DOCSLIB.ORG

Categories and Types for Axiomatic Domain Theory Adam Eppendahl

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Categories and Types for Axiomatic Domain Theory Adam Eppendahl Department of Computer Science Research Report No. RR-03-05 ISSN 1470-5559 December 2003 Categories and Types for Axiomatic Domain Theory Adam Eppendahl 1 Categories and Types for Axiomatic Domain Theory Adam Eppendahl Submitted for the degree of Doctor of Philosophy University of London 2003 2 3 Categories and Types for Axiomatic Domain Theory Adam Eppendahl Abstract Domain Theory provides a denotational semantics for programming languages and calculi con- taining fixed point combinators and other so-called paradoxical combinators. This dissertation presents results in the category theory and type theory of Axiomatic Domain Theory. Prompted by the adjunctions of Domain Theory, we extend Benton’s linear/nonlinear dual- sequent calculus to include recursive linear types and define a class of models by adding Freyd’s notion of algebraic compactness to the monoidal adjunctions that model Benton’s calculus. We observe that algebraic compactness is better behaved in the context of categories with structural actions than in the usual context of enriched categories. We establish a theory of structural algebraic compactness that allows us to describe our models without reference to en- richment. We develop a 2-categorical perspective on structural actions, including a presentation of monoidal categories that leads directly to Kelly’s reduced coherence conditions. We observe that Benton’s adjoint type constructors can be treated individually, semantically as well as syntactically, using free representations of distributors. We type various of fixed point combinators using recursive types and function types, which we consider the core types of such calculi, together with the adjoint types. We use the idioms of these typings, which include oblique function spaces, to give a translation of the core of Levy’s Call-By-Push-Value. The translation induces call-by-value and call-by-name translations of the core of Plotkin’s Fixed Point Calculus. Following Freyd, we construct a canonical fixed point operation from the algebras provided by the algebraic compactness of our models. Our analysis of Freyd’s construction exposes a remarkable property of morphisms from coalgebras to algebras: morphisms from Gp to s corre- spond one-for-one to morphisms from p to Hs, where p is a coalgebra for HG and s an algebra for GH. We give an application of this property to the transposition of recursive coalgebras in Taylor’s categorical theory of recursion where G is not left adjoint to H. We develop a theory of parametric transformations corresponding to the uniformity property characterizing canonical fixed points and use this to derive abstract conditions on categories of domains which ensure that the interpretation of fixed point combinators coincides with the canonical fixed point operation. Submitted for the degree of Doctor of Philosophy University of London 2003 4 Contents 1 Introduction 6 1.1 Background . 6 1.1.1 Axiomatic Domain Theory . 6 1.1.2 Fixed Points . 8 1.1.3 Categorical Semantics of Linear and Computation Types . 8 1.2 Main Results . 9 1.2.1 Recursive Morphisms . 9 1.2.2 Structural Algebraic Compactness . 9 1.2.3 Adjoint Types and Free Distributor Representations . 10 1.2.4 Quotient Relations . 11 1.3 Additional Contributions . 11 1.3.1 Minimal Coherence . 11 1.3.2 Fixed Point Algebras . 12 1.3.3 Exponential Structure . 12 1.3.4 Oblique Types . 13 1.3.5 Directors . 13 1.4 Overview . 13 1.4.1 Category Theory . 13 1.4.2 Type Theory . 14 2 Recursive Morphisms 16 2.1 Oblique Adjunctions for Free . 17 2.1.1 The Distributor of Recursive Morphisms . 17 2.1.2 Two Oblique Adjunctions . 18 2.2 Recursive Coalgebras and Corecursive Algebras . 20 2.2.3 Preservation of Recursive Coalgebras . 