Showing papers presented at "International Conference on Mathematical Foundations of Programming Semantics in 1993"

A Syntax for Linear Logic

Abstract: There is a standard syntax for Girard's linear logic, due to Abramsky, and a standard semantics, due to Seely. Alas, the former is incoherent with the latter: different derivations of the same syntax may be assigned different semantics. This paper reviews the standard syntax and semantics, and discusses the problem that arises and a standard approach to its solution. A new solution is proposed, based on ideas taken from Girard's Logic of Unity. The new syntax is based on pattern matching, allowing for concise expression of programs.

Time Abstracted Bisimiulation: Implicit Specifications and Decidability

Abstract: In the last few years a number of real-time process calculi have emerged with the purpose of capturing important quantitative aspects of real-time systems. In addition, a number of process equivalences sensitive to time-quantities have been proposed, among these the notion of timed (bisimulation) equivalence in [RR86, DS89, HR91, BB89, NRSV90, MT90, Wan91b].

Lifting Theorems for Kleisli Categories

Abstract: Monads, comonads and categories of algebras have become increasingly important tools in formulating and interpreting concepts in programming language semantics. A natural question that arises is how various categories of algebras for different monads relate functorially. In this paper we investigate when functors between categories with monads or comonads can be lifted to their corresponding Kleisli categories. Determining when adjoint pairs of functors can be lifted or inherited is of particular interest. The results lead naturally to various applications in both extensional and intensional semantics, including work on partial maps and data types and the work of Brookes/Geva on computational comonads.

Ultimately Periodic Words of Rational w-Languages

Abstract: In this paper we initiate the following program: Associate sets of finite words to Buchi-recognizable sets of infinite words, and reduce algorithmic problems on Buchi automata to simpler ones on automata on finite words. We know that the set of ultimately periodic words UP(L) of a rational language of infinite words L is sufficient to characterize L, since UP(L1)=UP(L2) implies L1=L2. We can use this fact as a test, for example, of the equivalence of two given Buchi automata. The main technical result in this paper is the construction of an automaton which recognizes the set of all finite words u · $ · v which naturally represent the ultimately periodic words of the form u ·554-01 in the language of infinite words recognized by a given Buchi automaton.

On the Symmetry of Sequentiality

Abstract: We offer a symmetric account of sequentiality, by means of symmetric algorithms, which are pairs of sequential functions, mapping input data to output data, and output exploration trees to input exploration trees, respectively. We use the framework of sequential data structures, a reformulation of a class of Kahn-Plotkin's concrete data structures. In sequential data structures, data are constructed by alternating questions and answers. Sequential data structures and symmetric algorithms are the objects and morphisms of a symmetric monoidal closed category, which is also cartesian, and is such that the unit is terminal. Our category is a full subcategory of categories of games considered by Lamarche, and by Abramsky-Jagadeesan, respectively.

Final Universes of Processes

Abstract: We describe the final universe approach to the characterisation of semantic universes and illustrate it by giving characterisations of the universes of CCS and CSP processes

Computational Adequacy via Mixed Inductive Definitions

Abstract: For programming languages whose denotational semantics uses fixed points of domain constructors of mixed variance, proofs of correspondence between operational and denotational semantics (or between two different denotational semantics) often depend upon the existence of relations specified as the fixed point of non-monotonic operators. This paper describes a new approach to constructing such relations which avoids having to delve into the detailed construction of the recursively defined domains themselves. The method is introduced by example, by considering the proof of computational adequacy of a denotational semantics for expression evaluation in a simple, untyped functional programming language.

On the Transformation between Direct and Continuation Semantics

Abstract: Proving the congruence between a direct semantics and a continuation semantics is often surprisingly complicated considering that direct-style λ-terms can be transformed into continuation style automatically. However, transforming the representation of a direct-style semantics into continuation style usually does not yield the expected representation of a continuation-style semantics (i.e., one written by hand).

An Investigation into Functions as Processes

Abstract: In [Mil90] Milner examines the encoding of the λ-calculus into the π-calculus [MPW92]. The former is the universally accepted basis for computations with functions, the latter aims at being its counterpart for computations with processes. The primary goal of this paper is to continue the study of Milner's encodings. We focus mainly on the lazy λ-calculus [Abr87]. We show that its encoding gives rise to a λ-model, in which a weak form of extensionality holds. However the model is not fully abstract: To obtain full abstraction, we examine both the restrictive approach, in which the semantic domain of processes is cut down, and the expansive approach, in which λ-calculus is enriched with constants to obtain a direct characterisation of the equivalence on λ-terms induced, via the encoding, by the behavioural equivalence adopted on the processes. Our results are derived exploiting an intermediate representation of Milner's encodings into the Higher-Order π-calculus, an ω-order extension of π-calculus where also agents may be transmitted. For this, essential use is made of the fully abstract compilation from the Higher-Order π-calculus to the π-calculus studied in [San92a].

