Minimization and Synthesis of the Tail in Sequential Compositions of Mealy Machines

TL;DR

This paper presents new methods for minimizing and synthesizing the tail component in sequential Mealy machine compositions, improving efficiency and analyzing complexity bounds.

Contribution

It introduces an exponential improvement for tail minimization and characterizes the exponential complexity in tail synthesis for known head systems.

Findings

New minimization method with exponential improvement

Polynomial-time decision for tail existence

Exponential size bounds for tail synthesis in certain cases

Abstract

We consider a system consisting of a sequential composition of Mealy machines, called head and tail. We study two problems related to these systems. In the first problem, models of both head and tail components are available, and the aim is to obtain a replacement for the tail with the minimum number of states. We introduce a minimization method for this context which yields an exponential improvement over the state of the art. In the second problem, only the head is known, and a desired model for the whole system is given. The objective is to construct a tail that causes the system to behave according to the given model. We show that, while it is possible to decide in polynomial time whether such a tail exists, there are instances where its size is exponential in the sizes of the head and the desired system. This shows that the complexity of the synthesis procedure is at least exponential, matching the upper bound in complexity provided by the existing methods for solving unknown component equations.