SAT-based Method Research Articles

SAT-based Decision Tree Learning for Large Data Sets

Decision trees of low depth are beneficial for understanding and interpreting the data they represent. Unfortunately, finding a decision tree of lowest complexity (depth or size) that correctly represents given data is NP-hard. Hence known algorithms either (i) utilize heuristics that do not minimize the depth or (ii) are exact but scale only to small or medium-sized instances. We propose a new hybrid approach to decision tree learning, combining heuristic and exact methods in a novel way. More specifically, we employ SAT encodings repeatedly to local parts of a decision tree provided by a standard heuristic, leading to an overall reduction in complexity. This allows us to scale the power of exact SAT-based methods to comparatively very large data sets. We evaluate our new approach experimentally on a range of real-world instances that contain up to several thousand samples. In almost all cases, our method successfully decreases the complexity of the initial decision tree; often, the decrease is significant.

Migrating Techniques from Search-based Multi-Agent Path Finding Solvers to SAT-based Approach

In the multi-agent path finding problem (MAPF) we are given a set of agents each with respective start and goal positions. The task is to find paths for all agents while avoiding collisions, aiming to minimize a given objective function. Many MAPF solvers were introduced in the past decade for optimizing two specific objective functions: sum-of-costs and makespan. Two prominent categories of solvers can be distinguished: search-based solvers and compilation-based solvers. Search-based solvers were developed and tested for the sum-of-costs objective, while the most prominent compilation-based solvers that are built around Boolean satisfiability (SAT) were designed for the makespan objective. Very little is known on the performance and relevance of solvers from the compilation-based approach on the sum-of-costs objective. In this paper, we start to close the gap between these cost functions in the compilation-based approach. Our main contribution is a new SAT-based MAPF solver called MDD-SAT, that is directly aimed to optimally solve the MAPF problem under the sum-of-costs objective function. Using both a lower bound on the sum-of-costs and an upper bound on the makespan, MDD-SAT is able to generate a reasonable number of Boolean variables in our SAT encoding. We then further improve the encoding by borrowing ideas from ICTS, a search-based solver. In addition, we show that concepts applicable in search-based solvers like ICTS and ICBS are applicable in the SAT-based approach as well. Specifically, we integrate independence detection, a generic technique for decomposing an MAPF instance into independent subproblems, into our SAT-based approach, and we design a relaxation of our optimal SAT-based solver that results in a bounded suboptimal SAT-based solver. Experimental evaluation on several domains shows that there are many scenarios where our SAT-based methods outperform state-of-the-art sum-of-costs search-based solvers, such as variants of the ICTS and ICBS algorithms.

SAT-based Decision Tree Learning for Large Data Sets

Decision trees of low depth are beneficial for understanding and interpreting the data they represent. Unfortunately, finding a decision tree of lowest depth that correctly represents given data is NP-hard. Hence known algorithms either (i) utilize heuristics that do not optimize the depth or (ii) are exact but scale only to small or medium-sized instances. We propose a new hybrid approach to decision tree learning, combining heuristic and exact methods in a novel way. More specifically, we employ SAT encodings repeatedly to local parts of a decision tree provided by a standard heuristic, leading to a global depth improvement. This allows us to scale the power of exact SAT-based methods to almost arbitrarily large data sets. We evaluate our new approach experimentally on a range of real-world instances that contain up to several thousand samples. In almost all cases, our method successfully decreases the depth of the initial decision tree; often, the decrease is significant.

An Efficient Procedure to Generate Highly Compact Diagnosis Patterns for Transition Faults

This article presents a diagnosis pattern generation procedure that not only can generate very compact diagnosis patterns to distinguish nonequivalent transition faults, but also can identify equivalent (EQ) transition faults efficiently. This procedure mainly consists of two major methods: 1) a user-defined-fault-based inactivation method that transforms the problem of distinguishing all transition faults into that of detecting a set of user-defined faults and then deals with all these faults at a time by using an ATPG tool and 2) a unified fault-pair transformation method that transforms the problem of distinguishing two transition faults into the problem of detecting a transition fault and then process all these faults also in one ATPG run. By these two methods, very compact diagnosis pattern sets can be obtained. For the very few fault pairs that cannot be handled by these two methods due to ATPG backtracking limit, we employ a SAT-based method to deal with these pairs and show that they are all EQ-fault pairs. Experimental results on ISCAS’89 and IWLS’05 benchmark circuits show that this is the first work that can distinguish all distinguishable transition faults and identify all EQ transition faults for ISCAS’89 and IWLS’05 benchmark circuits.

