Complete Partial Orders Research Articles

A HOL Basis for Reasoning about Functional Programs

Domain theory is the mathematical theory underlying denotational semantics. This thesis presents a formalization of domain theory in the Higher Order Logic (HOL) theorem proving system along with a mechanization of proof functions and other tools to support reasoning about the denotations of functional programs. By providing a fixed point operator for functions on certain domains which have a special undefined (bottom) element, this extension of HOL supports the definition of recursive functions which are not also primitive recursive. Thus, it provides an approach to the long-standing and important problem of defining non-primitive recursive functions in the HOL system.<br /> <br />Our philosophy is that there must be a direct correspondence between elements of complete partial orders (domains) and elements of HOL types, in order to allow the reuse of higher order logic and proof infrastructure already available in the HOL system. Hence, we are able to mix domain theoretic reasoning with reasoning in the set theoretic HOL world to advantage, exploiting HOL types and tools directly. Moreover, by mixing domain and set theoretic reasoning, we are able to eliminate almost all reasoning about the bottom element of complete partial orders that makes the LCF theorem prover, which supports a first order logic of domain theory, difficult and tedious to use. A thorough comparison with LCF is provided.<br /> <br />The advantages of combining the best of the domain and set theoretic worlds in the same system are demonstrated in a larger example, showing the correctness of a unification algorithm. A major part of the proof is conducted in the set theoretic setting of higher order logic, and only at a late stage of the proof domain theory is introduced to give a recursive definition of the algorithm, which is not primitive recursive. Furthermore, a total well-founded recursive unification function can be defined easily in pure HOL by proving that the unification algorithm (defined in domain theory) always terminates; this proof is conducted by a non-trivial well-founded induction. In such applications, where non-primitive recursive HOL functions are defined via domain theory and a proof of termination, domain theory constructs only appear temporarily.

Software merge

We present a language-independent semantic model of the process of combining changes to programs. This model extends the domains used in denotational semantics (complete partial orders) to Boolean algebras, and represents incompatible modifications as well as compatible extensions. The model is used to define the intended semantics of change-merging operations on programs and to establish some general properties of software merging. We determine conditions under which changes to subprograms of a software system can be merged independently and illustrate cases where this is not possible.

The comparison of a cpo-based semantics with a cms-based semantics for CSP

We study two semantics for CSP. Both semantics are compositional and interleaving. The first uses the concept of complete partial orders (cpo) for the handling of repetitive constructs and for the creation of suitable semantic domains. The second uses the concept of complete metric spaces (cms) for the same purposes. Our results show that the cms-semantics gives more detailed information about a program and that we can obtain the cpo-semantics from it. From this investigation we can draw conclusions concerning the influence of the mathematical discipline on the kind of semantics obtained.

Banach's Fixed-Point Theorem as a base for data-type equations

For a categoryK of data types, solutions of recursive data-type equationsX ℞T(X), whereT is an endofunctor ofK, can be constructed by iteratingT on the unique arrowT1 → 1. This is well-known forK enriched over complete posets and forT locally continuous (an application of the Kleene Fixed-Point Theorem). We prove this forK enriched over complete metric spaces and forT contracting (an application of the Banach Fixed-Point Theorem). Moreover, we prove that each such recursive equation has a unique solution. Our results generalize the approach of P. America and J. Rutten.

Set-theoretical and other elementary models of the λ-calculus

Part I (pp. 351–373) of this paper is the previously unpublished 1972 memorandum (Plotkin, 1972) with editorial changes and some minor corrections. Part II (pp. 373–409) presents what happened next, together with some further development of the material. The first part begins with an elementary set-theoretical model of the λβ-calculus. Functions are modelled in a similar way to that normally employed in set theory, by their graphs; difficulties are caused in this enterprise by the axiom of foundation. Next, based on that model, a model of the λβη-calculus is constructed by means of a natural deduction method. Finally, a theorem is proved giving some general properties of those nontrivial models of the λβη-calculus which are continuous complete lattices. The second part begins with a brief discussion of models of the λ-calculus in set theories with anti-foundation axioms. Next the model of the λβ-calculus of Part I and also the closely related — but different! — models of Scott (1976, 1980) and of Engeler (1981, 1988) are reviewed. Then general frameworks in which elementary constructions of models can be given are discussed. Following Longo (1982), one can employ certain Scott-Engeler algebras. Following Coppo et al. (1983), one can obtain filter models from their extended applicative type structures. An extended discussion is given of various ways of constructing models of the λβη-calculus, and the connections between them. Finally an extension of the theorem to complete partial orders is given. The theme of the paper is the consideration of means of constructing models. There is hardly any analysis of their properties; there is no discussion of their application.

