Relative Normalization in Orthogonal Expression Reduction Systems

We study reductions in orthogonal (left-linear and non-ambiguous) Expression Reduction Systems, a formalism for Term Rewriting Systems with bound variables and substitutions. To generalise the normalization theory of Huet and Levy, we introduce the notion of neededness with respect to a set of reductions π or a set of terms \(\mathcal{S}\) so that each existing notion of neededness can be given by specifying π or \(\mathcal{S}\). We imposed natural conditions on \(\mathcal{S}\), called stability, that are sufficient and necessary for each term not in \(\mathcal{S}\)-normal form (i.e., not in \(\mathcal{S}\)) to have at least one \(\mathcal{S}\)-needed redex, and repeated contraction of \(\mathcal{S}\)-needed redexes in a term t to lead to an \(\mathcal{S}\)-normal form of t whenever there is one. Our relative neededness notion is based on tracing (open) components, which are occurrences of contexts not containing any bound variable, rather than tracing redexes or subterms.

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