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A CurryHoward foundation for functional computation with control
 In Proceedings of ACM SIGPLANSIGACT Symposium on Principle of Programming Languages
, 1997
"... We introduce the type theory ¯ v , a callbyvalue variant of Parigot's ¯calculus, as a CurryHoward representation theory of classical propositional proofs. The associated rewrite system is ChurchRosser and strongly normalizing, and definitional equality of the type theory is consistent, com ..."
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We introduce the type theory ¯ v , a callbyvalue variant of Parigot's ¯calculus, as a CurryHoward representation theory of classical propositional proofs. The associated rewrite system is ChurchRosser and strongly normalizing, and definitional equality of the type theory is consistent, compatible with cut, congruent and decidable. The attendant callbyvalue programming language ¯pcf v is obtained from ¯ v by augmenting it by basic arithmetic, conditionals and fixpoints. We study the behavioural properties of ¯pcf v and show that, though simple, it is a very general language for functional computation with control: it can express all the main control constructs such as exceptions and firstclass continuations. Prooftheoretically the dual ¯ v constructs of naming and ¯abstraction witness the introduction and elimination rules of absurdity respectively. Computationally they give succinct expression to a kind of generic (forward) "jump" operator, which may be regarded as a unif...
Correspondence between Operational and Denotational Semantics
 Handbook of Logic in Computer Science
, 1995
"... This course introduces the operational and denotational semantics of PCF and examines the relationship between the two. Topics: Syntax and operational semantics of PCF, Activity Lemma, undefinability of parallel or; Context Lemma (first principles proof) and proof by logical relations Denotational ..."
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This course introduces the operational and denotational semantics of PCF and examines the relationship between the two. Topics: Syntax and operational semantics of PCF, Activity Lemma, undefinability of parallel or; Context Lemma (first principles proof) and proof by logical relations Denotational semantics of PCF induced by an interpretation; (standard) Scott model, adequacy, weak adequacy and its proof (by a computability predicate) Domain Theory up to SFP and Scott domains; non full abstraction of the standard model, definability of compact elements and full abstraction for PCFP (PCF + parallel or), properties of orderextensional (continuous) models of PCF, Milner's model and Mulmuley's construction (excluding proofs) Additional topics (time permitting): results on pure simplytyped lambda calculus, Friedman 's Completeness Theorem, minimal model, logical relations and definability, undecidability of lambda definability (excluding proof), dIdomains and stable functions Homepa...
Modified Realizability Toposes and Strong Normalization Proofs (Extended Abstract)
 Typed Lambda Calculi and Applications, LNCS 664
, 1993
"... ) 1 J. M. E. Hyland 2 C.H. L. Ong 3 University of Cambridge, England Abstract This paper is motivated by the discovery that an appropriate quotient SN 3 of the strongly normalising untyped 3terms (where 3 is just a formal constant) forms a partial applicative structure with the inherent appl ..."
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) 1 J. M. E. Hyland 2 C.H. L. Ong 3 University of Cambridge, England Abstract This paper is motivated by the discovery that an appropriate quotient SN 3 of the strongly normalising untyped 3terms (where 3 is just a formal constant) forms a partial applicative structure with the inherent application operation. The quotient structure satisfies all but one of the axioms of a partial combinatory algebra (pca). We call such partial applicative structures conditionally partial combinatory algebras (cpca). Remarkably, an arbitrary rightabsorptive cpca gives rise to a tripos provided the underlying intuitionistic predicate logic is given an interpretation in the style of Kreisel's modified realizability, as opposed to the standard Kleenestyle realizability. Starting from an arbitrary rightabsorptive cpca U , the tripostotopos construction due to Hyland et al. can then be carried out to build a modified realizability topos TOPm (U ) of nonstandard sets equipped with an equali...
Subject reduction and minimal types for higher order subtyping
 In Proceedings of the Second Chinese Language Processing Workshop
, 1997
"... We define the typed lambda calculus F ω ∧ , a natural generalization of Girard’s system F ω with intersection types and bounded polymorphism. A novel aspect of our presentation is the use of term rewriting techniques to present intersection types, which clearly splits the computational semantics (re ..."
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We define the typed lambda calculus F ω ∧ , a natural generalization of Girard’s system F ω with intersection types and bounded polymorphism. A novel aspect of our presentation is the use of term rewriting techniques to present intersection types, which clearly splits the computational semantics (reduction rules) from the syntax (inference rules) of the system. We establish properties such as ChurchRosser for the reduction relation on types and terms, and Strong Normalization for the reduction on types. We prove that types are preserved by computation (Subject Reduction property), and that the system satisfies the Minimal Types property. On the way to establishing these results, we define algorithms for type inference and subtype checking. 1
A paraconsistent higher order logic
 INTERNATIONAL WORKSHOP ON PARACONSISTENT COMPUTATIONAL LOGIC, VOLUME 95 OF ROSKILDE UNIVERSITY, COMPUTER SCIENCE, TECHNICAL REPORTS
, 2004
"... Classical logic predicts that everything (thus nothing useful at all) follows from inconsistency. A paraconsistent logic is a logic where an inconsistency does not lead to such an explosion, and since in practice consistency is difficult to achieve there are many potential applications of paracons ..."
