Chapter 13. Symmetry and Satisfiability

2021 ◽  
Author(s):  
Karem A. Sakallah
Keyword(s):  
Group Theory ◽  
Normal Form ◽  
Boolean Functions ◽  
Search Algorithm ◽  
Conjunctive Normal Form ◽  
Search Space ◽  
Decision Problems ◽  
Mathematical Language ◽  
Sat Solvers ◽  
Speed Up

Symmetry is at once a familiar concept (we recognize it when we see it!) and a profoundly deep mathematical subject. At its most basic, a symmetry is some transformation of an object that leaves the object (or some aspect of the object) unchanged. For example, a square can be transformed in eight different ways that leave it looking exactly the same: the identity “do-nothing” transformation, 3 rotations, and 4 mirror images (or reflections). In the context of decision problems, the presence of symmetries in a problem’s search space can frustrate the hunt for a solution by forcing a search algorithm to fruitlessly explore symmetric subspaces that do not contain solutions. Recognizing that such symmetries exist, we can direct a search algorithm to look for solutions only in non-symmetric parts of the search space. In many cases, this can lead to significant pruning of the search space and yield solutions to problems which are otherwise intractable. This chapter explores the symmetries of Boolean functions, particularly the symmetries of their conjunctive normal form (CNF) representations. Specifically, it examines what those symmetries are, how to model them using the mathematical language of group theory, how to derive them from a CNF formula, and how to utilize them to speed up CNF SAT solvers.

Improving Matsui’s Search Algorithm For The Best Differential/Linear Trails And Its Applications For DES, DESL And GIFT

2020 ◽  
Author(s):  
Fulei Ji ◽  
Wentao Zhang ◽  
Tianyou Ding
Keyword(s):  
Search Algorithm ◽  
Search Space ◽  
Search Methods ◽  
New Methods ◽  
Linear Layer ◽  
Speed Up ◽  
Automatic Search ◽  
The Cost ◽  
First Time ◽  
Improved Algorithm

Abstract Automatic search methods have been widely used for cryptanalysis of block ciphers, especially for the most classic cryptanalysis methods—differential and linear cryptanalysis. However, the automatic search methods, no matter based on MILP, SMT/SAT or CP techniques, can be inefficient when the search space is too large. In this paper, we propose three new methods to improve Matsui’s branch-and-bound search algorithm, which is known as the first generic algorithm for finding the best differential and linear trails. The three methods, named reconstructing DDT and LAT according to weight, executing linear layer operations in minimal cost and merging two 4-bit S-boxes into one 8-bit S-box, respectively, can efficiently speed up the search process by reducing the search space as much as possible and reducing the cost of executing linear layer operations. We apply our improved algorithm to DESL and GIFT, which are still the hard instances for the automatic search methods. As a result, we find the best differential trails for DESL (up to 14-round) and GIFT-128 (up to 19-round). The best linear trails for DESL (up to 16-round), GIFT-128 (up to 10-round) and GIFT-64 (up to 15-round) are also found. To the best of our knowledge, these security bounds for DESL and GIFT under single-key scenario are given for the first time. Meanwhile, it is the longest exploitable (differential or linear) trails for DESL and GIFT. Furthermore, benefiting from the efficiency of the improved algorithm, we do experiments to demonstrate that the clustering effect of differential trails for 13-round DES and DESL are both weak.

scholarly journals Accelerating backtrack search with a best-first-search strategy

2014 ◽  
Vol 24 (4) ◽  
pp. 901-916
Author(s):  
Zoltán Ádám Mann ◽  
Tamás Szép
Keyword(s):  
Search Strategy ◽  
Search Algorithm ◽  
Search Space ◽  
New Approach ◽  
Backtrack Search ◽  
Speed Up ◽  
Hard Problems ◽  
Best First Search ◽  
Problem Instances ◽  
Np Hard Problems

Abstract Backtrack-style exhaustive search algorithms for NP-hard problems tend to have large variance in their runtime. This is because “fortunate” branching decisions can lead to finding a solution quickly, whereas “unfortunate” decisions in another run can lead the algorithm to a region of the search space with no solutions. In the literature, frequent restarting has been suggested as a means to overcome this problem. In this paper, we propose a more sophisticated approach: a best-firstsearch heuristic to quickly move between parts of the search space, always concentrating on the most promising region. We describe how this idea can be efficiently incorporated into a backtrack search algorithm, without sacrificing optimality. Moreover, we demonstrate empirically that, for hard solvable problem instances, the new approach provides significantly higher speed-up than frequent restarting.

