Set Of Clauses Research Articles

Relaxation Search: A Simple Way of Managing Optional Clauses

A number of problems involve managing a set of optional clauses. For example, the soft clauses in a MAXSAT formula are optional—they can be falsified for a cost. Similarly, when computing a Minimum Correction Set for an unsatisfiable formula, all clauses are optional—some can be falsified in order to satisfy the remaining. In both of these cases the task is to find a subset of the optional clauses that achieves some criteria, and whose removal leaves a satisfiable formula. Relaxation search is a simple method of using a standard SAT solver to solve this task. Relaxation search is easy to implement, sometimes requiring only a simple modification of the variable selection heuristic in the SAT solver; it offers considerable flexibility and control over the order in which subsets of optional clauses are examined; and it automatically exploits clause learning to exchange information between the two phases of finding a suitable subset of optional clauses and checking if their removal yields satisfiability. We demonstrate how relaxation search can be used to solve MAXSAT and to compute Minimum Correction Sets. In both cases relaxation search is able to achieve state-of-the-art performance and solve some instances other solvers are not able to solve.

The New Democratic Revolution in Music during the Play Experience of the Ideological and Political Education Function

After a deep investigation on the maximum terms space of the clause set, the concept of the partial maximum terms space of the clause set, which the maximum terms of the clause set decomposed, is brought forward. By investigating the extension rule, this paper introduces the concept of the satisfiability and the unsatisfiability of the partial maximum terms space, and gives an algorithm determining the satisfiability of a partial space of the maximum terms - algorithm PSER (Partial Semi-Extension Rule). Then, the TP problem is decomposed into several sub-problems independent of each other, which can be solved by the given parallel computing method PPSER (Parallel Partial Semi-Extension Rule).

Improved Separations of Regular Resolution from Clause Learning Proof Systems

This paper studies the relationship between resolution and conflict driven clause learning (CDCL) without restarts, and refutes some conjectured possible separations. We prove that the guarded, xor-ified pebbling tautology clauses, which Urquhart proved are hard for regular resolution, as well as the guarded graph tautology clauses of Alekhnovich, Johannsen, Pitassi, and Urquhart have polynomial size pool resolution refutations that use only input lemmas as learned clauses. For the latter set of clauses, we extend this to prove that a CDCL search without restarts can refute these clauses in polynomial time, provided it makes the right choices for decision literals and clause learning. This holds even if the CDCL search is required to greedily process conflicts arising from unit propagation. This refutes the conjecture that the guarded graph tautology clauses or the guarded xor-ified pebbling tautology clauses can be used to separate CDCL without restarts from general resolution. Together with subsequent results by Buss and Kolodziejczyk, this means we lack any good conjectures about how to establish the exact logical strength of conflict-driven clause learning without restarts.

Tractable and intractable classes of propositional schemata

We study the complexity of the satisfiability problem for a class of logical formulae called iterated propositional schemata, modeling infinite sequences of structurally similar propositional formulae. We prove that the problem is EXPSPACE-complete in general and PSPACE-complete if the numbers occurring in the formula are polynomially bounded by the size of the schema. We then consider more restricted classes: We prove that the problem is still PSPACE-complete for the Horn class, but only NP-complete for the Krom class (sets of clauses of length 2). Finally, we devise a simple criterion ensuring that the satisfiability problem is in P.

Model Clauses for the Use of the UNIDROIT Principles of International Commercial Contracts

The Model Clauses for the Use of the Unidroit Principles of International Commercial Contracts, reproduced as an appendix to this note, were prepared by a working group set up by International Institute for the Unification of Private Law (Unidroit) and composed of renowned experts in the field of international contract law and arbitration.1 The working group was chaired by Don Wallace, Jr., while the author of this article was appointed Rapporteur. On the basis of a position paper prepared by the Rapporteur,2 the working group, at a meeting held in Rome in February 2013, after extensive discussion, approved a set of draft Model Clauses for the Use of the Unidroit Principles of International Commercial Contracts (Model Clauses) and asked the Rapporteur to prepare comments thereto. The draft Model Clauses and Comments were submitted to the Unidroit Governing Council, which adopted them, with minor...

Implicit Resolution

Let \Omega be a set of unsatisfiable clauses, an implicit resolution refutation of \Omega is a circuit \beta with a resolution proof {\alpha} of the statement "\beta describes a correct tree-like resolution refutation of \Omega". We show that such system is p-equivalent to Extended Frege. More generally, let {\tau} be a tautology, a [P, Q]-proof of {\tau} is a pair (\alpha,\beta) s.t. \alpha is a P-proof of the statement "\beta is a circuit describing a correct Q-proof of \tau". We prove that [EF,P] \leq p [R,P] for arbitrary Cook-Reckhow proof system P.

