# On the decidability of $k$-Block determinism

## TL;DR

This paper investigates the decidability of $k$-block determinism in regular languages, extending previous concepts of one-unambiguity, and provides a procedure to determine whether a language is $k$-block deterministic.

## Contribution

It proves that any $k$-block deterministic language can be recognized by a compact deterministic $k$-block automaton passing the BW-test and introduces a decidable enumeration procedure.

## Key findings

- Any $k$-block deterministic language has a compact recognizing automaton.
- A procedure to enumerate all compact deterministic $k$-block automata for a language.
- Decidability of testing whether a language is $k$-block deterministic.

## Abstract

Br\"uggemann-Klein and Wood define a one-unambiguous regular language as a language that can be recognized by a deterministic Glushkov automaton. They give a procedure performed on the minimal DFA, the BW-test, to decide whether a language is one-unambiguous. Block determinism is an extension of one-unambiguity while considering non-empty words as symbols and prefix-freeness as determinism. A block automaton is compact if it does not have two equivalent states (same right language). We showed that a language is $k$-block deterministic if it is recognized by some deterministic $k$-block automaton passing the BW-test. In this paper, we show that any $k$-block deterministic language is recognized by a compact deterministic $k$-block automaton passing the BW-test. We also give a procedure which enumerates, for a given language, the finite set of compact deterministic $k$-block automata. It gives us a decidable procedure to test whether a language is $k$-block deterministic.

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Source: https://tomesphere.com/paper/1705.10625