Historical introduction to “concrete domains” by G. Kahn and G.D. Plotkin

The original version of this paper appeared as a technical report 15 years ago [28], based on the work carried out at Edinburgh University in the Fall of 1975. The ideas and concepts introduced in this work have been widely referenced and have become the basis for a significant body of foundational research on semantics of programming languages. The purpose of this introduction is to sketch the background for this research and to mention briefly some of this later work, thus placing the paper in context. Christopher Strachey’s [40] work on the formal description of programming language concepts raised problems concerning the nature and existence of semantic models for /2-calculi. Dana Scott developed a theory of domains and continuous functions that established adequate mathematical foundations for Strachey’s semantic descriptions and formed the basis for the denotational approach to semantics, ad- vocated in [37] and later applied to a wide variety of programming languages [39,34,42,24]. Semantic domains were typically taken to be complete lattices [38] or more general kinds of complete partial order [38, 251, the precise combination of order-theoretic assumptions being chosen to suit the application and often justified by appealing to intuitive arguments concerning computability. A good survey of domain theory is provided in [24]. However, these semantic treatments typically make no intrinsic distinction between domains used to represent data and domains used to represent procedures or func- tions. For certain purposes it seems appropriate to make such distinctions: for

Parallel asynchronous iterations of least fixed points

Many problems on discrete data are modeled as systems of recursive equations on some posets, which can usually be solved by iterative algorithms. In a parallel system, asynchronous iterative algorithms are more efficient than synchronous ones because most synchronization overheads are eliminated. In this paper, the convergence of arbitrary asynchronous iterations for problems expressed as monotone equations on some ω −-complete posets is proved. An asynchronous iteration is characterized by representation functions, and the convergence of the iteration is then related to properties of the corresponding representation functions. As an example, the convergence of parallel asynchronous iterations of program flow analysis problems is shown.

A remark about algebraicity in complete partial orders

A characterization theorem for algebraic bounded complete cpos is proved similar to that for algebraic lattices.

The core of a chain complete poset with no one-way infinite fence and no tower

Like dismantling for finite posets, a perfect sequence Π = 〈P ξ :ξ≤λ〉 of a chain complete posetP represents a canonical procedure to produce a coreP λ. It has been proved that if the posetP contains no infinite antichain then this coreP λ is a retract ofP andP has the fixed point property iffP λ has this property. In this paper the condition of having no infinite antichain is replaced by a weaker one. We show that the same conclusion holds under the assumption thatP does not contain a one-way infinite fence or a tower.

A chain complete poset with no infinite antichain has a finite core

It is well known that dismantling a finite posetP leads to a retract, called the core ofP, which has the fixed-point property if and only ifP itself has this property. The PT-order, or passing through order, of a posetP is the quasi order ⊴ defined onP so thata⊴b holds if and only if every maximal chain ofP which passes througha also passes throughb. This leads to a generalization of the dismantling procedure which works for arbitrary chain complete posets which have no infinite antichain. We prove that such a poset also has a finite core, i.e. a finite retract which reflects the fixed-point property forP.

DI-Domains as prime information systems

This paper introduces prime information systems. Prime information systems determine dI-domains and stable approximable mappings give stable functions, and vice versa. The notion of rigid embedding is captured by a subsystem relation. Under this relation, prime information systems form a complete partial order (cpo). Constructions like lifting, sum, product, and function space are proposed which induce continuous functions on the cpo. In this way recursive prime information systems can be defined using fixed point theory.

An approximation theorem for topological lambda models and the topological incompleteness of lambda calculus

It is well known that a reflexive object in the Cartesian closed category of complete partial orders and Scott-continuous functions is a model of λ-calculus (briefly a topological model). A topological model, through the interpretation function, induces a λ-theory, i.e., a congruence relation on λ-terms closed under α- and β-reduction. It is natural to ask if all possible λ-theories are induced by a topological model, i.e., if topological models are complete w.r.t. λ-calculus. The authors prove an Approximation Theorem, which holds in all topological models. Using this theorem, they analyze some topological models and their induced λ-theories, and they exhibit a λ-theory which cannot be induced by a topological model. So they prove that topological models are not complete w.r.t. λ-calculus.

