Natural Semantics for Non-Determinism

We present a natural semantics for the untyped lazy -calculus plus McCarthy's amb, a nondeterministic choice operator. The natural semantics includes rules for both convergent behaviour (de ned inductively) and divergent behaviour (de ned co-inductively). This semantics is equivalent to a small step reduction semantics that corresponds closely to our operational intuitions about McCarthy's amb. We present equivalences for convergent and divergent behaviour based on the natural semantics and prove a Context Lemma for the convergence equivalence. We then give a -theory l8, based on the equivalences for convergent and divergent behaviour. Since it is able to distinguish between programs that di er only in their divergent behaviour, the -theory is more discriminating than equational theories based on current domain-theoretic models. It is therefore more suitable for reasoning about functional programs containing McCarthy's amb.