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Lambda Terms for Natural Deduction, Sequent Calculus and Cut Elimination
"... It is wellknown that there is an isomorphism between natural deduction derivations and typed lambda terms. Moreover normalising these terms corresponds to eliminating cuts in the equivalent sequent calculus derivations. Several papers have been written on this topic. The correspondence between sequ ..."
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It is wellknown that there is an isomorphism between natural deduction derivations and typed lambda terms. Moreover normalising these terms corresponds to eliminating cuts in the equivalent sequent calculus derivations. Several papers have been written on this topic. The correspondence between sequent calculus derivations and natural deduction derivations is, however, not a oneone map, which causes some syntactic technicalities. The correspondence is best explained by two extensionally equivalent type assignment systems for untyped lambda terms, one corresponding to natural deduction (N) and the other to sequent calculus (L). These two systems constitute different grammars for generating the same (type assignment relation for untyped) lambda terms. The second grammar is ambiguous, but the first one is not. This fact explains the manyone correspondence mentioned above. Moreover, the second type assignment system has a `cutfree' fragment (L cf ). This fragment generates exactly the typeable lambda terms in normal form. The cut elimination theorem becomes a simple consequence of the fact that typed lambda terms posses a normal form.
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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Cited by 7 (0 self)
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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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Cited by 6 (0 self)
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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.