The PT-order and the fixed point property

The PT-order, or passing through order, of a poset P is a quasi order ⊴ defined on P so that a⊴b holds if and only if every maximal chain of P which passes throug a also passes through b. We show that if P is chain complete, then it contains a subset X which has the properties that (i) each element of X is ⊴-maximal, (ii) X is a ⊴-antichain, and (iii) X is ⊴-dominating; we call such a subset a ⊴-good subset of P. A ⊴-good subset is a retract of P and any two ⊴-good subsets are order isomorphic. It is also shown that if P is chain complete, then it has the fixed point property if and only if a ⊴-good subset also has the fixed point property. Since a retract of a chain complete poset is also chain complete, the construction may be iterated transfinitely. This leads to the notion of the “core” of P (a ⊴-good subset of itself) which is the transfinite analogue of the core of a finite poset obtained by dismantling.

The fixed point theory of unbounded non-determinism

Abstract This paper presents a fixed point theorem for the infinite traces model of CSP. Unlike any other model of CSP, there is no complete partial order over the infinite traces model whose fixed point theory agrees with the operational semantics (A. W. Roscoe, Oxford University Computing Laboratory Technical Monograph PRG-67, 1988). This arises from the introduction of unbounded non-determinism. However, the subset of pre-deterministic processes, that is those which describe the behaviour of a process on some run of its implementation, do form a subset for which the usual order is complete. By requiring that each CSP operator has a monotonic implementation which preserves pre-determinism, it is possible to show that all CSP operators have a least fixed point. In effect, it is the requirement that all operators have a methodical implementation.

Using information systems to solve recursive domain equations

This paper aims to make the following main contribution: to show how to use the concrete nature of Scott's information systems to advantage in solving recursive domain equations. The method is based on the substructure relation between information systems. This essentially makes a complete partial order (cpo) of information systems. Standard domain constructions like function space can be made continous on this cpo so the solution of recursive domain equations reduces to the more familiar construction of forming the least fixed-point of a continuous function.

Finite antichain cutsets in posets

In 1985 I. Rival and N. Zaguia conjectured that in a chain complete poset having the finite cutset property every element belongs to an antichain cutset iff there is no alternating-cover cycle. In this paper a counterexample is given to show that the condition that there is no alternating-cover cycle is not sufficient and other two conditions on generalized alternating-cover paths are added to complete the characterization.

Fixed points for extrema of contractions

We consider a family of order-preserving contractions on a complete metric space equipped with a compatible, complete partial order. If an extremum (infimum or supremum) of the contractions exists as an order-preserving contraction, we show that the fixed point of the extremum contraction is less than or equal to the extremum of the fixed points of the individual contractions. We also give reasonable sufficient conditions for equality to hold. Given suitable starting points in the metric space, we show that the fixed point of the extremum can be monotonically approximated from above and below, thus yielding an algorithm for approximating this fixed point within a prescribed tolerance. We conclude by showing how our results may be applied to Markov decision models and families of integral equations.

Computable one-to-one enumerations of effective domains

In this paper ω-algebraic complete partial orders are considered the compact elements of which are not maximal in the partial order. Under the assumption that these elements are indexed such that their equality is decidable and the order is semi-decidable (completely enumerable) it is shown that the computable domain elements can be effectively enumerated without repetition. Computable one-to-one enumerations of the computable domain elements are minimal among all enumerations of these elements with respect to the reducibility of one enumeration to another. In computability studies of continuous complete partial orders one usually uses a generalization of Gödel numberings, called admissible numberings. They are maximal among the computable enumerations. As it is shown, each admissible numbering is recursively isomorphic to the directed sum of a computable family of computable one-to-one enumerations. Both results generalize well-known theorems of Friedberg and Schinzel respectively for the partial recursive functions. Their premise is satisfied by each type in the hierarchy of the Eršov-Scott higher type partial computable functionals, which means that any such type can be computably enumerated without repetition and any of its admissible indexings is recursively isomorphic to the directed sum of a computable family of computable one-to-one enumerations. The proofs use a priority argument.

Event structures and domains

In the theory of denotational semantics, we study event structures which generalize Kahn and Plotkin's concrete data structures and which model computational processes. With each event structure we associate canonically an event domain (a particular algebraic complete partial order), and conversely we derive a representation result for event domains. For a particular class of event structures, the canonical event structures, we obtain that any two canonical event structures are isomorphic iff they have order-isomorphic canonical domains.

Generalizations of Tarski's fixed point theorem for order varieties of complete meet semilattices

A poset P is μ-conditionally complete (μ a cardinal) if every set X\( \subseteq \)P all of whose subsets of cardinality <μ have an upper bound has a least upper bound. For μ ≤ σ we characterize the subposets of a μ-complete poset which can occur as the set of fixed points of some montonic function on P. This yields a generalization of Tarski's fixed point theorem. We also show that for every μ the class of μ-conditionally complete posets forms an order variety and we exhibit a simple generating poset for each such class.