Quantifiers closed under partial polymorphisms

TL;DR

This paper explores Lindstrom quantifiers with specific closure properties linked to polymorphisms in CSPs, introducing a pebble game to analyze their expressive power and demonstrating limitations in expressing certain linear algebra problems.

Contribution

It introduces a new class of quantifiers based on closure conditions from polymorphism equations and develops a pebble game to analyze their logical expressiveness.

Findings

Deciding solvability of linear equations in Z2 is not expressible with certain quantifiers.

The class of quantifiers closed under near-unanimity conditions is limited in expressive power.

The pebble game characterizes the distinguishing power of these quantifiers.

Abstract

We study Lindstrom quantifiers that satisfy certain closure properties which are motivated by the study of polymorphisms in the context of constraint satisfaction problems (CSP). When the algebra of polymorphisms of a finite structure B satisfies certain equations, this gives rise to a natural closure condition on the class of structures that map homomorphically to B. The collection of quantifiers that satisfy closure conditions arising from a fixed set of equations are rather more general than those arising as CSP. For any such conditions P, we define a pebble game that delimits the distinguishing power of the infinitary logic with all quantifiers that are P-closed. We use the pebble game to show that the problem of deciding whether a system of linear equations is solvable in Z2 is not expressible in the infinitary logic with all quantifiers closed under a near-unanimity condition.