Encoding deductive argumentation in quantified Boolean formulae

There are a number of frameworks for modelling argumentation in logic. They incorporate a formal representation of individual arguments and techniques for comparing conflicting arguments. A common assumption for logic-based argumentation is that an argument is a pair 〈Φ,α〉 where Φ is minimal subset of the knowledge-base such that Φ is consistent and Φ entails the claim α. Different logics provide different definitions for consistency and entailment and hence give us different options for argumentation. Classical propositional logic is an appealing option for argumentation but the computational viability of generating an argument is an issue. To better explore this issue, we use quantified Boolean formulae to characterise an approach to argumentation based on classical logic.     

Backdoor Sets of Quantified Boolean Formulas

We generalize the notion of backdoor sets from propositional formulas to quantified Boolean formulas (QBF). This allows us to obtain hierarchies of tractable classes of quantified Boolean formulas with the classes of quantified Horn and quantified 2CNF formulas, respectively, at their first level, thus gradually generalizing these two important tractable classes. In contrast to known tractable classes based on bounded treewidth, the number of quantifier alternations of our classes is unbounded. As a side product of our considerations we develop a theory of variable dependency which is of independent interest.     

Boolean Functions as Models for Quantified Boolean Formulas

In this paper, we introduce the notion of models for quantified Boolean formulas. For various classes of quantified Boolean formulas and various classes of Boolean functions, we investigate the problem of determining whether a model exists. Furthermore, we show for these classes the complexity of the model checking problem, which is to check whether a given set of Boolean functions is a model for a formula. For classes of Boolean functions, we establish some characterizations in terms of classes of quantified Boolean formulas that have such a model. This research has been supported in part by the Air Force Office of Scientific Research under grant FA9550-06-1-0050. This research has been supported in part by the NSFC under grants 60573011 and 10410638.     

Bounded-width QBF is PSPACE-complete

Tree-width and path-width are two well-studied parameters of structures that measure their similarity to a tree and a path, respectively. We show that QBF on instances with constant path-width, and hence constant tree-width, remains PSPACE-complete. This answers a question by Vardi. We also show that on instances with constant path-width and a very slow-growing number of quantifier alternations (roughly inverse-Ackermann many in the number of variables), the problem remains NP-hard. Additionally, we introduce a family of formulas with bounded tree-width that do have short refutations in Q-resolution, the natural generalization of resolution for quantified Boolean formulas.     

基于量化布尔公式的条件规划编码方式

介绍条件规划问题及其相关的求解系统,着重分析以逻辑为基拙的编码方式。针对基于量化布尔公式的转换方法进行详细分析,给出3种不同形式的量化布尔公式编码。最后,对这3种编码进行比较,分析基于命题逻辑公式与量化布尔公式这两种不同转换方式的优劣,讨论基于量化布尔公式的规划方法未来的研究方向和发展趋势。     

An Algorithm to Evaluate Quantified Boolean Formulae and Its Experimental Evaluation

The high computational complexity of advanced reasoning tasks such as reasoning about knowledge and planning calls for efficient and reliable algorithms for reasoning problems harder than NP. In this paper we propose Evaluate, an algorithm for evaluating quantified Boolean formulae (QBFs). Algorithms for evaluation of QBFs are suitable for experimental analysis of problems that belong to a wide range of complexity classes, a property not easily found in other formalisms. Evaluate is a generalization of the Davis–Putnam procedure for SAT and is guaranteed to work in polynomial space. Before presenting the algorithm, we discuss several abstract properties of QBFs that we singled out to make it more efficient. We also discuss various options that were investigated about heuristics and data structures and report the main results of the experimental analysis. In particular, Evaluate is orders of magnitude more efficient than a nested backtracking procedure that resorts to a Davis–Putnam algorithm for handling the innermost set of quantifiers. Moreover, experiments show that randomly generated QBFs exhibit regular patterns such as phase transition and easy-hard-easy distribution.     

A self-adaptive differential evolution algorithm for binary CSPs

A novel self-adaptive differential evolution (SADE) algorithm is proposed in this paper. SADE adjusts the mutation rate F and the crossover rate CR adaptively, taking account of the different distribution of population. In order to balance an individual’s exploration and exploitation capability for different evolving phases, F and CR are equal to two different self-adjusted nonlinear functions. Attention is concentrated on varying F and CR dynamically with each generation evolution. SADE maintains the diversity of population and improves the global convergence ability. It also improves the efficiency and success rate and avoids the premature convergence. Simulation and comparisons based on test-sets of CSPs demonstrate the effectiveness, efficiency and robustness of the proposed algorithm.     

