On modularity in infinitary term rewriting
We study modular properties in strongly convergent infinitary term rewriting. In particular, we show that: Confluence is not preserved across direct sum of a finite number of systems, even when these are non-collapsing. Confluence modulo equality of hypercollapsing subterms is not preserved across direct sum of a finite number of systems. Normalization is not preserved across direct sum of an infinite number of left-linear systems. Unique normalization with respect to reduction is not preserved across direct sum of a finite number of left-linear systems. Together, these facts constitute a radical departure from the Situation infinitary term rewriting. Positive results are: Confluence is preserved under the direct sum of an infinite number of left-linear systems iff at most one system contains a collapsing rule. Confluence is preserved under the direct sum of a finite number of non-collapsing systems if only terms of finite rank are considered. Top-termination is preserved under the direct sum of a finite number of left-linear systems. Normalization is preserved under the direct sum of a finite number of left-linear systems. All of the negative results above hold in the setting of weakly convergent rewriting as well, as do the positive results concerning modularity of top-termination and normalization for left-linear systems. (C) 2006 Elsevier Inc. All rights reserved.
modularity term rewriting confluence normalization infinitary rewriting strong convergence church-rosser property