Environmental bisimulations for higher-order languages

Abstract

Developing a theory of bisimulation in higher-order languages can be hard. Particularly challenging can be: (1) the proof of congruence, as well as enhancements of the bisimulation proof method with "up-to context" techniques, and (2) obtaining definitions and results that scale to languages with different features. To meet these challenges, we present environmental bisimulations, a form of bisimulation for higher-order languages, and its basic theory. We consider four representative calculi: pure λ-calculi (call-by-name and call-byvalue), call-by-value λ-calculus with higher-order store, and then Higher-Order π-calculus. In each case: we present the basic properties of environmental bisimilarity, including congruence; we show that it coincides with contextual equivalence; we develop some up-to techniques, including up-to context, as examples of possible enhancements of the associated bisimulation method. Unlike previous approaches (such as applicative bisimulations, logical relations, Sumii-Pierce-Koutavas-Wand), our method does not require induction/indices on evaluation derivation/steps (which may complicate the proofs of congruence, transitivity, and the combination with up-to techniques), or sophisticated methods such as Howe's for proving congruence. It also scales from the pure λ-calculi to the richer calculi with simple congruence proofs.

A Testing Theory for a Higher-Order Cryptographic Language

“…This work uses a strong assumption about the power of attackers: higher-order terms are sent over channels as decomposable syntax objects. The resulting proof methodology is significantly different from ours, employing environmental bisimulations and sophisticated up-to-context techniques [23]. However, bisimulations do not provide a complete proof technique for linear-time equivalences, such as safe equivalence.…”

“…This is similar in spirit to the environments used in environmental bisimulations [23,25] and in proof techniques for first-order cryptographic calculi [2,4]. Our LTS is first-order because it employs a symbolic treatment of higher-order values generated by the context, which obviates the need for quantification over functional contexts.…”

“…However it is unclear how the translation could be adapted to HOspi in a way that will be fully abstract and useful in proofs of equivalence. The second approach would be to make use of environmental bisimulations [23], which has been shown to provide a sound, complete and useful proof technique for reduction barbed congruence in Applied HOπ, a higher-order concurrent language with cryptographic primitives [25]. However, environmental bisimulations do not provide a complete proof principle for linear-time semantic equivalences, such as safe equivalence.…”

“…However, unlike [21,23,25], in order to keep the LTS first-order we severely restrict the ability of the observer to construct higher-order values. The soundness of our technique in Section 5 shows that this restriction does not compromise the possible observations made in the LTS.…”

“…We have given a characterisation of safety preservation in terms of novel set simulations, which provides a sound and complete proof methodology for process equivalence; the usefulness of the methodology has been demonstrated via simple illustrative examples. The LTS in our proof methodology makes extensive use of the current knowledge of observers, or adversaries, a generalisation of similar ideas for the first-order spi-calculus [2,4], and environmental bisimulations [23,25]. Sato and Sumii [25] defined a higher-order version of the applied π-calculus called the Applied HOπ calculus, developed a bisimulation method which is sound and complete with respect to reduction barbed congruence, and gave bisimulation proofs for secrecy properties of example higher-order systems.…”