Boolean Satisfiability Research Articles

Exact ASP Counting with Compact Encodings

Answer Set Programming (ASP) has emerged as a promising paradigm in knowledge representation and automated reason- ing owing to its ability to model hard combinatorial problems from diverse domains in a natural way. Building on advances in propositional SAT solving, the past two decades have wit- nessed the emergence of well-engineered systems for solv- ing the answer set satisfiability problem, i.e., finding mod- els or answer sets for a given answer set program. In re- cent years, there has been growing interest in problems be- yond satisfiability, such as model counting, in the context of ASP. Akin to the early days of propositional model count- ing, state-of-the-art exact answer set counters do not scale well beyond small instances. Exact ASP counters struggle with handling larger input formulas. The primary contribu- tion of this paper is a new ASP counting framework, called sharpASP, which counts answer sets avoiding larger input formulas. This relies on an alternative way of defining answer sets that allows lifting of key techniques developed in the con- text of propositional model counting. Our extensive empirical analysis over 1470 benchmarks demonstrates significant per- formance gain over current state-of-the-art exact answer set counters. Specifically, by using sharpASP, we were able to solve 1062 benchmarks with PAR2 score of 3082 whereas using prior state-of-the-art, we could only solve 895 bench- marks with PAR2 score of 4205, all other experimental con- ditions being the same.

Runtime vs. Extracted Proof Size: An Exponential Gap for CDCL on QBFs

Conflict-driven clause learning (CDCL) is the dominating algorithmic paradigm for SAT solving and hugely successful in practice. In its lifted version QCDCL, it is one of the main approaches for solving quantified Boolean formulas (QBF). In both SAT and QBF, proofs can be efficiently extracted from runs of (Q)CDCL solvers. While for CDCL, it is known that the proof size in the underlying proof system propositional resolution matches the CDCL runtime up to a polynomial factor, we show that in QBF there is an exponential gap between QCDCL runtime and the size of the extracted proofs in QBF resolution systems. We demonstrate that this is not just a gap between QCDCL runtime and the size of any QBF resolution proof, but even the extracted proofs are exponentially smaller for some instances. Hence searching for a small proof via QCDCL (even with non-deterministic decision policies) will provably incur an exponential overhead for some instances.

Improved (related-key) differential cryptanalysis on [formula omitted]

LBlock presented at ACNS 2011 is a lightweight block cipher. The length of the block and master key for LBlock is 64 and 80 bits, respectively. LBlock adopts the Feistel-SP network and has effective hardware implementation. The 18-round attack on LBlock is the most well-known single-key truncated differential attack to date. While using the MILP model in ASIACRYPT 2014, a 14-round related-key characteristic with probability 2−49 was discovered. In this paper, we investigate the (related-key) differential behavior of LBlock and implement the first 20-round differential attack against it under the single-key scenario. First, the related-key characteristics of LBlock are sought using the SAT (Boolean Satisfiability Problem) model with Matsui’s bounds constraints. Then, in order to increase the probabilities of distinguishers, a new algorithm is provided to search for characteristics with a clustering effect. Following that, the probability of a 17-round related-key differential which is the longest distinguisher is increased from 2−63 to 2−59.05. Additionally, several valid 15-round differentials found by the algorithm can be used to perform a 20-round differential attack. Finally, by expanding the 15-round distinguisher, we are able to perform the key recovery attack on 20-round LBlock. Data, time, and memory complexity are 260.98, 269.84, and 260.98, respectively. Compared to the existing single-key differential attack, our result covers two more rounds. The new algorithm can not only design block ciphers but also evaluate their security.

Efficient implementations of CRAFT cipher for Internet of Things

The rapid growth of the Internet of Things (IoT) highlights the importance of lightweight cryptography in maintaining security. However, enhancing performance while ensuring the same level of security remains a significant challenge. This paper presents two innovative architectures for the CRAFT lightweight block cipher, aiming to enhance performance without compromising security. The novel Serial and Unrolled architectures are introduced to achieve low area usage and high throughput, respectively. Specifically, the Serial architecture reduces the datapath from 64-bit to 4-bit, significantly decreasing the area. The Unrolled architecture, on the other hand, minimizes latency from 32 to 16. Additionally, Boolean satisfiability (SAT) solvers are employed to identify a lower-cost area implementation of the S-Box. The proposed designs underwent evaluation on three distinct FPGA platforms: Artix-7, Kintex-7, and Spartan-7. The results show that the low area design reduces area usage by 10.16% compared to the previous design. Additionally, the S-Box implementation achieves a significant area reduction of 28.9%. On the other hand, the unrolled design enhances the maximum throughput by 40.53% compared to the previous design. Therefore, the proposed designs could offer enhanced performance while maintaining security for IoT devices.

