On first-order definable operations on relational structures

2026-05-29

AI Summary

This work investigates the impact of unary and binary structure operations—such as disjoint union and Cartesian product—that are definable in first-order logic on the recognizability of classes of finite structures. By leveraging a backward translation theorem and a splitting theorem, it reduces first-order properties of output structures to finitely many first-order properties of input structures, preserving quantifier depth in the quantifier-free case and extending to logical fragments enriched with modulo-counting existential quantifiers. Combining first-order transductions with tree automata techniques over structures of bounded treewidth or cliquewidth, the paper establishes the recognizability of such finite structure classes under these operations and provides effective automata-based decision procedures, thereby forging a novel connection between structural recognizability and automata theory.

Abstract

We survey the definitions and main properties of first-order (FO) definable unary operations on relational structures, called FO-transductions, and of FO-definable binary operations based on disjoint union and Cartesian product. We focus our study on Backwards Translation Theorems and Splitting Theorems that permit to express FO properties of output structures in terms of finitely many FO properties of the corresponding input ones. In the particular cases where the operations are defined by quantifier-free (QF) formulas, the quantifier-heights of the obtained sentences are no larger than those of the input ones. It follows that the class of finite models of a FO sentence is recognizable with respect to the considered QF operations. Recognizability has interesting algorithmic properties based on finite automata on terms, for structures having bounded tree-width or clique-width. We extend our results to FO sentences constructed with modulo counting existential quantifiers.

Problem

• first-order definable operations
• relational structures
• Backwards Translation Theorems
• Splitting Theorems
• quantifier-free formulas

Innovation

• FO-transductions
• Backwards Translation Theorem
• Splitting Theorem
• quantifier-free definability
• recognizability