Probability logic and optimization SAT: The PSAT and CPA models

Both probabilistic satisfiability (PSAT) and the check of coherence of probability assessment (CPA) can be considered as probabilistic counterparts of the classical propositional satisfiability problem (SAT). Actually, CPA turns out to be a particular case of PSAT; in this paper, we compare the computational complexity of these two problems for some classes of instances. First, we point out the relations between these probabilistic problems and two well known optimization counterparts of SAT, namely Max SAT and Min SAT. We then prove that Max SAT with unrestricted weights is NP-hard for the class of graph formulas, where Min SAT can be solved in polynomial time. In light of the aforementioned relations, we conclude that PSAT is NP-complete for ideal formulas, where CPA can be solved in linear time.     

Easy Cases of Probabilistic Satisfiability

The Probabilistic Satisfiability problem (PSAT) can be considered as a probabilistic counterpart of the classical SAT problem. In a PSAT instance, each clause in a CNF formula is assigned a probability of being true; the problem consists in checking the consistency of the assigned probabilities. Actually, PSAT turns out to be computationally much harder than SAT, e.g., it remains difficult for some classes of formulas where SAT can be solved in polynomial time. A column generation approach has been proposed in the literature, where the pricing sub-problem reduces to a Weighted Max-SAT problem on the original formula. Here we consider some easy cases of PSAT, where it is possible to give a compact representation of the set of consistent probability assignments. We follow two different approaches, based on two different representations of CNF formulas. First we consider a representation based on directed hypergraphs. By extending a well-known integer programming formulation of SAT and Max-SAT, we solve the case in which the hypergraph does not contain cycles; a linear time algorithm is provided for this case. Then we consider the co-occurrence graph associated with a formula. We provide a solution method for the case in which the co-occurrence graph is a partial 2-tree, and we show how to extend this result to partial k-trees with k>2.     

Modelling and solving temporal reasoning as propositional satisfiability

Representing and reasoning about time dependent information is a key research issue in many areas of computer science and artificial intelligence. One of the best known and widely used formalisms for representing interval-based qualitative temporal information is Allen's interval algebra (IA). The fundamental reasoning task in IA is to find a scenario that is consistent with the given information. This problem is in general NP-complete.In this paper, we investigate how an interval-based representation, or IA network, can be encoded into a propositional formula of Boolean variables and/or predicates in decidable theories. Our task is to discover whether satisfying such a formula can be more efficient than finding a consistent scenario for the original problem. There are two basic approaches to modelling an IA network: one represents the relations between intervals as variables and the other represents the end-points of each interval as variables. By combining these two approaches with three different Boolean satisfiability (SAT) encoding schemes, we produced six encoding schemes for converting IA to SAT. In addition, we also showed how IA networks can be formulated into satisfiability modulo theories (SMT) formulae based on the quantifier-free integer difference logic (QF-IDL). These encodings were empirically studied using randomly generated IA problems of sizes ranging from 20 to 100 nodes. A general conclusion we draw from these experimental results is that encoding IA into SAT produces better results than existing approaches. More specifically, we show that the new point-based 1-D support SAT encoding of IA produces consistently better results than the other alternatives considered. In comparison with the six different SAT encodings, the SMT encoding came fourth after the point-based and interval-based 1-D support schemes and the point-based direct scheme. Further, we observe that the phase transition region maps directly from the IA encoding to each SAT or SMT encoding, but, surprisingly, the location of the hard region varies according to the encoding scheme. Our results also show a fixed performance ranking order over the various encoding schemes.     

一类可分离SAT问题的O(1.890n)精确算法

布尔可满足性问题（SAT）是指对于给定的布尔公式,是否存在一个可满足的真值指派.这是第1个被证明的NP完全问题,一般认为不存在多项式时间算法,除非P=NP.学者们大都研究了子句长度不超过k的SAT问题（k-SAT）,从全局搜索到局部搜索,给出了大量的相对有效算法,包括随机算法和确定算法.目前,最好算法的时间复杂度不超过O（（2-2/k）ⁿ）,当k=3时,最好算法时间复杂度为O（1.308ⁿ）.而对于更一般的与子句长度k无关的SAT问题,很少有文献涉及.引入了一类可分离SAT问题,即3-正则可分离可满足性问题（3-RSSAT）,证明了3-RSSAT是NP完全问题,给出了一般SAT问题3-正则可分离性的O（1.890ⁿ）判定算法.然后,利用矩阵相乘算法的研究成果,给出了3-RSSAT问题的O（1.890ⁿ）精确算法,该算法与子句长度无关.     

