On polynomial grammars extended with substitution

TL;DR

This paper explores the decidability of equivalence in extended register transducers with substitution, reducing the problem to polynomial grammar zeroness and identifying conditions for decidability and undecidability.

Contribution

It introduces a novel reduction of register transducer equivalence with substitution to polynomial grammar zeroness and delineates decidable and undecidable cases.

Findings

Decidability of equivalence under certain restrictions

Reduction to zeroness of polynomial grammars with substitution

Identification of a restricted model with undecidable equivalence

Abstract

We investigate decidability of equivalence of register transducers, also called copyful Streaming String Transducers in case of string input, extended with an operation of substituting a register for all occurrences of a given letter in another register. We reduce to zeroness of polynomial grammars (over ring of polynomials) extended with analogous substitution operation by encoding strings into polynomials; a similar method was used successfully by Seidl et al. in 2018. We give two restrictions under which register transducers with substitution have decidable equivalence. They seem to be very restrictive but on the other hand, they seem to be on the edge of the scope of this "polynomial" method, as in the third result we give a rather restricted model of polynomial grammars with substitution that has undecidable equivalence.