21 2.2.5 Freyd’s Square for Recursive Coalgebras . 21 2.2.9 Transposition of Corecursive Algebras and Recursive Coalgebras . 22 3 Monoidal Categories and Structural Actions 25 3.1 Lax Monoids and Comonoids . 25 3.1.1 Lax Monoid 0-Cells . 25 3.1.3 Lax Monoid 1-Cells . 26 3.1.8 Lax Comonoids . 32 3.1.10 Preservation of Internal Comonoids . 35 3.2 Monoidal Categories . 35 3.2.1 The Monoidal Category of Endofunctors . 35 3.2.2 The Transpose of Lax Monoid Structure . 36 3.2.3 Coherent Lax Monoids and Monoidal Categories . 36 3.3 Structural Actions . 37 3.3.3 Diagonal Structure . 38 3.3.6 The Indexed Category Construction . 38 5 4 Structural Algebraic Compactness 41 4.1 Algebraic Compactness in the Structural Setting . 42 4.1.3 The structurality of delivery functors. 42 4.1.12 The structural compactness of opposites and doubles. 44 4.2 Compactness in Various Settings . 45 4.2.1 Indexed Compactness . 45 4.2.4 Structural Compactness and Indexed Compactness . 46 4.2.8 Enriched Compactness and Indexed Compactness . 46 4.2.12 Structural Compactness Meets Enriched Compactness . 48 4.3 Structurally Algebraically Compact Categories of Domains . 48 4.3.1 Categories of Pointed Objects . 48 4.3.5 Categories of Partial Maps . 49 5 Structural Adjunctions 50 5.1 Monoidal Adjunctions as Structural Adjunctions . 50 5.1.3 Closed Structure . 51 5.1.8 Cartesian Structure . 51 5.1.13 Symmetric Structure . 52 5.2 Structural Adjunctions Proper . 52 5.2.2 Structurality and Balance . 53 5.2.6 Exponential Structure . 53 5.3 Structural Adjunctions on Categories of Domains . 55 5.3.1 Categories of Eilenberg-Moore Algebras . 55 5.3.4 Categories of Kleisli Maps . 55 6 Canonical Fixed Points 57 6.1 Corecursive Algebras and Unique Fixed Points . 57 6.1.2 Very Unique Fixed Points and Unique Fixed Generalized Points . 58 6.1.7 Comonoids and Parameterized Unique Fixed Points . 60 6.1.10 Comonads and Parameterized Unique Fixed Points . 61 6.2 Fixed Point Algebras . 62 6.2.3 Uniform Families of Recursive Morphisms . 62 6.2.8 Transposition of Fixed Point Algebras . 63 6.2.11 Fixed Point Objects as Fixed Point Algebras . 65 6.2.17 Freyd Algebras as Fixed Point Algebras . 66 6.3 Canonical Fixed Points in Compact Structural Adjunctions . 67 6.3.1 Ordinary Fixed Points . 67 6.3.2 Parameterized Fixed Points . 67 6.3.3 Internal Fixed Points . 68 6.3.4 Uniform Fixed Points . 68 7 Recursive Linear Types 70 7.1 A Recursive Linear/NonLinear Calculus . 70 7.1.1 Linear and Nonlinear Lambda Calculi . 71 7.1.2 Admissible Weakenings and Contractions . 72 7.1.3 Pairing and Units . 73 7.1.4 Recursive Linear Types . 73 7.2 Oblique Terms and Types . 78 7.2.1 The Term Distributor . 78 7.2.2 The Adjoint Fragments of LNL . 78 7.2.5 The Oblique Function Types . 80 6 Contents 8 Recursive Types for Fixed Point Combinators 83 8.1 Recursive Combinators in Recursive Types . 83 8.1.1 Recursive Combinators in FPC . 85 8.1.2 Recursive Combinators in RLNL . 89 8.2 A Linear Fixed Point Calculus . 98 8.2.1 Fixed Point Idioms . 98 8.2.2 Translating FPC into RLNL . 100 8.2.3 Parameterized Adjoint Types . 101 9 Quotient Relations and Parametric Models 106 9.1 Quotient Diparametricity . 106 9.1.1 Quotient Relations . 106 9.1.5 Push-me-pull-you’s . 108 9.1.10 Componentwise Liftings . 110 9.1.12 Uniformity as Quotient Diparametricity . 113 9.2 Domain Theoretic Models of Recursive Linear Types . 116 9.2.1 RLNL Models . 116 9.2.5 Recursive Types in Term Models . 117 9.2.7 Uniformity in RLNL Models . 119 9.2.9 Fixed Point Combinators in Domain Theoretic Models . 120 10 Directions for Future Work 121 10.1 Fox’s Construction and Lemma ?? . 121 10.2 The Free Adjunction on the Distributor Classifier . 121 10.3 Logical Relations . 121 10.4 Other Fixed Point Transformations . 122 10.5 Other Recursive Types . 122 10.6 Parametricity and Compactness . 123 10.7 Models of closed LNL. 123 10.8 2-Categories for Types . 123 Bibliography 125 A Distributors and Directors 128 A.1 Distributors.