Reducing the size of resolution proofs in linear time

DPLL-based SAT solvers progress by implicitly applying binary resolution. The resolution proofs that they generate are used, after the SAT solver’s run has terminated, for various purposes. Most notable uses in formal verification are: extracting an unsatisfiable core, extracting an interpolant, and detecting clauses that can be reused in an incremental satisfiability setting (the latter uses the proof only implicitly, during the run of the SAT solver). Making the resolution proof smaller can benefit all of these goals: it can lead to smaller cores, smaller interpolants, and smaller clauses that are propagated to the next SAT instance in an incremental setting. We suggest two methods that are linear in the size of the proof for doing so. Our first technique, called Recycle-Units uses each learned constant (unit clause) (x) for simplifying resolution steps in which x was the pivot, prior to when it was learned. Our second technique, called   simplifies proofs in which there are several nodes in the resolution graph, one of which dominates the others, that correspond to the same pivot. Our experiments with industrial instances show that these simplifications reduce the core by ≈5% and the proof by ≈13%. It reduces the core less than competing methods such as run- till- fix, but whereas our algorithms are linear in the size of the proof, the latter and other competing techniques are all exponential as they are based on SAT runs. If we consider the size of the proof (the resolution graph) as being polynomial in the number of variables (it is not necessarily the case in general), this gives our method an exponential time reduction comparing to existing tools for small core extraction. Our experiments show that this result is evident in practice more so for the second method: rarely it takes more than a few seconds, even when competing tools time out, and hence it can be used as a cheap proof post-processing procedure.     

Parallel Approximation Algorithms by Positive Linear Programming

Several sequential approximation algorithms for combinatorial optimization problems are based on the following paradigm: solve a linear or semidefinite programming relaxation, then use randomized rounding to convert fractional solutions of the relaxation into integer solutions for the original combinatorial problem. We demonstrate that such a paradigm can also yield parallel approximation algorithms by showing how to convert certain linear programming relaxations into essentially equivalent positive linear programming [LN] relaxations that can be near-optimally solved in NC. Building on this technique, and finding some new linear programming relaxations, we develop improved parallel approximation algorithms for Max Sat, Max Directed Cut, and Max k CSP. The Max Sat algorithm essentially matches the best approximation obtainable with sequential algorithms and has a fast sequential version. The Max k CSP algorithm improves even over previous sequential algorithms. We also show a connection between probabilistic proof checking and a restricted version of Max k CSP. This implies that our approximation algorithm for Max k CSP can be used to prove inclusion in P for certain PCP classes. Received November 1996; revised March 1997.     

Elimination of Boolean variables for probabilistic coherence

 In this paper we deal with the computational complexity problem of checking the coherence of a partial probability assessment (called CPA). The CPA problem, like its analogous PSAT, is NP-complete so we look for an heuristic procedure to make tractable reasonable instances of the problem. Starting from the characteristic feature of de Finetti's approach (i.e. the explicit distinction between the probabilistic assessment and the logical relations among the sentences) we introduce several rules for a sequential “elimination” of Boolean variables from the domain of the assessment. The procedure resembles the well-known Davis-Putnam rules for the satisfiability, however we have, as a drawback, the introduction of constraints (among real variables) whose satisfiability must be checked. In simple examples we test the efficiency of the procedure respect to the “traditional” approach of solving a linear system with a huge coefficient matrix built from the atoms generated by the domain of the assessment.     

