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08:30-09:00Coffee & Refreshments
09:00-10:30 Session 26G: Applications
Location: Ullmann 311
Towards an Efficient CNF Encoding of Block Ciphers

ABSTRACT. SAT-solvers are one of the primary tools to assess the security of block ciphers automatically. Common CNF encodings of s-boxes are based on algebraic representations (finding low-degree polynomials) or symbolic execution of considered function implementation. However, those methods are not strictly connected with algorithms used in efficient SAT-solvers. Therefore, we propose an application of minimal propagate complete encodings, which in their definition are tuned for modern satisfiability checking algorithms.

During the construction of the Boolean formula, there is often the problem of encoding linear XOR equations. The standard procedure includes a greedy shortening algorithm to decrease the size of the resulting encoding. Recently, the problem of a straight-line program has been successfully applied in obtaining efficient implementations of MDS matrices. In this paper, we propose to use the algorithm for finding a short straight-line program as a shortening procedure for a system of linear XOR equations.

As a result, we achieved a 20x speed-up of algebraic cryptanalysis of Small Scale AES block cipher to widely used algebraic representations by combining two approaches.

Calculating Sufficient Reasons for Random Forest Classifiers

ABSTRACT. In this paper, we formalize the implementations of decision tree and random forest classifiers of the Python Scikit-learn package, and we present a CNF encoding which can be used to calculate sufficient reasons a.k.a. prime implicants for them. Our decision tree encoding resembles a monotonic combinational circuit with pure input variables, of which we take advantage in our method for incremental enumeration of its prime implicants. Encoding the combination of several decision trees in a random forest would add a non-monotonic evaluation function to the root of the encoded circuit. In our approach, we solve an auxiliary SAT problem for enumerating valid leaf-node combinations of the random forest. Each valid leaf-node combination is used to incrementally update the original monotonic circuit encoding by extending the DNF at its root with a new term. Preliminary results show that enumeration of prime implicants by incrementally updating the encoding is by order of magnitudes faster than one-shot solving of the monolithic formula. We present preliminary runtime data, and some initial data about the size and number of samples found when translating the prime implicants back into database queries.

Adding Dual Variables to Algebraic Reasoning for Gate-Level Multiplier Verification
PRESENTER: Jakob Nordstrom

ABSTRACT. Algebraic reasoning has proven to be one of the most effective approaches for verifying gate-level integer multipliers, but it struggles with certain components, necessitating the complementary use of SAT solvers. For this reason validation certificates require proofs in two different formats. Approaches to unify the certificates are not scalable, meaning that the validation results can only be trusted up to the correctness of compositional reasoning. We show in this work that using dual variables in the algebraic encoding, together with a novel tail substitution and carry rewriting method, removes the need for SAT solvers in the verification flow and yields a single, uniform proof certificate.

This is a presentation-only submission for a paper previously published at DATE '22.

10:30-11:00Coffee Break
11:00-12:30 Session 31K: SAT and Parallel Solving
Location: Ullmann 311
Dinosat: A SAT Solver with Native DNF Support
PRESENTER: Markus Iser

ABSTRACT. In this paper we report our preliminary results with a new kind of SAT solver called Dinosat. Dinosat's input is a conjunction of clauses, at-most-one constraints and disjunctive normal form (DNF) formulas. The native support for DNF formulas is motivated by the application domain of SAT based product configuration. A DNF formula can also be viewed as a generalization of a clause, i.e., a clause (disjunction of literals) is special case of a DNF formula, where each term (conjunction of literals) has exactly one literal. Similarly, we can generalize the classical resolution rule and use it to resolve two DNF formulas. Based on that, the CDCL algorithm can be modified to work with DNF formulas instead of just clauses. Using randomly generated formulas (with DNFs) we experimentally show, that in certain relevant scenarios, it is more efficient to solve these formulas with Dinosat than translate them to CNF and use a state-of-the-art SAT solver. Another contribution of this paper is identifying the phase transition points for such formulas.

DPS: A Framework for Deterministic Parallel SAT Solvers

ABSTRACT. In this study, we propose a new framework for easily constructing efficient deterministic parallel SAT solvers, providing the delayed clause exchange technique introduced in ManyGlucose. This framework allows existing sequential SAT solvers to be parallelized with just as little effort as in the non-deterministic parallel solver framework such as PaInleSS. We show experimentally that parallel solvers built using this framework have performance comparable to state-of-the-art non-deterministic parallel SAT solvers while ensuring reproducible behavior.

