meSAT: multiple encodings of CSP to SAT

One approach for solving Constraint Satisfaction Problems (CSP) (and related Constraint Optimization Problems (COP)) involving integer and Boolean variables is reduction to propositional satisfiability problem (SAT). A number of encodings (e.g., direct, log, support, order) for this purpose exist as well as specific encodings for some constraints that are often encountered (e.g., cardinality constraints, global constraints). However, there is no single encoding that performs well on all classes of problems and there is a need for a system that supports multiple encodings. We present a system that translates specifications of finite linear CSP problems into SAT instances using several well-known encodings, and their combinations. We also present a methodology for selecting a suitable encoding based on simple syntactic features of the input CSP instance. Thorough evaluation has been performed on large publicly available corpora and our encoding selection method improves upon the efficiency of existing encodings and state-of-the-art tools used in comparison.     

Planning as satisfiability: parallel plans and algorithms for plan search

We address two aspects of constructing plans efficiently by means of satisfiability testing: efficient encoding of the problem of existence of plans of a given number t of time points in the propositional logic and strategies for finding plans, given these formulae for different values of t.For the first problem we consider three semantics for plans with parallel operator application in order to make the search for plans more efficient. The standard semantics requires that parallel operators are independent and can therefore be executed in any order. We consider a more relaxed definition of parallel plans which was first proposed by Dimopoulos et al., as well as a normal form for parallel plans that requires every operator to be executed as early as possible. We formalize the semantics of parallel plans emerging in this setting and present translations of these semantics into the propositional logic. The sizes of the translations are asymptotically optimal. Each of the semantics is constructed in such a way that there is a plan following the semantics exactly when there is a sequential plan, and moreover, the existence of a parallel plan implies the existence of a sequential plan with as many operators as in the parallel one.For the second problem we consider strategies based on testing the satisfiability of several formulae representing plans of n time steps for several values of n concurrently by several processes. We show that big efficiency gains can be obtained in comparison to the standard strategy of sequentially testing the satisfiability of formulae for an increasing number of time steps.     

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。     

On the computational complexity of satisfiability in propositional logics of programs

The satisfiability problems of propositional algorithmic logic and propositional dynamic logic are shown to be complete in the classes of languages accepted in polynomial space by the deterministic and alternating Turing machines respectively. Explicit upper and lower bounds on the space complexity are calculated. Exponential lower bounds on the space complexity of the satisfiability problems of these logics extended by adding a certain program connective are proved.     

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.     

智能规划中的动作互斥编码方式研究

以规划领域中的动作为对象,研究规划方法中的动作互斥编码方式。介绍基于规划图的动作互斥编码、利用提取领域相关信息生成动作效果的直接阻碍与间接阻碍编码,以及依赖于域转移图动作间的长距离互斥编码,说明每类动作互斥编码的构造方法及其削减搜索空间、提高求解效率的作用。     

命题编码中公理的组合与设计

近年来,基于可满足性的规划方法研究逐渐成为智能规划研究领域中的热点。提出3种基于Graphplan的编码方式中公理的改进:动作互斥的部分放松、动作互斥的完全放松方法、添加框架公理。基于SATPLAN2006规划系统分别实现上述3种改进的编码方式,并对国际规划竞赛中选用的标准后勤域与积木世界域的问题样例予以测试,分析不同编码方式的编码规模与求解效率,验证了基于Graphplan编码方式的改进在绝大多数情况下是有效的。最后,实现基于状态的编码方式,并对上述两个域进行测试,比较约简动作与约简状态这两种极端方式的求解效率和编码规模。实验结果表明,在后勤域的某些问题上基于状态的编码方式比基于动作的编码方式有效得多。上述的改进策略表明,可根据问题域的特性等来考虑该问题最适宜哪些公理组合的编码方式,而不固定使用某种特定的编码方式。     

