A Genetic-Based Approach for Satisfiability Problems

This dissertation is to illustrate the current state of the art of DNA computing achievements, especially of new approaches to solve theoretical 3-SAT problems and the hitting-set problem. Beginning with Adleman’s breakthrough which is an molecular algorithm for the solution of a NP-complete, combinatorial problem, the directed Hamiltonian path problem (HPP). Today, many researchers all over the world concentrate on proposing new methods to solve engineering or application problems with a DNA computing approach. Satisfiability problem is given a Boolean formula, and decide if a satisfying truth assignment exists. ( ) ( ) … ( ) ( ) is an example of Boolean formula. k-SAT problem means that each clause has exactly k literals. Not-All-Equal (NAE) 3-SAT problem and One-In-Three (1IN3) 3-SAT problem are both NP-complete problems. In this dissertation, we present molecular solutions to find all true assignments (3-SAT problem) and furthermore find Not-All-Equal (NAE) solutions and One-In-Three (1IN3) solutions in DNA-based Supercomputing. Hitting-set problem assume that there exists a collection C of subsets of a finite set S, and a positive integer K  |S|, and we need to know if there is a subset  S with | |  K such that contains at least one element of each subset in C. In other words, is the subset that intersects every subset in C and is called the hitting-set. In this dissertation, a DNA-based algorithm is proposed to solve the small hitting-set problem. A small hitting-set is a hitting-set with the smallest K value, i.e., the hitting-set with the smallest number of elements. Furthermore, an algorithm is introduced to find the number of ones from 2n combinations and minimum numbers of ones represents the small hitting-set since K is expected to be as small as possible. The complexity of all the presented DNA-based algorithms is also discussed. We describe time complexity and volume complexity of three Algorithms, numbers of test tube used and the longest library strand in solution space of all three Algorithms. Finally, the simulated experiment is applied to verify correctness of the proposed DNA-based algorithm for solving the One-In-Three (1IN3) 3-SAT problem, and simulation of Not-All-Equal (NAE) 3-SAT problem is similar. Also, another simulated experiment is applied to our proposed DNA-based algorithm 6-2, in order to solve the well-known hitting-set problem. This research has been motivated by the benefit and the application of DNA computing and gives new methods to solve two 3-SAT problems and the hitting-set problem which are NP-complete.

Propositional Satisfiability (SAT) is a keystone in the history of computer science. SAT was the first problem shown to be NP-complete in 1971 by Stephen Cook. Having passed more than 40 years from then, SAT is now a lively research field where theory and practice have a natural intermixing. In this talk, we overview the use of SAT in practical domains, where SAT is thought in a broad sense, i.e. including SAT extensions such as Maximum Satisfiability (MaxSAT), Pseudo-Boolean Optimization (PBO) and Quantified Boolean Formulas (QBF).

In this article we show the NP-completeness of some simple number-theoretic problems. Natural simplifications of these problems invariably are known to be in P. Our research was motivated by the question whether one could study non-deterministic computation without loss of generality on a restricted, number theoretically significant class of nondeterministic Turing machines, the nondeterministic diophantine machines defined below [1,2]. The results suggest this is true.Because of the relative difficulty of the reduction of the satisfiability problem used in the proof, and the distinctly number-theoretic character of the problems shown to be NP-complete, we hope that the NP-completeness of these problems will play a role in showing the NP-completeness of further problems of a numerical nature, much as the satisfiability problem has in showing the NP-completeness of combinatorial problems ([3],[5]).The results illustrate an intimate connection between problems in computational theory, such as “P=NP?”, and problems in number theory, e.g. about quadratic congruences in one unknown, which are just beyond the range in which efficient algorithms exist. Thus our work exposes an interface between the state-of-the-art in number theory and in the theory of computation. Also, our results can be seen as the solution to a natural and important revision of Hilbert's 10thproblem: Give a feasible algorithmic procedure to decide whether an arbitrary diophantine equation has solutions. Unless P=NP this is impossible, even for a class of quadratic diophantine equations in two unknowns for which a decision procedure in the original sense is in fact available. Certainly, these results present a striking rigorous demonstration that number theorists' intuitions about where problems about diophantine equations and quadratic congruences start to be truly difficult are justified.

Propositional satisfiability (SAT) is a fundamental problem of immense practical importance. While SAT is NP-complete when clauses can contain 3 literals or more, the problem can be solved in linear time when the given formula contains only binary clauses (2SAT). Many complete search algorithms for SAT solving have taken advantage of 2SAT information that occurs in the statement of the problem in order to simplify the solving process, only one that we are aware of uses 2SAT information that arises in the process of the search, as clauses are simplified. There are a number of possibilities for making use of 2SAT information to improve the SAT solving process: maintaining 2SAT satisfiability during search, detecting unit consequences of the 2SAT clauses, and Krom subsumption using 2SAT clauses. In this paper we investigate the tradeoffs of increasing complex 2SAT handling versus the search space reduction and execution time. We give experimental results illustrating that the SAT solver resulting from the best tradeoff is competitive with state of the art Davis-Putnam methods, on hard problems involving a substantial 2SAT component.