Value ordering for quantified CSPs

We investigate the use of value ordering in backtracking search for Quantified Constraint Satisfaction problems (QCSPs). We consider two approaches for ordering heuristics. The first approach is solution-focused and is inspired by adversarial search: on existential variables we prefer values that maximise the chances of leading to a solution, while on universal variables we prefer values that minimise those chances. The second approach is verification-focused, where we prefer values that are easier to verify whether or not they lead to a solution. In particular, we give instantiations of this approach using QCSP-Solve’s pure-value rule Gent et al. (QCSP-solve: A solver for quantified constraint satisfaction problems. In Proceedings of IJCAI, pp. 138–143, 2005). We show that on dense 3-block problems, using QCSP-Solve, the solution-focused adversarial heuristics are up to 50% faster than lexicographic ordering, while on sparse loose interleaved problems, the verification-focused pure-value heuristics are up to 50% faster. Both types are up to 50% faster on dense interleaved problems, with one pure-value heuristic approaching an order of magnitude improvement. This work was supported in part by Microsoft Research through the European PhD Scholarship Programme, and by the Embark Initiative of the Irish Research Council for Science, Engineering and Technology (IRCSET).     

Learning a subclass of k-quasi-Horn formulas with membership queries

Boolean formulas have been widely studied in the field of learning theory. We focus on the model of learning with queries, and study a restriction of the class of k-quasi-Horn formulas, that is, conjunctive normal form formulas where the number of unnegated literals per clause is at most k. This class is known to be as hard to learn as the general class CNF of conjunctive normal form formulas. By imposing some constraints, we define a fragment of this logic that can be learned using only membership queries. Also we prove that none of these constraints makes by itself the class learnable.     

Low-level dichotomy for quantified constraint satisfaction problems

Building on a result of Larose and Tesson for constraint satisfaction problems (CSPs), we uncover a dichotomy for the quantified constraint satisfaction problem QCSP(B), where B is a finite structure that is a core. Specifically, such problems are either in ALogtime or are L-hard. This involves demonstrating that if CSP(B) is first-order expressible, and B is a core, then QCSP(B) is in ALogtime.We show that the class of B such that CSP(B) is first-order expressible (indeed trivial) is a microcosm for all QCSPs. Specifically, for any B there exists a C — generally not a core — such that CSP(C) is trivial, yet QCSP(B) and QCSP(C) are equivalent under logspace reductions.     

一种基于布尔代数的秘密共享方案

秘密共享方案是在n个参与者之间共享秘密k的方法。将布尔代数中的与或逻辑引入秘密共享而提出的新方案运算速度快,并具有灵活的自适应能力和良好的可扩展性,便于软件编程和硬件固化,可以与经典的加密方法紧密结合,以提高其安全性。     

A Dichotomy Theorem for Learning Quantified Boolean Formulas

We consider the following classes of quantified boolean formulas. Fix a finite set of basic boolean functions. Take conjunctions of these basic functions applied to variables and constants in arbitrary ways. Finally quantify existentially or universally some of the variables. We prove the following dichotomy theorem: For any set of basic boolean functions, the resulting set of formulas is either polynomially learnable from equivalence queries alone or else it is not PAC-predictable even with membership queries under cryptographic assumptions. Furthermore, we identify precisely which sets of basic functions are in which of the two cases.     

Parallel FFT-based Poisson solver for isolated three-dimensional systems

We describe an implementation to solve Poisson?s equation for an isolated system on a unigrid mesh using FFTs. The method solves the equation globally on mesh blocks distributed across multiple processes on a distributed-memory parallel computer. Test results to demonstrate the convergence and scaling properties of the implementation are presented. The solver is offered to interested users as the library PSPFFT.

