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Relating Typability and Expressiveness in FiniteRank Intersection Type Systems (Extended Abstract)
 In Proc. 1999 Int’l Conf. Functional Programming
, 1999
"... We investigate finiterank intersection type systems, analyzing the complexity of their type inference problems and their relation to the problem of recognizing semantically equivalent terms. Intersection types allow something of type T1 /\ T2 to be used in some places at type T1 and in other places ..."
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Cited by 22 (9 self)
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We investigate finiterank intersection type systems, analyzing the complexity of their type inference problems and their relation to the problem of recognizing semantically equivalent terms. Intersection types allow something of type T1 /\ T2 to be used in some places at type T1 and in other places at type T2 . A finiterank intersection type system bounds how deeply the /\ can appear in type expressions. Such type systems enjoy strong normalization, subject reduction, and computable type inference, and they support a pragmatics for implementing parametric polymorphism. As a consequence, they provide a conceptually simple and tractable alternative to the impredicative polymorphism of System F and its extensions, while typing many more programs than the HindleyMilner type system found in ML and Haskell. While type inference is computable at every rank, we show that its complexity grows exponentially as rank increases. Let K(0, n) = n and K(t + 1, n) = 2^K(t,n); we prove that recognizing the pure lambdaterms of size n that are typable at rank k is complete for dtime[K(k1, n)]. We then consider the problem of deciding whether two lambdaterms typable at rank k have the same normal form, Generalizing a wellknown result of Statman from simple types to finiterank intersection types. ...
Calculi of Generalised βReduction and Explicit Substitutions: The TypeFree and Simply Typed Versions
, 1998
"... Extending the λcalculus with either explicit substitution or generalized reduction has been the subject of extensive research recently, and still has many open problems. This paper is the first investigation into the properties of a calculus combining both generalized reduction and explicit substit ..."
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Cited by 14 (7 self)
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Extending the λcalculus with either explicit substitution or generalized reduction has been the subject of extensive research recently, and still has many open problems. This paper is the first investigation into the properties of a calculus combining both generalized reduction and explicit substitutions. We present a calculus, gs, that combines a calculus of explicit substitution, s, and a calculus with generalized reduction, g. We believe that gs is a useful extension of the  calculus, because it allows postponement of work in two different but complementary ways. Moreover, gs (and also s) satisfies properties desirable for calculi of explicit substitutions and generalized reductions. In particular, we show that gs preserves strong normalization, is a conservative extension of g, and simulates fireduction of g and the classical calculus. Furthermore, we study the simply typed versions of s and gs, and show that welltyped terms are strongly normalizing and that other properties,...
Generalized BetaReduction and Explicit Substitutions
, 1996
"... Extending the calculus with either explicit substitution or generalised reduction has been the subject of extensive research recently which still has many open problems. Due to this reason, the properties of a calculus combining both generalised reduction and explicit substitutions have never been ..."
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Extending the calculus with either explicit substitution or generalised reduction has been the subject of extensive research recently which still has many open problems. Due to this reason, the properties of a calculus combining both generalised reduction and explicit substitutions have never been studied. This paper presents such a calculus sg and shows that it is a desirable extension of the calculus. In particular, we show that sg preserves strong normalisation, is sound and it simulates classical fireduction. Furthermore, we study the simply typed calculus extended with both generalised reduction and explicit substitution and show that welltyped terms are strongly normalising and that other properties such as subtyping and subject reduction hold. 1 Introduction 1.1 The calculus with generalised reduction In (( x : y :N)P )Q, the function starting with x and the argument P result in the redex ( x : y :N)P which when contracted will turn the function starting with y and Q i...
Calculi of Generalised betaReduction and Explicit Substitutions: The TypeFree and Simply Typed Versions
, 1997
"... Extending the calculus with either explicit substitution or generalised reduction has been the subject of extensive research recently and still has many open problems. This paper is the first investigation into the properties of a calculus combining both generalised reduction and explicit substitut ..."
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Extending the calculus with either explicit substitution or generalised reduction has been the subject of extensive research recently and still has many open problems. This paper is the first investigation into the properties of a calculus combining both generalised reduction and explicit substitutions. We present a calculus, gs, that combines a calculus of explicit substitution, s, and a calculus with generalized reduction, g. We believe that gs is a useful extension of the calculus because it allows postponment of work in two different but complementary ways. Moreover, gs (and also s) satisfies desirable properties of calculi of explicit substitutions and generalised reductions. In particular, we show that gs preserves strong normalisation, is a conservative extension of g, and simulates fireduction of g and the classical calculus. Furthermore, we study the simply typed versions of s and gs and show that well typed terms are strongly normalising and that other properties such as...
Redexes in Item Notation Classical Notation Item Notation
, 2009
"... • I(λx.B) = [x]I(B) and I(AB) = (I(B))I(A) • I((λx.(λy.xy))z) ≡ (z)[x][y](y)x. The items are (z), [x], [y] and (y). • applicator wagon (z) and abstractor wagon [x] occur NEXT to each other. • A term is a wagon followed by a term. • (β) ..."
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• I(λx.B) = [x]I(B) and I(AB) = (I(B))I(A) • I((λx.(λy.xy))z) ≡ (z)[x][y](y)x. The items are (z), [x], [y] and (y). • applicator wagon (z) and abstractor wagon [x] occur NEXT to each other. • A term is a wagon followed by a term. • (β)