Treewidth measures the tree-likeness of structures. Many NP-complete problems, e.g., propositional satisfiability, are tractable on bounded-treewidth structures. In this work, we study the impact of treewidth bounds on QBF, a canonical PSPACE-complete problem. This problem is known to be fixed-parameter tractable if both the treewidth and alternation depth are taken as parameters. We show here that the function bounding the complexity in the parameters is provably nonelementary (assuming P is different than NP). This yields a strict hierarchy of fixed-parameter tractability inside PSPACE. As a tool for proving this result, we first prove a similar hierarchy for model checking QPTL, quantified propositional temporal logic. Finally, we show that QBF, restricted to instances with a slowly increasing (log) treewidth, is still PSPACE-complete.

Tree-width is a well-studied parameter of structures that measures their similarity to a tree. Many important NP-complete problems, such as Boolean satisfiability (SAT), are tractable on bounded tree-width instances. In this paper we focus on the canonical PSPACE-complete problem QBF, the fully-quantified version of SAT. It was shown by Pan and Vardi [LICS 2006] that this problem is PSPACE-complete even for formulas whose tree-width grows extremely slowly. Vardi also posed the question of whether the problem is tractable when restricted to instances of bounded tree-width. We answer this question by showing that QBF on instances with constant tree-width is PSPACE-complete.

The exponential-time hypothesis (ETH) states that 3-SAT is not solvable in subexponential time, i.e. not solvable in O(c^n) time for arbitrary c > 1, where n denotes the number of variables. Problems like k-SAT can be viewed as special cases of the constraint satisfaction problem (CSP), which is the problem of determining whether a set of constraints is satisfiable. In this paper we study the worst-case time complexity of NP-complete CSPs. Our main interest is in the CSP problem parameterized by a constraint language Gamma (CSP(Gamma)), and how the choice of Gamma affects the time complexity. It is believed that CSP(Gamma) is either tractable or NP-complete, and the algebraic CSP dichotomy conjecture gives a sharp delineation of these two classes based on algebraic properties of constraint languages. Under this conjecture and the ETH, we first rule out the existence of subexponential algorithms for finite domain NP-complete CSP(Gamma) problems. This result also extends to certain infinite-domain CSPs and structurally restricted CSP(Gamma) problems. We then begin a study of the complexity of NP-complete CSPs where one is allowed to arbitrarily restrict the values of individual variables, which is a very well-studied subclass of CSPs. For such CSPs with finite domain D, we identify a relation SD such that (1) CSP({SD}) is NP-complete and (2) if CSP(Gamma) over D is NP-complete and solvable in O(c^n) time, then CSP({SD}) is solvable in O(c^n) time, too. Hence, the time complexity of CSP({SD}) is a lower bound for all CSPs of this particular kind. We also prove that the complexity of CSP({SD}) is decreasing when |D| increases, unless the ETH is false. This implies, for instance, that for every c>1 there exists a finite-domain Gamma such that CSP(Gamma) is NP complete and solvable in O(c^n) time.

Synthesis from examples enables non-expert users to generate programs by specifying examples of their behavior. A domain-specific form of such synthesis has been recently deployed in a widely used spreadsheet software product. In this paper we contribute to foundations of such techniques and present a complete algorithm for synthesis of a class of recursive functions defined by structural recursion over a given algebraic data type definition. The functions we consider map an algebraic data type to a string; they are useful for, e.g., pretty printing and serialization of programs and data. We formalize our problem as learning deterministic sequential top-down tree-to-string transducers with a single state (1STS). The first problem we consider is learning a tree-to-string transducer from any set of input/output examples provided by the user. We show that, given a set of input/output examples, checking whether there exists a 1STS consistent with these examples is NP-complete in general. In contrast, the problem can be solved in polynomial time under a (practically useful) closure condition that each subtree of a tree in the input/output example set is also part of the input/output examples. Because coming up with relevant input/output examples may be difficult for the user while creating hard constraint problems for the synthesizer, we also study a more automated active learning scenario in which the algorithm chooses the inputs for which the user provides the outputs. Our algorithm asks a worst-case linear number of queries as a function of the size of the algebraic data type definition to determine a unique transducer. To construct our algorithms we present two new results on formal languages. First, we define a class of word equations, called sequential word equations, for which we prove that satisfiability can be solved in deterministic polynomial time. This is in contrast to the general word equations for which the best known complexity upper bound is in linear space. Second, we close a long-standing open problem about the asymptotic size of test sets for context-free languages. A test set of a language of words L is a subset T of L such that any two word homomorphisms equivalent on T are also equivalent on L. We prove that it is possible to build test sets of cubic size for context-free languages, matching for the first time the lower bound found 20 years ago.

