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Lazy Lambda Calculus: Theories, Models and Local Structure Characterisation
 AUTOMATA, LANGUAGES AND PROGRAMMING, LNCS 623
, 1994
"... Lambda Calculus is commonly thought to be the basis for functional programming. However, there is a fundamental mismatch between the "standard" theory of sensible Lambda Calculus (as in e.g. [Bar84]) and the practice of lazy evaluation which is a distinctive feature of functional programming. This ..."
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Lambda Calculus is commonly thought to be the basis for functional programming. However, there is a fundamental mismatch between the "standard" theory of sensible Lambda Calculus (as in e.g. [Bar84]) and the practice of lazy evaluation which is a distinctive feature of functional programming. This paper proposes modification of a number of key notions in the sensible theory along the lines of laziness. Starting from the strongly unsolvables as the meaningless terms, we define and investigate properties of lazy (or weakly sensible) λtheories, lazy λmodels and a number of lazy behavioural preorders on λterms. In the second part, we show that all these notions have a natural place in a class of lazy psemodels. A major result of this paper is a new local structure theorem for lazy psemodels. This characterizes the ordering between denotations of λterms in the model by a new lazy behavioural preorder.
Information Categories
 Applied Categorical Structures
, 1991
"... \Information systems" have been introduced by Dana Scott as a convenient means of presenting a certain class of domains of computation, usually known as Scott domains. Essentially the same idea has been developed, if less systematically, by various authors in connection with other classes of dom ..."
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\Information systems" have been introduced by Dana Scott as a convenient means of presenting a certain class of domains of computation, usually known as Scott domains. Essentially the same idea has been developed, if less systematically, by various authors in connection with other classes of domains. In previous work, the present authors introduced the notion of an Icategory as an abstraction and enhancement of this idea, with emphasis on the solution of domain equations of the form D = F (D), with F a functor. An important feature of the work is that we are not conned to domains of computation as usually understood; other classes of spaces, more familiar to mathematicians in general, become also accessible. Here we present the idea in terms of what we call information categories, which are concrete Icategories in which the objects are structured sets of \tokens" and morphisms are relations between tokens. This is more in the spirit of information system work, and...
Towards Lambda Calculus OrderIncompleteness
 Workshop on Böhm theorem: applications to Computer Science Theory (BOTH 2001) Electronics Notes in Theoretical Computer Science
"... After Scott, mathematical models of the typefree lambda calculus are constructed by order theoretic methods and classified into semantics according to the nature of their representable functions. Selinger [47] asked if there is a lambda theory that is not induced by any nontrivially partially orde ..."
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After Scott, mathematical models of the typefree lambda calculus are constructed by order theoretic methods and classified into semantics according to the nature of their representable functions. Selinger [47] asked if there is a lambda theory that is not induced by any nontrivially partially ordered model (orderincompleteness problem). In terms of Alexandroff topology (the strongest topology whose specialization order is the order of the considered model) the problem of order incompleteness can be also characterized as follows: a lambda theory T is orderincomplete if, and only if, every partially ordered model of T is partitioned by the Alexandroff topology in an infinite number of connected components (= minimal upper and lower sets), each one containing exactly one element of the model. Towards an answer to the orderincompleteness problem, we give a topological proof of the following result: there exists a lambda theory whose partially ordered models are partitioned by the Alexandroff topology in an infinite number of connected components, each one containing at most one term denotation. This result implies the incompleteness of every semantics of lambda calculus given in terms of partially ordered models whose Alexandroff topology has a finite number of connected components (e.g. the Alexandroff topology of the models of the continuous, stable and strongly stable semantics is connected).
Algebraic models of sets and classes in categories of ideals
 In preparation
, 2006
"... We introduce a new sheaftheoretic construction called the ideal completion of a category and investigate its logical properties. We show that it satisfies the axioms for a category of classes in the sense of Joyal and Moerdijk [17], so that the tools of algebraic set theory can be applied to produc ..."
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We introduce a new sheaftheoretic construction called the ideal completion of a category and investigate its logical properties. We show that it satisfies the axioms for a category of classes in the sense of Joyal and Moerdijk [17], so that the tools of algebraic set theory can be applied to produce models of various elementary set theories. These results are then used to prove the conservativity of different set theories over various classical and constructive type theories. 1
Lambda calculus: models and theories
 Proceedings of the Third AMAST Workshop on Algebraic Methods in Language Processing (AMiLP2003), number 21 in TWLT Proceedings, pages 39–54, University of Twente, 2003. Invited Lecture
"... In this paper we give an outline of recent results concerning theories and models of the untyped lambda calculus. Algebraic and topological methods have been applied to study the structure of the lattice of λtheories, the equational incompleteness of lambda calculus semantics, and the λtheories in ..."
