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Abstract:

I outline a new proof of strong normalisation for system F, by direct calculation of the size of a normalisation tree. The system F itself is used as a formulism for representing these calculations, and any reasonably sensible model of the calculus can be used to verify that the correctness of the calculations. The method works for a variety of calculi, and seems reasonably general. Discussion Proofs of strong normalisation for typed-calculi, such as system F, are well known. The standard proof, developed by Tait and Girard using a notion called reducibility, proceeds by defining sets of certain normalising terms in such a way that it can be shown that every term is a member of the appropriate set. This proof is probably best understood in a model theoretic manner: the sets of normalising terms form a model of some sort. A term t may be interpreted in this model by some term ([t]), which is necessarily normalising, and either actually t = ([t]), or t and ([t]) are sufficiently similar so that we may infer that t is normalising. All previous proofs of normalisation for system F follow this pattern, there being some differences in the exact machinery used. We present here a novel proof, which does not rely on a model based on sets of terms. Instead, we present a translation t 7! ([t]) on terms, such that ([t]) is a program computing a witness for the \Sigma 0 1 statement that t is strongly normalising. By interpreting ([t]) in a model, or by otherwise extracting an actual value from ([t]), we can verify that t is in fact strongly normalising.

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