Convex Hull of Arithmetic Automata

TL;DR

This paper proves that the convex hull of arithmetic automata is rational polyhedral and provides an algorithm to compute its defining linear constraints, aiding in geometric analysis of complex solution sets.

Contribution

It establishes the rational polyhedral nature of convex hulls of arithmetic automata and introduces an algorithm to compute their linear constraints.

Findings

Convex hulls of arithmetic automata are rational polyhedral.

An algorithm for computing the linear constraints of these convex hulls is provided.

The results facilitate symbolic geometric analysis of solution sets.

Abstract

Arithmetic automata recognize infinite words of digits denoting decompositions of real and integer vectors. These automata are known expressive and efficient enough to represent the whole set of solutions of complex linear constraints combining both integral and real variables. In this paper, the closed convex hull of arithmetic automata is proved rational polyhedral. Moreover an algorithm computing the linear constraints defining these convex set is provided. Such an algorithm is useful for effectively extracting geometrical properties of the whole set of solutions of complex constraints symbolically represented by arithmetic automata.