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In this paper we give an outline of recent results concerning theories and models of the untyped lambda calculus. Algebraic and topological methods have been applied to study the structure of the lattice of λtheories, the equational incompleteness of lambda calculus semantics, and the λtheories induced by graph models of lambda calculus.
Applying Universal Algebra to Lambda Calculus
, 2007
"... The aim of this paper is double. From one side we survey the knowledge we have acquired these last ten years about the lattice of all λtheories ( = equational extensions of untyped λcalculus) and the models of lambda calculus via universal algebra. This includes positive or negative answers to se ..."
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The aim of this paper is double. From one side we survey the knowledge we have acquired these last ten years about the lattice of all λtheories ( = equational extensions of untyped λcalculus) and the models of lambda calculus via universal algebra. This includes positive or negative answers to several questions raised in these years as well as several independent results, the state of the art about the longstanding open questions concerning the representability of λtheories as theories of models, and 26 open problems. On the other side, against the common belief, we show that lambda calculus and combinatory logic satisfy interesting algebraic properties. In fact the Stone representation theorem for Boolean algebras can be generalized to combinatory algebras and λabstraction algebras. In every combinatory and λabstraction algebra there is a Boolean algebra of central elements (playing the role of idempotent elements in rings). Central elements are used to represent any combinatory and λabstraction algebra as a weak Boolean product of directly indecomposable algebras (i.e., algebras which cannot be decomposed as the Cartesian product of two other nontrivial algebras). Central elements are also used to provide applications of the representation theorem to lambda calculus. We show that the indecomposable semantics (i.e., the semantics of lambda calculus given in terms of models of lambda calculus, which are directly indecomposable as combinatory algebras) includes the continuous, stable and strongly stable semantics, and the term models of all semisensible λtheories. In one of the main results of the paper we show that the indecomposable semantics is equationally incomplete, and this incompleteness is as wide as possible.
An Epistemological Approach to the Design of Training Courses on Logic
"... Introduction Mathematical logic helps to form the rational basis of common sense and, at the same time, it clashes with it. This conflict can be explained by observing that results sistematically obtained by formal logic alter deeply the rationality categories socially accepted; in this respect, it ..."
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Introduction Mathematical logic helps to form the rational basis of common sense and, at the same time, it clashes with it. This conflict can be explained by observing that results sistematically obtained by formal logic alter deeply the rationality categories socially accepted; in this respect, it is worthwhile to note that these results often express innovations of natural science, already recognized by technology, that people uses without awareness. Thus, logic is a powerful educational tool to uptodate common sense rationality, that is to transfer new paradigms of thinking. Notwithstanding this fact, mathematical logic was not given a central role in the Italian high school curriculum till few time ago. At present, the situation is changing, as the diffusion of computer science and its technology is leading to the renewal of high school curricula. This renewal recognizes the increasing importance of logic in various scientific and humanistic fields and tak
Strict Geometry of Interaction Graph Models
, 2003
"... We study a class of \wavestyle" Geometry of Interaction (GoI) models based on the category Rel of sets and relations. Wave GoI models arise when Abramsky's GoI axiomatization, which generalizes Girard's original GoI, is applied to a traced monoidal category with the categorical product as tens ..."
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We study a class of \wavestyle" Geometry of Interaction (GoI) models based on the category Rel of sets and relations. Wave GoI models arise when Abramsky's GoI axiomatization, which generalizes Girard's original GoI, is applied to a traced monoidal category with the categorical product as tensor, using \countable power" as the traced strong monoidal functor !. Abramsky hinted that the category Rel is the basic setting for traditional denotational \static semantics". However, Rel, together with the cartesian product, apparently escapes Abramsky's original GoI construction. Here we show that Rel can be axiomatized as a strict GoI situation, i.e. a strict variant of Abramsky's GoI situation, which gives rise to a rich class of strict graph models. These are models of restricted calculi in the sense of [HL99], such as Church's Icalculus and the KN calculus.
to the following people for their corrections and suggestions: Arbob
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and suggestions for improvement. I also thank the following people for
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