Constraint satisfaction with infinite domains

Constraint satisfaction problems occur in many areas of computer science, most prominently in artificial intelligence including temporal or spacial reasoning, belief maintenance, machine vision, and scheduling (for an overview see [Kumar, 1992,Dechter, 2003]). Other areas are graph theory, boolean satisfiability, type systems for programming languages, database theory, automatic theorem proving, and, as for some of the problems discussed in this thesis, computational linguistics and computational biology. Many constraint satisfaction problems have a natural formulation as a homomorphism problem. For a fixed relational structure Γ we consider the following computational problem: Given a structure S with the same relational signature as Γ, is there a homomorphism from S to Γ? This problem is known as the constraint satisfaction problem CSP(Γ) for the template Γ and is intensively studied for relational structures Γ with a finite domain. However, many constraint satisfaction problems can not be formulated with a finite template. If we allow arbitrary infinite templates, constraint satisfaction is very expressive and e.g. contains undecidable problems. In this thesis, we impose two restrictions on the template. The first restriction is ω-categoricity, a natural and well-studied concept in model-theory. The computational complexity of CSP(Γ) is determined by the relations of Γ that have a primitive positive definition in Γ. For ω-categorical templates we can show that a relation is primitive positive definable in Γ if and only if it is preserved by the polymorphisms of Γ. This theorem is well-known for finite templates [Bodnarčuk et al., 1969, Geiger, 1968], and was the starting point of the algebraic approach to study the complexity of constraint satisfaction with finite templates, described e.g. in [Jeavons et al., 1997]. It shows that also for ω-categorical templates, the complexity of a constraint satisfaction problem is determined by the clone of polymorphisms of the template. One example where the existence of a certain polymorphism of the (finite or infinite) template implies tractability of the corresponding constraint satisfaction problem is the case where the polymorphism is a k-ary near-unanimity operation. In this case the problem can be solved by a Datalog-program of width k. For finite templates, [Feder and Vardi, 1999] proved that every constraint satisfaction problem that can be solved by a Datalog program of width k can also be solved by the canonical Datalog program of width k. This is another result we can generalize to ω-categorical templates. The second restriction is that the template Γ can be described by a finite set of forbidden induced substructures. In this case the constraint satisfaction problem for Γ is in monotone SNP, which is a fragment of existential second