In this chapter we generalize our notion of feature structure to allow for the possibility of having countably infinite collections of nodes. When we considered algebraic models in the last chapter, we implicitly allowed models with objects from which infinitely many distinct objects were accessible by iteratively applying feature value functions. In the case of feature structures as we have previously taken them, the set of substructures of a feature structure was always in one-to-one correspondence with the nodes of the feature structure and hence finite. In the case of finite feature structures, we were guaranteed joins or unifications for consistent finite sets of (finite) feature structures. We also saw examples of infinite sets of consistent finite feature structures which did not have a finite least upper bound. When we allow for the possibility of countably infinite feature structures, we have least upper bounds for arbitrary (possibly infinite) consistent sets of (possibly infinite) feature structures. In fact, the collection of feature structures with countable node sets turn out to form a predomain in the sense that when we factor out alphabetic variance, we are left with an algebraic countably based BCPO. Luckily, in the feature structure domain, the compact domain elements are just the finite feature structures, thus giving us a way to characterize arbitrary infinite feature structures as simple joins of finite feature structures. One benefit of such a move is that infinite or limit elements in our domains provide models for non-terminating inference procedures such as total type inference over an appropriateness specification with loops or extensionalization when non-maximal types are allowed to be extensional.

In this chapter we consider acyclic feature structures and their axiomatization. In standard first-order logic resolution theorem provers (see Wos et al. 1984 for an overview), it is necessary to make sure that when a variable X is unified with a term t, there is no occurrence of X in t. This is the so-called occurs check and dates back to Robinson's (1965) original algorithm for unification. Without the occurs check, resolution produces unsound inferences. The problem with the occurs check is that it is often expensive to compute in practical unification algorithms. Unification algorithms have been developed with built in occurs checks that are linear (Paterson and Wegman 1978) and quasi-linear (Martelli and Montanari 1982), but the data structures employed for computing the occurs check incur a heavy constant overhead (Jaffar 1984). Rather than carry out the occurs check, implementations of logic programming languages like Prolog simply omit it, leading to interpreters and compilers which are not sound with respect to the semantics of first-order logic and may furthermore cause the interpreters to hang during the processing of cyclic structures which are accidentally created. Rather than change the interpreters, the move made in Prolog II (Colmerauer 1984, 1987) was to change the semantics to allow for infinite rational trees, which correspond to the infinite unfoldings of the terms that result from unifications of a variable with a term that contains it. Considering the pointer-based implementation of unification (Jaffar 1984, Moshier 1988), infinite trees are more naturally construed as finite graphs containing cycles.

Abstract

Introduction

One of the cornerstones of programming language semantics is the correspondence between simply-typed λ-calculi and cartesian closed categories established by Lambek and Scott [LS86]. It provides a clean algebraic semantics for λ-terms, which helps us to reason effectively about the values of programs. Of equal interest to the implementer, however, is the reduction process which computes these values, i.e. the operational semantics. One of the most satisfying approaches to date is Plotkin's structural operational semantics [Plo75, Plo81] which uses a sequent calculus to describe the reduction process and provide a rigorous specification for computation.

In spite of their separate virtues, it is difficult to relate the logic (of operational semantics) to the categorical algebra (of denotational semantics); the usual approach, using computational adequacy, is rather unwieldy.

Introduction

In his seminal paper [Plo77], Plotkin introduced the functional language PCF (‘programming language for computable functions’) together with the standard denotational model of cpo's and continuous functions. He proved that this model is computationally adequate for PCF, but not fully abstract.

In order to obtain full abstraction he extended PCF by a parallel conditional operator. The problem with this operator is, that it changes the nature of the language. All computations in the original language PCF can be executed sequentially, but the new operator requires expressions to be evaluated in parallel.

Here we address the problem of full abstraction for the original sequential language PCF, i. e. instead of extending the language we try to improve the model. Our approach is to cut down the standard model with the aid of certain logical relations, which we call sequentiality relations. We give a semantic characterization of these relations and illustrate how they can be used to reason about sequential (i. e. PCF-definable) functions. Finally we prove that this style of reasoning is ‘complete’ for proving observational congruences between closed PCF-expressions of order ≤ 3. Technically, this completeness can be expressed as a full abstraction result for the sublanguage which consists of these expressions.

Sequential PCF

In [Plo77], PCF is defined as a simply typed λ-calculus over the ground types ι (of integers) and o (of Booleans).

Abstract

Fibrations have been widely used to model polymorphic λ-calculi. In this paper we describe the additional structure on a fibration that suffices to make it a model of a polymorphic λ-calculus with subtypes and bounded quantification; the basic idea is to single out a class of maps, the inclusions, in each fibre. Bounded quantification is made possible by a imposing condition that resembles local smallness. Since the notion of inclusion is not stable under isomorphism, some care must be taken to make everything strict.

We then show that PER models for λ-calculi with subtypes fit into this framework. In fact, not only PERs, but also any full reflective sub category of the category of modest sets (‘PERs’ in a realizability topos), provide a model; hence all the small complete categories of ‘synthetic domains’ found in various realizability toposes can be used to model subtypes.

