While developing a method for reasoning about programs, Pitts defined the [top ][top ]-closed relations as an alternative to the standard admissible relations. This paper reformulates and studies Pitts's operational concept of [top ][top ]-closure in a semantic framework. It investigates the non-trivial connection between [top ][top ]-closure and admissibility, showing that [top ][top ]-closure is strictly stronger than admissibility and that every [top ][top ]-closed relation corresponds to an admissible preorder.

Studies of the mathematical properties of impredicative polymorphic types have for the most part focused on the polymorphic lambda calculus of Girard–Reynolds, which is a calculus of total polymorphic functions. This paper considers polymorphic types from a functional programming perspective, where the partialness arising from the presence of fixpoint recursion complicates the nature of potentially infinite (‘lazy’) data types. An approach to Reynolds' notion of relational parametricity is developed that works directly on the syntax of a programming language, using a novel closure operator to relate operational behaviour to parametricity properties of types. Working with an extension of Plotkin's PCF with ∀-types, lazy lists and existential types, we show by example how the resulting logical relation can be used to prove properties of polymorphic types up to operational equivalence.

Higher Order Matching consists of solving finite sets of equations of the form u =βηv, where u, v are simply typed terms of λ-calculus, and v is a closed term. Whether this problem is decidable is an open question. In this paper its decidability is proved when the free variables of all equations are of order at most 4 – we actually extend this result to a more general situation in which in addition to equations, disequations are also considered.

The research reported in this paper exploits the view of constraint programming as computation in a logical system, namely constraint logic. The basic ingredients of constraint logic are: constraint models for the semantics (they form a comma-category over a fixed model of ‘built-ins’); generalized polynomials in the role of basic syntactic ingredient; and a constraint satisfaction relation between semantics and syntax. Category-based constraint logic means the development of the logic is abstract categorical rather than concrete set theoretical.

We show that (category-based) constraint logic is an institution, and we internalize the study of constraint logic to the abstract framework of category-based equational logic, thus opening the door for considering constraint logic programming over non-standard structures (such as CPO's, topologies, graphs, categories, etc.). By embedding category-based constraint logic into category-based equational logic, we integrate the constraint logic programming paradigm into (category-based) equational logic programming. Results include completeness of constraint logic deduction, a novel Herbrand theorem for constraint logic programming characterizing Herbrand models as initial models in constraint logic, and logical foundations for the modular combination of constraint solvers based on amalgamated sums of Herbrand models in the constraint logic institution.