Verification of Imperative Programs by Constraint Logic Program Transformation

We present a method for verifying partial correctness properties of imperative programs that manipulate integers and arrays by using techniques based on the transformation of constraint logic programs (CLP). We use CLP as a metalanguage for representing imperative programs, their executions, and their properties. First, we encode the correctness of an imperative program, say prog, as the negation of a predicate 'incorrect' defined by a CLP program T. By construction, 'incorrect' holds in the least model of T if and only if the execution of prog from an initial configuration eventually halts in an error configuration. Then, we apply to program T a sequence of transformations that preserve its least model semantics. These transformations are based on well-known transformation rules, such as unfolding and folding, guided by suitable transformation strategies, such as specialization and generalization. The objective of the transformations is to derive a new CLP program TransfT where the predicate 'incorrect' is defined either by (i) the fact 'incorrect.' (and in this case prog is not correct), or by (ii) the empty set of clauses (and in this case prog is correct). In the case where we derive a CLP program such that neither (i) nor (ii) holds, we iterate the transformation. Since the problem is undecidable, this process may not terminate. We show through examples that our method can be applied in a rather systematic way, and is amenable to automation by transferring to the field of program verification many techniques developed in the field of program transformation.

Reducing Higher-Order Theorem Proving to a Sequence of SAT Problems

We describe a complete theorem proving procedure for higher-order logic that uses SAT-solving to do much of the heavy lifting. The theoretical basis for the procedure is a complete, cut-free, ground refutation calculus that incorporates a restriction on instantiations. The refined nature of the calculus makes it conceivable that one can search in the ground calculus itself, obtaining a complete procedure without resorting to meta-variables and a higher-order lifting lemma. Once one commits to searching in a ground calculus, a natural next step is to consider ground formulas as propositional literals and the rules of the calculus as propositional clauses relating the literals. With this view in mind, we describe a theorem proving procedure that primarily generates relevant formulas along with their corresponding propositional clauses. The procedure terminates when the set of propositional clauses is unsatisfiable. We prove soundness and completeness of the procedure. The procedure has been implemented in a new higher-order theorem prover, Satallax, which makes use of the SAT-solver MiniSat. We also describe the implementation and give several examples. Finally, we include experimental results of Satallax on the higher-order part of the TPTP library.

Normalization Strategy of Logical Knowledge Representation for Text Document

This paper discussed the idea of the computer system capable of simulating understanding with respect to reading a text document. The research is concerned with the problem of generating sophisticated knowledge representation for the purpose of understanding the natural language. Due to that, a simplification form of logical-oriented model of knowledge representation called Pragmatic Skolem Clauses (PSC) is proposed to represent the semantic formalism for the computational linguistic. Each set of pragmatic skolem clauses containing at least one skolem constant, which shows the thematic role relationship between clauses. Semantically and pragmatically-accented approach will be discussed in this paper in the context of formal grammar and linguistic semantic.

REACHABILITY FOR FINITE-STATE PROCESS ALGEBRAS USING HORN CLAUSES

In this work we present an algorithm for solving the reachability problem in finite systems that are modelled with process algebras. Our method is based on Static Analysis, in particular, Data Flow Analysis, of the syntax of a process algebraic system with multi-way synchronisation. The results of the Data Flow Analysis are used in order to build a set of Horn clauses whose least model corresponds to an overapproximation of the reachable states. The computed model can be refined after each transition, and the algorithm runs until either a state whose reachability should be checked is encountered or it is not in the least model for all constructed states and thus is definitely unreachable. The advantages of the algorithm are that in many cases only a part of the Labelled Transition System will be built which leads to lower time and memory consumption. Also, it is not necessary to save all the encountered states which leads to further reduction of the memory requirements of the algorithm.

A Resolution Calculus for First-order Schemata

We devise a resolution calculus that tests the satisfiability of infinite families of clause sets, called clause set schemata. For schemata of propositional clause sets, we prove that this calculus is sound, refutationally complete, and terminating. The calculus is extended to first-order clauses, for which termination is lost, since the satisfiability problem is not semi-decidable for nonpropositional schemata. The expressive power of the considered logic is strictly greater than the one considered in our previous work.