Local and global symmetry breaking in itemset mining

The concept of symmetry has been extensively studied in the field of constraint programming and in the propositional satisfiability. Several methods for detection and removal of these symmetries have been developed, and their use in known solvers of these domains improved dramatically their effectiveness on a big variety of problems considered difficult to solve. The concept of symmetry may be exported to other areas where some structures can be exploited effectively. Particularly, in the area of data mining where some tasks can be expressed as constraints or logical formulas. We are interested here, by the detection and elimination of local and global symmetries in the item-set mining problem. Recent works have provided effective encodings as Boolean constraints for these data mining tasks and some idea on symmetry elimination in this area begin to appear, but still few and the techniques presented are often on global symmetry that is detected and eliminated statically in a preprocessing phase. In this work we study the notion of local symmetry and compare it to global symmetry for the itemset mining problem. We show how local symmetries of the boolean encoding can be detected dynamically and give some properties that allow to eliminate theses symmetries in SAT-based itemset mining solvers in order to enhance their efficiency.     

Uniform data encodings

Given two finite, directed, edge-labeled graphs, G (the guest) and H (the host) an encoding of G into H is an injection of guest-graph edges into host-graph paths that induces an injection of G vertices into H vertices. An encoding is uniform if like-labeled edges map to like-labeled paths. Encodings and uniform encodings are motivated by data structure representation issues. Although uniform encodings are economically expressed, three types of analysis indicate diseconomies introduced by the uniformity condition. First, space usage (i.e., the size of the host as a function of the size of the guest) can be exponential for ‘natural’ encodings such as uniform encodings of arrays into trees. By contrast, strongly connected hosts of size equal to the guests are adequate for nonuniform encodings. Secondly, the edge dilation under uniform encodings can be nonpolynomial in the size of the guest. By contrast, the same graphs can be nonuniformly encoded with edges mapping to paths of unit length. Thirdly, finding uniform encodings, even when the guest is a line, is PSPACE-complete. By contrast, there is a linear time algorithm for nonuniform encoding of lines in graphs. Additional upper and lower bounds amplify the limitations of uniform data encodings.     

Propositional semantics for disjunctive logic programs

In this paper we study the properties of the class of head-cycle-free extended disjunctive logic programs (HEDLPs), which includes, as a special case, all nondisjunctive extended logic programs. We show that any propositional HEDLP can be mapped in polynomial time into a propositional theory such that each model of the latter corresponds to an answer set, as defined by stable model semantics, of the former. Using this mapping, we show that many queries over HEDLPs can be determined by solving propositional satisfiability problems. Our mapping has several important implications: It establishes the NP-completeness of this class of disjunctive logic programs; it allows existing algorithms and tractable subsets for the satisfiability problem to be used in logic programming; it facilitates evaluation of the expressive power of disjunctive logic programs; and it leads to the discovery of useful similarities between stable model semantics and Clark's predicate completion.     

由一阶逻辑公式得到命题逻辑可满足性问题实例

命题逻辑可满足性(SAT)问题是计算机科学中的一个重要问题.近年来许多学者在这方面进行了大量的研究,提出了不少有效的算法.但是,很多实际问题如果用一组一阶逻辑公式来描述,往往更为自然.当解释的论域是一个固定大小的有限集合时,一阶逻辑公式的可满足性问题可以等价地归约为SAT问题.为了利用现有的高效SAT工具,提出了一种从一阶逻辑公式生成SAT问题实例的算法,并描述了一个自动的转换工具,给出了相应的实验结果.还讨论了通过增加公式来消除同构从而减小搜索空间的一些方法.实验表明,这一算法是有效的,可以用来解决数学研究和实际应用中的许多问题.     

A decomposition based algorithm for maximal contractions

This paper proposes a decomposition based algorithm for revision problems in classical propositional logic. A set of decomposing rules are presented to analyze the satisfiability of formulas. The satisfiability of a formula is equivalent to the satisfiability of a set of literal sets. A decomposing function is constructed to calculate all satisfiable literal sets of a given formula. When expressing the satisfiability of a formula, these literal sets are equivalent to all satisfied models of such formula. A revision algorithm based on this decomposing function is proposed, which can calculate maximal contractions of a given problem. In order to reduce the memory requirement, a filter function is introduced. The improved algorithm has a good performance in both time and space perspectives.     