The Boolean Satisfiability Problem (SAT) is a well known NP-Complete problem. While its complexity remains a source of many interesting questions for theoretical computer scientists, the problem has found many practical applications in recent years. The emergence of efficient SAT solvers which can handle large structured SAT instances has enabled the use of SAT solvers in diverse domains such as electronic design automation and artificial intelligence. These applications continue to motivate the development of faster and more robust SAT solvers. In this paper, we describe the popular SAT solver zchaff with a focus on recent developments.

This study is concerned with the Boolean satisfiability (SAT) problem and its solution in setting a hybrid computational intelligence environment of genetic and fuzzy computing. In this framework, fuzzy sets realize an embedding principle meaning that original two-valued (Boolean) functions under investigation are extended to their continuous counterparts resulting in the form of fuzzy (multivalued) functions. In the sequel, the SAT problem is reformulated for the fuzzy functions and solved using a genetic algorithm (GA). It is shown that a GA, especially its recursive version, is an efficient tool for handling multivariable SAT problems. Thorough experiments revealed that the recursive version of the GA can solve SAT problems with more than 1000 variables.

Quantum annealing device has shown a great potential in solving discrete problems that are theoretically and empirically hard. Boolean Satisfiability (SAT) problem, determining if there is an assignment of variables that satisfies a given Boolean function, is the first proven NP-complete problem widely used in various domains. Here, we present a novel mapping of the SAT problem to the quadratic unconstrained binary optimization problem (QUBO), and further develop a tool flow embedding the proposed QUBO to the architecture of the commercialized quantum computer D-Wave. By leveraging electronic design automation techniques including synthesis, placement and routing, this is not only the first work providing the detail flow that embeds the QUBO, but also a technique scalable for real world applications and some hard SAT problems with over 6000 variables in QUBO. Based on our results, we discuss the challenges in solving SAT using the current generation of annealing device, and explore the problem solving capability of future quantum annealing computers.

There is a decision problem from the discipline of Boolean logic which has assumed a significant position in the theoretical study of computer science. Indeed, it is the paradigm of an entire class of challenging problems, grouped together and given the name NP-Complete. This problem is usually referred to at the Boolean Satisfiability Problem or SAT for short. This paper presents an interesting method which shows some promise in the analysis of the Satisfiability Problem and attempts to characterize the instances of the problem to which the method can be applied in polynomial time. For a more detailed discussion of the theoretical importance of the Satisfiability Problem and its relation to the class called NP-Complete, the reader should consult references [1,2].

Today, resource waste is considered as one of the most important challenges in different industries. In this regard, the Rectangle Packing Problem (RPP) can affect noticeably both time and design issues in businesses. In this study, the main objective is to create a set of non-overlapping rectangles so that they have specific dimensions within a rectangular plate with a specified width and an unlimited height. The ensued challenge is an NP-complete problem. NP-complete problem, any of a class of computational problems that still there are no efficient solution for them. Most substantial computer-science problems such as the traveling salesman problem, satisfiability problems (sometimes called propositional satisfiability problem and abbreviated SAT or B-SAT), and graph-covering problems are belong to this class. Essentially, it is complicated to spot the best arrangement with the highest rate of resource utilization by emphasizing the linear computation time. This study introduces a time-efficient and exploratory algorithm for the RPP, including the lowest front-line strategy and a Best-Fit algorithm. The obtained results confirmed that the proposed algorithm can lead to a good performance with simplicity and time efficiency. Our evaluation shows that the proposed model with utilization rate about 94.37% outperforms others with 87.75%, 50.54%, and 87.17% utilization rate, respectively. Consequently, the proposed method is capable to of achieving much better utilization rate in comparison with other mentioned algorithms in just 0.023 s running-time, which is much faster than others.

The author transforms various NP-complete problems in layout, namely, two- and multilayer dogleg channel routing, two-way partitioning, one-dimensional and two-dimensional placement, into Boolean satisfiability problems. The transformations are efficient in that the number of inputs to the Boolean function, for which he has to find a satisfying assignment, only grows linearly or quasilinearly with the layout problem size. These transformations also produce a minimal-size Boolean function in order to speed up satisfiability check performance. >

Boolean Satisfiability (SAT) is a classical NP-complete problem with both theoretical and practical interests. This paper presents our work in developing an application-specific processor for SAT based on a commercial configurable processor core. We customize the processor configuration and design new instruction extensions based on the data structure and atomic operations used in SAT. The customized processor has achieved around 24 /spl times/ speedup at a very low hardware cost. The small size of the processor makes it possible to integrate multiple processors and other customized logic into a single chip for an application-specific multiprocessor solution for SAT. Our work shows the strength of application-specific processing in accelerating applications with complex control and dynamic data structures - an area that has traditionally not been targeted by application-specific processing. It also demonstrates that configurable processor cores can be used to cut the development time and cost for designing and building such application-specific processors.