Coherence graphs

We study the consistency of a number of probability distributions, which are allowed to be imprecise. To make the treatment as general as possible, we represent those probabilistic assessments as a collection of conditional lower previsions. The problem then becomes proving Walley's (strong) coherence of the assessments. In order to maintain generality in the analysis, we assume to be given nearly no information about the numbers that make up the lower previsions in the collection. Under this condition, we investigate the extent to which the above global task can be decomposed into simpler and more local ones. This is done by introducing a graphical representation of the conditional lower previsions that we call the coherence graph: we show that the coherence graph allows one to isolate some subsets of the collection whose coherence is sufficient for the coherence of all the assessments; and we provide a polynomial-time algorithm that finds the subsets efficiently. We show some of the implications of our results by focusing on three models and problems: Bayesian and credal networks, of which we prove coherence; the compatibility problem, for which we provide an optimal graphical decomposition; probabilistic satisfiability, of which we show that some intractable instances can instead be solved efficiently by exploiting coherence graphs.     

Dynamic symmetry-breaking for Boolean satisfiability

With impressive progress in Boolean Satisfiability (SAT) solving and several extensions to pseudo-Boolean (PB) constraints, many applications that use SAT, such as high-performance formal verification techniques are still restricted to checking satisfiability of certain conditions. However, there is also frequently a need to express a preference for certain solutions. Extending SAT-solving to Boolean optimization allows the use of objective functions to describe a desirable solution. Although recent work in 0–1 Integer Linear Programming (ILP) offers extensions that can optimize a linear objective function, this is often achieved by solving a series of SAT or ILP decision problems. Our work articulates some pitfalls of this approach. An objective function may complicate the use of any symmetry that might be present in the given constraints, even when the constraints are unsatisfiable and the objective function is irrelevant. We propose several new techniques that treat objective functions differently from CNF/PB constraints and accelerate Boolean optimization in many practical cases. We also develop an adaptive flow that analyzes a given Boolean optimization problem and picks the symmetry-breaking technique that is best suited to the problem characteristics. Empirically, we show that for non-trivial objective functions that destroy constraint symmetries, the benefit of static symmetry-breaking is lost but dynamic symmetry-breaking accelerates problem-solving in many cases. We also introduce a new objective function, Localized Bit Selection (LBS), that can be used to specify a preference for bit values in formal verification applications.     

Note on Max Lin-2 above Average

In the Max Lin-2 problem we are given a system S of m linear equations in n variables over F₂ in which equation j is assigned a positive integral weight wj for each j. We wish to find an assignment of values to the variables which maximizes the total weight of satisfied equations. This problem generalizes Max Cut. The expected weight of satisfied equations is W/2, where W=w₁+?+wm; W/2 is a tight lower bound on the optimal solution of Max Lin-2.Mahajan et al. (Parameterizing above or below guaranteed values, J. Comput. Syst. Sci. 75 (2009) 137-153) stated the following parameterized version of Max Lin-2: decide whether there is an assignment of values to the variables that satisfies equations of total weight at least W/2+k, where k is the parameter. They asked whether this parameterized problem is fixed-parameter tractable, i.e., can be solved in time f(k)(nm)^O(1), where f(k) is an arbitrary computable function in k only. Their question remains open, but using some probabilistic inequalities and, in one case, a Fourier analysis inequality, Gutin et al. (A probabilistic approach to problems parameterized above tight lower bound, in: Proc. IWPEC'09, in: Lect. Notes Comput. Sci., vol. 5917, 2009, pp. 234-245) proved that the problem is fixed-parameter tractable in three special cases.In this paper we significantly extend two of the three special cases using only tools from combinatorics. We show that one of our results can be used to obtain a combinatorial proof that another problem from Mahajan et al. (Parameterizing above or below guaranteed values, J. Comput. Syst. Sci. 75 (2009) 137-153), Max r-SAT above Average, is fixed-parameter tractable for each r?2. Note that Max r-SAT above Average has been already shown to be fixed-parameter tractable by Alon et al. (Solving MAX-r-SAT above a tight lower bound, in: Proc. SODA 2010, pp. 511-517), but the paper used the approach of Gutin et al. (A probabilistic approach to problems parameterized above tight lower bound, in: Proc. IWPEC'09, in: Lect. Notes Comput. Sci., vol. 5917, 2009, pp. 234-245).     