MathSAT: Tight Integration of SAT and Mathematical Decision Procedures

Recent improvements in propositional satisfiability techniques (SAT) made it possible to tackle successfully some hard real-world problems (e.g., model-checking, circuit testing, propositional planning) by encoding into SAT. However, a purely Boolean representation is not expressive enough for many other real-world applications, including the verification of timed and hybrid systems, of proof obligations in software, and of circuit design at RTL level. These problems can be naturally modeled as satisfiability in linear arithmetic logic (LAL), that is, the Boolean combination of propositional variables and linear constraints over numerical variables. In this paper we present MathSAT, a new, SAT-based decision procedure for LAL, based on the (known approach) of integrating a state-of-the-art SAT solver with a dedicated mathematical solver for LAL. We improve MathSAT in two different directions. First, the top‐level line procedure is enhanced and now features a tighter integration between the Boolean search and the mathematical solver. In particular, we allow for theory-driven backjumping and learning, and theory-driven deduction; we use static learning in order to reduce the number of Boolean models that are mathematically inconsistent; we exploit problem clustering in order to partition mathematical reasoning; and we define a stack-based interface that allows us to implement mathematical reasoning in an incremental and backtrackable way. Second, the mathematical solver is based on layering; that is, the consistency of (partial) assignments is checked in theories of increasing strength (equality and uninterpreted functions, linear arithmetic over the reals, linear arithmetic over the integers). For each of these layers, a dedicated (sub)solver is used. Cheaper solvers are called first, and detection of inconsistency makes call of the subsequent solvers superfluous. We provide a through experimental evaluation of our approach, by taking into account a large set of previously proposed benchmarks. We first investigate the relative benefits and drawbacks of each proposed technique by comparison with respect to a reference option setting. We then demonstrate the global effectiveness of our approach by a comparison with several state-of-the-art decision procedures. We show that the behavior of MathSAT is often superior to its competitors, both on LAL and in the subclass of difference logic. This work has been partly supported by ISAAC, a European-sponsored project, contract no. AST3-CT-2003-501848; by ORCHID, a project sponsored by Provincia Autonoma di Trento; and by a grant from Intel Corporation. The work of T. Junttila has also been supported by the Academy of Finland, project 53695. S. Schulz has also been supported by a grant of the Italian Ministero dell'Istruzione, dell'Università e della Ricerca and the University of Verona.     

Towards a unified approach to encodability and separation results for process calculi

We present a unified approach to evaluate the relative expressive power of process calculi. In particular, we identify a small set of criteria (that have already been somehow presented in the literature) that an encoding should satisfy to be considered a valid means for language comparison. We argue that the combination of such criteria is a valid proposal by noting that: (i) several well-known encodings appeared in the literature satisfy them; (ii) this notion is not trivial, because some known encodings do not satisfy all the criteria we have proposed; (iii) several well-known separation results can be formulated in terms of our criteria; and (iv) some widely believed (but never formally proved) separation results can be proved by using the criteria we propose. Moreover, the criteria defined induce general proof-techniques for separation results that can be easily instantiated to cover known case-studies.     

基于自动推理技术的智能规划方法

对几种智能规划方法中利用的逻辑演绎与推理技术予以分析,分别介绍利用命题逻辑的基于可满足性的规划方法与规划系统,利用模态逻辑与析取推理的Conformant规划方法与规划系统,利用非单调逻辑的规划方法和利用模糊描述逻辑的Flexible规划方法,并结合国际规划竞赛和相关论文等的实验结论说明上述方法的有效性和可行性.最后,提出目前基于自动推理技术的智能规划方法所面临的挑战、可能的处理方法以及与之相关的研究热点与趋势.     

