Locally Finite Constraint Satisfaction Problems

Abstract—First-order definable structures with atoms are infi-nite, but exhibit enough symmetry to be effectively manipulated. We study Constraint Satisfaction Problems (CSPs) where both the instance and the template are definable structures with atoms. As an initial step, we consider locally finite templates, which contain potentially infinitely many finite relations. We argue that such templates occur naturally in Descriptive Complexity Theory. We study CSPs over such templates for both finite and infinite, definable instances. In the latter case even decidability is not obvious, and to prove it we apply results from topological dynamics. For finite instances, we show that some central results from the classical algebraic theory of CSPs still hold: the complexity is determined by polymorphisms of the template, and the existence of certain polymorphisms, such as majority or Maltsev polymorphisms, guarantees the correctness of classical algorithms for solving finite CSP instances. Index Terms—Sets with atoms; Constraint Satisfaction Prob-lems Once and for all, fix a countably infinite set A = {1, 2, 3,...}, whose elements we call atoms