Load more

Recommended publications
  • Topology, Domain Theory and Theoretical Computer Science
    Draft URL httpwwwmathtulaneedumwmtopandcspsgz pages Top ology Domain Theory and Theoretical Computer Science Michael W Mislove Department of Mathematics Tulane University New Orleans LA Abstract In this pap er we survey the use of ordertheoretic top ology in theoretical computer science with an emphasis on applications of domain theory Our fo cus is on the uses of ordertheoretic top ology in programming language semantics and on problems of p otential interest to top ologists that stem from concerns that semantics generates Keywords Domain theory Scott top ology p ower domains untyp ed lamb da calculus Subject Classication BFBNQ Intro duction Top ology has proved to b e an essential to ol for certain asp ects of theoretical computer science Conversely the problems that arise in the computational setting have provided new and interesting stimuli for top ology These prob lems also have increased the interaction b etween top ology and related areas of mathematics such as order theory and top ological algebra In this pap er we outline some of these interactions b etween top ology and theoretical computer science fo cusing on those asp ects that have b een most useful to one particular area of theoretical computation denotational semantics This pap er b egan with the goal of highlighting how the interaction of order and top ology plays a fundamental role in programming semantics and related areas It also started with the viewp oint that there are many purely top o logical notions that are useful in theoretical computer science that
    [Show full text]
  • Domain Theory: an Introduction
    Domain Theory: An Introduction Robert Cartwright Rebecca Parsons Rice University This monograph is an unauthorized revision of “Lectures On A Mathematical Theory of Computation” by Dana Scott [3]. Scott’s monograph uses a formulation of domains called neighborhood systems in which finite elements are selected subsets of a master set of objects called “tokens”. Since tokens have little intuitive significance, Scott has discarded neighborhood systems in favor of an equivalent formulation of domains called information systems [4]. Unfortunately, he has not rewritten his monograph to reflect this change. We have rewritten Scott’s monograph in terms of finitary bases (see Cartwright [2]) instead of information systems. A finitary basis is an information system that is closed under least upper bounds on finite consistent subsets. This convention ensures that every finite answer is represented by a single basis object instead of a set of objects. 1 The Rudiments of Domain Theory Motivation Programs perform computations by repeatedly applying primitive operations to data values. The set of primitive operations and data values depends on the particular programming language. Nearly all languages support a rich collection of data values including atomic objects, such as booleans, integers, characters, and floating point numbers, and composite objects, such as arrays, records, sequences, tuples, and infinite streams. More advanced languages also support functions and procedures as data values. To define the meaning of programs in a given language, we must first define the building blocks—the primitive data values and operations—from which computations in the language are constructed. Domain theory is a comprehensive mathematical framework for defining the data values and primitive operations of a programming language.
    [Show full text]
  • Spectral Sets
    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector JOURNAL OF PURE AND APPLIED ALGEBRA ELSE&R Journal of Pure and Applied Algebra 94 (1994) lOlL114 Spectral sets H.A. Priestley Mathematical Institute, 24129 St. Giles, Oxford OXI 3LB, United Kingdom Communicated by P.T. Johnstone; received 17 September 1993; revised 13 April 1993 Abstract The purpose of this expository note is to draw together and to interrelate a variety of characterisations and examples of spectral sets (alias representable posets or profinite posets) from ring theory, lattice theory and domain theory. 1. Introduction Thanks to their threefold incarnation-in ring theory, in lattice theory and in computer science-the structures we wish to consider have been approached from a variety of perspectives, with rather little overlap. We shall seek to show how the results of these independent studies are related, and add some new results en route. We begin with the definitions which will allow us to present, as Theorem 1.1, the amalgam of known results which form the starting point for our later discussion. The theorem collects together ways of describing the prime spectra of commutative rings with identity or, equivalently, of distributive lattices with 0 and 1. A topology z on a set X is defined to be spectral (and (X; s) called a spectral space) if the following conditions hold: (i) ? is sober (that is, every non-empty irreducible closed set is the closure of a unique point), (ii) z is quasicompact, (iii) the quasicompact open sets form a basis for z, (iv) the family of quasicompact open subsets of X is closed under finite intersections.