Scalable Proof Producing Multi-Threaded SAT Solving with Gimsatul through Sharing instead of Copying Clauses
PRESENTER: Armin Biere

ABSTRACT. We give a first account of our new parallel SAT solver Gimsatul. Its key feature is to share clauses physically in memory instead of copying them, which is the method of other state-of-the-art multi-threaded SAT solvers to share clauses logically. Our approach keeps information about which literals are watched in a clause local to a solving thread but shares the actual immutable literals of a clause globally among all solving threads. This design gives quiet remarkable parallel scalability, allows aggressive clause sharing while keeping memory usage low and produces more compact proofs.

12:30-14:00Lunch Break

Lunches will be held in Taub hall and in The Grand Water Research Institute.

14:00-15:30 Session 34L: Proofs I
Location: Ullmann 311
TBUDDY: A Proof-Generating BDD Package

ABSTRACT. The TBUDDY library enables the construction and manipulation of reduced, ordered binary decision diagrams (BDDs). It extends the capabilities of the BUDDY BDD package to support trusted BDDs, where the generated BDDs are accompanied by proofs of their logical properties. These proofs are expressed in a standard clausal framework, for which a variety of proof checkers are available.

Building on TBUDDY via its application-program interface (API) enables developers to implement automated reasoning tools that generate correctness proofs for their outcomes. In some cases, BDDs serve as the core reasoning mechanism for the tool, while in other cases they provide a bridge from the core reasoner to proof generation. A Boolean satisfiability (SAT) solver based on TBUDDY can achieve polynomial scaling when generating unsatisfiability proofs for a number of problems that yield exponentially-sized proofs with standard solvers. It performs particularly well for formulas containing parity constraints, where it can employ Gaussian elimination to systematically simplify the constraints.

Combining CDCL, Gauss-Jordan Elimination, and Proof Generation

ABSTRACT. Traditional Boolean satisfiability (SAT) solvers based on the conflict-driven clause-learning (CDCL) framework fare poorly on formulas involving large numbers of parity constraints. The CryptoMiniSat solver augments CDCL with Gauss-Jordan elimination to greatly improve performance on these formulas. Integrating the TBUDDY proof-generating BDD library into CryptoMiniSat enables it to generate unsatisfiability proofs when using Gauss-Jordan elimination. These proofs are compatible with standard, clausal proof frameworks.

Towards the shortest DRAT proof of the Pigeonhole Principle
PRESENTER: Marijn Heule

ABSTRACT. The Pigeonhole Principle (PHP) has been heavily studied in automated reasoning, both theoretically and in practice. Most solvers have exponential runtime and proof length, while some specialized techniques achieve polynomial runtime and proof length. Several decades ago, Cook manually constructed O(n^4) extended resolution proofs, where n denotes the number of pigeons. Existing automated techniques only surpass Cook's proofs in similar proof systems for large n. We construct the shortest known proofs of PHP in the standard proof format of modern SAT solving, DRAT. Using auxiliary variables and by recursively decomposing the original program into smaller sizes, we manually obtain proofs having length O(n^3) and leading coefficient 5/2.

15:30-16:00Coffee Break
16:00-17:00 Session 37K: Proofs II
Location: Ullmann 311
SATViz: Real-Time Visualization of Clausal Proofs
PRESENTER: Tim Holzenkamp

ABSTRACT. Visual layouts of graphs representing SAT instances can highlight the community structure of SAT instances. The community structure of SAT instances has been associated with both instance hardness and known clause quality heuristics. Our tool SATViz visualizes CNF formulas using the variable interaction graph and a force-directed layout algorithm. With SATViz, clause proofs can be animated to continuously highlight variables that occur in a moving window of recently learned clauses. If needed, SATViz can also create new layouts of the variable interaction graph with the adjusted edge weights. In this paper, we describe the structure and feature set of SATViz. We also present some interesting visualizations created with SATViz.

Certified Symmetry and Dominance Breaking for Combinatorial Optimisation
PRESENTER: Jakob Nordstrom

ABSTRACT. Symmetry and dominance breaking can be crucial for solving hard combinatorial search and optimisation problems, but the correctness of these techniques sometimes relies on subtle arguments. For this reason, it is desirable to produce efficient, machine-verifiable certificates that solutions have been computed correctly. Building on the cutting planes proof system, we develop a certification method for optimisation problems in which symmetry and dominance breaking are easily expressible. Our experimental evaluation demonstrates that we can efficiently verify fully general symmetry breaking in Boolean satisfiability (SAT) solving, thus providing, for the first time, a unified method to certify a range of advanced SAT techniques that also includes XOR and cardinality reasoning. In addition, we apply our method to maximum clique solving and constraint programming as a proof of concept that the approach applies to a wider range of combinatorial problems.

This is a presentation-only submission of a paper that appeared at the 36th AAAI Conference on Artificial Intelligence (AAAI '22).

18:30-20:00Workshop Dinner (at the Technion, Taub Terrace Floor 2) - Paid event