Incomplete Transition Complexity of Basic Operations on Finite Languages

TL;DR

This paper investigates the incomplete state and transition complexity of basic operations on finite languages, providing tight bounds and correcting previous results for automata with incomplete transitions.

Contribution

It introduces tight upper bounds for incomplete automata on finite languages for key operations and corrects prior results on concatenation complexity.

Findings

Provided tight upper bounds for boolean operations, concatenation, star, and reversal.

Corrected the existing state complexity of concatenation for complete DFAs.

Proved tightness of bounds using family languages with variable alphabet sizes.

Abstract

The state complexity of basic operations on finite languages (considering complete DFAs) has been in studied the literature. In this paper we study the incomplete (deterministic) state and transition complexity on finite languages of boolean operations, concatenation, star, and reversal. For all operations we give tight upper bounds for both description measures. We correct the published state complexity of concatenation for complete DFAs and provide a tight upper bound for the case when the right automaton is larger than the left one. For all binary operations the tightness is proved using family languages with a variable alphabet size. In general the operational complexities depend not only on the complexities of the operands but also on other refined measures.