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66
Typing firstclass patterns
 In HigherOrder Rewriting, proceedings
, 2006
"... This extended abstract describes a small typed pattern calculus that is able to support four styles of polymorphism, namely, data (or parametric) polymorphism [6], structure polymorphism [4], path polymorphism and pattern polymorphism [3] (examples below). These combine to support generic queries, a ..."
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Cited by 18 (6 self)
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This extended abstract describes a small typed pattern calculus that is able to support four styles of polymorphism, namely, data (or parametric) polymorphism [6], structure polymorphism [4], path polymorphism and pattern polymorphism [3] (examples below). These combine to support generic queries, able to
A Structural Proof of Cut Elimination and Its Representation in a Logical Framework
, 1994
"... We present new proofs of cut elimination for intuitionistic and classical sequent calculi. In both cases the proofs proceed by three nested structural inductions, avoiding the explicit use of multisets and termination measures on sequent derivations. This makes them amenable to elegant and concise r ..."
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Cited by 17 (4 self)
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We present new proofs of cut elimination for intuitionistic and classical sequent calculi. In both cases the proofs proceed by three nested structural inductions, avoiding the explicit use of multisets and termination measures on sequent derivations. This makes them amenable to elegant and concise representations in LF, which are given in full detail. This work was supported by NSF Grant CCR9303383 The views and conclusions contained in this document are those of the author and should not be interpreted as representing the official policies, either expressed or implied, of NSF or the U.S. government. Keywords: Logic, Cut Elimination, Logical Framework Contents 1 Introduction 1 2 Intuitionistic Sequent Calculus 2 3 Proof Terms for the Sequent Calculus 8 4 Representing Sequent Derivations in LF 10 5 Admissibility of Cut 13 6 Extension to Classical Logic 18 7 Conclusion 24 A Detailed Admissibility Proofs for Cut 26 A.1 Intuitionistic Calculus : : : : : : : : : : : : : : : : : : :...
Cut Rules and Explicit Substitutions
, 2000
"... this paper deals exclusively with intuitionistic logic (in fact, only the implicative fragment), we require succedents to be a single consequent formula. Natural deduction systems, which we choose to call Nsystems, are symbolic logics generally given via introduction and elimination rules for the l ..."
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Cited by 15 (0 self)
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this paper deals exclusively with intuitionistic logic (in fact, only the implicative fragment), we require succedents to be a single consequent formula. Natural deduction systems, which we choose to call Nsystems, are symbolic logics generally given via introduction and elimination rules for the logical connectives which operate on the right, i.e., they manipulate the succedent formula. Examples are Gentzen's NJ and NK (Gentzen 1935). Logical deduction systems are given via leftintroduction and rightintroduction rules for the logical connectives. Although others have called these systems "sequent calculi", we call them Lsystems to avoid confusion with other systems given in sequent style. Examples are Gentzen's LK and LJ (Gentzen 1935). In this paper we are primarily interested in Lsystems. The advantage of Nsystems is that they seem closer to actual reasoning, while Lsystems on the other hand seem to have an easier proof theory. Lsystems are often extended with a "cut" rule as part of showing that for a given Lsystem and Nsystem, the derivations of each system can be encoded in the other. For example, NK proves the same as LK + cut (Gentzen 1935). Proof Normalization. A system is consistent when it is impossible to prove false, i.e., derive absurdity from zero assumptions. A system is analytic (has the analycity property) when there is an e#ective method to decompose any conclusion sequent into simpler premise sequents from which the conclusion can be obtained by some rule in the system such that the conclusion is derivable i# the premises are derivable (Maenpaa 1993). To achieve the goals of consistency and analycity, it has been customary to consider
Comparing approaches to generic programming in Haskell
 ICS, Utrecht University
, 2006
"... Abstract. The last decade has seen a number of approaches to datatypegeneric programming: PolyP, Functorial ML, ‘Scrap Your Boilerplate’, Generic Haskell, ‘Generics for the Masses’, etc. The approaches vary in sophistication and target audience: some propose fullblown programming languages, some s ..."
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Cited by 14 (4 self)
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Abstract. The last decade has seen a number of approaches to datatypegeneric programming: PolyP, Functorial ML, ‘Scrap Your Boilerplate’, Generic Haskell, ‘Generics for the Masses’, etc. The approaches vary in sophistication and target audience: some propose fullblown programming languages, some suggest libraries, some can be seen as categorical programming methods. In these lecture notes we compare the various approaches to datatypegeneric programming in Haskell. We introduce each approach by means of example, and we evaluate it along different dimensions (expressivity, ease of use, etc). 1
Pattern Matching as Cut Elimination
 In Logic in Computer Science
, 1999
"... We present typed pattern calculus with explicit pattern matching and explicit substitutions, where both the typing rules and the reduction rules are modeled on the same logical proof system, namely Gentzen sequent calculus for minimal logic. Our calculus is inspired by the CurryHoward Isomorphism, ..."
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Cited by 9 (2 self)
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We present typed pattern calculus with explicit pattern matching and explicit substitutions, where both the typing rules and the reduction rules are modeled on the same logical proof system, namely Gentzen sequent calculus for minimal logic. Our calculus is inspired by the CurryHoward Isomorphism, in the sense that types, both for patterns and terms, correspond to propositions, terms correspond to proofs, and term reduction corresponds to sequent proof normalization performed by cut elimination. The calculus enjoys subject reduction, confluence, preservation of strong normalization w.r.t a system with metalevel substitutions, and strong normalization for welltyped terms, and, as a consequence, can be seen as an implementation calculus for functional formalisms using metalevel operations for pattern matching and substitutions.
