Classifications of Computable Structures

Let K be a family of structures, closed under isomorphism, in a fixed computable language. We consider effective lists of structures from K such that every structure in K is isomorphic to exactly one structure on the list. Such a list is called a computable classification of K, up to isomorphism. Using the technique of Friedberg enumeration, we show that there is a computable classification of the family of computable algebraic fields, and that with a 0\u27-oracle, we can obtain similar classifications of the families of computable equivalence structures and of computable finite-branching trees. However, there is no computable classification of the latter, nor of the family of computable torsion-free abelian groups of rank 1, even though these families are both closely allied with computable algebraic fields

Infinite time Turing machines and an application to the hierarchy of equivalence relations on the reals

We describe the basic theory of infinite time Turing machines and some recent developments, including the infinite time degree theory, infinite time complexity theory, and infinite time computable model theory. We focus particularly on the application of infinite time Turing machines to the analysis of the hierarchy of equivalence relations on the reals, in analogy with the theory arising from Borel reducibility. We define a notion of infinite time reducibility, which lifts much of the Borel theory into the class $\bm{\Delta}^1_2$ in a satisfying way.Comment: Submitted to the Effective Mathematics of the Uncountable Conference, 200

Levelable Sets and the Algebraic Structure of Parameterizations

Asking which sets are fixed-parameter tractable for a given parameterization constitutes much of the current research in parameterized complexity theory. This approach faces some of the core difficulties in complexity theory. By focussing instead on the parameterizations that make a given set fixed-parameter tractable, we circumvent these difficulties. We isolate parameterizations as independent measures of complexity and study their underlying algebraic structure. Thus we are able to compare parameterizations, which establishes a hierarchy of complexity that is much stronger than that present in typical parameterized algorithms races. Among other results, we find that no practically fixed-parameter tractable sets have optimal parameterizations

A Trichotomy in the Complexity of Counting Answers to Conjunctive Queries

Conjunctive queries are basic and heavily studied database queries; in relational algebra, they are the select-project-join queries. In this article, we study the fundamental problem of counting, given a conjunctive query and a relational database, the number of answers to the query on the database. In particular, we study the complexity of this problem relative to sets of conjunctive queries. We present a trichotomy theorem, which shows essentially that this problem on a set of conjunctive queries is either tractable, equivalent to the parameterized CLIQUE problem, or as hard as the parameterized counting CLIQUE problem; the criteria describing which of these situations occurs is simply stated, in terms of graph-theoretic conditions

Scott Ranks of Classifications of the Admissibility Equivalence Relation

Let $\mathscr{L}$ be a recursive language. Let $S(\mathscr{L})$ be the set of $\mathscr{L}$-structures with domain $\omega$. Let $\Phi : {}^\omega 2 \rightarrow S(\mathscr{L})$ be a $\Delta_1^1$ function with the property that for all $x,y \in {}^\omega 2$, $\omega_1^x = \omega_1^y$ if and only if $\Phi(x) \approx_{\mathscr{L}} \Phi(y)$. Then there is some $x \in {}^\omega 2$ so that $\mathrm{SR}(\Phi(x)) = \omega_1^x + 1$