A method for counting models on grid Boolean formulas1

We present a novel algorithm based on combinatorial operations on lists for computing the number of models on two conjunctive normal form Boolean formulas whose restricted graph is represented by a grid graph Gm,n. We show that our algorithm is correct and its time complexity is O ( t · 1 . 618 t + 2 + t · 1 . 618 2 t + 4 ) , where t = n · m is the total number of vertices in the graph. For this class of formulas, we show that our proposal improves the asymptotic behavior of the time-complexity with respect of the current leader algorithm for counting models on two conjunctive form formulas of this kind.

CNF-SAT modelling for banyan-type networks and its application for assessing the rearrangeability

A banyan-type network is a switching network, which is constructed by placing unit switches with two inputs and two outputs in s (s > 1) stages. In each stage, 2 n – 1 (n > 1) unit switches are aligned. Past studies conjecture that this network becomes rearrangeable when s ≥ 2n-1. Although a considerable number of theoretical analyses have been done, the rearrangeability of the banyan-type network with 2n – 1 or more stages is not completely proved. As a tool to assess the rearrangeability, this study presents a CNF-SAT (conjunctive normal form - satisfiability) modelling scheme for banyan-type networks. In the proposed scheme, the routing is formulated to a SAT problem represented in CNF. By feeding the problem to a SAT solver, it is found whether the problem is satisfiable or unsatisfiable. If the problem is unsatisfiable for a certain request, the network is not rearrangeable. By contrast, if the problem is satisfiable for any requests, the network is rearrangeable. This study applies the CNF-SAT modelling scheme to various configurations of 2n – 1 stage banyan-type networks. These networks are assessed for rearrangeability by solving the SAT problems. The proposed method will be helpful to conduct future theoretical studies on banyan-type networks.

Boosting Intelligent Data Analysis in Smart Sensors by Integrating Knowledge and Machine Learning

The presented paper proposes a hybrid neural architecture that enables intelligent data analysis efficacy to be boosted in smart sensor devices, which are typically resource-constrained and application-specific. The postulated concept integrates prior knowledge with learning from examples, thus allowing sensor devices to be used for the successful execution of machine learning even when the volume of training data is highly limited, using compact underlying hardware. The proposed architecture comprises two interacting functional modules arranged in a homogeneous, multiple-layer architecture. The first module, referred to as the knowledge sub-network, implements knowledge in the Conjunctive Normal Form through a three-layer structure composed of novel types of learnable units, called L-neurons. In contrast, the second module is a fully-connected conventional three-layer, feed-forward neural network, and it is referred to as a conventional neural sub-network. We show that the proposed hybrid structure successfully combines knowledge and learning, providing high recognition performance even for very limited training datasets, while also benefiting from an abundance of data, as it occurs for purely neural structures. In addition, since the proposed L-neurons can learn (through classical backpropagation), we show that the architecture is also capable of repairing its knowledge.

Automatic Test Pattern Generation using Grover’s Algorithm

Quantum computing is an exciting new field in the intersection of computer science, physics and mathematics. It refines the central concepts from Quantum mechanics into its least difficult structures, peeling away the complications from the physical world. Any combinational circuit that has only one stuck at fault can be tested by applying a set of inputs that drive the circuit to verify the output response. The outputs of that circuit will be different from the one desired if the faults exist. This project describes a method of generating test patterns using the Boolean satisfaction method. First, the Boolean formula is constructed to express the Boolean difference between a fault-free circuit and a faulty circuit. Second, the Boolean satisfaction algorithm is applied to the formula in the previous step. The Grover algorithm is used to solve the Boolean satisfaction problem. The Boolean Satisfiability problem for Automatic Test Pattern Generation(ATPG) is implemented on IBM Quantum Experience. The Python program initially generates the boolean expression from the file and converts it into Conjunctive Normal Form(CNF) which is passed on to Grover Oracle and runs on IBM simulator and produces excellent results on combinational circuits for test pattern generation with a quadratic speedup. Grover’s Algorithm on this problem has a run time of O(√N).

