polymorphic lambda calculus and subtyping we present a denotational model for f the extension of second order lambda calculus with subtyping defined in cardelli and wegner acm comput surveys 17 4 1985 471 522 types are interpreted as arbitrary cpos and elements of types as natural transformations we prove the soundness of our model with respect to the equational theory of f cardelli et al internat conf on theoretical aspects of computer software lecture notes in computer science vol 526 springer berlin 1991 pp 750 770 and show coherence our model is of independent interest because it integrates ad hoc and parametric polymorphism in an elegant fashion admits nontrivial records and record update operations and formalizes an order faithfulness criterion for well behaved multiple subtyping

set theoretical and other elementary models of the l calculus abstract part i pp 351 373 of this paper is the previously unpublished 1972 memorandum plotkin 1972 with editorial changes and some minor corrections part ii pp 373 409 presents what happened next together with some further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 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 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 calculus which are continuous complete lattices the second part begins with a brief discussion of models of the calculus in set theories with anti foundation axioms next the model of the 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 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

on functions preserving levels of approximation a refined model construction for various lambda calculi abstract in this paper di domains are enriched by a family of projections which assign to each point a sequence of canonical approximations the morphisms are stable maps that preserve the levels of approximation generated by the projections for the computation of an approximation of a given level of the output they require in addition only information about the input of at most the same level of approximation it is shown that the category of these domains and maps is cartesian closed the set of morphisms between two such domains is a domain of the same kind but turns out not to be an exponent in the category domains d are constructed which are isomorphic to the exponent d d moreover it is proved that the space of retractions in an exponent d d is a retract of this both results together provide new models of amadio longo s extension p of the calculus as has been shown by amadio and longo strong type theories which incorporate the type type assumption can be syntactically interpreted in this calculus

universal coalgebra a theory of systems in the semantics of programming finite data types such as finite lists have traditionally been modelled by initial algebras later final coalgebras were used in order to deal with infinite data types coalgebras which are the dual of algebras turned out to be suited moreover as models for certain types of automata and more generally for transition and dynamical systems an important property of initial algebras is that they satisfy the familiar principle of induction such a principle was missing for coalgebras until the work of aczel 1988 on a theory of non wellfounded sets in which he introduced a proof principle nowadays called coinduction it was formulated in terms of bisimulation a notion originally stemming from the world of concurrent programming languages milner 1980 park 1981 using the notion of coalgebra homomorphism the definition of bisimulation on coalgebras can be shown to be formally dual to that of congruence on algebras aczel and mendler 1989 thus the three basic notions of universal algebra algebra homomorphism of algebras and congruence turn out to correspond to coalgebra homomorphism of coalgebras and bisimulation respectively in this paper the latter are taken as the basic ingredients of a theory called universal coalgebra some standard results from universal algebra are reformulated using the afore mentioned correspondence and proved for a large class of coalgebras leading to a series of results on e g the lattices of subcoalgebras and bisimulations simple coalgebras and coinduction and a covariety theorem for coalgebras similar to birkhoff s variety theorem

the category theoretic solution of recursive metric space equations it is well known that one can use an adaptation of the inverse limit construction to solve recursive equations in the category of complete ultrametric spaces we show that this construction generalizes to a large class of categories with metric space structure on each set of morphisms the exact nature of the objects is less important in particular the construction immediately applies to categories where the objects are ultrametric spaces with extra structure and where the morphisms preserve this extra structure the generalization is inspired by classical domain theoretic work by smyth and plotkin for many of the categories we consider there is a natural subcategory in which each set of morphisms is required to be a compact metric space our setting allows for a proof that such a subcategory always inherits solutions of recursive equations from the full category as another application we present a construction that relates solutions of generalized domain equations in the sense of smyth and plotkin to solutions of equations in our class of categories our primary motivation for solving generalized recursive metric space equations comes from recent and ongoing work on kripke style models in which the sets of worlds must be recursively defined we show a series of examples motivated by this line of work

