Effect-dependent transformations for concurrent programs

Abstract

We describe a denotational semantics for an abstract effect system for a higher-order, shared-variable concurrent programming language. We prove the soundness of a number of general effectbased program equivalences, including a parallelization equation that specifies sufficient conditions for replacing sequential composition with parallel composition. Effect annotations are relative to abstract locations specified by contracts rather than physical footprints allowing us in particular to show the soundness of some transformations involving fine-grained concurrent data structures, such as Michael-Scott queues, that allow concurrent access to different parts of mutable data structures.Our semantics is based on refining a trace-based semantics for first-order programs due to Brookes. By moving from concrete to abstract locations, and adding type refinements that capture the possible side-effects of both expressions and their concurrent environments, we are able to validate many equivalences that do not hold in an unrefined model. The meanings of types are expressed using a game-based logical relation over sets of traces. Two programs e 1 and e 2 are logically related if one is able to solve a two-player game: for any trace with result value v 1 in the semantics of e 1 (challenge) that the player presents, the opponent can present an (response) equivalent trace in the semantics of e 2 with a logically related result value v 2 .

Factorisation Systems for Logical Relations and Monadic Lifting in Type-and-effect System Semantics

“…We want to relate the free lifting to other lifting techniques, most notably ⊤⊤-, and codensity-, lifting. We would also like to relate Benton et al's [2] relational models to our construction. We want to apply this construction to more sophisticated computational effects, such as dynamic memory allocation [9].…”

“…Adequate denotational semantics is a natural technique for validating such equational transformations, and there is a long line of work validating type-and-effect-dependent transformations, starting with independent results by Tolmach [30], Wadler [31], and Benton et al [3], and continuing to this day [2]. In their most general form, the semantics for an effect system consists of a graded monad [15], a compatible family of monad-like structures T ε indexed by the effect annotations ε.…”

“…To give the model structure for an arbitrary Gifford-style type-and-effect system with n operation symbols, one would need to give the structure of 2 n different monad-like structures, n2 n−1 monad-like-morphisms, and commute more than the same amount of diagrams to discharge the relevant proof obligations. To circumvent this blow-up, for example, Benton et al give uniform bespoke definitions for each T ε , e.g., as in [2]. To avoid an ad-hoc definition for each collection of effects, Katsumata [15] constructs graded monads for Gifford-style systems when the effects in the language are free.…”

Relational cost analysis for functional-imperative programs

“…Unno et al [2017] present an automated approach to verification based on induction for Horn clauses, which can also be used for relational verification. Benton et al [Benton et al 2014, 2016 introduce abstract effects to reason about abstract locations. This is conceptually similar to the way our preconditions and postconditions allow us to reason about different independent locations.…”

Symbolic Timed Trace Equivalence

“…Various notions of observational equivalence have been proposed in the programming languages community as well as in concurrent systems [2], [7], [22], [28] using, for example, logical relations and bisimulation. Observational equivalence has also been proposed for protocol verification notably the work of Cortier and Delaune [14].…”