Superposition Decides the First-Order Logic Fragment Over Ground Theories

The hierarchic superposition calculus over a theory T, called SUP(T), enables sound reasoning on the hierarchic combination of a theory T with full first-order logic, FOL(T). If a FOL(T) clause set enjoys a sufficient completeness criterion, the calculus is even complete. Clause sets over the ground fragment of FOL(T) are not sufficiently complete, in general. In this paper we show that any clause set over the ground FOL(T) fragment can be transformed into a sufficiently complete one, and prove that SUP(T) terminates on the transformed clause set, hence constitutes a decision procedure provided the existential fragment of the theory T is decidable. Thanks to the hierarchic design of SUP(T), the decidability result can be extended beyond the ground case. We show SUP(T) is a decision procedure for the non-ground FOL fragment plus a theory T, if every non-constant function symbol from the underlying FOL signature ranges into the sort of the theory T, and every term of the theory sort is ground. Examples for T are in particular decidable fragments of arithmetic.

Learning compact Markov logic networks with decision trees

Statistical-relational learning combines logical syntax with probabilistic methods. Markov Logic Networks (MLNs) are a prominent model class that generalizes both first-order logic and undirected graphical models (Markov networks). The qualitative component of an MLN is a set of clauses and the quantitative component is a set of clause weights. Generative MLNs model the joint distribution of relationships and attributes. A state-of-the-art structure learning method is the moralization approach: learn a set of directed Horn clauses, then convert them to conjunctions to obtain MLN clauses. The directed clauses are learned using Bayes net methods. The moralization approach takes advantage of the high-quality inference algorithms for MLNs and their ability to handle cyclic dependencies. A weakness of moralization is that it leads to an unnecessarily large number of clauses. In this paper we show that using decision trees to represent conditional probabilities in the Bayes net is an effective remedy that leads to much more compact MLN structures. In experiments on benchmark datasets, the decision trees reduce the number of clauses in the moralized MLN by a factor of 5---25, depending on the dataset. The accuracy of predictions is competitive with the models obtained by standard moralization, and in many cases superior.

Piercing Politics: Religious Garb and Secularism in Public Schools

In Iacono v. Johnston County School Board, a 2010 case dealing with the suspension of a high school freshman, Ariana Iacono wore a nose piercing to her public high school in observance of her religious membership with the Church of Body Modification. In this recent case, a federal district court granted Ms. Iacono a temporary restraining order enjoining the school board from further suspending her, reasoning that Ms. Iacono’s claims had a strong likelihood of success on the merits, especially with regard to her claim that the policy violated her right to freely exercise her religious beliefs under the First Amendment of the United States Constitution. The Iacono case sets the stage for the remainder of the Article, which grapples with the question of where the line should be drawn between secularism and free exercise of religion in public schools. The Article discusses relevant religious garb cases in Free Exercise Clause jurisprudence and sets forth additional arguments ultimately not used by the Iacono plaintiffs, such as a potential violation of the Free Establishment Clause of the First Amendment. Following the review of jurisprudence and potential arguments in litigating in issue like that in Iacono, the Article widens the scope of the issue by discussing empirical findings relating to policies similar to the dress code employed by Johnston County. The empirical findings arise from a survey of dress code policies maintained by the largest public school district in each state in the U.S. Although the sampling size is limited, the data provides an understanding – albeit, a bird’s-eye view of the issue – of how strong the likelihood for a situation similar to the Iacono case to arise in other parts of the country. After relaying these empirical findings to establish the widespread nature of the religious apparel issue in public schools, the Article will discuss how schools may better arm themselves to tackle the issue of religious garb by revising dress code policies and proposes a model religious apparel exemption. Ultimately, the issue of secularism in public education pays homage to the struggle of the Founding Fathers when creating the United States, but it is one that still warrants careful consideration, especially in our schools- the “marketplace of ideas.”

Width and size of regular resolution proofs

This paper discusses the topic of the minimum width of a regular resolution refutation of a set of clauses. The main result shows that there are examples having small regular resolution refutations, for which any regular refutation must contain a large clause. This forms a contrast with corresponding results for general resolution refutations.

