Undecidability Results for Finite Interactive Systems

TL;DR

This paper proves several undecidability results for finite interactive systems (FISs), showing limits of algorithmic analysis, and establishes their equivalence with tile systems, thus characterizing their expressive power.

Contribution

It provides the first proofs of undecidability for key problems in FISs and clarifies their relation to tile systems, advancing theoretical understanding.

Findings

Undecidability of the emptiness problem for FISs

Undecidability of transition accessibility in FISs

Equivalence of FISs and tile systems

Abstract

A new approach to the design of massively parallel and interactive programming languages has been recently proposed using rv-systems (interactive systems with registers and voices) and Agapia programming. In this paper we present a few theoretical results on FISs (finite interactive systems), the underlying mechanism used for specifying control and interaction in these systems. First, we give a proof for the undecidability of the emptiness problem for FISs, by reduction to the Post Correspondence Problem. Next, we use the construction in this proof to get other undecidability results, e.g., for the accessibility of a transition in a FIS, or for the finiteness of the language recognized by a FIS. Finally, we present a simple proof of the equivalence between FISs and tile systems, making explicit that they precisely capture recognizable two-dimensional languages.