ion and Store Shapes . . . . . . . . . . 6 2 Two Variations on Algol 8 2.1 Idealized Algol . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 Name: P.W. O'Hearn Address: Department of Computer Science, Queen Mary and Westfield College, London E1 4NS, UK. e-mail: ohearn@dcs.qmw.ac.uk Name: J.C. Reynolds Address: School of Computer Science, Carnegie Mellon University, Pittsburgh PA 15213-3891, USA. e-mail: john.reynolds@cs.cmu.edu Permission to make digital or hard copies of part or all of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or direct commercial advantage and that copies show this notice on the first page or initial screen of a display along with the full citation. Copyrights for components of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, to republish, to post on servers, to redistribute to lists, or to use any component of th...

While the semantics of local variables in programming languages is by now well-understood, the semantics of pointer-addressed heap variables is still an outstanding issue. In particular, the commonly assumed relational reasoning principles for data representations have not been validated in a semantic model of heap variables. In this paper, we de ne a parametricity semantics for a Pascal-like language with pointers and heap variables which gives such reasoning principles. It is found that the correspondences between data representations are not simply relations between states, but more intricate correspondences that also need to keep track of visible locations whose pointers can be stored and leaked.

We introduce an axiomatic approach to logical relations and data refinement. We consider a programming language and the monad on the category of small categories generated by it. We identify abstract data types for the language with sketches for the associated monad, and define an axiomatic notion of "relation" between models of such a sketch in a semantic category. We then prove three results: (i) such models lift to the whole language together with the sketch; (ii) any such relation satisfies a soundness condition, and (iii) such relations compose. We do this for both equality of data representations and for an ordered version. Finally, we compare our formulation of data refinement with that of Hoare. This work has been done with the support of the MITI Cooperative Architecture Project. This author also acknowledges the support of Kaken-hi. y This author achnowledges the support of the MITI Cooperative Architecture Project. z This author acknowledges the support of EPSRC grant...

We present a simple and workable axiomatization of domain theory within intuitionistic set theory, in which predomains are (special) sets, and domains are algebras for a simple equational theory. We use the axioms to construct a relationally parametric set-theoretic model for a compact but powerful polymorphic programming language, given by a novel extension of intuitionistic linear type theory based on strictness. By applying the model, we establish the fundamental operational properties of the language. 1.

We present a domain-theoretic model of parametric polymorphism based on admissible per’s over a domain-theoretic model of the untyped lambda calculus. The model is shown to be a model of Abadi & Plotkin’s logic for parametricity, by the construction of an LAPL-structure as defined by the authors in [7, 5]. This construction gives formal proof of solutions to a large class of recursive domain equations, which we explicate. As an example of a computation in the model, we explicitly describe the natural numbers object obtained using parametricity. The theory of admissible per’s can be considered a domain theory for (impredicative) polymorphism. By studying various categories of admissible and chain complete per’s and their relations, we discover a picture very similar to that of domain theory. 1

polymorphic lambda calculus, F2. Girard additionally proved a Representation Theorem: every function on natural numbers that can be proved total in second-order intuitionistic predicate logic, P2, can be represented in F2. Reynolds additionally proved an Abstraction Theorem: every term in F2 satisfies a suitable notion of logical relation; and formulated a notion of parametricity satisfied by well-behaved models. We observe that the essence of Girard’s result is a projection from P2 into F2, and that the essence of Reynolds’s result is an embedding of F2 into P2, and that the Reynolds embedding followed by the Girard projection is the identity. We show that the inductive naturals are exactly those values of type natural that satisfy Reynolds’s notion of parametricity, and as a consequence characterize situations in which the Girard projection followed by the Reynolds embedding is also the identity. An earlier version of this paper used a logic over untyped terms. This version uses a logic over typed term, similar to ones considered by Abadi and Plotkin and by Takeuti, which better clarifies the relationship between F2 and P2. This paper uses colour to enhance its presentation. If the link below is not blue, follow it for the colour version.

Plotkin suggested using a polymorphic dual intuitionistic / linear type theory (PILLY) as a metalanguage for parametric polymorphism and recursion. In recent work the first two authors and R.L. Petersen have defined a notion of parametric LAPL-structure, which are models of PILLY, in which one can reason using parametricity and, for example, solve a large class of domain equations, as suggested by Plotkin. In this paper we show how an interpretation of a strict version of Bierman, Pitts and Russo’s language Lily into synthetic domain theory presented by Simpson and Rosolini gives rise to a parametric LAPL-structure. This adds to the evidence that the notion of LAPL-structure is a general notion suitable for treating many different parametric models, and it provides formal proofs of consequences of parametricity expected to hold for the interpretation. Finally, we show how these results in combination with Rosolini and Simpson’s computational adequacy result can be used to prove consequences of parametricity for Lily. In particular we show that one can solve domain equations in Lily up to ground contextual equivalence. 1

Types Peter W. O'Hearn Department of Computer Science Queen Mary & Westfield College (ohearn@dcs.qmw.ac.uk) October 7, 1998 1 Introduction Abstraction is one of the pillars of computer science. Of course, choosing the right concepts to emphasize, and details to suppress, is crucial in all of science, but in computer science the role of abstraction extends beyond that of sensible methodology. For in computing we have seen the emergence of an array of constructs and methods whose chief purpose is to provide general mechanisms for achieving, validating, or enforcing abstraction, rather than being specific instances or examples of it. Among these are programming concepts such as objects, procedures, abstract data types and modules, and mathematical methods such as simulation and logical relations. But what, more precisely, is the "abstraction" achieved by these constructs? In the case of procedural abstraction, a more or less satisfactory explanation can be given in terms of functions....

This paper presents a sound and complete category-theoretic notion of models for Linear Abadi & Plotkin Logic [Birkedal et al., 2006], a logic suitable for reasoning about parametricity in combination with recursion. A subclass of these called parametric LAPL structures can be seen as an axiomatization of domain theoretic models of parametric polymorphism, and we show how to solve general (nested) recursive domain equations in these. Parametric LAPL structures constitute a general notion of model of parametricity in a setting with recursion. In future papers we will demonstrate this by showing how many different models of parametricity and recursion give rise to parametric LAPL structures, including Simpson and Rosolini’s set theoretic models [Rosolini and Simpson, 2004], a syntactic model based on Lily [Pitts, 2000, Bierman et al., 2000] and a model based on admissible pers over a reflexive domain [Birkedal et al., 2007].

In this paper, we give the first relationally parametric model of the (extensional) calculus of constructions. Our model remains as simple as traditional PER models of dependent types, but unlike them, our model additionally permits relating terms at different implementation types. Using this model, we can validate the soundness of quotient types, as well as derive strong equality axioms for Church-encoded data, such as the eta-law for strong dependent pair types. 1.