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Perpetual Reductions in λCalculus
, 1999
"... This paper surveys a part of the theory of fireduction in calculus which might aptly be called perpetual reductions. The theory is concerned with perpetual reduction strategies, i.e., reduction strategies that compute infinite reduction paths from terms (when possible), and with perpetual red ..."
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This paper surveys a part of the theory of fireduction in calculus which might aptly be called perpetual reductions. The theory is concerned with perpetual reduction strategies, i.e., reduction strategies that compute infinite reduction paths from terms (when possible), and with perpetual redexes, i.e., redexes whose contraction in terms preserves the possibility (when present) of infinite reduction paths. The survey not only recasts classical theorems in a unified setting, but also offers new results, proofs, and techniques, as well as a number of applications to problems in calculus and type theory. 1. Introduction Considerable attention has been devoted to classification of reduction strategies in typefree calculus [4, 6, 7, 15, 38, 44, 81]see also [2, Ch. 13]. We are concerned with strategies differing in the length of reduction paths. This paper draws on several sources. In late 1994, van Raamsdonk and Severi [59] and Srensen [66, 67] independently developed ...
Perpetual Reductions in λCalculus
, 1999
"... This paper surveys a part of the theory of fireduction in λcalculus which might aptly be called perpetual reductions. The theory is concerned with perpetual reduction strategies, i.e., reduction strategies that compute infinite reduction paths from λterms (when possible), and with perpetual r ..."
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This paper surveys a part of the theory of fireduction in λcalculus which might aptly be called perpetual reductions. The theory is concerned with perpetual reduction strategies, i.e., reduction strategies that compute infinite reduction paths from λterms (when possible), and with perpetual redexes, i.e., redexes whose contraction in λterms preserves the possibility (when present) of infinite reduction paths. The survey not only recasts classical theorems in a unified setting, but also offers new results, proofs, and techniques, as well as a number of applications to problems in λcalculus and type theory.
Polynomial Interpretations for HigherOrder Rewriting ∗
"... The termination method of weakly monotonic algebras, which has been defined for higherorder rewriting in the HRS formalism, offers a lot of power, but has seen little use in recent years. We adapt and extend this method to the alternative formalism of algebraic functional systems, where the simply ..."
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The termination method of weakly monotonic algebras, which has been defined for higherorder rewriting in the HRS formalism, offers a lot of power, but has seen little use in recent years. We adapt and extend this method to the alternative formalism of algebraic functional systems, where the simplytyped λcalculus is combined with algebraic reduction. Using this theory, we define higherorder polynomial interpretations, and show how the implementation challenges of this technique can be tackled. A full implementation is provided in the termination tool WANDA.
Higherorder Lazy Narrowing Calculi in Perspective
, 2000
"... Higherorder lazy narrowing (HOLN for short) is a computational model for higherorder functional logic programming. It can be viewed as an extension of firstorder lazy narrowing with inference rules to solve equations involving lambdaabstractions and higherorder variables. A common feature of th ..."
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Higherorder lazy narrowing (HOLN for short) is a computational model for higherorder functional logic programming. It can be viewed as an extension of firstorder lazy narrowing with inference rules to solve equations involving lambdaabstractions and higherorder variables. A common feature of the HOLN calculi proposed so far is the high nondeterminism between the inference rules designed to solve equations which involve higherorder variables. In this paper we present various refinements of HOLN towards more deterministic versions. The refinements are defined for classes of higherorder functional logic programs which are useful for programming purposes. Our work draws on two sources: the calculus LN for pattern rewrite systems [Pre98] and the firstorder lazy narrowing calculus LNC and its deterministic refinements [MO98].