The syntactic nature of operational reasoning requires techniques to deal with term contexts, especially for reasoning about recursion. In this paper we study applicative bisimulation and a variant of Sands’ improvement theory for a small call-by-value functional language. We explore an indirect, relational approach for reasoning about contexts. It is inspired by Howe’s precise method for proving congruence of simulation orderings and by Pitts’ extension thereof for proving applicative bisimulation up to context. We illustrate this approach with proofs of the unwinding theorem and syntactic continuity and, more importantly, we establish analogues of Sangiorgi’s bisimulation up to context for applicative bisimulation and for improvement. Using these powerful bisimulation up to context techniques, we give concise operational proofs of recursion induction, the improvement theorem, and syntactic minimal invariance. Previous operational proofs of these results involve complex, explicit reasoning about contexts.
Abstract. Normal form bisimulation is a powerful theory of program equivalence, originally developed to characterize Lévy-Longo tree equivalence and Boehm tree equivalence. It has been adapted to a range of untyped, higher-order calculi, but types have presented a difficulty. In this paper, we present an account of normal form bisimulation for types, including recursive types. We develop our theory for a continuation-passing style calculus, Jump-With-Argument (JWA), where normal form bisimilarity takes a very simple form. We give a novel congruence proof, based on insights from game semantics. A notable feature is the seamless treatment of eta-expansion. We demonstrate the normal form bisimulation proof principle by using it to establish a syntactic minimal invariance result and the uniqueness of the fixed point operator at each type.
We present a new 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.
Abstract. Normal form bisimulation is a powerful theory of program equivalence, originally developed to characterize Lévy-Longo tree equivalence and Boehm tree equivalence. It has been adapted to a range of untyped, higher-order calculi, but types have presented a difficulty. In this paper, we present an account of normal form bisimulation for types, including recursive types. We develop our theory for a continuation-passing style calculus, Jump-With-Argument (JWA), where normal form bisimilarity takes a very simple form. We give a novel congruence proof, based on insights from game semantics. A notable feature is the seamless treatment of eta-expansion. We demonstrate the normal form bisimulation proof principle by using it to establish a syntactic minimal invariance result and the uniqueness of the fixed point operator at each type.
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