A Generic Solution to Register-bounded Synthesis with an Application to Discrete Orders

TL;DR

This paper introduces a generic approach for register-bounded synthesis in reactive systems with ordered data domains, extending decidability results to natural numbers with linear order and other data structures.

Contribution

It defines a regular approximability condition on data domains, enabling decidability of synthesis for ordered data beyond equality, and introduces a generic reducibility framework.

Findings

Decidability of register-bounded synthesis for $(N,<)$ and related domains.

A generic, language-theoretic method avoiding complex game-theoretic proofs.

Applicability to tuples of numbers and string prefix orders.

Abstract

We study synthesis of reactive systems interacting with environments using an infinite data domain. A popular formalism for specifying and modelling such systems is register automata and transducers. They extend finite-state automata by adding registers to store data values and to compare the incoming data values against stored ones. Synthesis from nondeterministic or universal register automata is undecidable in general. However, its register-bounded variant, where additionally a bound on the number of registers in a sought transducer is given, is known to be decidable for universal register automata which can compare data for equality, i.e., for data domain $(N,=)$. This paper extends the decidability border to the domain $(N,<)$ of natural numbers with linear order. Our solution is generic: we define a sufficient condition on data domains (regular approximability) for decidability of register-bounded synthesis. The condition is satisfied by natural data domains like $(N,<)$. It allows one to use simple language-theoretic arguments and avoid technical game-theoretic reasoning. Further, by defining a generic notion of reducibility between data domains, we show the decidability of synthesis in the domain $(N^d,<^d)$ of tuples of numbers equipped with the component-wise partial order and in the domain $(\Sigma^*,\prec)$ of finite strings with the prefix relation.