Publication/Citation: A Proof-Theoretic Approach to Mathematical Knowledge Management

There are many real-life examples of formal systems that support constructions or proofs, but that do not provide direct support for remembering them so that they can be recalled and reused in the future. In this paper we examine the operations of publication (remembering a proof) and citation (recalling a proof for reuse), regarding them as forms of common subexpression elimination on proof terms. We then develop this idea from a proof theoretic perspective, describing a simple complete proof system for universal Horn equational logic using three new proof rules, publish, cite, and forget. These rules can provide a proof-theoretic infrastructure for proof reuse in any system

Ornaments for Proof Reuse in Coq

Ornaments express relations between inductive types with the same inductive structure. We implement fully automatic proof reuse for a particular class of ornaments in a Coq plugin, and show how such a tool can give programmers the rewards of using indexed inductive types while automating away many of the costs. The plugin works directly on Coq code; it is the first ornamentation tool for a non-embedded dependently typed language. It is also the first tool to automatically identify ornaments: To lift a function or proof, the user must provide only the source type, the destination type, and the source function or proof. In taking advantage of the mathematical properties of ornaments, our approach produces faster functions and smaller terms than a more general approach to proof reuse in Coq

Categorical Abstract Logic: Hidden Multi-Sorted Logics as Multi-Term Institutions

Babenyshev and Martins proved that two hidden multi-sorted deductive systems are deductively equivalent if and only if there exists an isomorphism between their corresponding lattices of theories that commutes with substitutions. We show that the -institutions corresponding to the hidden multi-sorted deductive systems studied by Babenyshev and Martins satisfy the multi-term condition of Gil-Férez. This provides a proof of the result of Babenyshev and Martins by appealing to the general result of Gil-Férez pertaining to arbitrary multi-term -institutions. The approach places hidden multi-sorted deductive systems in a more general framework and bypasses the laborious reuse of well-known proof techniques from traditional abstract algebraic logic by using “off the shelf” tools

Towards correct-by-construction product variants of a software product line: GFML, a formal language for feature modules

Software Product Line Engineering (SPLE) is a software engineering paradigm that focuses on reuse and variability. Although feature-oriented programming (FOP) can implement software product line efficiently, we still need a method to generate and prove correctness of all product variants more efficiently and automatically. In this context, we propose to manipulate feature modules which contain three kinds of artifacts: specification, code and correctness proof. We depict a methodology and a platform that help the user to automatically produce correct-by-construction product variants from the related feature modules. As a first step of this project, we begin by proposing a language, GFML, allowing the developer to write such feature modules. This language is designed so that the artifacts can be easily reused and composed. GFML files contain the different artifacts mentioned above.The idea is to compile them into FoCaLiZe, a language for specification, implementation and formal proof with some object-oriented flavor. In this paper, we define and illustrate this language. We also introduce a way to compose the feature modules on some examples.Comment: In Proceedings FMSPLE 2015, arXiv:1504.0301

Impact assessment for vulnerabilities in open-source software libraries

Software applications integrate more and more open-source software (OSS) to benefit from code reuse. As a drawback, each vulnerability discovered in bundled OSS potentially affects the application. Upon the disclosure of every new vulnerability, the application vendor has to decide whether it is exploitable in his particular usage context, hence, whether users require an urgent application patch containing a non-vulnerable version of the OSS. Current decision making is mostly based on high-level vulnerability descriptions and expert knowledge, thus, effort intense and error prone. This paper proposes a pragmatic approach to facilitate the impact assessment, describes a proof-of-concept for Java, and examines one example vulnerability as case study. The approach is independent from specific kinds of vulnerabilities or programming languages and can deliver immediate results