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Cyclic Lambda Calculi
, 1997
"... . We precisely characterize a class of cyclic lambdagraphs, and then give a sound and complete axiomatization of the terms that represent a given graph. The equational axiom system is an extension of lambda calculus with the letrec construct. In contrast to current theories, which impose restrictio ..."
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Cited by 44 (5 self)
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. We precisely characterize a class of cyclic lambdagraphs, and then give a sound and complete axiomatization of the terms that represent a given graph. The equational axiom system is an extension of lambda calculus with the letrec construct. In contrast to current theories, which impose restrictions on where the rewriting can take place, our theory is very liberal, e.g., it allows rewriting under lambdaabstractions and on cycles. As shown previously, the reduction theory is nonconfluent. We thus introduce an approximate notion of confluence. Using this notion we define the infinite normal form or L'evyLongo tree of a cyclic term. We show that the infinite normal form defines a congruence on the set of terms. We relate our cyclic lambda calculus to the traditional lambda calculus and to the infinitary lambda calculus. Since most implementations of nonstrict functional languages rely on sharing to avoid repeating computations, we develop a variant of our calculus that enforces the ...
Normalization Results for Typeable Rewrite Systems
, 1997
"... In this paper we introduce Curryfied Term Rewriting Systems, and a notion of partial type assignment on terms and rewrite rules that uses intersection types with sorts and !. Three operations on types  substitution, expansion, and lifting  are used to define type assignment, and are proved to be ..."
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Cited by 30 (26 self)
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In this paper we introduce Curryfied Term Rewriting Systems, and a notion of partial type assignment on terms and rewrite rules that uses intersection types with sorts and !. Three operations on types  substitution, expansion, and lifting  are used to define type assignment, and are proved to be sound. With this result the system is proved closed for reduction. Using a more liberal approach to recursion, we define a general scheme for recursive definitions and prove that, for all systems that satisfy this scheme, every term typeable without using the typeconstant ! is strongly normalizable. We also show that, under certain restrictions, all typeable terms have a (weak) headnormal form, and that terms whose type does not contain ! are normalizable.
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"... In this paper we introduce Curryfied Term Rewriting Systems, and a notion of partial type assignment on terms and rewrite rules that uses intersection types with sorts and. Three operations on types – substitution, expansion, and lifting – are used to define type assignment, and are proved to be sou ..."
Abstract
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In this paper we introduce Curryfied Term Rewriting Systems, and a notion of partial type assignment on terms and rewrite rules that uses intersection types with sorts and. Three operations on types – substitution, expansion, and lifting – are used to define type assignment, and are proved to be sound. With this result the system is proved closed for reduction. Using a more liberal approach to recursion, we define a general scheme for recursive definitions and prove that, for all systems that satisfy this scheme, every term typeable without using the typeconstant is strongly normalizable. We also show that, under certain restrictions, all typeable terms have a (weak) headnormal form, and that terms whose type does not contain are normalizable. 1
AND
"... In this paper we introduce Curryfied Term Rewriting Systems, and a notion of partial type assignment on terms and rewrite rules that uses intersection types with sorts and ω. Three operations on types – substitution, expansion, and lifting – are used to define type assignment, and are proved to be s ..."
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In this paper we introduce Curryfied Term Rewriting Systems, and a notion of partial type assignment on terms and rewrite rules that uses intersection types with sorts and ω. Three operations on types – substitution, expansion, and lifting – are used to define type assignment, and are proved to be sound. With this result the system is proved closed for reduction. Using a more liberal approach to recursion, we define a general scheme for recursive definitions and prove that, for all systems that satisfy this scheme, every term typeable without using the typeconstant ω is strongly normalizable. We also show that, under certain restrictions, all typeable terms have a (weak) headnormal form, and that terms whose type does not contain ω are normalizable. 1