Showing papers presented at "International Conference on Mathematical Foundations of Programming Semantics in 1991"

Types, Abstractions, and Parametric Polymorphism, Part 2

Abstract: The concept of relations over sets is generalized to relations over an arbitrary category, and used to investigate the abstraction (or logical-relations) theorem, the identity extension lemma, and parametric polymorphism, for Cartesian-closed-category models of the simply typed lambda calculus and PL-category models of the polymorphic typed lambda calculus. Treatments of Kripke relations and of complete relations on domains are included.

An Upper Power Domain Construction in Terms of Strongly Compact Sets

Abstract: A novel upper power domain construction is defined by means of strongly compact sets. Its power domains contain less elements than the classical ones in terms of compact sets, but still admit all necessary operations, i.e. they contain less junk. The notion of strong compactness allows a proof of stronger properties than compactness would, e.g. an intrinsic universal property of the upper power construction, and its commutation with the lower construction.

An Algorithm for Analyzing Communicating Processes

Abstract: Developing, proving and compiling concurrent programs with communications poses some difficulties, especially due to the number of potential interactions between processes. Some static analysis methods have been developed to determine how processes are synchronised, but all the algorithms proposed until now are exponential. This paper presents, for CSP-like programs, a static and automatic analysis algorithm using abstract interpretation and based on counting the number of communications along each channel connecting two processes.

Correctness of Procedure Representations in Higher-Order Assembly Language

Abstract: Higher-order assembly language (HOAL) generalizes combinator-based target languages by allowing free variables in terms to play the role of registers. We introduce a machine model for which HOAL is the assembly language, and prove the correctness of a compiler from a tiny language into HOAL. We introduce the notion of a λ-representation, which is an abstract binding operation, show how some common representations of procedures and continuations can be expressed as λ-representations. Last, we prove the correctness of a typical procedure-calling convention in this framework.

From Operational to Denotational Semantics

Abstract: In this paper operational equivalence of simple functional programs is defined, and certain basic theorems proved thereupon. These basic theorems include congruence, least fixed-point, an analogue to continuity, and fixed-point induction. We then show how any ordering on programs for which these theorems hold can be easily extended to give a fully abstract cpo for the language, giving evidence that any operational semantics with these basic theorems proven is complete with respect to a denotational semantics. Furthermore, the mathematical tools used in the paper are minimal, the techniques should be applicable to a wide class of languages, and all proofs are constructive.

The Equivalence of Two Semantic Definitions for Inheritance in Object-Oriented Languages

Abstract: A simple language is presented which supports inheritance in object-oriented languages. Using this language, the definitions for the semantics of inheritance given in [CHC90] and [Mit90] are compared and shown to be equivalent. The equivalence is shown by presenting and comparing two denotational semantics of the simple language which capture the essence of each of the earlier semantics.

Information Links in Domain Theory

Abstract: This paper presents some ideas about the eventual shape of a general theory of information content and information flow. Such a theory will clearly need to encompass the insights into information from the denotational semantics of programming languages. But it will also need to provide a foundation for an understanding of the role information plays in ordinary human languages, like English, and other mundane uses of information. For the past decade or so, we have been working on situation theory, a theory of information that would apply to the analysis of natural languages. It is only over the past year or so that we have begun to see a clear relationship between situation theory and the work in domain theory. In this paper I would like to present some simple ideas in a way that is neutral with regard to basic theory. Here is a table that should allow anyone familiar with a little of both situation theory and domain theory to grasp the main point of this paper.

Decomposition of Domains

Abstract: The problem of decomposing domains into sensible factors is addressed and solved for the case of dI-domains. A decomposition theorem is proved which allows the represention of a large subclass of dI-domains in a product of flat domains. Direct product decompositions of Scott-domains are studied separately.

Typed Homomorphic Relations Extended with Sybtypes

Abstract: Typed homomorphic relations on heterogeneous algebras are generalized to allow relationships between elements in the carrier sets of different types. Such relations are needed for the model theory of incomplete, hierarchical specifications with subtypes. Typed logical relations are generalized similarly. These tools help give a simple model-theoretic account of subtyping among abstract data types as observed by terms of a simply-typed lambda-calculus with subtypes.