Local Consistency and SAT-Solvers

Local consistency techniques such as k-consistency are a key component of specialised solvers for constraint satisfaction problems. In this paper we show that the power of using k-consistency techniques on a constraint satisfaction problem is precisely captured by using a particular inference rule, which we call negative-hyper-resolution, on the standard direct encoding of the problem into Boolean clauses. We also show that current clause-learning SAT-solvers will discover in expected polynomial time any inconsistency that can be deduced from a given set of clauses using negative-hyper-resolvents of a fixed size. We combine these two results to show that, without being explicitly designed to do so, current clause-learning SAT-solvers efficiently simulate k-consistency techniques, for all fixed values of k. We then give some experimental results to show that this feature allows clause-learning SAT-solvers to efficiently solve certain families of constraint problems which are challenging for conventional constraint-programming solvers.

Maxterm Covering for Satisfiability

This paper presents a novel efficient satisfiability (SAT) algorithm based on maxterm covering. The satisfiability of a clause set is determined in terms of the number of relative maxterms of the empty clause with respect to the clause set. If the number of relative maxterms is zero, it is unsatisfiable, otherwise satisfiable. A set of synergic heuristic strategies are presented and elaborated. We conduct a number of experiments on 3-SAT and k-SAT problems at the phase transition region, which have been cited as the hardest group of SAT problems. Our experimental results on public benchmarks attest to the fact that, by incorporating our proposed heuristic strategies, our enhanced algorithm runs several orders of magnitude faster than the extension rule algorithm, and it also runs faster than zChaff and MiniSAT for most of k-SAT (k≥3) instances.

Complement clauses in spoken German and English: Syntax, deixis and discourse-pragmatics

This article examines features of spoken German and English complement dass/that-clauses. It focuses on clauses with and without a complementiser, correlative constructions and unintegrated complement clauses, as well as a related English copular construction. Differences between the languages are discussed in terms of the role of the complementiser in relation to word order, assertional vs presuppositional clauses, epistemic main clauses, deixis and discourse function. While German and English have a similar set of possible complement clauses, they are shown to differ in terms of what is typical in spoken language. The article underlines the need to examine spoken language structures in use on their own terms and further confirms that complementation by a finite-clause is not a unitary phenomenon within each language. The analysis is carried out from a usage-based perspective of language which works with the concept of constructions and sees a close relationship between lexis and grammar. The article also has implications for typological work on complementation in spoken language.

Computer Software-Related Litigation: Discovery and the Overly-Protective Order

Litigation involving allegations of intellectual property infringement concerning computer software is some of the most complex, time consuming, and expensive litigation in which private parties engage. Certain practices in discovery, including, most significantly, the use of poorly drafted discovery agreements that also include “overly protective” orders, increase that expense dramatically. Regardless of whether the allegation is patent infringement, copyright infringement, or trade secret misappropriation, prosecuting and defending the assertions in the case requires a probing analysis of the computer source code. In these types of cases, both parties will engage forensic software analysts to assist the lawyers in preparation for trial and to provide expert witness testimony for the court. The forensic software analysts will dissect the computer source code, often examining the source code of both parties, looking for signs of infringement or misappropriation as well as for technical explanations for similarities in the way the code is written or structured. But first, the computer source code must be disclosed to the opposing party. Such disclosure is almost always done pursuant to a protective order, typically stipulated to by the attorneys. Lawyers often agree to protective orders that significantly and unnecessarily increase the costs of discovery. Attorneys should pay careful attention to the provisions addressing the requirements of production and analysis. Additionally, attorneys must understand the consequences of the clauses contained in protective orders in these types of litigation. As described in this article, it is possible to provide robust protection for disclosed source code while at the same time not unnecessarily and dramatically increasing the cost of discovery by weaponizing the protective order. The goals of this article are three-fold. First, we seek to help lawyers understand the process of forensic software analysis. Second, we provide a set of model clauses aimed at avoiding pitfalls in the design of the discovery process, including model clauses for a protective order of appropriate scope and with appropriate protections from further disclosure of the source code that is produced. Third, for judges who are asked to intervene in discovery battles, including fights over the proper scope of a protective order, this article is meant to assist in evaluating the parties’ arguments.

Data TransfersWithin Europe: Contractual Data Protection Clauses in Practice

Abstract Other than in case of data transfers from Europe to countries without an adequate level of data protection, there is no set of standard contractual clauses - similar to the EU Model Clauses - for data transfers within countries with an adequate level data protection. In practice, this lack of such standard agreements sometimes raises questions and it (seemingly) also leaves more room to achieve compliance than in case of data transfers to countries without an adequate level of data protection. This article discusses and analyzes these questions and this “room”, with a focus on the practical implications of the use of “boilerplate” data protection clauses, i.e. generic contractual data protection clauses that are directly incorporated into commercial agreements.