Infinitary Combinatory Reduction Systems

Jeroen Ketema,Jakob Grue Simonsen

doi:10.1016/j.ic.2011.01.007

Jeroen Ketema, Jakob Grue Simonsen

Open Access

https://doi.org/10.1016/j.ic.2011.01.007

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Journal: Information and Control	Publication Date: Feb 4, 2011
Citations: 43	License type: publisher-specific-oa

Affiliation: Tohoku University, University of Copenhagen

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 λ -calculus are special cases. Furthermore, we generalise a number of known results from first-order infinitary rewriting and infinitary λ -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.

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