On modularity in infinitary term rewriting

TY - JOUR

T1 - On modularity in infinitary term rewriting

AU - Simonsen, Jakob Grue

PY - 2006/1/1

Y1 - 2006/1/1

N2 - 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 noncollapsing. • 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.

AB - 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 noncollapsing. • 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.

KW - Church-Rosser property

KW - Confluence

KW - Infinitary rewriting

KW - Modularity

KW - Normalization

KW - Strong convergence

KW - Term rewriting

UR - http://www.scopus.com/inward/record.url?scp=84855194957&partnerID=8YFLogxK

U2 - 10.1016/j.ic.2006.02.005

DO - 10.1016/j.ic.2006.02.005

M3 - Journal article

AN - SCOPUS:84855194957

SN - 0890-5401

VL - 204

SP - 957

EP - 988

JO - Information and Computation

JF - Information and Computation

IS - 6

ER -