On the van der Waerden numbers w(2;3,t)

We present results and conjectures on the van der Waerden numbers w(2;3,t) and on the new palindromic van der Waerden numbers pdw(2;3,t). We have computed the new number w(2;3,19) = 349, and we provide lower bounds for 20 <= t <= 39, where for t <= 30 we conjecture these lower bounds to be exact. The lower bounds for 24 <= t <= 30 refute the conjecture that w(2;3,t) <= t^2, and we present an improved conjecture. We also investigate regularities in the good partitions (certificates) to better understand the lower bounds. Motivated by such reglarities, we introduce *palindromic van der Waerden numbers* pdw(k; t_0,...,t_{k-1}), defined as ordinary van der Waerden numbers w(k; t_0,...,t_{k-1}), however only allowing palindromic solutions (good partitions), defined as reading the same from both ends. Different from the situation for ordinary van der Waerden numbers, these "numbers" need actually to be pairs of numbers. We compute pdw(2;3,t) for 3 <= t <= 27, and we provide lower bounds, which we conjecture to be exact, for t <= 35. All computations are based on SAT solving, and we discuss the various relations between SAT solving and Ramsey theory. Especially we introduce a novel (open-source) SAT solver, the tawSolver, which performs best on the SAT instances studied here, and which is actually the original DLL-solver, but with an efficient implementation and a modern heuristic typical for look-ahead solvers (applying the theory developed in the SAT handbook article of the second author).Comment: Second version 25 pages, updates of numerical data, improved formulations, and extended discussions on SAT. Third version 42 pages, with SAT solver data (especially for new SAT solver) and improved representation. Fourth version 47 pages, with updates and added explanation

Cube-and-Conquer approach for SAT solving on grids

Our goal is to develop techniques for using distributed computing re- sources to efficiently solve instances of the propositional satisfiability problem (SAT). We claim that computational grids provide a distributed computing environment suitable for SAT solving. In this paper we apply the Cube and Conquer approach to SAT solving on grids and present our parallel SAT solver CCGrid (Cube and Conquer on Grid) on computational grid infrastructure. Our solver consists of two major components. The master application runs march_cc, which applies a lookahead SAT solver, in order to partition the input SAT instance into work units distributed on the grid. The client application executes an iLingeling instance, which is a multi-threaded CDCL SAT solver. We use BOINC middleware, which is part of the SZTAKI Desktop Grid package and supports the Distributed Computing Application Programming Interface (DC-API). Our preliminary results suggest that our approach can gain significant speedup and shows a potential for future investigation and development. Keywords: grid, SAT, parallel SAT solving, lookahead, march_cc, iLingeling, SZTAKI Desktop Grid, BOINC, DC-AP

Some Results in Extremal Combinatorics

Extremal Combinatorics is one of the central and heavily contributed areas in discrete mathematics, and has seen an outstanding growth during the last few decades. In general, it deals with problems regarding determination and/or estimation of the maximum or the minimum size of a combinatorial structure that satisfies a certain combinatorial property. Problems in Extremal Combinatorics are often related to theoretical computer science, number theory, geometry, and information theory. In this thesis, we work on some well-known problems (and on their variants) in Extremal Combinatorics concerning the set of integers as the combinatorial structure. The van der Waerden number w(k;t_0,t_1,...,t_{k-1}) is the smallest positive integer n such that every k-colouring of 1, 2, . . . , n contains a monochromatic arithmetic progression of length t_j for some colour j in {0,1,...,k-1}. We have determined five new exact values with k=2 and conjectured several van der Waerden numbers of the form w(2;s,t), based on which we have formulated a polynomial upper-bound-conjecture of w(2; s, t) with fixed s. We have provided an efficient SAT encoding for van der Waerden numbers with k>=3 and computed three new van der Waerden numbers using that encoding. We have also devised an efficient problem-specific backtracking algorithm and computed twenty-five new van der Waerden numbers with k>=3 using that algorithm. We have proven some counting properties of arithmetic progressions and some unimodality properties of sequences regarding arithmetic progressions. We have generalized Szekeres’ conjecture on the size of the largest sub-sequence of 1, 2, . . . , n without an arithmetic progression of length k for specific k and n; and provided a construction for the lower bound corresponding to the generalized conjecture. A Strict Schur number S(h,k) is the smallest positive integer n such that every 2-colouring of 1,2,...,n has either a blue solution to x_1 +x_2 +···+x_{h-1} = x_h where x_1 < x_2 < ··· < x_h, or a red solution to x_1+x_2+···+x_{k-1} =x_k where x_1 <x_2 <···<x_k. We have proven the exact formula for S(3, k)