complete restrictions of the intersection type discipline abstract in this paper the intersection type discipline as defined in barendregt 1983 is studied we will present two different and independent complete restrictions of the intersection type discipline the first restricted system the strict type assignment system is presented in section 2 its major feature is the absence of the derivation rule and it is based on a set of strict types we will show that these together give rise to a strict filter lambda model that is essentially different from the one presented in barendregt we will show that the strict type assignment system is the nucleus of the full system i e for every derivation in the intersection type discipline there is a derivation in which is used only at the very end finally we will prove that strict type assignment is complete for inference semantics the second restricted system is presented in section 3 its major feature is the absence of the type we will show that this system gives rise to a filter i model and that type assignment without is complete for the i calculus finally we will prove that a lambda term is typeable in this system if and only if it is strongly normalizable

simulations in coalgebra a new approach to simulations is proposed within the theory of coalgebras by taking a notion of order on a functor as primitive such an order forms a basic building block for a lax relation lifting or relator as used by other authors simulations appear as coalgebras of this lifted functor and similarity as greatest simulation two way similarity is then similarity in both directions in general it is different from bisimilarity in the usual coalgebraic sense but a sufficient condition is formulated and illustrated to ensure that bisimilarity and two way similarity coincide also suitable conditions are identified which ensures that similarity on a final coalgebra forms an algebraic dcpo structure this involves a close investigation of the iterated applications fn 0 and fn 1 of a functor f with an order to the initial algebras and final objects

binary level set shape optimization model and algorithm for volumetric modulated arc therapy in radiotherapy treatment we introduce in this paper a variational framework for the construction of improved treatment plans in volumetric modulated arc therapy for cancer treatment this framework consists of a shape optimization component from the molding of beam shapes as well as strong constraints from the equipment involved on both beam shapes and intensities we apply the binary level set method to handle complex shape topologies and the fast sweeping method to handle beam intensity constraints the result is a fast flow based algorithm in a simplified setting that guarantees energy decrease we include numerical simulations of clinical cases to demonstrate the efficacy of the approach

on final coalgebras of continuous functors continuous endofunctors f of locally finitely presentable categories carry a natural metric on their final coalgebra whenever f 0 has an element this metric is proved to be a cauchy completion of the initial algebra of f this is illustrated on the poset of real numbers represented as a final coalgebra of an endofunctor of pos by pavlovic and pratt under additional assumptions on the locally finitely presentable category all finitary endofunctors are proved to have a final coalgebra constructed in steps of the natural iteration construction

infinite hypergraphs i basic properties abstract basic properties of the category of infinite directed hyperedge labelled hypergraphs are studied an algebraic structure is given which enables us to describe such hypergraphs by means of infinite expressions it is then shown that two expressions define the same hypergraphs if and only if they are congruent with respect to some rewriting system these results will be used in the second part of this paper to solve systems of recursive equations on hypergraphs and characterize their solutions

an irregular filter model in this paper we introduce a new filter model which is of a kind that has escaped investigation up to now it is induced by an intersection type theory generated in a non standard way by a preorder which puts into relation an atom with an arrow type without equating them we study the domain theoretic implications of this choice that are not trivial in order to describe this filter model a new category is introduced and a special purpose functor defined the filter model is then characterized as the initial algebra of the functor

the mixed powerdomain this paper introduces an operator m called the mixed powerdomain which generalizes the convex plotkin powerdomain the construction is based on the idea of representing partial information about a set of data items using a pair of sets one representing partial information in the manner of the upper smyth powerdomain and the other in the manner of the lower hoare powerdomain where the components of such pairs are required to satisfy a consistency condition this provides a richer family of meaningful partial descriptions than are available in the convex powerdomain and also makes it possible to include the empty set in a satisfactory way the new construct is given a rigorous mathematical treatment like that which has been applied to the known powerdomains it is proved that m is a continuous functor on bifinite domains which is left adjoint to the forgetful functor from a category of continuous structures called mix algebras for a domain d with a coherent scott topology elements of md can be represented as pairs u v where u d is a compact upper set v d is a closed set and the downward closure of u v is equal to v a stone dual characterization of m is also provided

