Complexity of computing with extended propositional logic programs

In this paper we introduce the notion of anF-program, whereF is a collection of formulas. We then study the complexity of computing withF-programs.F-programs can be regarded as a generalization of standard logic programs. Clauses (or rules) ofF-programs are built of formulas fromF. In particular, formulas other than atoms are allowed as building blocks ofF-program rules. Typical examples ofF are the set of all atoms (in which case the class of ordinary logic programs is obtained), the set of all literals (in this case, we get the class of logic programs with classical negation [9]), the set of all Horn clauses, the set of all clauses, the set of all clauses with at most two literals, the set of all clauses with at least three literals, etc. The notions of minimal and stable models [16, 1, 7] of a logic program have natural generalizations to the case ofF-programs. The resulting notions are called in this paperminimal andstable answer sets. We study the complexity of reasoning involving these notions. In particular, we establish the complexity of determining the existence of a stable answer set, and the complexity of determining the membership of a formula in some (or all) stable answer sets. We study the complexity of the existence of minimal answer sets, and that of determining the membership of a formula in all minimal answer sets. We also list several open problems.This work was partially supported by National Science Foundation under grant IRI-9012902.This work was partially supported by National Science Foundation under grant CCR-9110721.     

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.     

Nested expressions in logic programs

We extend the answer set semantics to a class of logic programs with nested expressions permitted in the bodies and heads of rules. These expressions are formed from literals using negation as failure, conjunction (,) and disjunction (;) that can be nested arbitrarily. Conditional expressions are introduced as abbreviations. The study of equivalent transformations of programs with nested expressions shows that any such program is equivalent to a set of disjunctive rules, possibly with negation as failure in the heads. The generalized answer set semantics is related to the Lloyd–Topor generalization of Clark’s completion and to the logic of minimal belief and negation as failure. This revised version was published online in June 2006 with corrections to the Cover Date.     

Semantics and complexity of recursive aggregates in answer set programming

The addition of aggregates has been one of the most relevant enhancements to the language of answer set programming (ASP). They strengthen the modelling power of ASP in terms of natural and concise problem representations. Previous semantic definitions typically agree in the case of non-recursive aggregates, but the picture is less clear for aggregates involved in recursion. Some proposals explicitly avoid recursive aggregates, most others differ, and many of them do not satisfy desirable criteria, such as minimality or coincidence with answer sets in the aggregate-free case.In this paper we define a semantics for programs with arbitrary aggregates (including monotone, antimonotone, and nonmonotone aggregates) in the full ASP language allowing also for disjunction in the head (disjunctive logic programming — DLP). This semantics is a genuine generalization of the answer set semantics for DLP, it is defined by a natural variant of the Gelfond–Lifschitz transformation, and treats aggregate and non-aggregate literals in a uniform way. This novel transformation is interesting per se also in the aggregate-free case, since it is simpler than the original transformation and does not need to differentiate between positive and negative literals. We prove that our semantics guarantees the minimality (and therefore the incomparability) of answer sets, and we demonstrate that it coincides with the standard answer set semantics on aggregate-free programs.Moreover, we carry out an in-depth study of the computational complexity of the language. The analysis pays particular attention to the impact of syntactical restrictions on programs in the form of limited use of aggregates, disjunction, and negation. While the addition of aggregates does not affect the complexity of the full DLP language, it turns out that their presence does increase the complexity of normal (i.e., non-disjunctive) ASP programs up to the second level of the polynomial hierarchy. However, we show that there are large classes of aggregates the addition of which does not cause any complexity gap even for normal programs, including the fragment allowing for arbitrary monotone, arbitrary antimonotone, and stratified (i.e., non-recursive) nonmonotone aggregates. The analysis provides some useful indications on the possibility to implement aggregates in existing reasoning engines.     

适于大数目输入变量的逻辑综合启发式算法

本文给出了适用于大数目输入变量的逻辑综合启发式算法。该方法采用自顶向下的逼近，通过包含的文字产生蕴涵项，包含文字的选择是基于启发式方法，根据文字发生的频率进行的。这个算法已经通过标准benchmarks例题和随机产生的大数目输入变量问题的测试。实验证明该方法是有效的，特别对大数目输入变量的函数，具有较好的综合效果。     

Outlier detection using default reasoning

Default logics are usually used to describe the regular behavior and normal properties of domain elements. In this paper we suggest, conversely, that the framework of default logics can be exploited for detecting outliers. Outliers are observations expressed by sets of literals that feature unexpected semantical characteristics. These sets of literals are selected among those explicitly embodied in the given knowledge base. Hence, essentially we perceive outlier detection as a knowledge discovery technique. This paper defines the notion of outlier in two related formalisms for specifying defaults: Reiter's default logic and extended disjunctive logic programs. For each of the two formalisms, we show that finding outliers is quite complex. Indeed, we prove that several versions of the outlier detection problem lie over the second level of the polynomial hierarchy. We believe that a thorough complexity analysis, as done here, is a useful preliminary step towards developing effective heuristics and exploring tractable subsets of outlier detection problems.     