Complete SAT based Cryptanalysis of RC5 Cipher

Keeping the proper security level of ciphers used in communication networks is today a very important problem. Cryptanalysts ensure a constant need for improvement complexity and ciphers' security by trying to break them. Sometimes they do not instantly try to break the strongest version of the cipher, but they are looking for weaknesses by splitting it and independently checking all algorithm components. Often cryptanalysts also attempt to break cipher by using its weaker version or configuration. There are plenty of mechanisms and approaches to cryptanalysis to solve those challenges. One of them is SAT-based method, that uses logical encoding. In this article, we present our wide analysis and new experimental results of SAT-based, direct cryptanalysis of the RC5 cipher. To perform such actions on the given cipher, we initially create a propositional logical formula, that describes and represents the entire RC5 algorithm. The second step is to randomly generate key and plaintext. Then we determine the ciphertext. In the last step of our computations, we use SAT-solvers. They are particularly designed tools for checking the satisfiability of the Boolean formulas. In our research, we make cryptanalysis of RC5 cipher in the case with plaintext and ciphertext. To get the best result, we compared many SAT-solvers and choose several. Some of them were relatively old, but still very efficient and some were modern and popular.

Old techniques in new ways: Clause weighting, unit propagation and hybridization for maximum satisfiability

Maximum Satisfiability (MaxSAT) is a basic and important constraint optimization problem. When dealing with both hard and soft constraints, the MaxSAT problem is referred to as Partial MaxSAT, which has been used to effectively solve many combinatorial optimization problems in real world. The local search method and the SAT-based method are two popular methods for Partial MaxSAT. Nevertheless, local search algorithms have been dominated by SAT-based algorithms on industrial benchmarks. This work develops an effective local search algorithm for industrial benchmarks, which for the first time is competitive with SAT-based algorithms on industrial MaxSAT benchmarks.We propose a local search algorithm, called SATLike, which exploits the structure of Partial MaxSAT by a novel clause weighting scheme. Then we improve SATLike by integrating a unit propagation based decimation algorithm to generate initial solutions, leading to an improved algorithm, called SATLike 2.0. We also refine SATLike 2.0 and obtain SATLike 3.0. Furthermore, we apply SATLike 3.0 to improve two SAT-based solvers, leading to two hybrid MaxSAT solvers that push the state of the art in MaxSAT solving.Experiments on benchmarks from the MaxSAT Evaluations 2017 and 2018 show that SATLike significantly outperforms previous local search solvers on all the benchmarks. More importantly, SATLike 3.0 has better performance than state-of-the-art SAT-based solvers on unweighted industrial benchmarks. The two hybridized solvers obtain significant improvements on both unweighted and weighted benchmarks.

Deterministic finite automata learning using counterexample guided abstraction refinement

Subject of Research. The paper studies minimum-sized deterministic finite automata inferring problem. A hybrid method is developed and implemented reducing the given problem to Boolean satisfiability (SAT) technique and at the same time applying a counterexample guided abstraction refinement approach. Method. It is proposed to use not all given behavior examples as training data but start with some subset of them and build a consistent automaton, using SAT- based automata inferring method. Then the built automaton is checked against the complete set of behavior examples. The examples not consistent with the automaton are counterexamples. Some subset of counterexamples is added to the current training data and the process is being repeated. Main Results. The proposed method is implemented as a part of deterministic finite automata inferring tool in Python language. Experimental comparison of the developed method and the SAT-based method without abstraction refinement is carried out. Practical Relevance. Experimental research has shown that the developed method is reasonable for application if the number of behavior examples is large enough, at least, two hundred times exceeds the number of the automaton states, and, therefore, the Boolean formula being created contains tens and hundreds of millions of clauses.

Parallelizing SAT-based de-camouflaging attacks by circuit partitioning and conflict avoiding

As one of the most effective proactive countermeasures against reverse engineering, circuit camouflaging has emerged to be a hot research topic and it is becoming a mature technology with the development of various de-camouflaging attacks. Among them, the SAT-based method is the most powerful one to defeat circuit camouflaging. However, SAT-based attacks have scalability problem due to the complexity of the underlying SAT solvers, and straightforward approach to parallelize SAT-based attacks will fail. In this article, we propose a novel parallelization framework for SAT-based attacks, which consists of a two-level partition method (independent module partitioning and k-medoids clustering), together with a novel conflict avoidance strategy. Specifically, we first break down the camouflaged netlist into multiple independent modules that can be solved in parallel. However, any good circuit camouflaging approach should produce one or multiple large modules. To solve this problem, we further apply the k-medoids algorithm to partition the large modules into multiple “high cohesion” and “low coupling” clusters. Utilizing the relative independence of these clusters, we propose a two-stage attack method to avoid conflicts among clusters. Experimental results on OpenSparc T1 microprocessor controller demonstrate that our approach can on average reduce the scales of the SAT formulas by more than 50%, reduce the attack iteration number by 45%, and achieves on average 3.6× and maximum 10× speed up over the best-known SAT-based de-camouflaging tool.