Simulating Circuit-Level Simplifications on CNF

Boolean satisfiability (SAT) and its extensions have become a core technology in many application domains, such as planning and formal verification, and continue finding various new application domains today. The SAT-based approach divides into three steps: encoding, preprocessing, and search. It is often argued that by encoding arbitrary Boolean formulas in conjunctive normal form (CNF), structural properties of the original problem are not reflected in the CNF. This should result in the fact that CNF-level preprocessing and SAT solver techniques have an inherent disadvantage compared to related techniques applicable on the level of more structural SAT instance representations such as Boolean circuits. Motivated by this, various simplification techniques and intricate CNF encodings for circuit-level SAT instance representations have been proposed. On the other hand, based on the highly efficient CNF-level clause learning SAT solvers, there is also strong support for the claim that CNF is sufficient as an input format for SAT solvers. In this work we study the effect of CNF-level simplification techniques, focusing on SatElite-style variable elimination (VE) and what we call blocked clause elimination (BCE). We show that BCE is surprisingly effective both in theory and in practice on CNF formulas resulting from a standard CNF encoding for circuits: without explicit knowledge of the underlying circuit structure, it achieves the same level of simplification as a combination of circuit-level simplifications and previously suggested polarity-based CNF encodings. We also show that VE can achieve many of the same effects as BCE, but not all. On the other hand, it turns out that VE and BCE are indeed partially orthogonal techniques. We also study the practical effects of combining BCE and VE for reducing the size of formulas and on the running times of state-of-the-art SAT solvers. Furthermore, we address the problem of how to construct original witnesses to satisfiable CNF formulas when applying a combination of BCE and VE.     

Bounded Model Checking of CTL

Bounded Model Checking has been recently introduced as an efficient verification method for reactive systems. This technique reduces model checking of linear temporal logic to propositional satisfiability. In this paper we first present how quantified Boolean decision procedures can replace BDDs. We introduce a bounded model checking procedure for temporal logic CTL＊ which reduces model checking to the satisfiability of quantified Boolean formulas. Our new technique avoids the space blow up of BDDs, and extends the concept of bounded model checking.     

Relating chemical activity to structure: An examination of ILP successes

Problems concerned with learning the relationships between molecular structure and activity have been important test-beds for Inductive Logic programming (ILP) systems. In this paper we examine these applications and empirically evaluate the extent to which a first-order representation was required. We compared ILP theories with those constructed using standard linear regression and a decision-tree learner on a series of progressively more difficult problems. When a propositional encoding is feasible for the feature-based algorithms, we show that such algorithms are capable of matching the predictive accuracies of an ILP theory. However, as the complexity of the compounds considered increased, propositional encodings becomes intractable. In such cases, our results show that ILP programs can still continue to construct accurate, understandable theories. Based on this evidence, we propose future work to realise fully the potential of ILP in structure-activity problem.     

Structural plan similarity based on refinements in the space of partial plans

Plan similarity measures play a key role in many areas of artificial intelligence, such as case‐based planning, plan recognition, ambient intelligence, or digital storytelling. In this paper, we present 2 novel structural similarity measures to compare plans based on a search process in the space of partial plans. Partial plans are compact representations of sets of plans with some common structure and can be organized in a lattice so that the most general partial plans are above the most specific ones. To compute our similarity measures, we traverse this space of partial plans from the most general to the most specific using successive refinements. Our first similarity measure is designed for propositional plan formalisms, and the second is designed for classical planning formalisms (including variables and types). We also introduce 2 novel refinement operators used to traverse the space of plans: an ideal downward refinement operator for propositional partial plans and a finite and complete downward refinement operator for classical partial plans. Finally, we evaluate our similarity measures in the context of a nearest neighbor classifier using 2 datasets commonly used in the plan recognition literature (Linux and Monroe), showing good results in both synthetic and real data.     

Generalized modal satisfiability

It is well known that modal satisfiability is PSPACE-complete (Ladner (1977) [21]). However, the complexity may decrease if we restrict the set of propositional operators used. Note that there exist an infinite number of propositional operators, since a propositional operator is simply a Boolean function. We completely classify the complexity of modal satisfiability for every finite set of propositional operators, i.e., in contrast to previous work, we classify an infinite number of problems. We show that, depending on the set of propositional operators, modal satisfiability is PSPACE-complete, coNP-complete, or in P. We obtain this trichotomy not only for modal formulas, but also for their more succinct representation using modal circuits. We consider both the uni-modal and the multi-modal cases, and study the dual problem of validity as well.