CLIN-S - A Semantically Guided First-Order Theorem Prover

CLIN-S is an instance-based, clause-form first-order theorem prover. CLIN-S employs three inference procedures: semantic hyper-linking, which uses semantics to guide the proof search and performs well on non-Horn parts of the proofs involving small literals, rough resolution, which removes large literals in the proofs, and UR resolution, which proves the Horn parts of the proofs. A semantic structure for the input clauses is given as input. During the search for the proof, ground instances of the input clauses are generated and new semantic structures are built based on the input semantics and a model of the ground clause set. A proof is found if the ground clause set is unsatisfiable. In this article, we describe the system architecture and major inference rules used in CLIN-S.     

Cooperation of Background Reasoners in Theory Reasoning by Residue Sharing

We propose a general way of combining background reasoners in theory reasoning. Using a restricted version of the Craig interpolation lemma, we show that background reasoner cooperation can be achieved as a form of constraint propagation, much in the spirit of existing combination methods for decision procedures. In this case, constraint information is propagated across reasoners eexchanging residues that are, in essence, disjunctions of ground literals over a common signature. As an application of our approach, we describe a multitheory version of the semantic tableau calculus, and we prove it sound and complete.     

Link inheritance in abstract clause graphs

Clause graphs, as they were defined in the 1970s, are graphs representing first order formulas in conjunctive normal form: the nodes are labelled with literals and the edges (links) connect complementary unifiable literals, i.e. they provide an explicit representation of the resolution possibilities. This report describes a generalization of this concept, called abstract clause graphs. The nodes of abstract clause graphs are still labelled with literals, the links however connect literals that are unifiable relative to a given relation between literals. This relation is not explicitely defined, only certain abstract properties are required. For instance the existence of a special purpose unification algorithm is assumed, which computes substitutions, the application of which makes the relation hold for two literals.When instances of already existing literals are added to the graph (e.g. due to resolution or factoring), the links to the new literals are derived from the links of their ancestors. An inheritance mechanism for such links is presented which operates only on the attached substitutions and does not have to unify the literals. The definition of abstract clause graphs and the theory about link inheritance is general enough to provide a framework so that as new ideas are proposed for graph based theorem provers, the methodology for both implementing links and proving their properties will be readily available.This research was supported by the Sonderforschungsbereich 314, Künstliche Intelligenz, of the German Research Agency.     

Integrating advanced reasoning into a SAT solver

1 Introduction The SAT (Satisfiability) problem is one of the basic NP problems that have been widely researched. Many problems in EDA (Electronics Design Automation) domain such as ATPG (Automatic Test Pattern Generation), Logic Synthesis, Equivalence Checking[1] and Model Checking[2] can be reduced to the SAT problem. Typical algorithms for SAT can be classified into two categories: incomplete and complete ones. The former, including GSAT[3] and WalkSAT[4], are based on local…     

Support vector machines of interval-based features for time series classification

In previous works, a time series classification system has been presented. It is based on boosting very simple classifiers, formed only by one literal. The used literals are based on temporal intervals.The obtained classifiers were simply a linear combination of literals, so it is natural to expect some improvements in the results if those literals were combined in more complex ways. In this work we explore the possibility of using the literals selected by the boosting algorithm as new features, and then using a SVM with these metafeatures. The experimental results show the validity of the proposed method.     

t-DeLP: an argumentation-based Temporal Defeasible Logic Programming framework

The aim of this paper is to propose an argumentation-based defeasible logic, called t-DeLP, that focuses on forward temporal reasoning for causal inference. We extend the language of the DeLP logical framework by associating temporal parameters to literals. A temporal logic program is a set of basic temporal facts and (strict or defeasible) durative rules. Facts and rules combine into durative arguments representing temporal processes. As usual, a dialectical procedure determines which arguments are undefeated, and hence which literals are warranted, or defeasibly follow from the program. t-DeLP, though, slightly differs from DeLP in order to accommodate temporal aspects, like the persistence of facts. The output of a t-DeLP program is a set of warranted literals, which is first shown to be non-contradictory and be closed under sub-arguments. This basic framework is then modified to deal with programs whose strict rules encode mutex constraints. The resulting framework is shown to satisfy stronger logical properties like indirect consistency and closure.     

