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Descriptional Complexity of Finite Automata -- Selected Highlights

Arto Salomaa, Kai Salomaa, Taylor J. Smith

Abstract

The state complexity, respectively, nondeterministic state complexity of a regular language $L$ is the number of states of the minimal deterministic, respectively, of a minimal nondeterministic finite automaton for $L$. Some of the most studied state complexity questions deal with size comparisons of nondeterministic finite automata of differing degree of ambiguity. More generally, if for a regular language we compare the size of description by a finite automaton and by a more powerful language definition mechanism, such as a context-free grammar, we encounter non-recursive trade-offs. Operational state complexity studies the state complexity of the language resulting from a regularity preserving operation as a function of the complexity of the argument languages. Determining the state complexity of combined operations is generally challenging and for general combinations of operations that include intersection and marked concatenation it is uncomputable.

Descriptional Complexity of Finite Automata -- Selected Highlights

Abstract

The state complexity, respectively, nondeterministic state complexity of a regular language is the number of states of the minimal deterministic, respectively, of a minimal nondeterministic finite automaton for . Some of the most studied state complexity questions deal with size comparisons of nondeterministic finite automata of differing degree of ambiguity. More generally, if for a regular language we compare the size of description by a finite automaton and by a more powerful language definition mechanism, such as a context-free grammar, we encounter non-recursive trade-offs. Operational state complexity studies the state complexity of the language resulting from a regularity preserving operation as a function of the complexity of the argument languages. Determining the state complexity of combined operations is generally challenging and for general combinations of operations that include intersection and marked concatenation it is uncomputable.
Paper Structure (4 sections, 4 theorems, 1 equation)

This paper contains 4 sections, 4 theorems, 1 equation.

Table of Contents

  1. Introduction
  2. Non-recursive trade-offs
  3. Ambiguity of NFAs and state complexity
  4. Operational state complexity

Key Result

Theorem 2.1

Let $S_1$ and $S_2$ be descriptional systems for recursive languages. The trade-off between $S_1$ and $S_2$ is non-recursive if the following conditions hold. There exists a descriptional system $S_3$ and a property $P$ that is not semi-decidable for languages with a representation in $S_3$ such tha

Theorems & Definitions (5)

  • Theorem 2.1: Hartmanis1983Kutrib2005
  • Theorem 3.1: Leung1998
  • Theorem 3.2: Hromkovic2011
  • Definition 4.1
  • Theorem 4.2: Salomaa2013