Inverse Design of Self-Folding 3D Shells.

Self-folding is an emerging paradigm for the inverse design of three-dimensional structures. While most efforts have concentrated on the shape of the net, our approach introduces a new design dimension-bond specificity between the edges. We transform this design process into a Boolean satisfiability problem to derive solutions for various target structures. This method significantly enhances the yield of the folding process. Furthermore, by linearly combining independent solutions, we achieve designs for shape-shifting nets wherein the dominant structure evolves with varying external conditions. This approach is demonstrated through coarse-grained simulations on two examples of triangular and square nets capable of folding into multiple target shapes.

A genetic algorithm for the Resource-Constrained Project Scheduling Problem with Alternative Subgraphs using a boolean satisfiability solver

This study evaluates a new solution approach for the Resource-Constrained Project Scheduling with Alternative Subgraphs (RCPSP-AS) in case that complex relations (i.e. nested and linked alternatives) are considered. In the RCPSP-AS, the project activity structure is extended with alternative activity sequences. This implies that only a subset of all activities should be scheduled, which corresponds with a set of activities in the project network that model an alternative execution mode for a work package. Since only the selected activities should be scheduled, the RCPSP-AS comes down to a traditional RCPSP problem when the selection subproblem is solved. It is known that the RCPSP and, hence, its extension to the RCPSP-AS is NP-hard. Since similar scheduling and selection subproblems have already been successfully solved by satisfiability (SAT) solvers in the existing literature, we aim to test the performance of a GA-SAT approach that is derived from the literature and adjusted to be able to deal with the problem-specific constraints of the RCPSP-AS. Computational results on small- and large-scale instances (both artificial and empirical) show that the algorithm can compete with existing metaheuristic algorithms from the literature. Also, the performance is compared with an exact mathematical solver and learning behaviour is observed and analysed. This research again validates the broad applicability of SAT solvers as well as the need to search for better and more suited algorithms for the RCPSP-AS and its extensions.

A Novel, Efficient and Generalised Approach to Boolean Satisfiability

One question which has occupied mathematicians, engineers and scientists of this age is the problem of Boolean satisfiability (SAT) which asks whether a given CNF formula has at least a satisfying assignment. The goal of this work is to present a novel, efficient and general method of deciding SAT and finding all the satisfying assignments of CNF formulas. This method is powerful as it employs resolution identities in transforming clauses without a particular variable into clauses with that variable.

Puzzle and Dragons is hard

Candy Crush Saga (CCS) is a mobile game released in 2012. Based on a specific game-play, it was proven to be NP-complete in 2014. The 15-slide puzzle is also a classic game that has been proven to be NP-complete. In 2012, a brand new ball-spinning game called Puzzle and Dragons emerged. Puzzle and Dragons (PAD) combines the three-elimination game-play of Candy Crush Saga and the 15-puzzle. This study developed a construction and proved that under certain rules, PAD can be reduced to a one-in-three positive Boolean satisfiability problem (1-3-SAT+), which also proves that PAD is NP-Hard.

Monitoring of spatio-temporal properties with nonlinear SAT solvers

The automotive industry is increasingly dependent on computing systems with different critical requirements. The verification and validation methods for these systems are now leveraging complex AI methods, for which the decision algorithms introduce non-determinism, especially in autonomous driving. This paper presents a runtime verification technique agnostic to the target system, which focuses on monitoring spatio-temporal properties that abstract the evolution of objects’ behavior in their spatial and temporal flow. First, a formalization of three known traffic rules (from the Vienna convention on road traffic) is presented, where a spatio-temporal logic fragment is used. Then, these logical expressions are translated to a monitoring model written in first-order logic, where they are processed by a non-linear satisfiability solver. Finally, the translation allows the solver to check the validity of the encoded properties according to an instance of a specific traffic scenario (a trace). The results obtained from our tool, which automatically generates a monitor from a formula, show that our approach is feasible for online monitoring in a real-world environment.

