A New Technique for Reachability of States in Concatenation Automata

TL;DR

This paper introduces a novel technique to simplify proving state reachability in concatenation automata, aiding the analysis of their state complexity without relying on complex induction methods.

Contribution

The paper presents a new approach that captures the core of induction arguments, making reachability proofs in concatenation automata more straightforward and less reliant on induction.

Findings

Simplifies reachability proofs in concatenation automata

Provides a unified framework capturing induction-based arguments

Facilitates analysis of state complexity in automata concatenation

Abstract

We present a new technique for demonstrating the reachability of states in deterministic finite automata representing the concatenation of two languages. Such demonstrations are a necessary step in establishing the state complexity of the concatenation of two languages, and thus in establishing the state complexity of concatenation as an operation. Typically, ad-hoc induction arguments are used to show particular states are reachable in concatenation automata. We prove some results that seem to capture the essence of many of these induction arguments. Using these results, reachability proofs in concatenation automata can often be done more simply and without using induction directly.