Completing Herbelin’s programme
"... In 1994 Herbelin started and partially achieved the programme of showing that, for intuitionistic implicational logic, there is a CurryHoward interpretation of sequent calculus into a variant of the λcalculus, specifically a variant which manipulates formally “applicative contexts” and inverts t ..."
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Cited by 8 (4 self)
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In 1994 Herbelin started and partially achieved the programme of showing that, for intuitionistic implicational logic, there is a CurryHoward interpretation of sequent calculus into a variant of the λcalculus, specifically a variant which manipulates formally “applicative contexts” and inverts the associativity of “applicative terms”. Herbelin worked with a fragment of sequent calculus with constraints on left introduction. In this paper we complete Herbelin’s programme for full sequent calculus, that is, sequent calculus without the mentioned constraints, but where permutative conversions necessarily show up. This requires the introduction of a lambdalike calculus for full sequent calculus and an extension of natural deduction that gives meaning to “applicative contexts” and “applicative terms”. Such extension is a calculus with modus ponens and primitive substitution that refines von Plato’s natural deduction; it is also a “coercion calculus”, in the sense of Cervesato and Pfenning. The prooftheoretical outcome is noteworthy: the puzzling relationship between cut and substitution is settled; and cutelimination in sequent calculus is proven isomorphic to normalisation in the proposed natural deduction system. The isomorphism is the mapping that inverts the associativity of applicative terms.
Programming with heterogeneous structures: Manipulating xml data using bondi
 In Vladimir EstivillCastro and Gillian Dobbie, editors, TwentyNinth Australasian Computer Science Conference (ACSC2006), volume 48(1) of Australian Computer Science Communications
, 2006
"... Copyright c○2005 F.Y. Huang and C.B. Jay and D.B. Skillicorn Manipulating semistructured data, such as XML, does not fit well within conventional programming languages. A typical manipulation requires finding all occurrences of a structure whose context may be different in different places, and both ..."
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Cited by 8 (7 self)
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Copyright c○2005 F.Y. Huang and C.B. Jay and D.B. Skillicorn Manipulating semistructured data, such as XML, does not fit well within conventional programming languages. A typical manipulation requires finding all occurrences of a structure whose context may be different in different places, and both aspects cause difficulty. Most approaches to these problems of data access are addressed by specialpurpose languages which then pass their results to other, generalpurpose languages, creating the impedance mismatch problem. Alternative approaches add a few new means of data access to existing languages without being able to achieve the desired expressive power. We show how an existing programming language, bondi, can be used as is to handle XML data, since data access patterns, called signposts, are firstclass expressions which can be passed as parameters to functions. Programming with Heterogeneous Structures: Manipulating XML data using bondi
Conservative extensions of the λcalculus for the computational interpretation of sequent calculus
, 2002
"... ..."
A DomainSpecific Language for Incremental and Modular Design of LargeScale VerifiablySafe Flow Networks (Preliminary Report)
"... We define a domainspecific language (DSL) to inductively assemble flow networks from small networks or modules to produce arbitrarily large ones, with interchangeable functionallyequivalent parts. Our small networks or modules are “small ” only as the building blocks in this inductive definition ( ..."
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Cited by 5 (4 self)
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We define a domainspecific language (DSL) to inductively assemble flow networks from small networks or modules to produce arbitrarily large ones, with interchangeable functionallyequivalent parts. Our small networks or modules are “small ” only as the building blocks in this inductive definition (there is no limit on their size). Associated with our DSL is a type theory, a system of formal annotations to express desirable properties of flow networks together with rules that enforce them as invariants across their interfaces, i.e., the rules guarantee the properties are preserved as we build larger networks from smaller ones. A prerequisite for a type theory is a formal semantics, i.e., a rigorous definition of the entities that qualify as feasible flows through the networks, possibly restricted to satisfy additional efficiency or safety requirements. This can be carried out in one of two ways, as a denotational semantics or as an operational (or reduction) semantics; we choose the first in preference to the second, partly to avoid exponentialgrowth rewriting in the operational approach. We set up a typing system and prove its soundness for our DSL. 1
A COMBINATORY ACCOUNT OF INTERNAL STRUCTURE
"... Abstract. Traditional combinatory logic is able to represent all Turing computable functions on natural numbers, but there are effectively calculable functions on the combinators themselves that cannot be so represented, because they have direct access to the internal structure of their arguments. S ..."
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Cited by 5 (4 self)
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Abstract. Traditional combinatory logic is able to represent all Turing computable functions on natural numbers, but there are effectively calculable functions on the combinators themselves that cannot be so represented, because they have direct access to the internal structure of their arguments. Some of this expressive power is captured by adding a factorisation combinator. It supports structural equality, and more generally, a large class of generic queries for updating of, and selecting from, arbitrary structures. The resulting combinatory logic is structure complete in the sense of being able to represent patternmatching functions, as well as simple abstractions. §1. Introduction. Traditional combinatory logic [21, 4, 10] is computationally equivalent to pure λcalculus [3] and able to represent all of the Turing computable functions on natural numbers [23], but there are effectively calculable functions on the combinators themselves that cannot be so represented, as they examine the internal structure of their arguments.