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The Longest Perpetual Reductions in Orthogonal Expression Reduction Systems
 In: Proc. of the 3 rd International Conference on Logical Foundations of Computer Science, LFCS'94, A. Nerode and Yu.V. Matiyasevich, eds., Springer LNCS
, 1994
"... We consider reductions in Orthogonal Expression Reduction Systems (OERS), that is, Orthogonal Term Rewriting Systems with bound variables and substitutions, as in the calculus. We design a strategy that for any given term t constructs a longest reduction starting from t if t is strongly normaliza ..."
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We consider reductions in Orthogonal Expression Reduction Systems (OERS), that is, Orthogonal Term Rewriting Systems with bound variables and substitutions, as in the calculus. We design a strategy that for any given term t constructs a longest reduction starting from t if t is strongly normalizable, and constructs an infinite reduction otherwise. The Conservation Theorem for OERSs follows easily from the properties of the strategy. We develop a method for computing the length of a longest reduction starting from a strongly normalizable term. We study properties of pure substitutions and several kinds of similarity of redexes. We apply these results to construct an algorithm for computing lengths of longest reductions in strongly persistent OERSs that does not require actual transformation of the input term. As a corollary, we have an algorithm for computing lengths of longest developments in OERSs. 1 Introduction A strategy is perpetual if, given a term t, it constructs an infinit...
Coinductive Characterizations of Applicative Structures
 MATH. STRUCTURES IN COMP. SCI. 9(4):403–435
, 1998
"... We discuss new ways of characterizing, as maximal fixed points of monotone operators, observational congruences on terms and, more in general, equivalences on applicative structures. These characterizations naturally induce new forms of coinduction principles, for reasoning on program equivalences, ..."
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We discuss new ways of characterizing, as maximal fixed points of monotone operators, observational congruences on terms and, more in general, equivalences on applicative structures. These characterizations naturally induce new forms of coinduction principles, for reasoning on program equivalences, which are not based on Abramsky's applicative bisimulation. We discuss in particular, what we call, the cartesian coinduction principle, which arises when we exploit the elementary observation that functional behaviours can be expressed as cartesian graphs. Using the paradigm of final semantics, the soundness of this principle over an applicative structure can be expressed easily by saying that the applicative structure can be construed as a strongly extensional coalgebra for the functor (P( \Theta )) \Phi (P( \Theta )). In this paper, we present two general methods for showing the soundenss of this principle. The first applies to approximable applicative structures. Many c.p.o. models in...