An Open Challenge Problem Repository for Systems Supporting Binders

A variety of logical frameworks support the use of higher-order abstract syntax in representing formal systems; however, each system has its own set of benchmarks. Even worse, general proof assistants that provide special libraries for dealing with binders offer a very limited evaluation of such libraries, and the examples given often do not exercise and stress-test key aspects that arise in the presence of binders. In this paper we design an open repository ORBI (Open challenge problem Repository for systems supporting reasoning with BInders). We believe the field of reasoning about languages with binders has matured, and a common set of benchmarks provides an important basis for evaluation and qualitative comparison of different systems and libraries that support binders, and it will help to advance the field.Comment: In Proceedings LFMTP 2015, arXiv:1507.0759

Computation-as-deduction in Abella: work in progress

International audienceThe Abella theorem prover is based on a logic in which relations, and not functions, are defined by induction (and coinduction). Of course, many relations do, in fact, define functions and there is real value in separating functional computation (marked by determinism) from more general deduction (marked by nondeterminism and backtracking). Recent work on focused proof systems for the logic underlying Abella is used in this paper to motivate the design of various extensions to the Abella system. With these extensions to the system (which do not extend the logic), it is possible to fully automate functional computations within the relational setting as soon as a proof is provided that a given relation does, in fact, capture a total function. In this way, we can use Abella to compute functions even when data structures contain bindings

POPLMark reloaded: Mechanizing proofs by logical relations

We propose a new collection of benchmark problems in mechanizing the metatheory of programming languages, in order to compare and push the state of the art of proof assistants. In particular, we focus on proofs using logical relations (LRs) and propose establishing strong normalization of a simply typed calculus with a proof by Kripke-style LRs as a benchmark. We give a modern view of this well-understood problem by formulating our LR on well-typed terms. Using this case study, we share some of the lessons learned tackling this problem in different dependently typed proof environments. In particular, we consider the mechanization in Beluga, a proof environment that supports higher-order abstract syntax encodings and contrast it to the development and strategies used in general-purpose proof assistants such as Coq and Agda. The goal of this paper is to engage the community in discussions on what support in proof environments is needed to truly bring mechanized metatheory to the masses and engage said community in the crafting of future benchmarks

Author manuscript, published in "Principles and Practice of Declarative Programming (2013)" DOI: 10.1145/2505879.2505889 Reasoning About Higher-Order Relational Specifications

The logic of hereditary Harrop formulas (HH) has proven useful for specifying a wide range of formal systems that are commonly presented via syntax-directed rules that make use of contexts and side-conditions. The two-level logic approach, as implemented in the Abella theorem prover, embeds the HH specification logic within a rich reasoning logic that supports inductive and co-inductive definitions, an equality predicate, and generic quantification. Properties of the encoded systems can then be proved through the embedding, with special benefit being extracted from the transparent correspondence between HH derivations and those in the encoded formal systems. The versatility of HH relies on the free use of nested implications, leading to dynamically changing assumption sets in derivations. Realizing an induction principle in this situation is nontrivial and the original Abella system uses only a subset of HH for this reason. We develop a method here for supporting inductive reasoning over all of HH. Our approach relies on the ability to characterize dynamically changing contexts through finite inductive definitions, and on a modified encoding of backchaining for HH that allows these finite characterizations to be used in inductive arguments. We demonstrate the effectiveness of our approach through examples of formal reasoning on specifications with nested implications in an extended version of Abella.