Partial Relations Research Articles

An application of PER models to program extraction

Type theory allows us to extract from a constructive proof that a specification is satisfiable a program that satisfies the specification. Algorithms for optimization of such programs are currently the object of research.In this paper we consider one such algorithm, which was described in Beeson (1985) and which we will call ‘Harrop’. This algorithm greatly simplifies programs extracted from proofs in the Pure Construction Calculus. We use a Partial Equivalence Relation model for higher order lambda calculus, to check that t and Harrop(t) return the same outputs from the same inputs, i.e. that they are extensionally equal.As a corollary, we show that it is correct (and, of course, useful) to replace a program t with Harrop(t). Such a correctness result has already been proved by Möhring (Möhring 1989a, 1989b) using realizability semantics, but we obtain it as a corollary of a new result, the extensional equality between t and Harrop(t). Also the semantic method we use is interesting in its own right.

Least fixpoints of endofunctors of cartesian closed categories

Least fixpoints are constructed for finite coproducts of definable endofunctors of Cartesian closed categories that have weak polynomial products and joint equalizers of arbitrary families of pairs of parallel arrows. Both conditions hold in PER, the category whose objects are partial equivalence relations on N, and whose arrows are partial recursive functions. Weak polynomial products exist in any cartesian closed category with a finite number of objects as well as in any model of second order polymorphic lambda calculus: that is, in the proof theory of any second order positive intuitionistic propositional calculus, but such a category need not have equalizers. However, any finite coproduct of definable endofunctors of a cartesian closed category with weak polynomial products will have a least fixpoint in a larger category with equalizers whose objects are right ideals (or sieves) of modulo certain congruence relations, and whose arrows are induced from .

Layered Predicates

We review the concept of logical relations and how they interact with structural induction, furthermore we give examples of their use, and of particular interest is the combination with the PER-idea (partial equivalence relations). This is then generalized to Kripke logical relations; the major application is to show that in combination with the PER-idea this solves the problem of establishing a substitution property in a manner oonducive to structural induction. Finally we introduce the concept of Kripke-layered predicates; this allows a modular definition of predicates and supports a methodology of ''proofs in stages'' where each stage forcuses on only one aspect and thus is more manageable. All of these techniques have been tested and refined in ''realistic applications'' that have been documented elsewhere.

Building domains from graph models

In this paper we study partial equivalence relations (PERs) over graph models of the λcalculus. We define categories of PERs that behave like predomains, and like domains. These categories are small and complete; so we can solve domain equations and construct polymorphic types inside them. Upper, lower and convex powerdomain constructions are also available, as well as interpretations of subtyping and bounded quantification. Rather than performing explicit calculations with PERs, we work inside the appropriate realizability topos: this is a model of constructive set theory in which PERs, can be regarded simply as special kinds of sets. In this framework, most of the definitions and proofs become quite smple and attractives. They illustrative some general technicques in ‘synthetic domain theory’ that rely heavily on category theory; using these methods, we can obtain quite powerful results about classes of PERs, even when we know very little about their internal structure.

Constructing type systems over an operational semantics

Type theories in the sense of Martin-Löf and the NuPRL system are based on taking as primitive a type-free programming language given by an operational semantics, and defining types as partial equivalence relations on the set of closed terms. The construction of a type system is based on a general form of inductive definition that may either be taken as acceptable in its own right, or further explicated in terms of other patterns of induction. One such account, based on a general theory of inductively-defined relations, was given by Allen. An alternative account, based on an essentially set-theoretic argument, is presented.

Extensional PERs

A class of Partial Equivalence Relations (PERs) is described such that the resulting full subcategory of the realizability universe has the expected properties of a good category of CPOs; that is, it is a Cartesian closed category and every endomorphism has a canonical fixed point. Moreover, the reflection functor into the subcategory of strict maps (usually called the “lifting operation”) yields a good notion of “partial map,” a necessary condition if one wishes to maintain a connection between strict maps and partial maps. It is shown that all functors that arise in practice have canonical invariant objects; hence a host of domain equations are guaranteed to have solutions.

A semantic basis for Quest

AbstractQuest is a programming language based on impredicative type quantifiers and subtyping within a three-level structure of kinds, types and type operators, and values.The semantics of Quest is rather challenging. In particular, difficulties arise when we try to model simultaneously features such as contravariant function spaces, record types, subtyping, recursive types and fixpoints.In this paper we describe in detail the type inference rules for Quest, and give them meaning using a partial equivalence relation model of types. Subtyping is interpreted as in previous work by Bruce and Longo (1989), but the interpretation of some aspects – namely subsumption, power kinds, and record subtyping – is novel. The latter is based on a new encoding of record types.We concentrate on modelling quantifiers and subtyping; recursion is the subject of current work.

Recursive types for Fun

The language Fun [13] is a typed polymorphic lambda calculus with a notion of subtyping and quantifiers ranging over subtypes of a given type. In this paper we show that it is consistent to allow recursive type definitions in Fun, by constructing an interpretation of types as partial equivalence relations of a special kind, terms being interpreted as equivalence classes, modulo such relations, of elements of a model for an underlying untyped language.

Self complementary topologies and preorders

A topology on a set X is self complementary if there is a homeomorphic copy on the same set that is a complement in the lattice of topologies on X. The problem of characterizing finite self complementary topologies leads us to redefine the problem in terms of preorders (i.e. reflexive, transitive relations). A preorder P on a set X is self complementary if there is an isomorphic copy P′ of P on X that is arc disjoint to P (except for loops) and with the property that P∪P′ is strongly connected. We characterize here self complementary finite partial orders and self complementary finite equivalence relations.

A modest model of records, inheritance, and bounded quantification

The authors give a formal semantics for the language Bounded Fun, which supports both parametric and subtype polymorphism. They show how to use partial equivalence relations to model inheritance in this language, which supports the notion of subtype and record types. A generalization of partial equivalence relations, known as omega -sets, is used in combination with modest sets to provide the first known model of Bounded Fun (with explicit polymorphism). Connections with previous work on the semantics of explicit parametric polymorphism is established by noting that the semantics of polymorphic types presented here (using dependent products) is isomorphic to that given by the intersection interpretation of polymorphism. >