    [Show full text]
  • Domain Theory Corrected and Expanded Version Samson Abramsky1 and Achim Jung2
    Domain Theory Corrected and expanded version Samson Abramsky1 and Achim Jung2 This text is based on the chapter Domain Theory in the Handbook for Logic in Computer Science, volume 3, edited by S. Abramsky, Dov M. Gabbay, and T. S. E. Maibaum, published by Clarendon Press, Oxford in 1994. While the numbering of all theorems and definitions has been kept the same, we have included comments and corrections which we have received over the years. For ease of reading, small typo- graphical errors have simply been corrected. Where we felt the original text gave a misleading impression, we have included additional explanations, clearly marked as such. If you wish to refer to this text, then please cite the published original version where possible, or otherwise this on-line version which we try to keep available from the page http://www.cs.bham.ac.uk/∼axj/papers.html We will be grateful to receive further comments or suggestions. Please send them to [email protected] So far, we have received comments and/or corrections from Francesco Consentino, Joseph D. Darcy, Mohamed El-Zawawy, Weng Kin Ho, Klaus Keimel, Olaf Klinke, Xuhui Li, Homeira Pajoohesh, Dieter Spreen, and Dominic van der Zypen. 1Computing Laboratory, University of Oxford, Wolfson Building, Parks Road, Oxford, OX1 3QD, Eng- land. 2School of Computer Science, University of Birmingham, Edgbaston, Birmingham, B15 2TT, England. Contents 1 Introduction and Overview 5 1.1 Origins ................................. 5 1.2 Ourapproach .............................. 7 1.3 Overview ................................ 7 2 Domains individually 10 2.1 Convergence .............................. 10 2.1.1 Posetsandpreorders .. .... .... ... .... .... 10 2.1.2 Notation from order theory .
    [Show full text]
  • Domain Theory Corrected and Expanded Version Samson Abramsky1 and Achim Jung2
    Domain Theory Corrected and expanded version Samson Abramsky1 and Achim Jung2 This text is based on the chapter Domain Theory in the Handbook of Logic in Com- puter Science, volume 3, edited by S. Abramsky, Dov M. Gabbay, and T. S. E. Maibaum, published by Clarendon Press, Oxford in 1994. While the numbering of all theorems and definitions has been kept the same, we have included comments and corrections which we have received over the years. For ease of reading, small typo- graphical errors have simply been corrected. Where we felt the original text gave a misleading impression, we have included additional explanations, clearly marked as such. If you wish to refer to this text, then please cite the published original version where possible, or otherwise this on-line version which we try to keep available from the page http://www.cs.bham.ac.uk/˜axj/papers.html We will be grateful to receive further comments or suggestions. Please send them to [email protected] So far, we have received comments and/or corrections from Liang-Ting Chen, Francesco Consentino, Joseph D. Darcy, Mohamed El-Zawawy, Miroslav Haviar, Weng Kin Ho, Klaus Keimel, Olaf Klinke, Xuhui Li, Homeira Pajoohesh, Dieter Spreen, and Dominic van der Zypen. 1Computing Laboratory, University of Oxford, Wolfson Building, Parks Road, Oxford, OX1 3QD, Eng- land. 2School of Computer Science, University of Birmingham, Edgbaston, Birmingham, B15 2TT, England. Contents 1 Introduction and Overview 5 1.1 Origins ................................. 5 1.2 Ourapproach .............................. 7 1.3 Overview ................................ 7 2 Domains individually 10 2.1 Convergence .............................
    [Show full text]
  • Domain Theory I: Semantic Algebras
    Domain Theory I: Semantic Algebras Before we can study the semantics of programming languages, we must establish a suitable collection of meanings for programs. We employ a framework called domain theory: the study of ``structured sets'' and their operations. A programmer might view domain theory as ``data structures for semantics.'' Nonetheless, domain theory is a formal branch of (computing-related) mathematics and can be studied on its own. The fundamental concept in domain theory is a semantic domain, a set of elements grouped together because they share some common property or use. The set of natural numbers is a useful semantic domain; its elements are structurally similar and share common use in arithmetic. Other examples are the Greek alphabet and the diatonic (musical) scale. Domains may be nothing more than sets, but there are situations in which other structures such as lattices or topologies are used instead. We can use domains without worrying too much about the underlying mathematics. Sets make good domains, and you may safely assume that the structures de®ned in this chapter are nothing more than the sets discussed in Chapter 2. Chapter 6 presents reasons why domains other than sets might be necessary. Accompanying a domain is a set of operations. The operations are functions that need arguments from the domain to produce answers. Operations are de®ned in two parts. First, the operation's domain and codomain are given by an expression called the operation's func- . → tionality. For an operation f, its functionality f: D1× D2× × Dn A says that f needs an argument from domain D1 and one from D2, .
