Infinitary Combinatory Reduction Systems

  • Authors:

  • Jeroen Ketema;Jakob Grue Simonsen

  • Affiliations:

  • Research Institute of Electrical Communication, Tohoku University, 2-1-1 Katahira, Aoba-ku, Sendai 980-8577, Japan;Department of Computer Science, University of Copenhagen (DIKU), Universitetsparken 1, 2100 Copenhagen Ø, Denmark

  • Venue:
  • Information and Computation
  • Year:
  • 2011

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Abstract

We define infinitary Combinatory Reduction Systems (iCRSs), thus providing the first notion of infinitary higher-order rewriting. The systems defined are sufficiently general that ordinary infinitary term rewriting and infinitary @l-calculus are special cases. Furthermore, we generalise a number of known results from first-order infinitary rewriting and infinitary @l-calculus to iCRSs. In particular, for fully-extended, left-linear iCRSs we prove the well-known compression property, and for orthogonal iCRSs we prove that (1) if a set of redexes U has a complete development, then all complete developments of U end in the same term and that (2) any tiling diagram involving strongly convergent reductions S and T can be completed iff at least one of S/T and T/S is strongly convergent. We also prove an ancillary result of independent interest: a set of redexes in an orthogonal iCRS has a complete development iff the set has the so-called finite jumps property.