Introduction

What this paper is about

At the core of object-oriented programming, and related approaches to programming, are the notions of subtyping and inheritance. These have proved to be very powerful tools for structuring programs, and they appear—in one form or another—in a wide variety of modern programming languages.

One way of studying these notions formally is to use the framework of typed λ-calculus. That is, we start with a formal system (say, a version of the polymorphic λ-calculus) and extend it by adding a notion of type inclusion, together with suitable rules; we obtain a larger system. We can then use the methods of mathematical logic to study the properties of the system.

Abstract

We explore some foundational issues in the development of a theory of intensional semantics, in which program denotations may convey information about computation strategy in addition to the usual extensional information. Beginning with an “extensional” category C, whose morphisms we can think of as functions of some kind, we model a notion of computation using a comonad with certain extra structure and we regard the Kleisli category of the comonad as an intensional category. An intensional morphism, or algorithm, can be thought of as a function from computations to values, or as a function from values to values equipped with a computation strategy. Under certain rather general assumptions the underlying category C can be recovered from the Kleisli category by taking a quotient, derived from a congruence relation that we call extensional equivalence. We then focus on the case where the underlying category is cartesian closed. Under further assumptions the Kleisli category satisfies a weak form of cartesian closure: application morphisms exist, currying and uncurrying of morphisms make sense, and the diagram for exponentiation commutes up to extensional equivalence. When the underlying category is an ordered category we identify conditions under which the exponentiation diagram commutes up to an inequality. We illustrate these ideas and results by introducing some notions of computation on domains and by discussing the properties of the corresponding categories of algorithms on domains.

Introduction

Most existing denotational semantic treatments of programming languages are extensional, in that they abstract away from computational details and ascribe essentially extensional meanings to programs.

Introduction

There has been considerable recent interest in the use of algebraic methodologies to define and elucidate constructions in fixed point semantics [B], [FMRS], [Mu2]. In this paper we present recent results utilizing categorical methods, particularly strong monads, algebras and dinatural transformations to build general fixed point operators. The approach throughout is to evolve from the specific to the general case by eventually discarding the particulars of domains and continuous functions so often used in this setting. Instead we rely upon the structure of strong monads and algebras to provide a general algebraic framework for this discussion. This framework should provide a springboard for further investigations into other issues in semantics.

By way of background, the issues raised in this paper find their origins in several different sources. In [Mu2] the formal role of iteration in a cartesian closed category (ccc) with fixed points was investigated. This was motivated by the observation in [H-P] that the presence of a natural number object (nno) was inconsistent with ccc's and fixed points. This author introduced the notion of onno (ordered nno) which in semantic categories played a role as iterator and was precisely the initial T-algebra for T the strong lift monad. Using the onno a factorization of fix was produced and further it was shown fix was in fact a dinatural transformation. This was accomplished by avoiding the traditional projection/embedding approach to semantics. Similar results were extended to order-enriched and effective settings as well.

Turning to monads, their role in computation is not new. In particular, it was emphasized early in the development of topos theory that the partial map classifier was a strong monad.

Abstract

Domain theoretic understanding of databases as elements of powerdomains is modified to allow multisets of records instead of sets. This is related to geometric theories and classifying toposes, and it is shown that algebraic base domains lead to algebraic categories of models in two cases analogous to the lower (Hoare) powerdomain and Gunter's mixed powerdomain.

Terminology

Throughout this paper, “domain” means algebraic poset – not necessarily with bottom, nor second countable. The information system theoretic account of algebraic posets fits very neatly with powerdomain constructions. Following Vickers [90], it may be that essentially the same methods work for continuous posets; but we defer treating those until we have a better understanding of the necessary generalizations to topos theory.

More concretely, a domain is a preorder (information system) (D, ⊆) of tokens, and associated with it are an algebraic poset pt D of points (ideals of D; one would normally think of pt D as the domain), and a frame ΩD of opens (upper closed subsets of D; ΩD is isomorphic to the Scott topology on pt D).

“Topos” always means “Grothendieck topos”, and not “elementary topos” morphisms between toposes are understood to be geometric morphisms.

S, italicized, denotes the category of sets.

We shall follow, usually without comment, the notation of Vickers [89], which can be taken as our standard reference for the topological and localic notions used here.

Abstract

Introduction

The straightforward way to describe the simply typed λ-calculus (denoted here by λ1) categorically is in terms of cartesian closed categories (CCC's), see [10]. On the type theoretic side this caused some discomfort, because one commonly uses only exponent types without assuming cartesian product types — let alone unit (i.e. terminal) types. The typical reply from category theory is that one needs cartesian products in order to define exponents. Below we give a categorical description of exponent types without assuming cartesian product types. We do use cartesian products of contexts; these always exist by concatenation. Thus both sides can be satisfied by carefully distinguishing between cartesian products of types and cartesian products of contexts. We introduce an appropriate categorical structure for doing so.

In [6] one can find a general method to describe typed λ-calculi categorically. One of the main organizing principles used there is formulated below. It deserves the status of a slogan.

