Efficient Certified Resolution Proof Checking

We present a novel propositional proof tracing format that eliminates complex processing, thus enabling efficient (formal) proof checking. The benefits of this format are demonstrated by implementing a proof checker in C, which outperforms a state-of-the-art checker by two orders of magnitude. We then formalize the theory underlying propositional proof checking in Coq, and extract a correct-by-construction proof checker for our format from the formalization. An empirical evaluation using 280 unsatisfiable instances from the 2015 and 2016 SAT competitions shows that this certified checker usually performs comparably to a state-of-the-art non-certified proof checker. Using this format, we formally verify the recent 200 TB proof of the Boolean Pythagorean Triples conjecture

Extracting verified decision procedures: DPLL and Resolution

This article is concerned with the application of the program extraction technique to a new class of problems: the synthesis of decision procedures for the classical satisfiability problem that are correct by construction. To this end, we formalize a completeness proof for the DPLL proof system and extract a SAT solver from it. When applied to a propositional formula in conjunctive normal form the program produces either a satisfying assignment or a DPLL derivation showing its unsatisfiability. We use non-computational quantifiers to remove redundant computational content from the extracted program and translate it into Haskell to improve performance. We also prove the equivalence between the resolution proof system and the DPLL proof system with a bound on the size of the resulting resolution proof. This demonstrates that it is possible to capture quantitative information about the extracted program on the proof level. The formalization is carried out in the interactive proof assistant Minlog

A Verified SAT Solver Framework with Learn, Forget, Restart, and Incrementality

International audienceWe developed a formal framework for CDCL (conflict-driven clause learning) in Isabelle/HOL. Through a chain of refinements, an abstract CDCL calculus is connected to a SAT solver expressed in a functional programming language, with total correctness guarantees. The framework offers a convenient way to prove metatheorems and experiment with variants. Compared with earlier SAT solver verifications, the main novelties are the inclusion of rules for forget, restart, and incremental solving and the application of refinement

Formalizing the Metatheory of Logical Calculi and Automatic Provers in Isabelle/HOL (Invited Talk)

International audienceIsaFoL (Isabelle Formalization of Logic) is an undertaking that aims at developing formal theories about logics, proof systems, and automatic provers, using Isabelle/HOL. At the heart of the project is the conviction that proof assistants have become mature enough to actually help researchers in automated reasoning when they develop new calculi and tools. In this paper, I describe and reflect on three verification subprojects to which I contributed: a first-order resolution prover, an imperative SAT solver, and generalized term orders for λ-free higher-order logic

Self-Formalisation of Higher-Order Logic: Semantics, Soundness, and a Verified Implementation

This is the final version of the article. It first appeared from Springer via http://dx.doi.org/10.1007/s10817-015-9357-xWe present a mechanised semantics for higher-order logic (HOL), and a proof of soundness for the inference system, including the rules for making definitions, implemented by the kernel of the HOL Light theorem prover. Our work extends Harrison’s verification of the inference system without definitions. Soundness of the logic extends to soundness of a theorem prover, because we also show that a synthesised implementation of the kernel in CakeML refines the inference system. Apart from adding support for definitions and synthesising an implementation, we improve on Harrison’s work by making our model of HOL parametric on the universe of sets, and we prove soundness for an improved principle of constant specification in the hope of encouraging its adoption. Our semantics supports defined constants directly via a context, and we find this approach cleaner than our previous work formalising Wiedijk’s Stateless HOL.The first author was supported by the Gates Cambridge Trust. The second author was funded in part by the EPSRC (grant number EP/K503769/1). The third author was partially supported by the Royal Society UK and the Swedish Research Council

A verified prover based on ordered resolution

International audienceThe superposition calculus, which underlies first-order theorem provers such as E, SPASS, and Vampire, combines ordered resolution and equality reasoning. As a step towards verifying modern provers, we specify, using Isabelle/HOL, a purely functional first-order ordered resolution prover and establish its soundness and refutational completeness. Methodologically, we apply stepwise refinement to obtain, from an abstract nondeterministic specification, a verified de-terministic program, written in a subset of Isabelle/HOL from which we extract purely functional Standard ML code that constitutes a semidecision procedure for first-order logic

A Machine-Checked, Type-Safe Model of Java Concurrency : Language, Virtual Machine, Memory Model, and Verified Compiler

The Java programming language provides safety and security guarantees such as type safety and its security architecture. They distinguish it from other mainstream programming languages like C and C++. In this work, we develop a machine-checked model of concurrent Java and the Java memory model and investigate the impact of concurrency on these guarantees. From the formal model, we automatically obtain an executable verified compiler to bytecode and a validated virtual machine