Stable States of Boolean Regulatory Networks Composed Over Hexagonal Grids

Cellular processes are governed by complex molecular regulatory networks. To understand the dynamics emerging from these networks, a popular approach relies on a Boolean abstraction. These Boolean regulatory networks define qualitative models with discrete dynamics in which properties of interest relate to the so-called attractors and their reachability. When considering multi-cellular systems, cell-cell communication must be accounted by properly inter-connecting cellular network models. This is done through logical composition rules that define cell-cell communication, leading to a (composed) model of the regulatory control of the whole.This work focuses on Boolean models composed over hexagonal grids, which suitably represent simple epithelia. Stable states embody stable patterns of the grid, i.e., ensembles of differentiated cell types, each characterized by a specific pattern of gene expression. Identification of these stable states is a challenging problem due to the combinatorial explosion of the dimensions of the model state space and to the potentially huge number of solutions.Following the formalisation of model composition, we present a SAT-based method to identify the stable states of Boolean models composed over hexagonal grids. This approach is applied to a few prototypical case studies, illustrating counter-intuitive dependencies of the number of stable states on: the cellular model, the composition rule and the grid characteristics (i.e., dimensions and border conditions).

Analysis of SET Reconvergence and Hardening in the Combinational Circuit Using a SAT-Based Method

The reconvergence-phenomenon is common in modern circuit design. It occurs when the signal reconvenes at a certain point through multiple sensitized paths. It affects the soft error estimation and hardening. It is complicated to analyze the reconvergence-phenomenon accurately. In this paper, we propose a method based on the satisfiability problem (SAT) to analyze the single event transient (SET) reconvergence-phenomenon in combinational circuits. Our method translates the SET reconvergence issue into an SAT. An SAT solver is used to determine whether the reconvergence-phenomenon could occur and the conditions for its occurrence. According to the results of the solutions, the sensitive nodes in the circuit can be defined. The results show that compared with random hardening, the average soft error rate is reduced 20% by hardening the node selected by our method.

Scalable SMT-Based Equivalence Checking of Nested Loop Pipelining in Behavioral Synthesis

In this article, we present a novel methodology based on SMT-solvers to verify equality of a high-level described specification and a pipelined RTL implementation produced by a high-level synthesis tool. The complex transformations existing in the high-level synthesis process, such as nested loop pipelining, cause the conventional methods of equivalence checking to be inefficient. The proposed equivalence checking method simultaneously attacks the two problems in this context: (1) state space explosion and (2) complex high-level synthesis transformations. To show the scalability and efficiency of the proposed method, the verification results of large designs are compared with those of the SAT-based method, including three different state-of-the-art SAT-solvers: the SMT-based procedure, the modular Horner expansion diagram (M-HED)-based method, and the M-HED partitioning approach. The results show 2470×, 2540×, and 142× average memory usage reduction and 252×, 28×, and 914× speedup in comparison with M-HED, M-HED partitioning, and SMT-solver without using the proposed method, respectively.

Exact finite-state machine identification from scenarios and temporal properties

Finite-state models, such as finite-state machines (FSMs), aid software engineering in many ways. They are often used in formal verification and also can serve as visual software models. The latter application is associated with the problems of software synthesis and automatic derivation of software models from specification. Smaller synthesized models are more general and are easier to comprehend, yet the problem of minimum FSM identification has received little attention in previous research. This paper presents four exact methods to tackle the problem of minimum FSM identification from a set of test scenarios and a temporal specification represented in linear temporal logic. The methods are implemented as an open-source tool. Three of them are based on translations of the FSM identification problem to SAT or QSAT problem instances. Accounting for temporal properties is done via counterexample prohibition. Counterexamples are either obtained from previously identified FSMs, or based on bounded model checking. The fourth method uses backtracking. The proposed methods are evaluated on several case studies and on a larger number of randomly generated instances of increasing complexity. The results show that the Iterative SAT-based method is the leader among the proposed methods. The methods are also compared with existing inexact approaches, i.e. the ones which do not necessarily identify the minimum FSM, and these comparisons show encouraging results.

A SAT Approach to Clique-Width

Clique-width is a graph invariant that has been widely studied in combinatorics and computational logic. Computing the clique-width of a graph is an intricate problem, because the exact clique-width is not known even for very small graphs. We present a new method for computing clique-width via an encoding to propositional satisfiability (SAT), which is then evaluated by a SAT solver. Our encoding is based on a reformulation of clique-width in terms of partitions that utilizes an efficient encoding of cardinality constraints. Our SAT-based method is the first to discover the exact clique-width of various small graphs, including famous named graphs from the literature as well as random graphs of various density. With our method, we determined the smallest graphs that require a small predescribed clique-width. We further show how our method can be modified to compute the linear clique-width of graphs, a variant of clique-width that has recently received considerable attention. In an appendix, we provide certificates for tight upper bounds for the clique-width and linear clique-width of famous named graphs.

