Alexander Ivrii

On computing minimal independent support and its applications to sampling and counting

Constrained sampling and counting are two fundamental problems arising in domains ranging from artificial intelligence and security, to hardware and software testing. Recent approaches to approximate solutions for these problems rely on employing SAT solvers and universal hash functions that are typically encoded as XOR constraints of length n/2 for an input formula with n variables. As the runtime performance of SAT solvers heavily depends on the length of XOR constraints, recent research effort has been focused on reduction of length of XOR constraints. Consequently, a notion of Independent Support was proposed, and it was shown that constructing XORs over independent support (if known) can lead to a significant reduction in the length of XOR constraints without losing the theoretical guarantees of sampling and counting algorithms. In this paper, we present the first algorithmic procedure (and a corresponding tool, called MIS) to determine minimal independent support for a given CNF formula by employing a reduction to group minimal unsatisfiable subsets (GMUS). By utilizing minimal independent supports computed by MIS, we provide new tighter bounds on the length of XOR constraints for constrained counting and sampling. Furthermore, the universal hash functions constructed from independent supports computed by MIS provide two to three orders of magnitude performance improvement in state-of-the-art constrained sampling and counting tools, while still retaining theoretical guarantees.

Computing interpolants without proofs

We describe an incremental algorithm for computing interpolants for a pair ϕA, ϕB of formulas in propositional logic. In contrast with the common approaches, our method does not require a proof of unsatisfiability of ϕA∧ϕB, and can be realized using any SAT solver as a black box. We achieve this by combining model enumeration with the ability to easily generate interpolants in the special case that one of the formulas is a cube.

Pushing to the top

IC3 is undoubtedly one of the most successful and important recent techniques for unbounded model checking. Understanding and improving IC3 has been a subject of a lot of recent research. In this regard, the most fundamental questions are how to choose Counterexamples to Induction (CTIs) and how to generalize them into (blocking) lemmas. Answers to both questions influence performance of the algorithm by directly affecting the quality of the lemmas learned. In this paper, we present a new IC3-based algorithm, called QUIP1, that is designed to more aggressively propagate (or push) learned lemmas to obtain a safe inductive invariant faster. QUIP modifies the recursive blocking procedure of IC3 to prioritize pushing already discovered lemmas over learning of new ones. However, a naive implementation of this strategy floods the algorithm with too many useless lemmas. In QUIP, we solve this by extending IC3 with may-proof-obligations (corresponding to the negations of learned lemmas), and by using an under-approximation of reachable states (i.e., states that witness why a may-proof-obligation is satisfiable) to prune non-inductive lemmas. We have implemented QUIP on top of an industrial-strength implementation of IC3. The experimental evaluation on HWMCC benchmarks shows that the QUIP is a significant improvement (at least 2x in runtime and more properties solved) over IC3. Furthermore, the new reasoning capabilities of QUIP naturally lead to additional optimizations and new techniques that can lead to further improvements in the future.

Perfect hashing and CNF encodings of cardinality constraints

We study the problem of encoding cardinality constraints (threshold functions) on Boolean variables into CNF. Specifically, we propose new encodings based on (perfect) hashing that are efficient in terms of the number of clauses, auxiliary variables, and propagation strength. We compare the properties of our encodings to known ones, and provide experimental results evaluating their practical effectiveness.

Small Inductive Safe Invariants

Computing minimal (or even just small) certificates is a central problem in automated reasoning and, in particular, in automated formal verification. For example, Minimal Unsatisfiable Subsets (MUSes) have a wide range of applications in verification ranging from abstraction and generalization to vacuity detection and more. In this paper, we study the problem of computing minimal certificates for safety properties. In this setting, a certificate is a set of clauses Inυ such that each clause contains initial states, and their conjunction is safe (no bad states) and inductive. A certificate is minimal, if no subset of Inυ is safe and inductive. We propose a two-tiered approach for computing a Minimal Safe Inductive Subset (MSIS) of Inv. The first tier is two efficient approximation algorithms that under-and over-approximate MSIS, respectively. The second tier is an optimized reduction from MSIS to a sequence of computations of Maximal Inductive Subsets (MIS). We evaluate our approach on the HWMCC benchmarks and certificates produced by our variant of IC3. We show that our approach is several orders of magnitude more effective than the naive reduction of MSIS to MIS.

On efficient computation of variable MUSes

In this paper we address the following problem: given an unsatisfiable CNF formula

Learning support sets in IC3 and Quip: The good, the bad, and the ugly

{mathcal{F}}

Speeding up MUS Extraction with Preprocessing and Chunking

, find a minimal subset of variables of

Mining Backbone Literals in Incremental SAT

{mathcal{F}}

Finding All Minimal Safe Inductive Sets

that constitutes the set of variables in some unsatisfiable core of