tail recursion through universal invariants abstract tail recursive constructions suggest a new semantics for datatypes which allows a direct match between specifications and tail recursive programs the semantics focuses on loops their fixpoints invariants and convergence convergent models of the natural numbers and lists are examined in detail and under very mild conditions are shown to be equivalent to the corresponding initial algebra models

universal homogeneous event structures and domains abstract in the theory of denotational semantics of programming languages several authors established the existence of particular kinds of universal domains here we use a general model theoretic result to show that there exists a unique countable universal homogeneous event structure from this we deduce that the category of all event domains with stable embedding projection pairs as morphisms contains a universal object similarly we also obtain a universal di domain we also show that the category of all event domains is closed under inverse limits similar results are derived for kahn and plotkin s concrete data structures and concrete domains

a 2 categorical pasting theorem in order to facilitate the study of 2 categories with structure we state and prove an n categorical pasting theorem this is based upon a new definition of n pasting scheme that generalises johnson s definition of a well formed loop free pasting scheme by weakening his no direct loops condition we define n pasting prove the theorem and show that for n 3 it incorporates all possible composites of n cells we show that that is not true for higher n we define the horizontal n category of an n 1 category to generalise that of a 2 category we define horizontal and vertical composition for an n 1 category and we state and prove an interchange law we also study further conditions on a pasting diagram and their impact upon how one may evaluate a composite and we express street s free n categories in terms of left adjoints

the comparison of a cpo based semantics with 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

quantitative domains and infinitary algebras abstract we describe a class of models of the lambda calculus that generalize and simplify the quantitative domains of girard 1988 their salient feature is that these domains do not have enough points in the sense that a map of domains is not entirely determined by its value on the points i e the morphisms from the terminal object of the category we construct a universal domain whose structure is particularly simple and describe some of the features of the untyped lambda calculus associated to it finally we show how linear logic can be interpreted in these domains and construct another category of domains that supports linear logic

on proving limiting completeness we give two proofs of wadsworth s classic approximation theorem for the pure lambda calculus one of these illustrates a new method utilising a certain kind of intermediate semantics for proving correspondences between denotational and operational semantics the other illustrates a direct technique of milne employing recursively specified inclusive relations

the category theoretic solution of recursive domain equations recursive specifications of domains plays a crucial role in denotational semantics as developed by scott and strachey and their followers the purpose of the present paper is to set up a categorical framework in which the known techniques for solving these equations find a natural place the idea is to follow the well known analogy between partial orders and categories generalizing from least fixed points of continuous functions over cpos to initial ones of continuous functors over omega categories to apply these general ideas we introduce wand s bf o categories where the morphism sets have a partial order structure and which include almost all the categories occurring in semantics the idea is to find solutions in a derived category of embeddings and we give order theoretic conditions which are easy to verify and which imply the needed categorical ones the main tool is a very general form of the limit colimit coincidence remarked by scott in the concluding section we outline how compatibilit

completeness of rewrite rules and rewrite strategies for fp this paper treats languages whose operational semantics is given by a set of rewrite rules for such languages it is important to be able to determine that there are enough rules to be able to compute the correct meaning of all expressions but not so many that the system of rules is inconsistent a formal framework is developed in which to give a precise treatment of these completeness and soundness issues which are then investigated in the context of an extended version of the functional programming language fp the rewrite rules of fp are shown to be sound and complete with respect to three different notions of completeness the latter half of the paper considers rewrite strategies in order to implement a language based on rewrite rules it does not suffice to know that there are enough rules in the language a good strategy for determining the order in which to apply them is also needed but what is good corresponding to each notion of completeness there is a notion of a good rewrite strategy these notions of goodness are examined and characterized and examples of a number of natural good strategies are given although these results are presented in the context of fp the techniques some of which are nontrivial extensions of techniques first used in the context of calculus should apply well beyond the realm of fp rewriting systems