Program summary

Program title: PSPFFTCatalogue identifier: AEJK_v1_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AEJK_v1_0.htmlProgram obtainable from: CPC Program Library, Queen?s University, Belfast, N. IrelandLicensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.htmlNo. of lines in distributed program, including test data, etc.: 110 243No. of bytes in distributed program, including test data, etc.: 16 332 181Distribution format: tar.gzProgramming language: Fortran 95Computer: Any architecture with a Fortran 95 compiler, distributed memory clustersOperating system: Linux, UnixHas the code been vectorized or parallelized?: Yes, using MPI. An arbitrary number of processors may be used (subject to some constraints). The program has been tested on from 1 up to ∼ 13 000 processors. RAM: Depends on the problem size, approximately 170 MBytes for 48³ cells per process.Classification: 4.3, 6.5External routines: MPI (http://www.mcs.anl.gov/mpi/), FFTW (http://www.fftw.org), Silo (https://wci.llnl.gov/codes/silo/) (only necessary for running test problem).Nature of problem: Solving Poisson?s equation globally on unigrid mesh distributed across multiple processes on distributed memory system.Solution method: Numerical solution using multidimensional discrete Fourier Transform in a parallel Fortran 95 code.Unusual features: This code can be compiled as a library to be readily linked and used as a blackbox Poisson solver with other codes.Running time: Depends on the size of the problem, but typically less than 1 second per solve.     

State feedback stabilization for probabilistic Boolean networks

A probabilistic Boolean network (PBN) is a discrete-time system composed of a family of Boolean networks (BNs) between which the PBN switches in a stochastic fashion. Studying control-related problems in PBNs may provide new insights into the intrinsic control in biological systems and enable us to develop strategies for manipulating complex biological systems using exogenous inputs. This paper investigates the problem of state feedback stabilization for PBNs. Based on the algebraic representation of logic functions, a necessary and sufficient condition is derived for the existence of a globally stabilizing state feedback controller, and a control design method is proposed when the presented condition holds. It is shown that the controller designed via the proposed procedure can simultaneously stabilize a collection of PBNs that are composed of the same constituent BNs.     

A parallel direct solver for the self-adaptive hp Finite Element Method

In this paper we present a new parallel multi-frontal direct solver, dedicated for the hp Finite Element Method (hp-FEM). The self-adaptive hp-FEM generates in a fully automatic mode, a sequence of hp-meshes delivering exponential convergence of the error with respect to the number of degrees of freedom (d.o.f.) as well as the CPU time, by performing a sequence of hp refinements starting from an arbitrary initial mesh. The solver constructs an initial elimination tree for an arbitrary initial mesh, and expands the elimination tree each time the mesh is refined. This allows us to keep track of the order of elimination for the solver. The solver also minimizes the memory usage, by de-allocating partial LU factorizations computed during the elimination stage of the solver, and recomputes them for the backward substitution stage, by utilizing only about 10% of the computational time necessary for the original computations. The solver has been tested on 3D Direct Current (DC) borehole resistivity measurement simulations problems. We measure the execution time and memory usage of the solver over a large regular mesh with 1.5 million degrees of freedom as well as on the highly non-regular mesh, generated by the self-adaptive hp$hp$-FEM, with finite elements of various sizes and polynomial orders of approximation varying from p=1$p=1$ to p=9$p=9$. From the presented experiments it follows that the parallel solver scales well up to the maximum number of utilized processors. The limit for the solver scalability is the maximum sequential part of the algorithm: the computations of the partial LU factorizations over the longest path, coming from the root of the elimination tree down to the deepest leaf.     

A two-level self-adaptive variable neighborhood search algorithm for the prize-collecting vehicle routing problem

This paper investigates the prize-collecting vehicle routing problem (PCVRP), which has a strong background in practical industries. In the PCVRP, the capacities of all available vehicles are not sufficient to satisfy the demands of all customers. Consequently it is not a compulsory requirement that all customers should be visited. However, a prize can be collected once a customer is visited. In addition, it is required that the total demands of visited customers should reach a pre-specified value at least. The objective is to establish a schedule of vehicle routes so as to minimize the total transportation cost and at the same time maximize the prize collected by all vehicles. The total transportation cost consists of the total distance of vehicle routes and the sum of vehicles used in the schedule. To solve the PCVRP, a two-level self-adaptive variable neighborhood search (TLSAVNS) algorithm is developed according to the two levels of decisions in the PCVRP, namely the selection of customers to visit and the visiting sequence of selected customers in each vehicle route. The proposed TLSAVNS algorithm is self-adaptive because the neighborhoods and their search sequence are determined automatically by the algorithm itself based on the analysis of its search history. In addition, a graph extension method is adopted to obtain the lower bound for PCVRP by transforming the proposed mixed integer programming model of PCVRP into an equivalent traveling salesman problem (TSP) model, and the obtained lower bound is used to evaluate the proposed TLSAVNS algorithm. Computational results on benchmark problems show that the proposed TLSAVNS algorithm is efficient for PCVRP.