Program Logics for Sequential Higher-Order Control

Abstract

We introduce a Hoare logic for call-by-value higher-order functional languages with control operators such as callcc. The key idea is to build the assertion language and proof rules around an explicit logical representation of jumps and their dual 'places-to-jump-to'. This enables the assertion language to capture precisely the intensional and extensional effects of jumping by internalising rely/guarantee reasoning, leading to simple proof rules for higher-order functions with callcc. We show that the logic can reason easily about non-trivial uses of callcc. The logic matches exactly with the operational semantics of the target language (observational completeness), is relatively complete in Cook's sense and allows efficient generation of characteristic formulae.

Program Logics for Homogeneous Meta-programming

“…The logic smoothly generalises previous work on axiomatic semantics for the ML family of languages [1,2,9,11,12,21]. The logic is for total correctness (for partial correctness see [4]).…”

“…Our logic is a Hoare logic with pre-and post-conditions in the tradition of logics for ML-like languages [1,2,11,12]. Expressions, ranged over by e, e , ... and formulae A, B, ... of the logic are given by the grammar below, using the types and variables of PCF.…”

“…-First, developing logics for non-meta-programming programming languages is already a hard problem, and only recently have satisfactory solutions been found for reasoning about programs with higher-order functions, state, pointers, continuations or concurrency [1,14,21]. Since reasoning about meta-programs contains reasoning about normal programs as a special case, program logics for meta-programming are at least as complicated as logics for normal languages.…”

Hoare-style reasoning with (algebraic) continuations

“…In contrast, in HTTcc, we allow the full expressiveness of CIC, including Σ-types over specifications, which, as we illustrated, is required for specifying closures that return captured continuations. Berger [5] presents a first-order Hoare logic for callcc in an otherwise purely functional language. One of the main features of the logic is the polarity distinction between the types of programs that perform jumps ('jumping-to') and the types of labels for jumps ('being-jumped-to').…”

Hoare-style reasoning with (algebraic) continuations

“…The presence of dynamic state significantly complicates matters, and differentiates our work from recent Hoare-style logics for higher-order jumps [5,10]. From the semantic point of view, supporting dynamic state requires building a model in which executing a continuation does not roll back the mutable state to the point at which the continuation is captured.…”