Certifying proofs for SAT-based model checking

Abstract

In the context of formal verification, certifying proofs are evidences of the correctness of a model in a deduction system produced automatically as outcome of the verification. They are quite appealing for high-assurance systems because they can be verified independently by proof checkers, which are usually simpler to certify than the proof-generating tools. Model checking is one of the most prominent approaches to formal verification of temporal properties and is based on an algorithmic search of the system state space. Although modern algorithms integrate deductive methods, the generation of proofs is typically restricted to invariant properties only. Moreover, it assumes that the verification produces an inductive invariant of the original system, while model checkers usually involve a variety of complex pre-processing simplifications. In this paper we show how, exploiting the k-liveness algorithm, to extend proof generation capabilities for invariant checking to cover full linear-time temporal logic (LTL) properties, in a simple and efficient manner, with essentially no overhead for the model checker. Besides the basic k-liveness algorithm, we integrate in the proof generation a variety of widely used pre-processing techniques such as temporal decomposition, model simplification via computation of equivalences with ternary simulation, and the use of stabilizing constraints. These techniques are essential in many cases to prove that a property holds, both for invariant and for LTL model checking, and thus need to be considered within the proof. We implemented the proof generation techniques on top of IC3 engines, and show the feasibility of the approach on a variety of benchmarks taken from the literature and from the Hardware Model Checking Competition. Our results confirm that proof generation results in negligible overhead for the model checker.

A Derivation proofs

In this section we detail the derivation of the rules derived from the system defined in [26] and used in the proof generated as described in the previous sections. Each derived rule corresponds to a valid formula, which has been proved valid also with the nuXmv model checker. However, to avoid circularity in the proof generation, we cannot exploit the model checker result and we given an explicit version of the deduction proofs.

1.1 A.1 Basic lemmas

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1.2 A.2 Proofs for k-liveness

The main step to prove the k-liveness derivation is the following deduction:

which is derived as follows:

Combining the $\textsc {klb}$ in a chain, we obtain a full derivation of a proof for the $\textsc {kl}[k]$ rule:

1.3 A.3 Proofs for stabilizing constraints

The following derived rules are used in the proofs exploiting stabilizing constraints:

These rules are derived as follows:

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1.4 A.4 Proof for simplified transition relation

The following derived rule is used to simplify T with some invariant $\xi$:

which is derived as follows:

1.5 A.5 Proof for temporal decomposition

The derived rule used with temporal decomposition for liveness properties is the following:

which is derived as follows.

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Similarly for invariant properties, we have the following derived rule:

which is derived as follows.

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1.6 A.6 Proofs for LTL encoding

Here we use the equivalence $(\phi \leftrightarrow Exp(\phi )) \leftrightarrow ((\phi \rightarrow Exp(\phi )) \wedge (Exp(\phi )\leftrightarrow \phi ))$ to match the structure of the $\textsc {and-el}$. Moreover, given the specific construction of $M_{\lnot \phi }$, $Enc^{-1}(T_{\lnot \phi })$ and $\mathbf{G }\mathbf{F }Enc^{-1}(f_i)$ are always valid formulae. Moreover, we leverage that $Enc^{-1}(I_{\lnot \phi }) =Exp(\lnot \phi )=\lnot Exp(\phi )$, and the fact that $Enc^{-1}(T_{\lnot \phi })$ is in the form $\bigwedge _\beta \mathbf{X }\beta \leftrightarrow Next(Exp(\beta ))$.

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1.7 A.7 Overall proof for LTL

1.8 A.8 Overall proof for invariant