 
Summary: Cyclic Lambda Calculi
Zena M. Ariola 1 and Stefan Blom 2
1 Department of Computer & Information Sciences
University of Oregon. Eugene, OR 97401, USA
email: ariola@cs.uoregon.edu
2 Department of Mathematics and Computer Science
Vrije Universiteit, De Boelelaan 1081a, 1081 HV Amsterdam
email: sccblom@cs.vu.nl
Abstract. 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 theo
ries, 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 no
tion 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.
