic finite automata on level 1 by default, and we prove that the hierarchy corresponds to alternating (boolean) finite automata on level 2. We show that there exists an 2-state level 3 finite automaton with generalized acceptance, such that its equivalent minimal deterministic finite automaton has more that 2 2 2 states.