Deep Graph Learning for Circuit Deobfuscation

Circuit obfuscation is a recently proposed defense mechanism to protect the intellectual property (IP) of digital integrated circuits (ICs) from reverse engineering. There have been effective schemes, such as satisfiability (SAT)-checking based attacks that can potentially decrypt obfuscated circuits, which is called deobfuscation. Deobfuscation runtime could be days or years, depending on the layouts of the obfuscated ICs. Hence, accurately pre-estimating the deobfuscation runtime within a reasonable amount of time is crucial for IC designers to optimize their defense. However, it is challenging due to (1) the complexity of graph-structured circuit; (2) the varying-size topology of obfuscated circuits; (3) requirement on efficiency for deobfuscation method. This study proposes a framework that predicts the deobfuscation runtime based on graph deep learning techniques to address the challenges mentioned above. A conjunctive normal form (CNF) bipartite graph is utilized to characterize the complexity of this SAT problem by analyzing the SAT attack method. Multi-order information of the graph matrix is designed to identify the essential features and reduce the computational cost. To overcome the difficulty in capturing the dynamic size of the CNF graph, an energy-based kernel is proposed to aggregate dynamic features into an identical vector space. Then, we designed a framework, Deep Survival Analysis with Graph (DSAG), which integrates energy-based layers and predicts runtime inspired by censored regression in survival analysis. Integrating uncensored data with censored data, the proposed model improves the standard regression significantly. DSAG is an end-to-end framework that can automatically extract the determinant features for deobfuscation runtime. Extensive experiments on benchmarks demonstrate its effectiveness and efficiency.

GRCircuit - Functional Macro Blocks Circuit Recovering Tool

Considering that the more information you can gather about a particular circuit, you can address problems more accurately in the Eletronic Design Automation (EDA) eld, therefore, many tools focus on obtaining the maximum amount of information about the input to which it is provided in order to determine which are the best algorithms to each instance. Some of these tools are the Boolean Satisfiability (SAT) problem solvers; which, for the most part, receive formulas described in Conjunctive Normal Form (CNF) as input. The circuits encoding process to the CNF format, unfortunately, destroy much of the information that could have been used to optimize SAT solvers, as part of this informations must be recovered to avoid applying generic algorithms in the solution of SAT problems. One of the difficult aspects of retrieving this information corresponds to the matching of clauses to its respective logic gates, as well as which sets of logic gates correlate to a functional block. The present work makes use of subgraph isomorphism algorithms to recover circuits encoded in CNF-DIMACS maximimizing the number of clauses handled, both at the level of logic gates as well as more complex structural blocks, which allow their identification at higher levels of abstraction. Our tool was able to successfully recover all circuits

A Structural Entropy Measurement Principle of Propositional Formulas in Conjunctive Normal Form

The satisfiability (SAT) problem is a core problem in computer science. Existing studies have shown that most industrial SAT instances can be effectively solved by modern SAT solvers while random SAT instances cannot. It is believed that the structural characteristics of different SAT formula classes are the reasons behind this difference. In this paper, we study the structural properties of propositional formulas in conjunctive normal form (CNF) by the principle of structural entropy of formulas. First, we used structural entropy to measure the complex structure of a formula and found that the difficulty solving the formula is related to the structural entropy of the formula. The smaller the compressing information of a formula, the more difficult it is to solve the formula. Secondly, we proposed a λ-approximation strategy to approximate the structural entropy of large formulas. The experimental results showed that the proposed strategy can effectively approximate the structural entropy of the original formula and that the approximation ratio is more than 92%. Finally, we analyzed the structural properties of a formula in the solution process and found that a local search solver tends to select variables in different communities to perform the next round of searches during a search and that the structural entropy of a variable affects the probability of the variable being flipped. By using these conclusions, we also proposed an initial candidate solution generation strategy for a local search for SAT, and the experimental results showed that this strategy effectively improves the performance of the solvers CCAsat and Sparrow2011 when incorporated into these two solvers.

Chapter 13. Symmetry and Satisfiability

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.

Chapter 9. Preprocessing in SAT Solving

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.