A Profunctorial Scott Semantics

In this paper, we study the bicategory of profunctors with the free finite coproduct pseudo-comonad and show that it constitutes a model of linear logic that generalizes the Scott model. We formalize the connection between the two models as a change of base for enriched categories which induces a pseudo-functor that preserves all the linear logic structure. We prove that morphisms in the co-Kleisli bicategory correspond to the concept of strongly finitary functors (sifted colimits preserving functors) between presheaf categories. We further show that this model provides solutions of recursive type equations which provides 2-dimensional models of the pure lambda calculus and we also exhibit a fixed point operator on terms

Decision problems and profinite completions of groups

We consider pairs of finitely presented, residually finite groups P\hookrightarrow\G for which the induced map of profinite completions \hat P\to \hat\G is an isomorphism. We prove that there is no algorithm that, given an arbitrary such pair, can determine whether or not $P$ is isomorphic to \G. We construct pairs for which the conjugacy problem in \G can be solved in quadratic time but the conjugacy problem in $P$ is unsolvable. Let $\mathcal J$ be the class of super-perfect groups that have a compact classifying space and no proper subgroups of finite index. We prove that there does not exist an algorithm that, given a finite presentation of a group \G and a guarantee that \G\in\mathcal J, can determine whether or not \G\cong\{1\}. We construct a finitely presented acyclic group \H and an integer $k$ such that there is no algorithm that can determine which $k$-generator subgroups of \H are perfect

Degree spectra and computable dimensions in algebraic structures

AbstractWhenever a structure with a particularly interesting computability-theoretic property is found, it is natural to ask whether similar examples can be found within well-known classes of algebraic structures, such as groups, rings, lattices, and so forth. One way to give positive answers to this question is to adapt the original proof to the new setting. However, this can be an unnecessary duplication of effort, and lacks generality. Another method is to code the original structure into a structure in the given class in a way that is effective enough to preserve the property in which we are interested. In this paper, we show how to transfer a number of computability-theoretic properties from directed graphs to structures in the following classes: symmetric, irreflexive graphs; partial orderings; lattices; rings (with zero-divisors); integral domains of arbitrary characteristic; commutative semigroups; and 2-step nilpotent groups. This allows us to show that several theorems about degree spectra of relations on computable structures, nonpreservation of computable categoricity, and degree spectra of structures remain true when we restrict our attention to structures in any of the classes on this list. The codings we present are general enough to be viewed as establishing that the theories mentioned above are computably complete in the sense that, for a wide range of computability-theoretic nonstructure type properties, if there are any examples of structures with such properties then there are such examples that are models of each of these theories