Grid based propositional satisfiability solving

This work studies how grid and cloud computing can be applied to efficiently solving propositional satisfiability problem (SAT) instances. Propositional logic provides a convenient language for expressing real-world originated problems such as AI planning, automated test pattern generation, bounded model checking and cryptanalysis. The interest in SAT solving has increased mainly due to improvements in the solving algorithms, which recently have increasingly focused on using parallelism offered by multi-CPU computers. Partly orthogonally to these improvements this work studies several novel approaches to parallel solving of SAT instances in a grid of widely distributed "virtual" computers instead of workstations or supercomputers. Two types of parallel SAT solving approaches are analyzed and used as building blocks for more complex systems: using several solvers which compete to solve a given instance in parallel, and splitting the search space of the instance and solving the resulting partitions in parallel. The work presents several efficient partitioning functions, critical in successful splitting according to an analytical result, and presents novel solving systems that are less dependent on the partitioning function efficiency. Finally, the work studies combining clause learning, a key technique in modern SAT solvers, with the novel types of parallel solvers. Different heuristics are studied for filtering clauses learned in parallel, and the work proposes techniques which allow exchanging the clauses between different splits. The practical significance of the results are studied using large, standard benchmark sets from SAT competitions. Some of the approaches are able to solve several instances that have either not been solved at all by any other solver, or which are significantly slower to solve with other solvers

Incremental Satisfiability Solving and its Applications

The propositional logic satisfiability problem (SAT) is a computationally hard decision problem. Despite its theoretical hardness, decision procedures for solving instances of this problem have become surprisingly efficient in recent years. These procedures, known as SAT solvers, are able to solve large instances originating from real-life problem domains, such as artificial intelligence and formal verification. Such real-life applications often require solving several related instances of SAT. Therefore, modern solvers posses an incremental interface that allows the input of sequences of incrementally encoded instances of SAT. When solving these instances sequentially the solver can reuse some of the information it has gathered across related consecutive instances. This dissertation contains six publications. The two focus areas of the combined work are incremental usage of SAT solvers, and the usage of parallelism in applications of SAT solvers. It is shown in this work that these two seemingly contradictory concepts form a natural combination. Moreover, this dissertations unifies, analyzes, and extends the results of the six publications, for example, by studying information propagation in incremental solvers through graphical visualizations. The concrete contributions made by the work in this dissertation include, but are not limited to: Improvements to algorithms for MUS finding, the use of graphical visualizations to understand information propagation in incremental solvers, asynchronous incremental solving, and concurrent clause strengthening

CDCL(Crypto) and Machine Learning based SAT Solvers for Cryptanalysis

Over the last two decades, we have seen a dramatic improvement in the efficiency of conflict-driven clause-learning Boolean satisfiability (CDCL SAT) solvers over industrial problems from a variety of applications such as verification, testing, security, and AI. The availability of such powerful general-purpose search tools as the SAT solver has led many researchers to propose SAT-based methods for cryptanalysis, including techniques for finding collisions in hash functions and breaking symmetric encryption schemes. A feature of all of the previously proposed SAT-based cryptanalysis work is that they are \textit{blackbox}, in the sense that the cryptanalysis problem is encoded as a SAT instance and then a CDCL SAT solver is invoked to solve said instance. A weakness of this approach is that the encoding thus generated may be too large for any modern solver to solve it efficiently. Perhaps a more important weakness of this approach is that the solver is in no way specialized or tuned to solve the given instance. Finally, very little work has been done to leverage parallelism in the context of SAT-based cryptanalysis. To address these issues, we developed a set of methods that improve on the state-of-the-art SAT-based cryptanalysis along three fronts. First, we describe an approach called \cdcl (inspired by the CDCL($T$) paradigm) to tailor the internal subroutines of the CDCL SAT solver with domain-specific knowledge about cryptographic primitives. Specifically, we extend the propagation and conflict analysis subroutines of CDCL solvers with specialized codes that have knowledge about the cryptographic primitive being analyzed by the solver. We demonstrate the power of this framework in two cryptanalysis tasks of algebraic fault attack and differential cryptanalysis of SHA-1 and SHA-256 cryptographic hash functions. Second, we propose a machine-learning based parallel SAT solver that performs well on cryptographic problems relative to many state-of-the-art parallel SAT solvers. Finally, we use a formulation of SAT into Bayesian moment matching to address heuristic initialization problem in SAT solvers

Proceedings of the 21st Conference on Formal Methods in Computer-Aided Design – FMCAD 2021

The Conference on Formal Methods in Computer-Aided Design (FMCAD) is an annual conference on the theory and applications of formal methods in hardware and system verification. FMCAD provides a leading forum to researchers in academia and industry for presenting and discussing groundbreaking methods, technologies, theoretical results, and tools for reasoning formally about computing systems. FMCAD covers formal aspects of computer-aided system design including verification, specification, synthesis, and testing