FROM XSAT TO SAT BY EXHIBITING BOOLEAN FUNCTIONS

2009 ◽  
Vol 18 (05) ◽  
pp. 783-799
Author(s):  
RICHARD OSTROWSKI ◽  
LIONEL PARIS
Keyword(s):  
Graph Coloring ◽  
Boolean Functions ◽  
Structural Information ◽  
Conjunctive Normal Form ◽  
Satisfiability Problem ◽  
Boolean Formula ◽  
Truth Assignment ◽  
Coloring Problem ◽  
Graph Coloring Problem ◽  
Speed Up

Given a Boolean formula in conjunctive normal form (CNF), the Exact Satisfiability problem (XSAT), a variant of the Satisfiability problem (SAT), consists in finding an assignment to the variables such that each clause contains exactly one satisfied literal. Best algorithms to solve this problem run in [Formula: see text] ([Formula: see text] for X3SAT). Another possibility is to transform each clause in a set of equivalent clauses for the Satisfiability problem and to use modern and powerful solvers (zChaff, Berkmin, MiniSat, RSat etc.) to find such truth assignment. In this paper we introduce three new encodings from XSAT instances to SAT instances that lead to a lot of structural information (equivalency gates and and gates) which is naturally hidden in the pairwise transformation. Some solvers (lsat,march_dl,eqsatz) can take into account this kinds of structural information to make simplifications as pretreatment and speed-up the resolution. Then we show the interest of dealing with the XSAT formalism by introducing an encoding of binary CSP and graph coloring problem into XSAT instances. Preliminary results on real-world binary CSP and graph coloring problem show the importance of exhibiting equivalencies for the XSAT problem.

scholarly journals FourierSAT: A Fourier Expansion-Based Algebraic Framework for Solving Hybrid Boolean Constraints

2020 ◽  
Vol 34 (02) ◽  
pp. 1552-1560
Author(s):  
Anastasios Kyrillidis ◽  
Anshumali Shrivastava ◽  
Moshe Vardi ◽  
Zhiwei Zhang
Keyword(s):  
Boolean Functions ◽  
Continuous Optimization ◽  
Conjunctive Normal Form ◽  
Cardinality Constraints ◽  
Sat Solvers ◽  
Boolean Satisfiability Problem ◽  
Algebraic Framework ◽  
Clause Learning ◽  
Sat Solver ◽  
Analysis Of Boolean Functions

The Boolean SATisfiability problem (SAT) is of central importance in computer science. Although SAT is known to be NP-complete, progress on the engineering side—especially that of Conflict-Driven Clause Learning (CDCL) and Local Search SAT solvers—has been remarkable. Yet, while SAT solvers, aimed at solving industrial-scale benchmarks in Conjunctive Normal Form (CNF), have become quite mature, SAT solvers that are effective on other types of constraints (e.g., cardinality constraints and XORs) are less well-studied; a general approach to handling non-CNF constraints is still lacking. In addition, previous work indicated that for specific classes of benchmarks, the running time of extant SAT solvers depends heavily on properties of the formula and details of encoding, instead of the scale of the benchmarks, which adds uncertainty to expectations of running time.To address the issues above, we design FourierSAT, an incomplete SAT solver based on Fourier analysis of Boolean functions, a technique to represent Boolean functions by multilinear polynomials. By such a reduction to continuous optimization, we propose an algebraic framework for solving systems consisting of different types of constraints. The idea is to leverage gradient information to guide the search process in the direction of local improvements. Empirical results demonstrate that FourierSAT is more robust than other solvers on certain classes of benchmarks.

Chapter 9. Preprocessing in SAT Solving

2021 ◽  
Author(s):  
Armin Biere ◽  
Matti Järvisalo ◽  
Benjamin Kiesl
Keyword(s):  
Normal Form ◽  
Conjunctive Normal Form ◽  
Boolean Satisfiability ◽  
Sat Solving ◽  
Speed Up ◽  
Sat Solver ◽  
Main Ideas ◽  
Simplification Techniques

Preprocessing has become a key component of the Boolean satisfiability (SAT) solving workflow. In practice, preprocessing is situated between the encoding phase and the solving phase, with the aim of decreasing the total solving time by applying efficient simplification techniques on SAT instances to speed up the search subsequently performed by a SAT solver. In this chapter, we overview key preprocessing techniques proposed in the literature. While the main focus is on techniques applicable to formulas in conjunctive normal form (CNF), we also selectively cover main ideas for preprocessing structural and higher-level SAT instance representations.

Chapter 2. CNF Encodings

2021 ◽  
Author(s):  
Steven Prestwich
Keyword(s):  
Normal Form ◽  
Case Studies ◽  
Search Algorithm ◽  
Empirical Work ◽  
Combinatorial Problem ◽  
Conjunctive Normal Form ◽  
Search Algorithms ◽  
File Format ◽  
Sources Of Error ◽  
Algorithm Implementation

Before a combinatorial problem can be solved by current SAT methods, it must usually be encoded in conjunctive normal form, which facilitates algorithm implementation and allows a common file format for problems. Unfortunately there are several ways of encoding most problems and few guidelines on how to choose among them, yet the choice of encoding can be as important as the choice of search algorithm. This chapter reviews theoretical and empirical work on encoding methods, including the use of Tseitin encodings, the encoding of extensional and intensional constraints, the interaction between encodings and search algorithms, and some common sources of error. Case studies are used for illustration.

