On modularity in infinitary term rewriting

  • Authors:

  • Jakob Grue Simonsen

  • Affiliations:

  • Department of Computer Science, University of Copenhagen (DIKU), Universitetsparken 1, DK-2100 Copenhagen Ø, Denmark

  • Venue:
  • Information and Computation
  • Year:
  • 2006

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Abstract

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 in finitary 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.