Trade-Offs in True Concurrency: Pomsets and Mazurkiewicz Traces

Abstract: We compare finite pomsets and Mazurkiewicz traces, two models of true concurrency which generalize strings We show that Mazurkiewicz traces are equivalent to a restricted class of pomsets The restrictions lead to more algebraic structure satisfying additional properties

Continuous Functions and Parallel Algorithms on Concrete Data Structures

Abstract: We report progress in two closely related lines of research: the semantic study of sequentially and parallelism, and the development of a theory of intensional semantics. We generalize Kahn and Plotkin's concrete data structures to obtain a cartesian closed category of generalized concrete data structures and continuous functions. The generalized framework continues to support a definition of sequential functions. Using this ccc as an extensional framework, we define an intensional framework — a ccc of generalized concrete data structures and parallel algorithms. This construction is an instance of a more general and more widely applicable category-theoretic approach to intensional semantics, encapsulating a notion of intensional behavior as a computational comonad, and employing the co-Kleisli category as an intensional framework. We discuss the relationship between parallel algorithms and continuous functions, and supply some operational intuition for the parallel algorithms. We show that our parallel algorithms may be seen as a generalization of Berry and Curien's sequential algorithms.

On Relating Concurency and Nondeterminism

Abstract: We present an intuitive preorder for a simple CCS-like language whose semantic theory allows us to relate concurrency and nondeterminism without reducing the former to the latter. The preorder over processes is induced by using an equationally defined preorder over computations in a bisimulation-like protocol. The relationships of the proposed preorder with pomset bisimulation and standard strong bisimulation equivalence are studied in detail. Moreover, we give an axiomatization of the preorder over recursion-free processes.

Primitive Recursive Functionals with Dependent Types

Abstract: This paper presents an extension of an ML style type system to support dependent product types for programs. The study is based on the lambda calculus with primitive recursion and a simple type system with primitive recursive sequences of types and products dependent on natural numbers. I show that it is possible in this system, called Tπ, to type a sensible primitive recursive function that is not typable in ML. The paper presents typing rules, a semantic model with soundness and completeness theorems, and examples of how to type representations of tuples and projections as products dependent on the tuple width. I discuss a method for type reconstruction that extends ML style type reconstruction to dependent types. The type system is a representation of an earlier system by Tait [9] and Martin Lof

Program Correctness and Matricial Iteration Theories

Abstract: The analysis of partial correctness assertions given in [Bea] is specialized to matricial iteration theories A system of rules for partial correctness is given and a completeness theorem is stated It is shown how to formulate total correctness in all iteration theories as an equation, and some total correctness rules are considered

Cartesian Closed Categories of Domains and the Space Proj(D)

Abstract: This paper studies algebraic dcpos and their space \(\left[ {D\underrightarrow {pr}D} \right]\)of Scottcontinuous projections. Let ALG⊥ be the category of algebraic dcpos with bottom and Scott-continuous maps as arrows. If C is a full cartesian closed subcategory of ALG⊥ such that C is closed under D → \(\left[ {D\underrightarrow {pr}D} \right]\), then every object D is projection-stable, i.e., im(p) is algebraic for all p ∃ \(\left[ {D\underrightarrow {pr}D} \right]\). This is equivalent to assuming that all order-dense chains in K(D) are degenerate. If C contains an isomorphic copy of the flat natural numbers, then C is not closed under finitary Scott-continuous retractions. In this case, C contains an object D, such that K(D) is not a lower set in D. This puts serious constraints on the existence of cartesian closed categories closed under D → \(\left[ {D\underrightarrow {pr}D} \right]\). The cartesian closed category of dI-domains, however, is closed under D → Retr(D) and D → Proj(D), where Retr(D) is the dI-domain of all stable Scott-continuous retractions in the stable order. The dI-domain Proj(D) consists of all p ∃ Retr(D) which are below the identity in the stable order. All dI-domains are projection-stable, and Proj(D) is isomorphic to the Hoare power domain of the ideal completion of the poset of complete primes of D. It is therefore a completely distributive bialgebraic lattice and all maps f: D → E into complete lattices E preserving all existing suprema have unique extensions to Proj(D), where λc.λx.x; ∏ c is the embedding of D into Proj(D).