Conditional and composite temporal CSPs

Constraint Satisfaction Problems (CSPs) have been widely used to solve combinatorial problems. In order to deal with dynamic CSPs where the information regarding any possible change is known a priori and can thus be enumerated beforehand, conditional constraints and composite variables have been studied in the past decade. Indeed, these two concepts allow the addition of variables and their related constraints in a dynamic manner during the resolution process. More precisely, a conditional constraint restricts the participation of a variable in a feasible scenario while a composite variable allows us to express a disjunction of variables where only one will be added to the problem to solve. In order to deal with a wide variety of real life applications under temporal constraints, we present in this paper a unique temporal CSP framework including numeric and symbolic temporal information, conditional constraints and composite variables. We call this model, a Conditional and Composite Temporal CSP (or CCTCSP). To solve the CCTCSP we propose two methods respectively based on Stochastic Local Search (SLS) and constraint propagation. In order to assess the efficiency in time of the solving methods we propose, experimental tests have been conducted on randomly generated CCTCSPs. The results demonstrate the superiority of a variant of the Maintaining Arc Consistency (MAC) technique (that we call MAX+) over the other constraint propagation strategies, Forward Checking (FC) and its variants, for both consistent and inconsistent problems. It has also been shown that, in the case of consistent problems, MAC+ outperforms the SLS method Min Conflict Random Walk (MCRW) for highly constrained CCTCSPs while both methods have comparable time performance for under and middle constrained problems. MCRW is, however, the method of choice for highly constrained CCTCSPs if we decide to trade search time for the quality of the solution returned (number of solved constraints).     

Sorting,linear time and the satisfiability problem

Let DLIN denote the class of problems that are computable withinO(n/logn) uniform time on a RAM which can read its input (of lengthn) in blocks and which only uses integersO(n/logn). We prove that the sorting problem belongs to DLIN and formulate the Linear Time Thesis: Each problem computable by a linear time bounded algorithm (in an intuitive sense) belongs to DLIN. We also study the subclass DTIME_SORT(n) of problems computable within linear time on a Turing machine using in addition a fixed number of sortings, and we show how the reductions of this class, so-called sort-lin reductions, are useful to classify NP problems: e.g. there is a sort-lin reduction from SAT to 3-SAT (using no sorting) and a sort-lin reduction from the NP-complete graph problem KERNEL to SAT (but we do not know of any similar reduction in DTIME(n)). Similarly, problem 2-SAT is linearly equivalent to the problem of Horn renaming (via sort-lin reductions). Finally, SAT is compared with many other combinatorial problems. A problemA is SAT-easy (resp. SAT-hard) if there is a sort-lin reduction fromA to SAT (resp. SAT toA). A SAT-equivalent problem is a problem both SAT-easy and SAT-hard. It is shown that the class of SAT-easy (resp. SAT-equivalent) problems is very large and that its generalization to so-called special clauses or, more generally, regular clauses, does not enlarge it. Moreover, we justify our opinion that problem SAT is, in some sense, a minimal NP-complete problem.     

数据流管理系统中Max、Min聚集算子的示例概要算法

Max和Min是数据流管理系统中重要聚集算子。应用基于滑动窗口下的示例概要法在实时数据流场景下计算Max和Min。在本方法中不需要保存所有落入滑动窗口中数据元组,这意味着可以极大地减小存储空间。由于存储元组的减少,系统的处理时间也显著地减少。实验结果表明基于滑动窗口的示例概要法显著降低了时间和空间的开销。     

A SAT approach to query optimization in mediator systems

Mediator systems integrate distributed, heterogeneous and autonomous data sources, but their effective use requires the solution of hard query optimization problems. This is usually done in two phases: the selection of a set of data sources is similar to a set covering problem, and their ordering into a feasible and efficient query is a capability restricted join order problem. However, a two-phase approach is unlikely to find optimum queries. We describe a new single-phase approach that, under a simple cost model, can be encoded and solved as a SAT problem. Results on artificial benchmarks indicate that this is an interesting problem from the encoding and search viewpoints, and we use them to address three of the ten SAT challenges posed by Selman, Kautz and McAllester in 1997.     