    [Show full text]
  • Domain Theory for Concurrency
    Domain Theory for Concurrency Mikkel Nygaard Glynn Winskel BRICS∗ Computer Laboratory University of Aarhus University of Cambridge Abstract A simple domain theory for concurrency is presented. Based on a categorical model of linear logic and associated comonads, it highlights the role of linear- ity in concurrent computation. Two choices of comonad yield two expressive metalanguages for higher-order processes, both arising from canonical construc- tions in the model. Their denotational semantics are fully abstract with respect to contextual equivalence. One language derives from an exponential of linear logic; it supports a straightforward operational semantics with simple proofs of soundness and adequacy. The other choice of comonad yields a model of affine-linear logic, and a process language with a tensor operation to be under- stood as a parallel composition of independent processes. The domain theory can be generalised to presheaf models, providing a more refined treatment of nondeterministic branching. The article concludes with a discussion of a broader programme of research, towards a fully fledged domain theory for concurrency. 1 Introduction Denotational semantics and domain theory of Scott and Strachey provide a global mathematical setting for sequential computation, and thereby place programming languages in connection with each other; connect with the mathematical worlds of algebra, topology and logic; and inspire programming languages, type disciplines and methods of reasoning. In concurrent/distributed/interactive computation that global mathematical gui- dance is missing, and domain theory has had little direct influence on theories of concurrent computation. One reason is that classical domain theory has not scaled up to the more intricate models used there.
    [Show full text]
  • Notes on Domain Theory and Topology
    Notes on domain theory and topology Fr´ed´ericBlanqui 17 August 2012 These notes have been written to complement the study of the very nice book of Gilles Dowek on \Proofs and Algorithms - An Introduction to Logic and Computability" [2] by Ying Jiang's group of students at the Institute of Software of the Chinese Academy of Sciences (ISCAS) that met more or less every Tuesday from May to August 2012. It introduces the notion of directed-complete partial order (dCPO) that is the basis of domain theory [1] and topology. 1 Directed-complete partial orders Let ≤ be an ordering relation on a set E. The first chapter of Dowek's book starts by introducing the following notions: Definition 1 (Limit, complete ordering, continuity) An element l 2 E is the limit of an increasing sequence u0 ≤ u1 ≤ ::: if l is the least upper bound (lub) of the set fu0; u1;:::g. The ordering ≤ is weakly complete if every increasing sequence has a limit. It is strongly complete if every subset of E has a least upper bound. A function f : E ! E is increasing if for all x; y 2 E, x ≤ y ) f(x) ≤ f(y). In a weakly complete ordering ≤, a function f : E ! E is continuous if it is increasing and if, for every increasing sequence (ui)i≥0, lubff(ui)gi≥0 = f(lubfuigi≥0). A set E equipped with an ordering relation ≤ is also called a partially ordered set or poset. In a totally ordered set, every two elements are comparable. An increasing function is more usually called monotone, monotonic or iso- tone (it is a morphism in the category of posets).
    [Show full text]
  • A Categorical View on Algebraic Lattices in Formal Concept Analysis
    Wright State University CORE Scholar Computer Science and Engineering Faculty Publications Computer Science & Engineering 2006 A Categorical View on Algebraic Lattices in Formal Concept Analysis Pascal Hitzler [email protected] Markus Krotzsch Guo-Qiang Zhang Follow this and additional works at: https://corescholar.libraries.wright.edu/cse Part of the Computer Sciences Commons, and the Engineering Commons Repository Citation Hitzler, P., Krotzsch, M., & Zhang, G. (2006). A Categorical View on Algebraic Lattices in Formal Concept Analysis. Fundamenta Informaticae, 74 (2/3), 301-328. https://corescholar.libraries.wright.edu/cse/165 This Article is brought to you for free and open access by Wright State University’s CORE Scholar. It has been accepted for inclusion in Computer Science and Engineering Faculty Publications by an authorized administrator of CORE Scholar. For more information, please contact [email protected]. Fundamenta Informaticae XX (2006) 1–28 1 IOS Press A Categorical View on Algebraic Lattices in Formal Concept Analysis Pascal Hitzler, Markus Krötzsch Institut AIFB Universität Karlsruhe Karlsruhe, Germany Guo-Qiang Zhang Department of Electrical Engineering and Computer Science Case Western Reserve University Cleveland, Ohio, U.S.A. Abstract. Formal concept analysis has grown from a new branch of the mathematical field of lattice theory to a widely recognized tool in Computer Science and elsewhere. In order to fully benefit from this theory, we believe that it can be enriched with notions such as approximation by computation or representability. The latter are commonly studied in denotational semantics and domain theory and captured most prominently by the notion of algebraicity, e.g. of lattices.
