Decidability of definability

TL;DR

This paper proves the decidability of the definability problem for relations over certain infinite structures, using advanced algebraic, model theoretic, and combinatorial techniques, including Ramsey theory and topological dynamics.

Contribution

It establishes the decidability of primitive positive and related definability problems for structures definable in ordered homogeneous Ramsey classes, expanding the scope of computability results in logic.

Findings

Decidability of primitive positive definability for structures with ordered homogeneous Ramsey properties.

Decidability results extend to existential positive and existential definability.

Application to structures like the rationals order, random graph, and homogeneous posets.

Abstract

For a fixed countably infinite structure \Gamma\ with finite relational signature \tau, we study the following computational problem: input are quantifier-free \tau-formulas \phi_0,\phi_1,...,\phi_n that define relations R_0,R_1,...,R_n over \Gamma. The question is whether the relation R_0 is primitive positive definable from R_1,...,R_n, i.e., definable by a first-order formula that uses only relation symbols for R_1,..., R_n, equality, conjunctions, and existential quantification (disjunction, negation, and universal quantification are forbidden). We show decidability of this problem for all structures \Gamma\ that have a first-order definition in an ordered homogeneous structure \Delta\ with a finite relational signature whose age is a Ramsey class and determined by finitely many forbidden substructures. Examples of structures \Gamma\ with this property are the order of the rationals, the random graph, the homogeneous universal poset, the random tournament, all homogeneous universal C-relations, and many more. We also obtain decidability of the problem when we replace primitive positive definability by existential positive, or existential definability. Our proof makes use of universal algebraic and model theoretic concepts, Ramsey theory, and a recent characterization of Ramsey classes in topological dynamics.