Revisiting Elementary Denotational Semantics

We present a semantics for an applied call-by-value λ-calculus that is compositional, extensional, and elementary. We present four di‚erent views of the semantics: 1) as a relational (big-step) semantics that is not operational but instead declarative, 2) as a denotational semantics that does not use domain theory, 3) as a non-deterministic interpreter, and 4) as a variant of the intersection type systems of the Torino group. We prove that the semantics is correct by showing that it is sound and complete with respect to operational semantics on programs and that is sound with respect to contextual equivalence. We have not yet investigated whether it is fully abstract. We demonstrate that this approach to semantics is useful with three case studies. First, we use the semantics to prove correctness of a compiler optimization that inlines function application. Second, we adapt the semantics to the polymorphic λ-calculus extended with general recursion and prove semantic type soundness. Œird, we adapt the semantics to the call-by-value λ-calculus with mutable references. All of the de€nitions and proofs in this paper are mechanized in Isabelle in under 3,000 lines.