The Relationship Between Separation Logic and Implicit Dynamic Frames

TL;DR

This paper establishes a formal connection between separation logic and implicit dynamic frames, defining a unified semantics and demonstrating how separation logic can be encoded into implicit dynamic frames for automated verification.

Contribution

It introduces a unified logic encompassing both separation logic and implicit dynamic frames, with a novel semantics based on minimal state extension, enabling encoding and tool support.

Findings

Defined a total heap semantics for the combined logic

Proved equivalence of semantics for separation logic subsyntax

Showed encoding of separation logic into implicit dynamic frames preserves semantics

Abstract

Separation logic is a concise method for specifying programs that manipulate dynamically allocated storage. Partially inspired by separation logic, Implicit Dynamic Frames has recently been proposed, aiming at first-order tool support. In this paper, we precisely connect the semantics of these two logics. We define a logic whose syntax subsumes both that of a standard separation logic, and that of implicit dynamic frames as sub-syntaxes. We define a total heap semantics for our logic, and, for the separation logic subsyntax, prove it equivalent the standard partial heaps model. In order to define a semantics which works uniformly for both subsyntaxes, we define the novel concept of a minimal state extension, which provides a different (but equivalent) definition of the semantics of separation logic implication and magic wand connectives, while also giving a suitable semantics for these connectives in implicit dynamic frames. We show that our resulting semantics agrees with the existing definition of weakest pre-condition semantics for the implicit dynamic frames fragment. Finally, we show that we can encode the separation logic fragment of our logic into the implicit dynamic frames fragment, preserving semantics. For the connectives typically supported by tools, this shows that separation logic can be faithfully encoded in a first-order automatic verification tool (Chalice).