Proof assistants are increasingly used to verify hardware and software and to formalize mathematics. However, despite the success stories, they remain very laborious to use. The situation has improved with the integration of first-order automatic theorem provers—superposition provers and SMT (satisfiability modulo theories) solvers—through middleware such as Sledgehammer for Isabelle/HOL and HOLyHammer for HOL Light and HOL4; but this research has now reached the point of diminishing returns. Only so much can be done when viewing automatic provers as black boxes.
To make interactive verification more cost-effective, we propose to deliver very high levels of automation to users of proof assistants by fusing and extending two lines of research: automatic and interactive theorem proving. This is our grand challenge. Our starting point is that first-order (FO) automatic provers are the best tools available for performing most of the logical work. Our approach will be to enrich superposition and SMT with higher-order (HO) reasoning in a careful manner, to preserve their desirable properties. We will design proof rules and strategies, guided by representative benchmarks from interactive verification.
With higher-order superposition and higher-order SMT in place, we will develop highly automatic provers building on modern superposition provers and SMT solvers, following a novel stratified architecture. To reach end users, these new provers will be integrated in proof assistants and will be available as backends to more specialized verification tools. The users of proof assistants and similar tools stand to experience substantial productivity gains: From 2010 to 2016, the success rate of automatic provers on interactive proof obligations from a representative benchmark suite called Judgment Day has risen from 47% to 77%; with this project, we aim at 90%–95% proof automation.
The use of advanced methods to solve practical and industrially relevant problems by computers has a long history. Whereas Symbolic Computation is concerned with the algorithmic determination of exact solutions to complex mathematical problems, more recent developments in the area of Satisfiability Checking tackle similar problems but with different algorithmic and technological solutions. Though both communities have made remarkable progress in the last decades, they still need to be strengthened to tackle practical problems of rapidly increasing size and complexity. Their separate tools (computer algebra systems and SMT solvers) are urgently needed to examine prevailing problems with a direct effect to our society. For example, Satisfiability Checking is an essential backend for assuring the security and the safety of computer systems. In various scientific areas, Symbolic Computation enables dealing with large mathematical problems out of reach of pencil and paper developments. Currently the two communities are largely disjoint and unaware of the achievements ofeach other, despite strong reasons for them to discuss and collaborate, as they share many central interests. However, researchers from these two communities rarely interact, and also their tools lack common, mutual interfaces for unifiying their strengths. Bridges between the communities in the form of common platforms and roadmaps are necessary to initiate an exchange, and to support and to direct their interaction. These are the main objectives of this CSA. We will initiate a wide range of activities to bring the two communities together, identify common challenges, offer global events and bilateral visits, propose standards, and so on. We believe that these activities will initiate cross-fertilisation of both fields and bring mutual improvements. Combining the knowledge, experience and the technologies in these communities will enable the development of radically improved software tools.
The SMArT project, funded by ANR and DFG within the Programme Blanc, aims at providing new solutions in the field of satisfiability modulo theories (SMT) in combination with non-linear arithmetic. The focus of the project lies on both, new theoretical insights as well as a state-of-the-art software implementations. These ideas are put into practice by a multilateral cooperation between several groups including established research teams and industrial partners. Pascal Fontaine and his team at Inria/Loria in Nancy, France, are the designers of the award-winning SMT-solver veriT. The non-linear arithmetic facilities for veriT are provided by our researchers in Saarbrücken under the supervision of Thomas Sturm, co-developer of one of the most mature computer logic systems, Redlog. Additional support is given by researchers at DIMAp/UFRN, Natal, Brazil and our industrial partner Systerel, Aix-en-Provence, France, that allows us to direct our developement towards real-world application scenarios.
The VeriDis project aims to exploit and further develop the advances and integration of interactive and automated theorem proving applied to the area of concurrent and distributed systems. The goal of the project is to assist algorithm and system designers to carry out formally proved developments, where proofs of relevant properties as well as bugs can be found fully automatically.
VeriDis is a joint research project of Inria Nancy and the Max Planck Institute for Informatics.