A SAT-based Method for Solving the Two-dimensional Strip Packing Problem

We propose a satisfiability testing (SAT) based exact approach for solving the two-dimensional strip packing problem (2SPP). In this problem, we are given a set of rectangles and one large rectangl...

A SAT-based Method for Solving the Two-dimensional Strip Packing Problem

We propose a satisfiability testing (SAT) based exact approach for solving the two-dimensional strip packing problem (2SPP). In this problem, we are given a set of rectangles and one large rectangle called a strip. The goal of the problem is to pack all rectangles without overlapping, into the strip by minimizing the overall height of the packing. Although the 2SPP has been studied in Operations Research, some instances are still hard to solve. Our method solves the 2SPP by translating it into a SAT problem through a SAT encoding called order encoding. The translated SAT problems tend to be large; thus, we apply several techniques to reduce the search space by symmetry breaking and positional relations of rectangles. To solve a 2SPP, that is, to compute the minimum height of a 2SPP, we need to repeatedly solve similar SAT problems. We thus reuse learned clauses and assumptions from the previously solved SAT problems. To evaluate our approach, we obtained results for 38 instances from the literature and made comparisons with a constraint satisfaction solver and an ad-hoc 2SPP solver.

Haplotype inference with pseudo-Boolean optimization

The fast development of sequencing techniques in the recent past has required an urgent development of efficient and accurate haplotype inference tools. Besides being a crucial issue in genetics, haplotype inference is also a challenging computational problem. Among others, pure parsimony is a viable modeling approach to solve the problem of haplotype inference and also an interesting NP-hard problem in itself. Recently, the introduction of SAT-based methods, including pseudo-Boolean optimization (PBO) methods, has produced very efficient solvers. This paper provides a detailed description of RPoly, a PBO approach for the haplotype inference by pure parsimony (HIPP) problem. Moreover, an extensive evaluation of existent HIPP solvers, on a comprehensive set of instances, confirms that RPoly is currently the most efficient and robust HIPP approach.

Editor’s Introduction to the Special Volume on Application of Constraints to Formal Verification

Editor’s Introduction to the Special Volume on Application of Constraints to Formal Verification

Boosting Haplotype Inference with Local Search

A very challenging problem in the genetics domain is to infer haplotypes from genotypes. This process is expected to identify genes affecting health, disease and response to drugs. One of the approaches to haplotype inference aims to minimise the number of different haplotypes used, and is known as haplotype inference by pure parsimony (HIPP). The HIPP problem is computationally difficult, being NP-hard. Recently, a SAT-based method (SHIPs) has been proposed to solve the HIPP problem. This method iteratively considers an increasing number of haplotypes, starting from an initial lower bound. Hence, one important aspect of SHIPs is the lower bounding procedure, which reduces the number of iterations of the basic algorithm, and also indirectly simplifies the resulting SAT model. This paper describes the use of local search to improve existing lower bounding procedures. The new lower bounding procedure is guaranteed to be as tight as the existing procedures. In practice the new procedure is in most cases considerably tighter, allowing significant improvement of performance on challenging problem instances.

Search techniques for SAT-based Boolean optimization

The last few years have seen significant advances in Boolean satisfiability (SAT) solving. This has lead to the successful deployment of SAT solvers in a wide range of problems in Engineering and Computer Science. In general, most SAT solvers are applied to Boolean decision problems that are expressed in conjunctive normal form (CNF). While this input format is applicable to some engineering tasks, it poses a significant obstacle to others. One of the main advances in SAT is generalizing SAT solvers to handle stronger representation of constraints. Specifically, pseudo-Boolean (PB) constraints which are efficient in representing counting constraints and can replace an exponential number of CNF constraints. Another significant advantage of PB constraints is its ability to express Boolean optimization problems. This allows for new applications that were never handled by SAT solvers before. In this paper, we describe two methods to solve Boolean optimization problems using SAT solvers. Both methods are implemented and evaluated using the SAT solver PBS. We develop an adaptive flow that analyzes a given Boolean optimization problem and selects the solving method that best fits the problem characteristics. Empirical evidence on a variety of benchmarks shows that both methods are competitive. The results also show that SAT-based methods are very competitive with generic integer linear programming (ILP) solvers.

SAT-Based Methods for Sequential Hardware Equivalence Verification without Synchronization

The BDD- and SAT-based model checking and verification methods normally require an initial state. Here we are concerned with sequential hardware verification, where an initial state must be one of the reset states. In practice, a reset state is not always given by the designer, and computing a reset state of a circuit is a hard problem. In this paper we propose a method allowing usage of SAT-based verification methods without a need for a user-given or a computed initial state. The idea is to employ a binary encoding of 3-valued modeling of circuits, and use the undefined state X as a reset state.