Probabilistic Power Domains, Information Systems, and Locales

Abstract: The probabilistic power domain construction of Jones and Plotkin [6, 7] is defined by a construction on dcpo's. We present alternative definitions in terms of information systems a la Vickers [12], and in terms of locales. On continuous domains, all three definitions coincide.

Linear Domains and Linear Maps

Abstract: We study the symmetric monoidal closed category LIN of linear domains. Its objects are inverse limits of finite, bounded complete posets with respect to projection-embedding pairs preserving all suprema. The full reflective subcategory LL of linear lattices is a denotational model of linear logic; the negation is A ↦ A op and !(A) is the lattice of all Scott-closed sets of A. The Scott-continuous function space [A → B] models intuitionistic implication. Prime-algebraic lattices are linear and ℘ equals ⊗ for these lattices; in general, ℘ ≠ ⊗ in LL. Distributive, linear domains are exactly the prime-algebraic ones.

Holomorhpic Models of Exponential Types in Linear Logic

Abstract: In this paper we describe models of several fragments of linear logic with the exponential operator ! (called Of course) in categories of linear spaces. We model ! by the Fock space construction in Banach (or Hilbert) spaces, a notion originally introduced in the context of quantum field theory. Several variants of this construction are presented, and the representation of Fock space as a space of holomorphic functions is described. This also suggests that the “non-linear” functions we arrive at via! are not merely continuous, but analytic.

Mechanizing Logical Relations

Abstract: We give an algorithm for deciding whether there exists a definable element of a finite model of an applied typed lambda calculus that passes certain tests, in the special case when all the constants and test arguments are of order at most one. When there is such an element, the algorithm outputs a term that passes the tests; otherwise, the algorithm outputs a logical relation that demonstrates the nonexistence of such an element. Several example applications of the C implementation of this algorithm are considered.

An Operational Semantics for TOOPLE: A Statically-Typed Object-Oriented Programming Language

Abstract: In this paper we present an operational semantics for the language TOOPLE, a statically-typed functional object-oriented programming language which has a number of desirable properties. The operational semantics, given in the form of a natural semantics, is significantly simpler than the previous denotational semantics for the language. A “subject reduction” theorem for the natural semantics provides a proof that the language is type-safe. We also show that the natural semantics is consistent with the denotational semantics of the language.

Some Quasi-Varieties of Iteration Theories

Abstract: All known structures involving a constructively obtainable fixed point (or iteration) operation satisfy the equational laws defining iteration theories. Hence, there seems to be a general equational theory of iteration. This paper provides evidence that there is no general implicational theory of iteration. In particular, the quasi-variety generated by the continuous ordered theories, in which fixed point equations have least solutions, is incomparable with the quasi-variety generated by the pointed iterative theories, in which fixed point equations have unique solutions.

A chemical abstract machine for graph reduction extended abstract

Abstract: Graph reduction is an implementation technique for the lazy λ-calculus. It has been used to implement many non-strict functional languages, such as lazy ml, Gofer and Miranda. Parallel graph reduction allows for concurrent evaluation. In this paper, we present parallel graph reduction as a Chemical Abstract Machine, and show that the resulting testing semantics is adequate wrt testing equivalence for the lazy λ - calculus. We also present a π-calculus implementation of the graph reduction machine, and show that the resulting testing semantics is also adequate.

Timewise Refinement for Communicating Processes

Abstract: A theory of timewise refinement is presented. This allows the translation of specifications and proofs of correctness between semantic models, permitting each stage in the verification of a system to take place at the appropriate level of abstraction. The theory is presented within the context of CSP. A denotational characterisation is given in terms of relations between behaviours at different levels of abstraction, and various properties for the preservation of refinement through parallel composition are discussed. An operational characterisation is also given in terms of timed and untimed tests, and observed to coincide with the denotational characterisation.

A Strucutral Co-Induction Theorem

Abstract: The Structural Induction Theorem (Lehmann and Smyth, 1981; Plotkin, 1981) characterizes initial F-algebras of locally continuous functors F on the category of cpo's with strict and continuous maps. Here a dual of that theorem is presented, giving a number of equivalent characterizations of final coalgebras of such functors. In particular, final coalgebras are order strongly-extensional (sometimes called internal full abstractness): the order is the union of all (ordered) F-bisimulations. (Since the initial fixed point for locally continuous functors is also final, both theorems apply.) Further, a similar co-induction theorem is given for a category of complete metric spaces and locally contracting functors.

