On modularity in infinitary term rewriting

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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.

Keywords

Term rewriting
Modularity
Infinitary rewriting
Strong convergence
Church-rosser property
Confluence
Normalization

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