Solving the SAT problem with the string multiset rewriting calculus

In this paper, we develop computing machinery within the framework of the String Multiset Rewriting calculus (SMSR), as defined by Barbuti et al. [4], to solve the SAT problem in linear time regarding the number of variables of a given conjunctive normal form. This shows that SMSR can be considered a computational model capable of significantly reducing the time requirement of classical decision problems.

Fully CMOS‐Based p‐Bits with a Bistable Resistor for Probabilistic Computing

AbstractProbabilistic computing can solve complex combinatorial optimization problems more efficiently than conventional deterministic computing. A probabilistic bit (p‐bit) with an n‐p‐n bistable resistor (biristor) is demonstrated for probabilistic computing. It is fabricated on an 8‐inch wafer with complementary metal–oxide–semiconductor (CMOS) compatible technologies. Its stochastic behavior of threshold switching, which is based on the phenomenon of a single transistor latch, provides output with a Boltzmann distribution. The p‐bit is composed of a biristor, a serial resistor, and a comparator. The output probability of the biristor‐based p‐bits shows a sigmoidal relationship with the input voltage, showing typical p‐bit characteristics. Invertible Boolean logic operations with p‐bits are demonstrated, and weighted maximum Boolean satisfiability problems are solved with high energy efficiency and accuracy. The biristor‐based p‐bits with perfect CMOS compatibility show sufficient device stability, demonstrating the possibility of large‐scale integration with a p‐bit array for complex optimization solvers.

A circuit area optimization of MK-3 S-box

In MILCOM 2015, Kelly et al. proposed the authentication encryption algorithm MK-3, which applied the 16-bit S-box. This paper aims to implement the 16-bit S-box with less circuit area. First, we classified the irreducible polynomials over F2n\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {F}_{2^n}$$\\end{document} into three kinds. Then we compared the logic gates required for multiplication over the finite field constructed by the three types of irreducible polynomials. According to the comparison result, we constructed the composite fields, F(24)2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {F}_{(2^4)^2}$$\\end{document} and F(28)2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {F}_{(2^8)^2}$$\\end{document}. Based on the isomorphism of finite fields, the operations over F216\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {F}_{2^{16}}$$\\end{document} can be conducted over F(28)2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {F}_{(2^8)^2}$$\\end{document}. Similarly, elements over F28\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {F}_{2^8}$$\\end{document} can be mapped to the corresponding elements over F(24)2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {F}_{(2^4)^2}$$\\end{document}. Next, the SAT solver was used to optimize the operations over smaller field F24\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {F}_{2^4}$$\\end{document}. At last, the architecture of the optimized MK-3 S-box was worked out. Compared with the implementation proposed by the original designer, the circuit area of the MK-3 S-box in this paper is reduced by at least 55.9%.

Solving the Cubic Monotone 1-in-3 SAT Problem in Polynomial Time

The exact 3-satisfiability problem (X3SAT) is known to remain NP-complete when restricted to expressions where every variable has exactly three occurrences, even in the absence of negated variables (Cubic Monotone 1-in-3 SAT Problem). The present paper shows that the Cubic Monotone 1-in-3 SAT Problem can be solved in polynomial time and, therefore prove that the conjecture P=NP holds.

Modeling of 3 SAT discrete Hopfield neural network optimization using genetic algorithm optimized K-modes clustering