    [Show full text]
  • An Introduction to Domain Theory Notes for a Short Course
    Draft (26 May 2003) 33 pages An Introduction to Domain Theory Notes for a Short Course Michael Mislove 1 Department of Mathematics, Tulane University, New Orleans, LA 70118 Abstract This set of notes arises from a short course given at the Universit¶a Degli Studi Udine, given in May, 2003. The aim of the short course is to introduce both undergraduate and PhD students to domain theory, providing some of its history, as well as some of its most recent advances. 1 Outline of the Course The aim of this course is to present some of the basic results about domain theory. This will include the historically most signi¯cant contribution of the theory { the use of domains to provide a model of the untyped ¸-calculus of Church and Curry. The course will proceed with the following sections: (i) Some aspects of the ¸-calculus, for motivation. (ii) Some results from universal algebra and category theory for background. (iii) !-complete partial orders. (iv) Solving domain equations, and, in particular, building a model of the untyped ¸-calculus. The last part of the course will include lectures that survey additional portions of domain theory, including: ² Continuous domains and topology, where we discover alternative views of domains. ² Measurements, in which we see how to generalize some of the fundamental results of domain theory, and ² Probabilistic models, where we discover the domain-theoretic approach to probabilistic computation. 1 The author gratefully acknowledges the support of the National Science Foundation and the US O±ce of Naval Research. Domain Theory Notes This outline is being set down at the beginning of the course, and it isn't clear how much of this program we actually will accomplish.
    [Show full text]
  • Chapter 10 DOMAIN THEORY and FIXED-POINT SEMANTICS
    Chapter 10 DOMAIN THEORY AND FIXED-POINT SEMANTICS lthough we did not stress the point in Chapter 9, the notation of denotational semantics is built upon that of the lambda calculus. The A purpose of denotational semantics is to provide mathematical descrip- tions of programming languages independent of their operational behavior. The extended lambda calculus serves as a mathematical formalism, a metalanguage, for denotational definitions. As with all mathematical formal- isms, we need to know that the lambda calculus has a model to ensure that the definitions are not meaningless. Furthermore, denotational definitions, as well as programming languages in general, rely heavily on recursion, a mechanism whose description we de- ferred in the discussion of the lambda calculus in Chapter 5. Normally a user of a programming language does not care about the logical foundations of declarations, but we maintain that serious questions can be raised concern- ing the validity of recursion. In this chapter we justify recursive definitions to guarantee that they actually define meaningful objects. 10.1 CONCEPTS AND EXAMPLES Programmers use recursion to define functions and procedures as subpro- grams that call themselves and also to define recursive data structures. Most imperative programming languages require the use of pointers to declare recursive data types such as (linked) lists and trees. In contrast, many func- tional programming languages allow the direct declaration of recursive types. Rather than investigating recursive types in an actual programming language, we study recursive data declarations in a wider context. In this introductory section we consider the problems inherent in recursively defined functions and data and the related issue of nontermination.
    [Show full text]
  • Prime Algebraicity
    Prime Algebraicity Glynn Winskel, University of Cambridge Computer Laboratory, England April 27, 2009 Abstract A prime algebraic lattice can be characterised as isomorphic to the downwards-closed subsets, ordered by inclusion, of its complete primes. It is easily seen that the downwards-closed subsets of a par- tial order form a completely distributive algebraic lattice when ordered by inclusion. The converse also holds; any completely distributive al- gebraic lattice is isomorphic to such a set of downwards-closed subsets of a partial order. The partial order can be recovered from the lattice as the order of the lattice restricted to its complete primes. Conse- quently prime algebraic lattices are precisely the completely distribu- tive algebraic lattices. The result extends to Scott domains. Several consequences are explored briefly: the representation of Berry's dI- domains by event structures; a simplified form of information systems for completely distributive Scott domains; and a simple domain theory for concurrency. Introduction It is 30 years since Mogens Nielsen, Gordon Plotkin and I introduced prime algebraic lattices, and the more general prime algebraic domains, as an inter- mediary in relating Petri nets and Scott domains [19]. The recognition that prime algebraic lattices were well-known in another guise, that of completely distributive algebraic lattices, came a little later, partly while I was a postdoc visiting Mogens in Aarhus, with the final pieces falling into place early after my move to CMU in 1982. The first part of this article is essentially based 1 on a CMU research report [25] from my time there.1 Since their introduction prime algebraic domains have come to play a significant role in several other areas and have broader relevance today.
    [Show full text]