Invariant Synthesis for Incomplete Verification Engines

We propose a framework for synthesizing inductive invariants for incomplete verification engines, which soundly reduce logical problems in undecidable theories to decidable theories. Our framework is based on the counter-example guided inductive synthesis principle (CEGIS) and allows verification engines to communicate non-provability information to guide invariant synthesis. We show precisely how the verification engine can compute such non-provability information and how to build effective learning algorithms when invariants are expressed as Boolean combinations of a fixed set of predicates. Moreover, we evaluate our framework in two verification settings, one in which verification engines need to handle quantified formulas and one in which verification engines have to reason about heap properties expressed in an expressive but undecidable separation logic. Our experiments show that our invariant synthesis framework based on non-provability information can both effectively synthesize inductive invariants and adequately strengthen contracts across a large suite of programs

Predicate Abstraction with Indexed Predicates

Predicate abstraction provides a powerful tool for verifying properties of infinite-state systems using a combination of a decision procedure for a subset of first-order logic and symbolic methods originally developed for finite-state model checking. We consider models containing first-order state variables, where the system state includes mutable functions and predicates. Such a model can describe systems containing arbitrarily large memories, buffers, and arrays of identical processes. We describe a form of predicate abstraction that constructs a formula over a set of universally quantified variables to describe invariant properties of the first-order state variables. We provide a formal justification of the soundness of our approach and describe how it has been used to verify several hardware and software designs, including a directory-based cache coherence protocol.Comment: 27 pages, 4 figures, 1 table, short version appeared in International Conference on Verification, Model Checking and Abstract Interpretation (VMCAI'04), LNCS 2937, pages = 267--28

Bounded Refinement Types

We present a notion of bounded quantification for refinement types and show how it expands the expressiveness of refinement typing by using it to develop typed combinators for: (1) relational algebra and safe database access, (2) Floyd-Hoare logic within a state transformer monad equipped with combinators for branching and looping, and (3) using the above to implement a refined IO monad that tracks capabilities and resource usage. This leap in expressiveness comes via a translation to "ghost" functions, which lets us retain the automated and decidable SMT based checking and inference that makes refinement typing effective in practice.Comment: 14 pages, International Conference on Functional Programming, ICFP 201

Generating Property-Directed Potential Invariants By Backward Analysis

This paper addresses the issue of lemma generation in a k-induction-based formal analysis of transition systems, in the linear real/integer arithmetic fragment. A backward analysis, powered by quantifier elimination, is used to output preimages of the negation of the proof objective, viewed as unauthorized states, or gray states. Two heuristics are proposed to take advantage of this source of information. First, a thorough exploration of the possible partitionings of the gray state space discovers new relations between state variables, representing potential invariants. Second, an inexact exploration regroups and over-approximates disjoint areas of the gray state space, also to discover new relations between state variables. k-induction is used to isolate the invariants and check if they strengthen the proof objective. These heuristics can be used on the first preimage of the backward exploration, and each time a new one is output, refining the information on the gray states. In our context of critical avionics embedded systems, we show that our approach is able to outperform other academic or commercial tools on examples of interest in our application field. The method is introduced and motivated through two main examples, one of which was provided by Rockwell Collins, in a collaborative formal verification framework.Comment: In Proceedings FTSCS 2012, arXiv:1212.657

SMT-based Model Checking for Recursive Programs

We present an SMT-based symbolic model checking algorithm for safety verification of recursive programs. The algorithm is modular and analyzes procedures individually. Unlike other SMT-based approaches, it maintains both "over-" and "under-approximations" of procedure summaries. Under-approximations are used to analyze procedure calls without inlining. Over-approximations are used to block infeasible counterexamples and detect convergence to a proof. We show that for programs and properties over a decidable theory, the algorithm is guaranteed to find a counterexample, if one exists. However, efficiency depends on an oracle for quantifier elimination (QE). For Boolean Programs, the algorithm is a polynomial decision procedure, matching the worst-case bounds of the best BDD-based algorithms. For Linear Arithmetic (integers and rationals), we give an efficient instantiation of the algorithm by applying QE "lazily". We use existing interpolation techniques to over-approximate QE and introduce "Model Based Projection" to under-approximate QE. Empirical evaluation on SV-COMP benchmarks shows that our algorithm improves significantly on the state-of-the-art.Comment: originally published as part of the proceedings of CAV 2014; fixed typos, better wording at some place

Invariant Synthesis for Incomplete Verification Engines

We propose a framework for synthesizing inductive invariants for incomplete verification engines, which soundly reduce logical problems in undecidable theories to decidable theories. Our framework is based on the counter-example guided inductive synthesis principle (CEGIS) and allows verification engines to communicate non-provability information to guide invariant synthesis. We show precisely how the verification engine can compute such non-provability information and how to build effective learning algorithms when invariants are expressed as Boolean combinations of a fixed set of predicates. Moreover, we evaluate our framework in two verification settings, one in which verification engines need to handle quantified formulas and one in which verification engines have to reason about heap properties expressed in an expressive but undecidable separation logic. Our experiments show that our invariant synthesis framework based on non-provability information can both effectively synthesize inductive invariants and adequately strengthen contracts across a large suite of programs

Counterexample-Guided Prophecy for Model Checking Modulo the Theory of Arrays

We develop a framework for model checking infinite-state systems by automatically augmenting them with auxiliary variables, enabling quantifier-free induction proofs for systems that would otherwise require quantified invariants. We combine this mechanism with a counterexample-guided abstraction refinement scheme for the theory of arrays. Our framework can thus, in many cases, reduce inductive reasoning with quantifiers and arrays to quantifier-free and array-free reasoning. We evaluate the approach on a wide set of benchmarks from the literature. The results show that our implementation often outperforms state-of-the-art tools, demonstrating its practical potential.Comment: 23 pages, 1 figure, 1 table, extended version of paper to be published in International Conference on Tools and Algorithms for the Construction and Analysis of Systems 202