<p>The discrete Hopfield neural network 3-satisfiability (DHNN-3SAT) model represents an innovative application of deep learning techniques to the Boolean SAT problem. Existing research indicated that the DHNN-3SAT model demonstrated significant advantages in handling 3SAT problem instances of varying scales and complexities. Compared to traditional heuristic algorithms, this model converged to local minima more rapidly and exhibited enhanced exploration capabilities within the global search space. However, the model faced several challenges and limitations. As constraints in SAT problems dynamically increased, decreased, or changed, and as problem scales expanded, the model's computational complexity and storage requirements may increase dramatically, leading to reduced performance in handling large-scale SAT problems. To address these challenges, this paper first introduced a method for designing network synaptic weights based on fundamental logical clauses. This method effectively utilized the synaptic weight information from the original SAT problem within the DHNN network, thereby significantly reducing redundant computations. Concrete examples illustrated the design process of network synaptic weights when constraints were added, removed, or updated, offering new approaches for managing the evolving constraints in SAT problems. Subsequently, the paper presented a DHNN-3SAT model optimized by genetic algorithms combined with K-modes clustering. This model employed genetic algorithm-optimized K-modes clustering to effectively cluster the initial space, significantly reducing the search space. This approach minimized the likelihood of redundant searches and reduced the risk of getting trapped in local minima, thus improving search efficiency. Experimental tests on benchmark datasets showed that the proposed model outperformed traditional DHNN-3SAT models, DHNN-3SAT models combined with genetic algorithms, and DHNN-3SAT models combined with imperialist competitive algorithms across four evaluation metrics. This study not only broadened the application of DHNN in solving 3SAT problems but also provided valuable insights and guidance for future research.</p>

A novel processor for dynamic evolution of constrained SAT problems: The dynamic evolution variant of the discrete Hopfield neural network satisfiability model

The Boolean satisfiability problem, a renowned NP-complete challenge in computer science, has recently garnered interest in the Discrete Hopfield Neural Network - Satisfiability model. This model adeptly integrates logical rules into Hopfield networks, excelling in locating global minima for traditional SAT problems. However, it faces efficiency challenges when dealing with SAT problems characterized by dynamic evolution constraints due to its static network architecture. During dynamic evolution iterations, there is a significant exponential increase in the computational costs due to redundant and repetitive computations.In order to address this challenge, this paper introduces a dynamic evolution variant of the Discrete Hopfield Neural Network - Satisfiability model. In extensive simulation experiments, we progressively augmented the number of constraint clauses from 1 to 1500, seeking global minima for CNF problems. The proposed model exhibited congruent performance with the traditional model, achieving a Global Minimum Ratio of 1 and a Hamming distance of 0. Crucially, the proposed model minimized CPU utilization and neared zero error metrics, while the traditional model experienced exponential CPU and error metric escalation. These outcomes affirm the proposed model's robust global search capabilities and high precision, aligning with the traditional model. Furthermore, owing to this model's incorporation of not only temporal constraint increment operator but also innovative real-time learning techniques and clever integration methods, along with the establishment of a novel real-time decision mechanism, the proposed model effectively addresses the issues of redundancy and repeated calculations inherent in traditional models. This results in a stable and significantly improved computational speed. Additionally, this model's dynamic evolution network architecture is specifically designed to accommodate arbitrary and efficient extensions of dynamic constraints, while also featuring a built-in preprocessing function to filter out duplicate variables. In the future, as a dynamic evolution processor, this model exhibits immense potential for solving large-scale SAT problems with dynamic evolution constraints and holds broad application prospects in emerging fields such as automated reasoning, intelligent recommendation systems, and machine learning.

Amplitude amplification-inspired QAOA: improving the success probability for solving 3SAT

The Boolean satisfiability problem (SAT), in particular 3SAT with its bounded clause size, is a well-studied problem since a wide range of decision problems can be reduced to it. The Quantum Approximate Optimization Algorithm (QAOA) is a promising candidate for solving 3SAT for Noisy Intermediate-Scale Quantum devices in the near future due to its simple quantum ansatz. However, although QAOA generally exhibits a high approximation ratio, there are 3SAT problem instances where the algorithm’s success probability when obtaining a satisfying variable assignment from the approximated solution drops sharply compared to the approximation ratio. To address this problem, in this paper, we present variants of the algorithm that are inspired by the amplitude amplification algorithm to improve the success probability for 3SAT. For this, (i) three amplitude amplification-inspired QAOA variants are introduced and implemented, (ii) the variants are experimentally compared with a standard QAOA implementation, and (iii) the impact on the success probability and ansatz complexity is analyzed. The experiment results show that an improvement in the success probability can be achieved with only a moderate increase in circuit complexity.