Forbidding and enforcing in membrane computing

Motivated by biochemistry and the non-deterministic reactions between molecules,the authors in (Ehrenfeucht and Rozenberg, 2003) introduced the concept of forbidding-enforcing systems(fe-systems) that define families of languages. Using the same concept we propose to study forbidding and enforcing within membrane systems. Two approaches are presented; in the first case the membrane system generates families of languages and in the second casethe membrane system generates a single language. We show that by using forbidding-enforcing in membranes, families of languages that cannot be defined by any fe-system can be generated. When a single language is generated, we show that SAT can be solved in a constant time (at price of using an exponential space). Also we show an example of a context-free language that can be generated without any forbidders.     

Accelerating Bounded Model Checking of Safety Properties

Bounded Model Checking based on SAT methods has recently been introduced as a complementary technique to BDD-based Symbolic Model Checking. The basic idea is to search for a counterexample in executions whose length is bounded by some integer k. The BMC problem can be efficiently reduced to a propositional satisfiability problem, and can therefore be solved by SAT methods rather than BDDs. SAT procedures are based on general-purpose heuristics that are designed for any propositional formula. We show how the unique characteristics of BMC invariant formulas (G p) can be exploited for a variety of optimizations in the SAT checking procedure. Experiments with these optimizations on real designs prove their efficiency in many of the hard test cases, in comparison to both the standard SAT procedure and a BDD-based model checker.     

Complete problems for space bounded subclasses of NP

Summary Several problems are shown to be log space complete, when restricted to bandwidth f(n), for the subclass of NP defined by nondeterministic polynomial time bounded Turing machines with a simultaneous f(n) space restriction, denoted by NTISP(poly, f(n)). These problems are NOT-ALL-EQUAL 3SAT, MONOCHROMATIC TRIANGLE, CUBIC SUBGRAPH, DOMINATING SET, ONE-IN-THREE 3SAT and MONOTONE 3SAT. The problems DOMATIC NUMBER, PARTITION INTO FORESTS and DISJOINT CONNECTING PATHS restricted to bandwidth f(n) are shown to be log space hard for NTISP(poly, f(n)). Their membership in the class NTISP(poly, f(n)) is currently open. As one application of these results, we note that the first six of the problems mentioned are examples of NSPACE(log n) complete problems when restricted to bandwidth log n.All of the authors were partially supported by NSF grants MCS 79-08919 and MCS 81-09280     

A Classification of Time and/or Probability Dependent Security Properties

In multilevel systems it is important to avoid unwanted indirect information flow from higher levels to lower levels, namely the so called covert channels. Initial studies of information flow analysis were performed by abstracting away from time and probability. It is already known that systems that are considered to be secure may turn out to be insecure when time or probability are considered. Recently, work has been done in order to consider also aspects either of time or of probability, but not both. In this paper we propose a general framework, based on Probabilistic Timed Automata, where both probabilistic and timing covert channels can be studied. We define a Non-Interference security property that allows one to express information flow in a timed and probabilistic setting, and we compare the property with analogous properties defined in settings where either time or probability or none of them are taken into account. This allows to classify properties depending on their discerning power.     

Hard constraint satisfaction problems have hard gaps at location 1

An instance of the maximum constraint satisfaction problem (Max CSP) is a finite collection of constraints on a set of variables, and the goal is to assign values to the variables that maximises the number of satisfied constraints. Max CSP captures many well-known problems (such as Maxk-SAT and Max Cut) and is consequently NP-hard. Thus, it is natural to study how restrictions on the allowed constraint types (or constraint language) affect the complexity and approximability of Max CSP. The PCP theorem is equivalent to the existence of a constraint language for which Max CSP has a hard gap at location 1; i.e. it is NP-hard to distinguish between satisfiable instances and instances where at most some constant fraction of the constraints are satisfiable. All constraint languages, for which the CSP problem (i.e., the problem of deciding whether all constraints can be satisfied) is currently known to be NP-hard, have a certain algebraic property. We prove that any constraint language with this algebraic property makes Max CSP have a hard gap at location 1 which, in particular, implies that such problems cannot have a PTAS unless P=NP. We then apply this result to Max CSP restricted to a single constraint type; this class of problems contains, for instance, Max Cut and Max DiCut. Assuming P≠NP, we show that such problems do not admit PTAS except in some trivial cases. Our results hold even if the number of occurrences of each variable is bounded by a constant. Finally, we give some applications of our results.