scholarly journals A Cardinal Improvement to Pseudo-Boolean Solving

2020 ◽  
Vol 34 (02) ◽  
pp. 1495-1503
Author(s):  
Jan Elffers ◽  
Jakob Nordstr”m
Keyword(s):  
Normal Form ◽  
Building Blocks ◽  
Conjunctive Normal Form ◽  
Conflict Analysis ◽  
Cardinality Constraints ◽  
Sat Solvers ◽  
Boolean Formulas ◽  
Clause Learning

Pseudo-Boolean solvers hold out the theoretical potential of exponential improvements over conflict-driven clause learning (CDCL) SAT solvers, but in practice perform very poorly if the input is given in the standard conjunctive normal form (CNF) format. We present a technique to remedy this problem by recovering cardinality constraints from CNF on the fly during search. This is done by collecting potential building blocks of cardinality constraints during propagation and combining these blocks during conflict analysis. Our implementation has a non-negligible but manageable overhead when detection is not successful, and yields significant gains for some SAT competition and crafted benchmarks for which pseudo-Boolean reasoning is stronger than CDCL. It also boosts performance for some native pseudo-Boolean formulas where this approach helps to improve learned constraints.

The decision problem for formulas in prenex conjunctive normal form with binary disjunctions

1970 ◽  
Vol 35 (2) ◽  
pp. 210-216 ◽  
Author(s):  
M. R. Krom
Keyword(s):  
Normal Form ◽  
Positive Solution ◽  
Decision Problem ◽  
Conjunctive Normal Form ◽  
Decision Problems ◽  
First Order ◽  
Reduction Class ◽  
Image Position ◽  
Positive Integers ◽  
Order Predicate Calculus

In [8] S. J. Maslov gives a positive solution to the decision problem for satisfiability of formulas of the formin any first-order predicate calculus without identity where h, k, m, n are positive integers, αi, βi are signed atomic formulas (atomic formulas or negations of atomic formulas), and ∧, ∨ are conjunction and disjunction symbols, respectively (cf. [6] for a related solvable class). In this paper we show that the decision problem is unsolvable for formulas that are like those considered by Maslov except that they have prefixes of the form ∀x∃y1 … ∃yk∀z. This settles the decision problems for all prefix classes of formulas for formulas that are in prenex conjunctive normal form in which all disjunctions are binary (have just two terms). In our concluding section we report results on decision problems for related classes of formulas including classes of formulas in languages with identity and we describe some special properties of formulas in which all disjunctions are binary including a property that implies that any proof of our result, that a class of formulas is a reduction class for satisfiability, is necessarily indirect. Our proof is based on an unsolvable combinatorial tag problem.

scholarly journals An Information-Motivated Exploration Agent to Locate Stationary Persons with Wireless Transmitters in Unknown Environments

Sensors ◽  
2021 ◽  
Vol 21 (22) ◽  
pp. 7695
Author(s):  
Daniel Barry ◽  
Andreas Willig ◽  
Graeme Woodward
Keyword(s):  
Information Gain ◽  
Search Algorithm ◽  
A Priori ◽  
Search Space ◽  
Wireless Transmitters ◽  
Multiple Sensors ◽  
Single Target ◽  
Information Exploration ◽  
Speed Up ◽  
Priori Information

Unmanned Aerial Vehicles (UAVs) show promise in a variety of applications and recently were explored in the area of Search and Rescue (SAR) for finding victims. In this paper we consider the problem of finding multiple unknown stationary transmitters in a discrete simulated unknown environment, where the goal is to locate all transmitters in as short a time as possible. Existing solutions in the UAV search space typically search for a single target, assume a simple environment, assume target properties are known or have other unrealistic assumptions. We simulate large, complex environments with limited a priori information about the environment and transmitter properties. We propose a Bayesian search algorithm, Information Exploration Behaviour (IEB), that maximizes predicted information gain at each search step, incorporating information from multiple sensors whilst making minimal assumptions about the scenario. This search method is inspired by the information theory concept of empowerment. Our algorithm shows significant speed-up compared to baseline algorithms, being orders of magnitude faster than a random agent and 10 times faster than a lawnmower strategy, even in complex scenarios. The IEB agent is able to make use of received transmitter signals from unknown sources and incorporate both an exploration and search strategy.

On Starting Population Selection for GSAT

2013 ◽  
Vol 365-366 ◽  
pp. 190-193 ◽  
Author(s):  
Anna Gorbenko ◽  
Vladimir Popov
Keyword(s):  
Normal Form ◽  
Normal Forms ◽  
Search Algorithm ◽  
Conjunctive Normal Form ◽  
Random Functions ◽  
Population Selection ◽  
Selection For ◽  
Artificial Physics

GSAT is a well-known satisfiability search algorithm for conjunctive normal forms. GSAT uses some random functions. One of such functions is a function of starting population of truth assignments for the variables of conjunctive normal form. In this paper, we consider a method of artificial physics optimization for computing a function of starting population.

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