2-SAT discrete Hopfield neural networks optimization via Crow search and fuzzy dynamical clustering approach

<abstract> <p>Within the swiftly evolving domain of neural networks, the discrete Hopfield-SAT model, endowed with logical rules and the ability to achieve global minima of SAT problems, has emerged as a novel prototype for SAT solvers, capturing significant scientific interest. However, this model shows substantial sensitivity to network size and logical complexity. As the number of neurons and logical complexity increase, the solution space rapidly contracts, leading to a marked decline in the model's problem-solving performance. This paper introduces a novel discrete Hopfield-SAT model, enhanced by Crow search-guided fuzzy clustering hybrid optimization, effectively addressing this challenge and significantly boosting solving speed. The proposed model unveils a significant insight: its uniquely designed cost function for initial assignments introduces a quantification mechanism that measures the degree of inconsistency within its logical rules. Utilizing this for clustering, the model utilizes a Crow search-guided fuzzy clustering hybrid optimization to filter potential solutions from initial assignments, substantially narrowing the search space and enhancing retrieval efficiency. Experiments were conducted with both simulated and real datasets for 2SAT problems. The results indicate that the proposed model significantly surpasses traditional discrete Hopfield-SAT models and those enhanced by genetic-guided fuzzy clustering optimization across key performance metrics: Global minima ratio, Hamming distance, CPU time, retrieval rate of stable state, and retrieval rate of global minima, particularly showing statistically significant improvements in solving speed. These advantages play a pivotal role in advancing the discrete Hopfield-SAT model towards becoming an exemplary SAT solver. Additionally, the model features exceptional parallel computing capabilities and possesses the potential to integrate with other logical rules. In the future, this optimized model holds promise as an effective tool for solving more complex SAT problems.</p> </abstract>

Novel Optimized Implementations of Lightweight Cryptographic S-Boxes via SAT Solvers

Novel Optimized Implementations of Lightweight Cryptographic S-Boxes via SAT Solvers

Automatic Search of Differential Characteristics and Improved Differential Cryptanalysis for PRINCE, QARMA, and MANTIS

Reflection structure has a significant advantage that realizing decryption and encryption results in minimum additional costs, and many block ciphers tend to adopt such structure to achieve the requirement of low overhead. PRINCE, MANTIS, QARMA, and PRINCEv2 are lightweight block ciphers with reflection feature proposed in recent years. In this paper, we consider the automatic differential cryptanalysis of reflection block ciphers based on Boolean satisfiability (SAT) method. Since reflection block ciphers have different round functions, we extend forward and backward from the middle structure and achieve to accelerate the search of the optimal differential characteristics for such block ciphers with the Matsui’s bounding conditions. As a result, we present the optimal differential characteristics for PRINCE up to 12 rounds (full round), and they are also the optimal characteristics for PRINCEv2. We also find the optimal differential characteristics for MANTIS, QARMA‐64, and QARMA‐128 up to 10, 12, and 8 rounds, respectively. To mount an efficient differential attack on such block ciphers, we present a uniform SAT model by combining the differential characteristic searching process and the key recovery process. With this model, we find two sets of 7‐round differential characteristics for PRINCE with less guessed key bits and use them to present a multiple differential attack against 11‐round PRINCE, which improves the known single‐key attack on PRINCE by one round to our knowledge.

Quantum computing with time-travelling quantum gates

Quantum computing can solve problems that are difficult to solve in classical computing, expanding the range of problems that can be effectively computed within the allowable range of classical physical principles, and posing a challenge to the extended Church-Turing thesis in classical computing. Here, we discuss an interesting question: how to achieve more powerful computers by breaking through the limitations of physical principles, further enhancing the capabilities of quantum computers. To extend quantum computing, novel operations related to relativistic physics are a crucial candidate. Among them, the concept of closed time-like curve has aroused widespread interest, and it introduces the ability for time travel. Mathematically, quantum state along the closed time-like curve is determined through self-consistent equations, which has been demonstrated in simulations. Here, we consider a novel manipulation capability that allows quantum computing to achieve time-travelling quantum control gate. This is an intuitive extension of the graphical language of quantum circuits. Explaining quantum circuits as tensor networks, we first explain how to experimentally simulate the output of such a circuit in a system without time-travel capability. Then, we take an example to demonstrate an extended quantum algorithm that can efficiently solve SAT problems, indicating that with the involvement of time-travelling quantum gates, the computational complexity class P = NP. We also anticipate that the time-travelling quantum gates will play a facilitating role in accomplishing other quantum tasks, including achieving deterministic non-orthogonal quantum state discrimination, and quantum state cloning. Our results contribute to a more in-depth understanding of the relationship between computation and physical principles.