Abstract

Introduction

There are many situations in logic, theoretical computer science, and category theory where two binary operations, “tensor products” (though one may be a “sum”), play a key role. The multiplicative fragment of linear logic is a particularly interesting example as it is a Gentzen style sequent calculus in which the structural rules of contraction, thinning, and (sometimes) exchange are dropped. The fact that these rules are omitted considerably simplifies the derivation of the cut elimination theorem. Furthermore, the proof theory of this fragment is interesting and known [Se89] to correspond to *-autonomous categories as introduced by Ban in [Ba79].

In the study of categories with two tensor products one usually assumes a distributivity condition, particularly in the case when one of these is either the product or sum.

Up to this point, the semantics of the commands is determined by the relation between precondition and postcondition. This point of view is too restricted for the treatment of concurrent programs and reactive systems. The usual example is that of an operating system which is supposed to perform useful tasks without ever reaching a postcondition.

For this purpose, the semantics of commands must be extended by consideration of conditions at certain moments during execution. We do not want to be forced to consider all intermediate states or to formalize sequences of intermediate states. We have chosen the following level of abstraction. To every procedure name h, a predicate z.h is associated. The temporal semantic properties of a command q depend on the values of z.h.x for the procedure calls, say of procedure h in state x, induced by execution of command q. The main properties are ‘always’ and ‘eventually’, which are distinguished by the question whether z.h.x should hold for all induced calls or for at least one induced call. The concept of ‘always’ is related to stability and safety. The concept of ‘eventually’ is related to progress and liveness.

In this chapter, we regard nontermination of simple commands as malfunctioning and nontermination of procedures as potentially useful infinite behaviour. We therefore use wp for the interpretation of simple commands and wlp for procedure calls.

This book is about programs as mathematical objects. We focus on one of the aspects of programs, namely their functionality, their meaning or semantics. Following Dijkstra we express the semantics of a program by the weakest precondition of the program as a function of the postcondition. Of course, programs have other aspects, like syntactic structure, executability and (if they are executable) efficiency. In fact, perhaps surprisingly, for programming methodology it is useful to allow a large class of programs, many of which are not executable but serve as partially implemented specifications.

Weakest preconditions are used to define the meanings of programs in a clean and uniform way, without the need to introduce operational arguments. This formalism allows an effortless incorporation of unbounded nondeterminacy. Now programming methodology poses two questions. The first question is, given a specification, to design a general program that is proved to meet the specification but need not be executable or efficient, and the second question is to transform such a program into a more suitable one that also meets the specification.

We do not address the methodological question how to design, but we concentrate on the mathematical questions concerning semantic properties of programs, semantic equality of programs and the refinement relation between programs. We provide a single formal theory that supports a number of different extensions of the basic theory of computation. The correctness of a program with respect to a specification is for us only one of its semantic properties.

This chapter is devoted to the formal definition of the semantics of sequential composition, unbounded choice and recursion and to the proofs of the properties of commands that were introduced and used in the previous chapters. The semantics of the simple commands is taken for granted, but otherwise the reader should not rely on old knowledge but only use facts that have already been justified in the new setting. At the end of the chapter, the foundations of the previous chapters will be complete.

Some examples of the theory are given in the exercises at the end of the chapter. The text of the chapter has almost no examples. One reason is that the chapters 1, 2 and 3 may be regarded as examples of the theory. On the other hand, every nontrivial example tends to constitute additional theory.

In Section 4.1, we introduce complete lattices and investigate the lattice of the predicate transformers and some important subsets. Section 4.2 contains our version of the theorem of Knaster–Tarski. A syntactic formalism for commands with unbounded choice is introduced in Section 4.3.

Section 4.4 contains the main definition. From the definition on simple commands, the functions wp and wlp are extended to procedure names and command expressions. In Sections 4.5 and 4.6, the healthiness laws, which are postulated for simple commands, are extended to procedure names and command expressions.

In this chapter we fulfil the remaining proof obligations of Chapter 12. Section 13.1 contains a strengthened version of Theorem 4(8), our version of the theorem of Knaster-Tarski. In Section 13.2, we provide the basic set-up, in which we need not yet distinguish between wp and wlp. Section 13.3 contains the construction of the strong preorder and the proofs of rule 12(4) and a variation of rule 12(5). In this way, the proof of the accumulation rule 12(5) is reduced to the verification of two technical conditions: sup-safety (for wp) and inf-safety (for wlp). These conditions comprise the base case of the induction and a continuity property.

In Section 13.4, the base case is reduced to a condition on function abort⊙. Section 13.5 contains the proof for inf-safety. Section 13.6 contains the definition of the set Lia and the proof for sup-safety. In Sections 13.7 and 13.8 we justify the rules for Lia stated in Section 11.2.

It may seem unsatisfactory that, in the presence of unbounded nondeterminacy, computational induction needs such a complicated theory. The examples in Sections 11.7 and 12.4, however, show that the accumulation rules 11(6) and 12(5) need their complicated conditions. Therefore, corresponding complications must occur in the construction or in the proofs.