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Classical logic predicts that everything (thus nothing useful at all) follows from inconsistency. A paraconsistent logic is a logic where an inconsistency does not lead to such an explosion, and since in practice consistency is difficult to achieve there are many potential applications of paraconsistent logics in knowledgebased systems, logical semantics of natural language, etc. Higher order logics have the advantages of being expressive and with several automated theorem provers available. Also the type system can be helpful. We present a concise description of a paraconsistent higher order logic with countable infinite indeterminacy, where each basic formula can get its own indeterminate truth value (or as we prefer: truth code). The meaning of the logical operators is new and rather different from traditional manyvalued logics as well as from logics based on bilattices. The adequacy of the logic is examined by a case study in the domain of medicine. Thus we try to build a bridge between the HOL and MVL communities. A sequent calculus is proposed based on recent work by Muskens.
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 programm ..."
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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.
Bottomup βreduction: Uplinks and λDAGs
 Proceedings of the 14th European Symposium on Programming (ESOP 2005), number 3444 in LNCS
, 2005
"... Abstract. Representing a λcalculus term as a DAG rather than a tree allows us to represent the sharing that arises from βreduction, thus avoiding combinatorial explosion in space. By adding uplinks from a child to its parents, we can efficiently implement βreduction in a bottomup manner, thus av ..."
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Abstract. Representing a λcalculus term as a DAG rather than a tree allows us to represent the sharing that arises from βreduction, thus avoiding combinatorial explosion in space. By adding uplinks from a child to its parents, we can efficiently implement βreduction in a bottomup manner, thus avoiding combinatorial explosion in time required to search the term in a topdown fashion. We present an algorithm for performing βreduction on λterms represented as uplinked DAGs; discuss its relation to alternate techniques such as Lamping graphs, explicitsubstitution calculi and director strings; and present some timings of an implementation. Besides being both fast and parsimonious of space, the algorithm is particularly suited to applications such as compilers, theorem provers, and typemanipulation systems that may need to examine terms inbetween reductions—i.e., the “readback ” problem for our representation is trivial. Like Lamping graphs, and unlike director strings or the suspension λcalculus, the algorithm functions by sideeffecting the term containing the redex; the representation is not a “persistent ” one. The algorithm additionally has the charm of being quite simple: a complete implementation of the core data structures and algorithms is 180 lines of SML. 1
NonDeterminism in a Functional Setting (Extended Abstract)
 In Proceedings 8th LICS
, 1993
"... ) C.H. Luke Ong Computer Laboratory, University of Cambridge, Pembroke Street, CB2 3QG England. Email: Luke.Ong@cl.cam.ac.uk and discs, National University of Singapore. Abstract The pure untyped Lambda Calculus augmented with an (erratic) choice operator is considered as an idealised nondeter ..."
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) C.H. Luke Ong Computer Laboratory, University of Cambridge, Pembroke Street, CB2 3QG England. Email: Luke.Ong@cl.cam.ac.uk and discs, National University of Singapore. Abstract The pure untyped Lambda Calculus augmented with an (erratic) choice operator is considered as an idealised nondeterministic functional language. Both the "may" and the "must" modalities of convergence are of interest to us. Following Abramsky's work on domain theory in logical form, we identify the denotational type that captures our computational situation: ffi = P[[ffi ! ffi ] ? ] where P[\Gamma] is the Plotkin powerdomain functor. We then carry out a systematic programme which hinges on three distinct interpretations of ffi , namely, processtheoretic, denotational and logical. The main theme of our programme is the complementarity of the various interpretations of ffi . This work may be seen as a step towards a reapprochement between the algebraic theory of processes in Concurrency on the one hand, ...
Determinacy in a synchronous πcalculus ∗
, 2008
"... The Sπcalculus is a synchronous πcalculus which is based on the SL model. The latter is a relaxation of the Esterel model where the reaction to the absence of a signal within an instant can only happen at the next instant. In the present work, we study the notions of determinacy and (local) conflu ..."
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The Sπcalculus is a synchronous πcalculus which is based on the SL model. The latter is a relaxation of the Esterel model where the reaction to the absence of a signal within an instant can only happen at the next instant. In the present work, we study the notions of determinacy and (local) confluence for the Sπcalculus and we introduce a typing system that guarantees determinacy. 1
Lambda Calculus
, 1997
"... Recursive functions are representable as lambda terms, and de nability in the calculus may be regarded as a de nition of computability. This forms part of the standard foundations of computer science. Lambda calculus is the commonly accepted basis of functional programming languages; and it is folkl ..."
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Recursive functions are representable as lambda terms, and de nability in the calculus may be regarded as a de nition of computability. This forms part of the standard foundations of computer science. Lambda calculus is the commonly accepted basis of functional programming languages; and it is folklore that the calculus is the prototypical functional language in puri ed form. The course investigates the syntax and semantics of lambda calculus both as a theory of functions from a foundational point of view, and as a minimal programming language.