A necessary and sufficient condition in order that a Herbrand interpretation be expressive relative to recursive programs

It is proved that a recursive program (without counters) is able to enumerate all elements in any Herbrand interpretation. It follows that all recursive program domains in a Herbrand interpretation can be defined by first-order formulas iff there are first-order formulas expressing integer arithmetic in that interpretation

Kripke Semantics for Intersection Formulas

We propose a notion of the Kripke-style model for intersection logic. Using a game interpretation, we prove soundness and completeness of the proposed semantics. In other words, a formula is provable (a type is inhabited) if and only if it is forced in every model. As a by-product, we obtain another proof of normalization for the Barendregt–Coppo–Dezani intersection type assignment system

Using Inhabitation in Bounded Combinatory Logic with Intersection Types for Composition Synthesis

We describe ongoing work on a framework for automatic composition synthesis from a repository of software components. This work is based on combinatory logic with intersection types. The idea is that components are modeled as typed combinators, and an algorithm for inhabitation {\textemdash} is there a combinatory term e with type tau relative to an environment Gamma? {\textemdash} can be used to synthesize compositions. Here, Gamma represents the repository in the form of typed combinators, tau specifies the synthesis goal, and e is the synthesized program. We illustrate our approach by examples, including an application to synthesis from GUI-components.Comment: In Proceedings ITRS 2012, arXiv:1307.784

On the Mints Hierarchy in First-Order Intuitionistic Logic

We stratify intuitionistic first-order logic over $(\forall,\to)$ into fragments determined by the alternation of positive and negative occurrences of quantifiers (Mints hierarchy). We study the decidability and complexity of these fragments. We prove that even the $\Delta_2$ level is undecidable and that $\Sigma_1$ is Expspace-complete. We also prove that the arity-bounded fragment of $\Sigma_1$ is complete for co-Nexptime

On the Mints Hierarchy in First-Order Intuitionistic Logic

We stratify intuitionistic first-order logic over $(\forall,\to)$ intofragments determined by the alternation of positive and negative occurrences ofquantifiers (Mints hierarchy). We study the decidability and complexity of these fragments. We prove thateven the $\Delta_2$ level is undecidable and that $\Sigma_1$ isExpspace-complete. We also prove that the arity-bounded fragment of $\Sigma_1$is complete for co-Nexptime

On the Mints Hierarchy in First-Order Intuitionistic Logic

We stratify intuitionistic first-order logic over $(\forall,\to)$ into fragments determined by the alternation of positive and negative occurrences of quantifiers (Mints hierarchy). We study the decidability and complexity of these fragments. We prove that even the $\Delta_2$ level is undecidable and that $\Sigma_1$ is Expspace-complete. We also prove that the arity-bounded fragment of $\Sigma_1$ is complete for co-Nexptime