SATCHMORE: SATCHMO with RElevancy

We introduce a relevancy detection algorithm to be used in conjunction with the SATCHMO prover. The version of SATCHMO considered here is essentially a bidirectional prover, utilizing Prolog (back chaining) on Horn clauses and forward chaining on non-Horn clauses. Our extension, SATCHMORE (SATCHMO with RElevancy), addresses the major weakness of SATCHMO: the uncontrolled use of forward chaining. By marking potentially relevant clause head literals, and then requiring that all the head literals be marked relevant (be totally relevant) before a clause is used for forward chaining, SATCHMORE is able to guide the use of these rules. Furthermore, the relevancy testing is performed without extending the proof search beyond what is done in SATCHMO. A simple implementation of the extended SATCHMO can be written in Prolog. We describe our relevancy testing approach, present the implementation, prove soundness and completeness, and provide examples that demonstrate the power of relevancy testing.This research was partially supported by NSF Grants IRI-8805696 and CCR-9116203. This paper is a major revision of Wilson and Loveland (1989).     

Index conditions of resolution

1 Introduction Automated reasoning mechanisms[1,2] are widely used, for example to solve constraint satisfaction prob- lems. The exploration of e?cient automated reasoning strategies is therefore important. If more strategies are incorporated into such mechanisms, resolution based deduction becomes stricter. This results in the gen- eration of a lower number of unuseable clauses. Un- fortunately however, some strategies are incompatible with each other. For example, deletion strategy can dest…     

Defeasible inheritance with doubt index and its axiomatic characterization

This article introduces and uses a representation of defeasible inheritance networks where links in the network are viewed as propositions, and where defeasible links are tagged with a quantitative indication of the proportion of exceptions, called the doubt index. This doubt index is used for restricting the length of the chains of inference.The representation also introduces the use of defeater literals that disable the chaining of subsumption links. The use of defeater literals replaces the use of negative defeasible inheritance links, expressing “most A are not B”. The new representation improves the expressivity significantly.Inference in inheritance networks is defined by a combination of axioms that constrain the contents of network extensions, a heuristic restriction that also has that effect, and a nonmonotonic operation of minimizing the set of defeater literals while retaining consistency.We introduce an underlying semantics that defines the meaning of literals in a network, and prove that the axioms are sound with respect to this semantics. We also discuss the conditions for obtaining completeness.Traditional concepts, assumptions and issues in research on nonmonotonic or defeasible inheritance are reviewed in the perspective of this approach.     

NP-completeness of small conflict set generation for congruence closure

The efficiency of satisfiability modulo theories (SMT) solvers is dependent on the capability of theory reasoners to provide small conflict sets, i.e. small unsatisfiable subsets from unsatisfiable sets of literals. Decision procedures for uninterpreted symbols (i.e. congruence closure algorithms) date back from the very early days of SMT. Nevertheless, to the best of our knowledge, the complexity of generating smallest conflict sets for sets of literals with uninterpreted symbols and equalities had not yet been determined, although the corresponding decision problem was believed to be NP-complete. We provide here an NP-completeness proof, using a simple reduction from SAT.     

Signed Systems for Paraconsistent Reasoning

We present a novel approach to paraconsistent reasoning, that is, to reasoning from inconsistent information. The basic idea is the following. We transform an inconsistent theory into a consistent one by renaming all literals occurring in the theory. Then, we restore some of the original contents of the theory by introducing progressively formal equivalences linking the original literals to their renamings. This is done as long as consistency is preserved. The restoration of the original contents of the theory is done by appeal to default logic. The overall approach provides us with a family of paraconsistent consequence relations.Our approach is semantical because it works at the level of the propositions; it deals with the semantical link between a proposition and its negation. The approach is therefore independent of the combination of the connectives that are actually applied to the propositions in order to form entire formulas.     

基于信息增益的中文文本关联分类

关联分类是一种通过挖掘训练集中的关联规则,并利用这些规则预测新数据类属性的分类技术。最近的研究表明,关联分类取得了比传统的分类方法如C4.5更高的准确率。现有的基于支持度-置信度架构的关联分类方法仅仅是选择频繁文字构建分类规则,忽略了文字的分类有效性。本文提出一种新的ACIG算法,结合信息增益与FoilGain在中文文本中选择规则的文字,以提高文字的分类有效性。实验结果表明,ACIG算法比其他关联分类算法(CPAR)有更高的准确率。