Logic, computation and constraint satisfaction

We study a class of non-deterministic program schemes with while loops: firstly, augmented with a priority queue for memory; secondly, augmented with universal quantification; and, thirdly, augmented with universal quantification and a stack for memory. We try to relate these respective classes of program schemes to well-known complexity classes and logics. We study classes of structure on which path system logic coincides with polynomial time P. We examine the complexity of generalisations of non-uniform boolean constraint satisfaction problems, where the inputs may have a bounded number of quantifier alternations (as opposed to the purely existential quantification of the CSP). We prove, for all bounded-alternation prefixes that have some universal quantifiers to the outside of some existential quantifiers (i.e. Π2 and above), that this generalisation of boolean CSP respects the same dichotomy as that for the non-uniform boolean quantified constraint satisfaction problem. We study the non-uniform QCSP, especially on digraghs, through a combinatorial analog – the alternating-homomorphism problem – that sits in relation to the QCSP exactly as the homomorphism problem sits with the CSP. We establish a trichotomy theorem for the non-uniform QCSP when the template is restricted to antireflexive, undirected graphs with at most one cycle. Specifically, such templates give rise to QCSPs that are either tractable, NP-complete or Pspace-complete. We study closure properties on templates that respect QCSP hardness or QCSP equality. Our investigation leads us to examine the properties of first-order logic when deprived of the equality relation. We study the non-uniform QCSP on tournament templates, deriving sufficient conditions for tractability, NP-completeness and Pspace-completeness. In particular, we prove that those tournament templates that give rise to tractable CSP also give rise to tractable QCSP.