On abstract normalisation beyond neededness

CARLOS LOMBARDI; ALEJANDRO RÍOS; EDUARDO BONELLI; ALEJANDRO RÍOS; EDUARDO BONELLI; DELIA KESNER; DELIA KESNER; CARLOS LOMBARDI

THEORETICAL COMPUTER SCIENCE

ELSEVIER SCIENCE BV

Lugar: Amsterdam; Año: 2017 vol. 672 p. 36 - 63

0304-3975

We study normalisation of multistep strategies, strategies that reduce a set of redexes at a time, focussing on the notion of necessary sets, those which contain at least one redex that cannot be avoided in order to reach a normal form. This is particularly appealing in the setting of non-sequential rewrite systems, in which terms that are not in normal form may not have any needed redex. We first prove a normalisation theorem for abstract rewrite systems or ARS, a general rewriting framework encompassing many rewriting systems developed by P-A.Melliès. The theorem states that multistep strategies reducing so called necessary and non-gripping sets of redexes at a time are normalising in any ARS. Gripping refers to an abstract property reflecting the behavior of higher-order substitution. We then apply this result to the particular case of the Pure Pattern Calculus, a calculus of patterns; and to the lambda-calculus with parallel-or.