The impact of higher-order state and control effects on local relational reasoning

Abstract

Reasoning about program equivalence is one of the oldest problems in semantics. In recent years, useful techniques have been developed, based on bisimulations and logical relations, for reasoning about equivalence in the setting of increasingly realistic languages-languages nearly as complex as ML or Haskell. Much of the recent work in this direction has considered the interesting representation independence principles enabled by the use of local state, but it is also important to understand the principles that powerful features like higher-order state and control effects disable. This latter topic has been broached extensively within the framework of game semantics, resulting in what Abramsky dubbed the "semantic cube": fully abstract game-semantic characterizations of various axes in the design space of ML-like languages. But when it comes to reasoning about many actual examples, game semantics does not yet supply a useful technique for proving equivalences.In this paper, we marry the aspirations of the semantic cube to the powerful proof method of step-indexed Kripke logical relations. Building on recent work of Ahmed, Dreyer, and Rossberg, we define the first fully abstract logical relation for an ML-like language with recursive types, abstract types, general references and call/cc. We then show how, under orthogonal restrictions to the expressive power of our language-namely, the restriction to first-order state and/or the removal of call/cc-we can enhance the proving power of our possible-worlds model in correspondingly orthogonal ways, and we demonstrate this proving power on a range of interesting examples. Central to our story is the use of state transition systems to model the way in which properties of local state evolve over time.

Verifying an Open Compiler Using Multi-language Semantics

“…Our logical relation extends that of Dreyer et al [15] with the ability to handle multi-language type abstraction. We give an overview of the logical relation and a more detailed discussion of its novel features in the technical report [9].…”

Abstract effects and proof-relevant logical relations

“…Therefore, if f has effect of writing on the region r, it will set x to one. Callback with Lock Example We now show equivalence of the following programs, also due to Dreyer et al [14]:…”

“…Similarly, σ 1 when taken to the low point of the square, that is, where the locations in w r are forgotten, the resulting heap is equivalent to σ 2 . Modified Awkward Example Consider now the following variant of the Awkward example, due to Dreyer et al [14]:…”

A Contextual Equivalence Checker for IMJ*

“…We evaluated CONEQCT on examples drawn from the literature around contextual equivalence for high-level languages [1,3,5,6,8,12,13], adapted to IMJ * syntax. (The website of the tool contains a more detailed listing.)…”