Algorithmic Problems for Computation Trees

TL;DR

This paper investigates the computational complexity and decidability of three problems related to computation trees: optimization, solvability, and satisfiability, analyzing their interdependencies and providing examples with varying decidability.

Contribution

It introduces a detailed study of the relationships between the decidability of optimization, solvability, and satisfiability problems for computation trees, including new examples.

Findings

Decidability of optimization depends on solvability and satisfiability.

Examples of both decidable and undecidable cases are provided.

The paper clarifies the interdependence of these problems' decidability.

Abstract

In this paper, we study three algorithmic problems involving computation trees: the optimization, solvability, and satisfiability problems. The solvability problem is concerned with recognizing computation trees that solve problems. The satisfiability problem is concerned with recognizing sentences that are true in at least one structure from a given set of structures. We study how the decidability of the optimization problem depends on the decidability of the solvability and satisfiability problems. We also consider various examples with both decidable and undecidable solvability and satisfiability problems.