Hereditary undecidability of fragments of some elementary theories

TL;DR

This paper demonstrates hereditary undecidability results for certain fragments of elementary theories by constructing interpretations between classes of finite structures, showing how undecidability propagates through these interpretations.

Contribution

It constructs specific interpretations between classes of finite structures to establish hereditary undecidability of certain theory fragments, revealing optimal decidability boundaries.

Findings

Hereditary undecidability of the $ ext{Σ}_2$-theory of pairs of equivalence relations

Hereditary undecidability of the $ ext{Σ}_2$-theory of pairs of a linear order and an equivalence relation

Decidability of the $ ext{Π}_2$-theories of the classes considered

Abstract

It is well known that whenever a class of structures $\mathcal{K}_1$ is interpretable in a class of structures $\mathcal{K}_2$, then the hereditary undecidability of (a fragment of) the theory of $\mathcal{K}_1$ implies the hereditary undecidability of (a suitable fragment of) the theory of $\mathcal{K}_2$. In the present paper, we construct a $\Sigma_1$-interpretation of the class of all finite bipartite graphs in the class of all pairs of equivalence relations on the same finite domain; from this we obtain the hereditary undecidability of the $\Sigma_2$-theory of the second class. Next, we construct a $\Sigma_1$-interpretation of the class of all pairs of equivalence relations on the same finite domain in the class of all pairs consisting of a linear ordering and an equivalence relation on the same finite domain; this gives us the hereditary undecidability of the $\Sigma_2$-theory of the second class. The corresponding results are, in a sense, optimal, since the $\Pi_2$-theories of the classes under consideration are decidable. Keywords: undecidability, elementary theories, prefix fragments