Liminf Progress Measures

Abstract: Consider a program P that satisfies a specification S. It is natural to think that every single step of a computation of P somehow contributes to making the computation closer to satisfying S. For specifications that define liveness properties, methods that directly quantify such a notion of progress or convergence are often quite limited in scope. Instead many approaches, such as those that deal with termination under fairness, rely on program transformations, since there was no known way of directly expressing progress towards fair termination.

An Exper Model for Quest

Abstract: We have applied various results about the synthetic approach to domain theory to describe a PER-like model of the language Quest. Besides, we have presented a different model for Quest which allows the interpretation of the recursive operator at all types. This seems to be the first such in the literature, and we are certain that it could be given thanks to the tools provided by categorical logic. It also seems possible to relate the first model we have given with that of good PER's of Abadi and Plotkin, and that the two are essentially different. But we believe that the good PER's may provide a new interpretation for Quest in the spirit of our second model.

On Continuous Time Agents

Abstract: Continuous time agents are studied in an enriched categorical framework that allows for a comprehensive treatment of both the interleaving and the true concurrent paradigms in parallelism. The starting point is a paper by Cardelli, where actions have a duration in a (dense) time domain. More recent works are also briefly considered and some possible directions towards timed “true concurrent” processes are indicated.

Nonwellfounded Sets and Programming Language Semantics

Abstract: For a large class of transition systems that are defined by specifications in the SOS style, it is shown how these induce a compositional semantics. The main difference with earlier work on this subject is the use of a nonstandard set theory that is based on Aczel's anti-foundation-axiom. Solving recursive domain equations in this theory leads to solutions that contain nonwellfounded elements. These are particularly useful for justifying recursive definitions, both of semantic operators and semantic models. The use of nonwellfounded sets further allows for the construction of compositional models for a larger class of transition systems than in the setting of complete metric spaces, which was used before.

Call-by-Value Combinatory Logic and the Lambda-Value Calculus

Abstract: Since it is unsound to reason about call-by-value languages using call-by name equational theories, we present two by-value combinatory logics and translations from the λ-value (λv) calculus to the logics. The first by-value logic is constructed in a manner similar to the λv-calculus: it is based on the byname combinatory logic, but the combinators are strict. The translation is non-standard to account for the strictness of the input program. The second by-value logic introduces laziness to K terms so that the translation can preserve the structure of functions that do not use their argument. Both logics include constants and delta rules, and we prove their equivalence with the λv-calculus.

A Monoidal Closed Category of Event Structures

Abstract: This paper introduces the following new constructions on stable event structures: the tensor product, the linear function space, and the exponential. It results in a monoidal closed category of stable event structures which can be used to interpret intuitionistic linear logic.

Primitive recursive functional with dependent types

Abstract: This paper presents an extension of an ML style type system to support dependent product types for programs. The study is based on the lambda calculus with primitive recursion and a simple type system with primitive recursive sequences of types and products dependent on natural numbers. I show that it is possible in this system, called Tπ, to type a sensible primitive recursive function that is not typable in ML. The paper presents typing rules, a semantic model with soundness and completeness theorems, and examples of how to type representations of tuples and projections as products dependent on the tuple width. I discuss a method for type reconstruction that extends ML style type reconstruction to dependent types. The type system is a representation of an earlier system by Tait [9] and Martin Lof

HSP Type Theorems in the Category of Posets

Abstract: This paper introduces the notion of a theory based on posets and HSP or Birkhoff subcategories thereof. It turns out that the nature of these subcategories depends on what is meant by subobject. The correspondence between subcategories does not hold as it does in sets, primarily because the axiom of choice (in the form that epimorphisms split) totally fails in posets. Although equations, suitably generalized to include inequalities, determine HSP subcategories, the converse fails and it may be necessary to iterate the process of forming the subcategories by means of equations and inequalities.

Simultaneous Substitution in the Typed Lambda Calculus

Abstract: The syntax and operational semantics of the named-variable form of the pure typed lambda calculus with simultaneous substitutions is described together with its denotational semantics in a cartesian closed category. It is shown that operationally equivalent terms have the same denotational interpretations and that the expected rule for substituting a composition of simultaneous substitutions is valid.