trees and semantics abstract several types of trees are proposed in the literature to model the behaviour of nondeterministic processes the purpose of this paper is to clarify the distinction between the various types of trees and to discuss their use as semantic domains trees in the sense of milner s derivation trees and bisimulation equivalence classes of trees are considered in three different settings in the category of sets in the metric setting and in the partial order setting each type of tree is characterized by a domain equation special attention is given to the use of trees as models for processes which allow for unbounded nondeterminism as for instance milners ccs processes in this case one has to deal with infinitely branching trees which cause problems to give denotational semantics that are fully abstract w r t a transition system based operational semantics

non well founded sets modeled as ideal fixed points abstract motivated by ideas from the study of abstract data types we show how to interpret non well founded sets as fixed points of continuous transformations of an initial continuous algebra we conisder a preordered structure closely related to the set hf of well founded hereditarily finite sets by taking its ideal completion we obtain an initial continuous algebra in which we are able to solve all of the usual systems of equations that characterize hereditarily finite non well founded sets in this way we are able to obtain a structure which is isomorphic to hf 1 the non well founded analog of hf

notions of computation and monads abstract the calculus is considered a useful mathematical tool in the study of programming languages since programs can be identified with terms however if one goes further and uses conversion to prove equivalence of programs then a gross simplification is introduced programs are identified with total functions from values to values that may jeopardise the applicability of theoretical results in this paper we introduce calculi based on a categorical semantics for computations that provide a correct basis for proving equivalence of programs for a wide range of notions of computation

a co induction principle for recursively defined domains abstract this paper establishes a new property of predomains recursively defined using the cartesian product disjoint union partial function space and convex powerdomain constructors we prove that the partial order on such a recursive predomain d is the greatest fixed point of a certain monotone operator associated to d this provides a structurally defined family of proof principles for these recursive predomains to show that one element of d approximates another it suffices to find a binary relation containing the two elements that is a post fixed point for the associated monotone operator the statement of the proof principles is independent of any of the various methods available for explicit construction of recursive predomains following milner and tofte 1991 the method of proof is called co induction it closely resembles the way bisimulations are used in concurrent process calculi milner 1989 two specific instances of the co induction principle already occur in the work of abramsky 1990 1991 in the form of internal full abstraction theorems for denotational semantics of sccs and lazy lambda calculus in the first case post fixed binary relations are precisely abramsky s partial bisimulations whereas in the second case they are his applicative bisimulations the co induction principle also provides an apparently useful tool for reasoning about equality of elements of recursively defined datatypes in strict or lazy higher order functional programming languages

the largest cartesian closed category of domains the main result of this paper provides a necessary and sufficient condition that the function space d d of a domain algebraic cpo d be again a domain the condition confirming a conjecture of plotkin is that d be strongly algebraic sfp the importance of this is that it enables us to conclude that sfp is the largest category of domains that is closed under the constructions of interest in semantics in the final section we briefly consider how our results may need to be extended if our admittedly rather restrictive notion of category of domains is relaxed in certain ways

universal profinite domains abstract we introduce a bicartesian closed category of what we call profinite domains study of these domains is carried out through the use of an equivalent category of pre orders in a manner similar to the information systems approach advocated by dana scott and others a class of universal profinite domains is defined and used to derive sufficient conditions for the profinite solution of domain equations involving continuous operators as a special instance of this construction a universal domain for the category sfp is demonstrated necessary conditions for the existence of solutions for domain equations over the profinites are also given and used to derive results about solutions of some equations a new universal bounded complete domain is also demonstrated using an operator which has bounded complete domains as its fixed points