Multi-Sequential Word Relations

TL;DR

This paper extends the concept of sequential functions to multi-sequential relations, providing a polynomial-time decision procedure and construction method based on the weak twinning property for relations defined by non-deterministic finite transducers.

Contribution

It introduces the weak twinning property to decide multi-sequentiality of relations and generalizes the determinisation procedure for NFT to arbitrary relations.

Findings

Decidable in PTime whether an NFT relation is multi-sequential.

Provides an effective construction of input-deterministic transducers for multi-sequential relations.

Generalizes the determinisation procedure to arbitrary NFT relations.

Abstract

Rational relations are binary relations of finite words that are realised by non-deterministic finite state transducers (NFT). A particular kind of rational relations is the sequential functions. Sequential functions are the functions that can be realised by input-deterministic transducers. Some rational functions are not sequential. However, based on a property on transducers called the twinning property, it is decidable in PTime whether a rational function given by an NFT is sequential. In this paper, we investigate the generalisation of this result to multi-sequential relations, i.e. relations that are equal to a finite union of sequential functions. We show that given an NFT, it is decidable in PTime whether the relation it defines is multi-sequential, based on a property called the weak twinning property. If the weak twinning property is satisfied, we give a procedure that effectively constructs a finite set of input-deterministic transducers whose union defines the relation. This procedure generalises to arbitrary NFT the determinisation procedure of functional NFT.