On abstract normalisation beyond neededness

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 (ARS), a general rewriting framework encompassing many rewriting systems developed by P-A.Melli\`es in his PhD thesis. The theorem states that multistep strategies reducing so called necessary and never-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 PPC, a calculus of patterns and to the lambda-calculus with parallel-or.

[1]  Richard Kennaway.  Sequential Evaluation Strategies for Parallel-Or and Related Reduction Systems , 1989, Ann. Pure Appl. Log..

[2]  William C. Frederick,et al.  A Combinatory Logic , 1995 .

[3]  Henk Barendregt,et al.  The Lambda Calculus: Its Syntax and Semantics , 1985 .

[4]  Aart Middeldorp,et al.  A Sequential Reduction Strategy , 1996, Theor. Comput. Sci..

[5]  Jan Willem Klop,et al.  Combinatory reduction systems , 1980 .

[6]  Jean-Jacques Lévy,et al.  Computations in Orthogonal Rewriting Systems, II , 1991, Computational Logic - Essays in Honor of Alan Robinson.

[7]  J. Davenport.  Editor , 1960 .

[8]  Alejandro Ríos,et al.  An abstract normalisation result with applications to non-sequential calculi , 2014, ArXiv.

[9]  Vincent van Oostrom,et al.  Normalisation in Weakly Orthogonal Rewriting , 1999, RTA.

[10]  R. Lathe.  Phd by thesis , 1988, Nature.

[11]  Barry Jay.  Pattern Calculus - Computing with Functions and Structures , 2009 .

[12]  Michael J. O'Donnell,et al.  Computing in systems described by equations , 1977, Lecture Notes in Computer Science.

[13]  Ugo Dal Lago,et al.  (Leftmost-Outermost) Beta Reduction is Invariant, Indeed , 2016, Log. Methods Comput. Sci..

[14]  Vincent van Oostrom,et al.  On equal μ-terms , 2011, Theor. Comput. Sci..

[15]  R. Stephenson.  A and V , 1962, The British journal of ophthalmology.

[16]  Delia Kesner,et al.  Pure Pattern Calculus , 2006, ESOP.

[17]  Thibaut Balabonski,et al.  Optimality for Dynamic Patterns Extended Abstract , 2010 .

[18]  G Boudol.  Computational semantics of term rewriting systems , 1986 .

[19]  Delia Kesner,et al.  A nonstandard standardization theorem , 2014, POPL.

[20]  Alejandro Ríos,et al.  Normalisation for Dynamic Pattern Calculi , 2012, RTA.

[21]  Femke van Raamsdonk,et al.  Outermost-Fair Rewriting , 1997, TLCA.

[22]  Delia Kesner,et al.  First-class patterns , 2009, Journal of Functional Programming.

[23]  Thibaut Balabonski,et al.  On the Implementation of Dynamic Patterns , 2011, HOR.

[24]  F. Raamsdonk.  Confluence and Normalisation of Higher-Order Rewriting , 1996 .

[25]  Chang Liu,et al.  Term rewriting and all that , 2000, SOEN.

[26]  V. V. Oostrom.  The dynamic pattern calculus as a higher-order pattern rewriting system , 2015 .