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A Complete, Co-inductive Syntactic Theory of Sequential Control and State

2009; Springer Science+Business Media; Linguagem: Inglês

10.1007/978-3-642-04164-8_17

1611-3349

Kristian Støvring, Søren B. Lassen,

Logic, Reasoning, and Knowledge

We present a co-inductive syntactic theory, eager normal form bisimilarity, for the untyped call-by-value lambda calculus extended with continuations and mutable references. We demonstrate that the associated bisimulation proof principle is easy to use and that it is a powerful tool for proving equivalences between recursive imperative higher-order programs. The theory is modular in the sense that eager normal form bisimilarity for each of the calculi extended with continuations and/or mutable references is a fully abstract extension of eager normal form bisimilarity for its sub-calculi. For each calculus, we prove that eager normal form bisimilarity is a congruence and is sound with respect to contextual equivalence. Furthermore, for the calculus with both continuations and mutable references, we show that eager normal form bisimilarity is complete: it coincides with contextual equivalence. Eager normal form bisimilarity is inspired by Böhm-tree equivalence in the pure lambda calculus. We clarify the associated proof principle by reviewing properties of Böhm trees and surveying previous work on normal form bisimulation.