Reasoning About Higher-Order Relational Specifications

TL;DR

This paper introduces a method for supporting inductive reasoning over the full hereditary Harrop formulas logic within the Abella theorem prover, enabling more comprehensive reasoning about formal systems with dynamic assumptions.

Contribution

It develops a novel approach leveraging focusing properties of HH to facilitate inductive reasoning over all of HH in the Abella system.

Findings

Successfully implemented in Abella with several reasoning examples

Enables inductive reasoning with dynamically changing assumptions

Improves the expressiveness of formal specifications in Abella

Abstract

The logic of hereditary Harrop formulas (HH) has proven useful for specifying a wide range of formal systems. This logic includes a form of hypothetical judgment that leads to dynamically changing sets of assumptions and that is key to encoding side conditions and contexts that occur frequently in structural operational semantics (SOS) style presentations. Specifications are often useful in reasoning about the systems they describe. The Abella theorem prover supports such reasoning by explicitly embedding the specification logic within a rich reasoning logic; specifications are then reasoned about through this embedding. However, realizing an induction principle in the face of dynamically changing assumption sets is nontrivial and the original Abella system uses only a subset of the HH specification logic for this reason. We develop a method here for supporting inductive reasoning over all of HH. Our approach takes advantage of a focusing property of HH to isolate the use of an assumption and the ability to finitely characterize the structure of any such assumption in the reasoning logic. We demonstrate the effectiveness of these ideas via several specification and meta-theoretic reasoning examples that have been implemented in an extended version of Abella.