Hierarchical Nominal Terms and Their Theory of Rewriting

Nominal rewriting introduced a novel method of specifying rewriting on syntax-with-binding. We extend this treatment of rewriting with hierarchy of variables representing increasingly 'meta-level' variables, e.g. in hierarchical nominal term rewriting the meta-level unknowns (representing unknown terms) in a rewrite rule can be 'folded into' the syntax itself (and rewritten). To the extent that rewriting is a mathematical meta-framework for logic and computation, and nominal rewriting is a framework with native support for binders, hierarchical nominal term rewriting is a meta-to-the-omega level framework for logic and computation with binders.

[1]  Mark R. Shinwell,et al.  Fresh Objective Caml user manual , 2005 .

[2]  Murdoch James Gabbay.  A new calculus of contexts , 2005, PPDP '05.

[3]  Terese.  Term rewriting systems , 2003, Cambridge tracts in theoretical computer science.

[4]  Jon Barwise,et al.  An Introduction to First-Order Logic , 1977 .

[5]  Ian A. Mason,et al.  Using typed lambda calculus to implement formal systems on a machine , 1992, Journal of Automated Reasoning.

[6]  M. Gabbay.  Nominal Algebra , 2006 .

[7]  Tobias Nipkow,et al.  Higher-Order Rewrite Systems and Their Confluence , 1998, Theor. Comput. Sci..

[8]  Christian Urban,et al.  alpha-Prolog: A Logic Programming Language with Names, Binding and a-Equivalence , 2004, ICLP.

[9]  J. Benthem,et al.  Higher-Order Logic , 2001 .

[10]  Maribel Fernández,et al.  Nominal rewriting , 2007, Inf. Comput..

[11]  Dov M. Gabbay,et al.  Handbook of Philosophical Logic Vol. 10 , 2001 .

[12]  Pierre Lescanne.  From Lambda-sigma to Lambda-upsilon a Journey Through Calculi of Explicit Substitutions. , 1994 .

[13]  Christian Urban,et al.  Nominal unification , 2004, Theor. Comput. Sci..

[14]  H. Keisler,et al.  Handbook of mathematical logic , 1977 .

[15]  B. Russell,et al.  Principia Mathematica Vol. I , 1910 .

[16]  Tobias Nipkow,et al.  The Isabelle Reference Manual , 2007 .

[17]  Cj Roel Bloo,et al.  Preservation of termination for explicit substitution , 1997 .

[18]  Robin Milner,et al.  A Calculus of Mobile Processes, II , 1992, Inf. Comput..

[19]  Bengt Nordström,et al.  Programming in Martin-Löf's Type Theory , 1990 .

[20]  Murdoch James Gabbay,et al.  Capture-avoiding substitution as a nominal algebra , 2007, Formal Aspects of Computing.

[21]  Robin Milner,et al.  A Calculus of Mobile Processes, II , 1992, Inf. Comput..

[22]  J. V. Tucker,et al.  Basic Simple Type Theory , 1997 .

[23]  Pierre Lescanne,et al.  From λσ to λν: a journey through calculi of explicit substitutions , 1994, POPL '94.

[24]  Bengt Nordström,et al.  Programming in Martin-Lo¨f's type theory: an introduction , 1990 .