Syntactic definitions of undefined: On defining the undefined
In the λ-calculus, there is a standard notion of what terms should be considered to be “undefined”: the unsolvable terms. There are various equivalent characterisations of this property of terms. We attempt to find a similar notion for orthogonal term rewrite systems. We find that in general the properties of terms analogous to the various characterisations of solvability differ.
We give two axioms that a notion of undefinedness should satisfy, and explore some of their consequences. The axioms lead to a concept analogous to the Böhm trees of the λ-calculus. Each term has a unique Böhm tree, and the set of such trees forms a domain which provides a denotational semantics for the rewrite system. We consider several particular notions of undefinedness satisfying the axioms, and compare them.
AMS subject Classification68Q42
CR subject ClassificationF1.1 F4.1 F4.2
Keywords & phrasesOrthogonal term rewriting systems Böhm Trees undefined terms lambda calculus solvability
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