Topological Models for Higher Ordr Control Flow

Abstract: Semantic models are presented for two simple imperative languages with higher order constructs. In the first language the interesting notion is that of second order assignment x:=s, for x a procedure variable and s a statement. The second language extends this idea by a form of higher order communication, with statements c! s and c? x, for c a channel. We develop operational and denotational models for both languages, and study their relationships. Both in the definitions and the comparisons of the semantic models, convenient use is made of some tools from (metric) topology. The operational models are based on (SOS-style) transition systems; the denotational definitions use domains specified as solutions of domain equations in a category of 1-bounded complete ultrametric spaces. In establishing the connection between the two kinds of models, fruitful use is made of Rutten's processes as terms technique. Another new tool consists in the use of metric transition systems, with a metric defined on the configurations of the system. In addition to higher order programming notions, we use higher order definitional techniques, e.g., in defining the semantic mappings as fixed points of (contractive) higher order operators. By Banach's theorem, such fixed points are unique, yielding another important proof principle for our paper.

Axiomatising Real-Time Processes

Abstract: In this paper, we present a relativised compositional proof system for real-timed processes. The proof system allows us to derive statements of the form A ⊢ E = F, where processes E, F may contain free time variables and A is a formula of the first order theory of time domain. The formula A ⊢ E = F means that A is a condition for process E to be bisimilar to process F. The proof system is sound and is independent of the choice of time domain, allowing time to be discrete or dense. It is complete for finite terms, i.e. terms without recursion, over dense time domains. It is also shown complete for a sublanguage over discrete time domains. We discuss how to restrict occurrences of time variables to obtain the sublanguage. We finally discuss extensions of the proof system for recursively defined processes.

Three Metric Domains of Processes for Bisimulation

Abstract: A new metric domain of processes is presented. This domain is located in between two metric process domains introduced by De Bakker and Zucker. The new process domain characterizes the collection of image finite processes. This domain has as advantages over the other process domains that no complications arise in the definitions of operators like sequential composition and parallel composition, and that image finite language constructions like random assignment can be modelled in an elementary way. As in the other domains, bisimilarity and equality coincide in this domain.

Compositional Process Semantics of Petri Boxes

Abstract: The Petri Box algebra defines a linear notation to express a structured class of Petri nets which can be seen as a modification and generalisation of Milner's CCS. The calculus has been designed as an intermediate stage in the compositional translation of higher level concurrent programming notations into Petri nets. This paper defines the notion of a ‘Box process’ intended to capture the (Petri net) partial order semantics of the Box algebra. The main result is the equivalence of the direct compositional semantics so defined, and the indirect non-compositional semantics which uses processes of Petri nets, for a class of expressions.

Another Approach to Sequentiality: Kleene's Unimonotone Functions

Abstract: We show that Kleene's theory of unimonotone functions strictly relates to the theory of sequentiality originated by the full abstraction problem for PCF. Unimonotone functions are defined via a class of oracles, which turn out to be alternative descriptions of a subclass of Berry-Curien's sequential algorithms.

A Predicative Semantics for the Refinement of Real-Time Systems

Abstract: A formal framework for a calculus of real-time systems is presented. Specifications and program statements are combined into a single language called TAM (the Temporal Agent Model), that allows the user to express both functional and timing properties. A specification-oriented semantics for TAM is given, along with the definition of a refinement relation and a calculus which is sound with respect to that relation. A simple real-time program is also developed using the calculus.

Universal Quasi-Prime Algebraic Domains

Abstract: This paper demonstrates the existence of a saturated quasi-prime algebraic domain. It also presents a cpo of quasi-prime generated information systems for solving domain equations.

Sequential Functions on Indexed Domains and Full Abstraction for a Sub-Language of PCF

Abstract: We present a general semantic framework of sequential functions on domains equipped with a parameterized notion of incremental sequential computation. Under the simplifying assumption that computation over function spaces proceeds by successive application to constants, we construct a sequential semantic model for a non-trivial sub-language of PCF with a corresponding syntactic restriction --- that variables of function type may only be applied to closed terms. We show that the model is fully abstract for the sub-language, with respect to the usual notion of program behavior.

A Categorical Interpretation of Landin's Correspondence Principle

Abstract: Many programming languages can be studied by desugaring them into an intermediate language, namely, the simply-typed λ-calculus. In this manner Landin and Tennent discovered a “correspondence” between the semantics of definition bindings and parameter bindings such that the semantics of free identifiers becomes independent of their mode of definition.