A characterization of answer sets for logic programs

Checking if a program has an answer set, and if so, compute its answer sets are just some of the important problems in answer set logic programming. Solving these problems using Gelfond and Lifschitz's original definition of answer sets is not an easy task. Alternative characterizations of answer sets for nested logic pro- grams by Erdem and Lifschitz, Lee and Lifschitz, and You et al. are based on the completion semantics and various notions of tightness. However, the notion of tightness is a local notion in the sense that for different answer sets there are, in general, different level mappings capturing their tightness. This makes it hard to be used in the design of algorithms for computing answer sets. This paper proposes a characterization of answer sets based on sets of generating rules. From this char- acterization new algorithms are derived for computing answer sets and for per- forming some other reasoning tasks. As an application of the characterization a sufficient and necessary condition for the equivalence between answer set seman- tics and completion semantics has been proven, and a basic theorem is shown on computing answer sets for nested logic programs based on an extended notion of loop formulas. These results on tightness and loop formulas are more general than that in You and Lin's work.     

On computing logic programs

In this paper we present and compare some classical problem-solving methods for computing the stable models of logic programs with negation. Using a graph theoretic representation of logic programs and their stable models, we discuss and compare linear programming, propositional satisfiability, constraint satisfaction, and graph methods.     

On nonmonotonic reasoning with the method of sweeping presumptions

Reasoning almost always occurs in the face of incomplete information. Such reasoning is nonmonotonic in the sense that conclusions drawn may later be withdrawn when additional information is obtained. There is an active literature on the problem of modeling such nonmonotonic reasoning, yet no category of method-let alone a single method-has been broadly accepted as the right approach. This paper introduces a new method, called sweeping presumptions, for modeling nonmonotonic reasoning. The main goal of the paper is to provide an example-driven, nontechnical introduction to the method of sweeping presumptions, and thereby to make it plausible that sweeping presumptions can usefully be applied to the problems of nonmonotonic reasoning. The paper discusses a representative sample of examples that have appeared in the literature on nonmonotonic reasoning, and discusses them from the point of view of sweeping presumptions.     

Answer set based design of knowledge systems

The aim of this paper is to demonstrate that A-Prolog is a powerful language for the construction of reasoning systems. In fact, A-Prolog allows to specify the initial situation, the domain model, the control knowledge, and the reasoning modules. Moreover, it is efficient enough to be used for practical tasks and can be nicely integrated with programming languages such as Java. An extension of A-Prolog (CR-Prolog) allows to further improve the quality of reasoning by specifying requirements that the solutions should satisfy if at all possible. The features of A-Prolog and CR-Prolog are demonstrated by describing in detail the design of USA-Advisor, an A-Prolog based decision support system for the Space Shuttle flight controllers.     

On the relationship between annotated logic programs and nonmonotonic formalisms

Abstract

In the past we developed a semantics for a restricted annotated logic language for inheritance reasoning. Here we generalize it to annotated Horn logic programs. We first provide a formal account of the language, describe its semantics, and provide an interpreter written in Prolog for it. We then investigate its relationship to Belnap's 4-valued logic, Gelfond and Lifschitz's semantics for logic programs with negation, Brewka's prioritized default logics and other annotated logics due to Kifer et al.     

Possibilistic uncertainty handling for answer set programming

In this work, we introduce a new framework able to deal with a reasoning that is at the same time non monotonic and uncertain. In order to take into account a certainty level associated to each piece of knowledge, we use possibility theory to extend the non monotonic semantics of stable models for logic programs with default negation. By means of a possibility distribution we define a clear semantics of such programs by introducing what is a possibilistic stable model. We also propose a syntactic process based on a fix-point operator to compute these particular models representing the deductions of the program and their certainty. Then, we show how this introduction of a certainty level on each rule of a program can be used in order to restore its consistency in case of the program has no model at all. Furthermore, we explain how we can compute possibilistic stable models by using available softwares for Answer Set Programming and we describe the main lines of the system that we have developed to achieve this goal.     

A logic programming system for nonmonotonic reasoning

The evolution of logic programming semantics has included the introduction of a new explicit form of negation, beside the older implicit (or default) negation typical of logic programming. The richer language has been shown adequate for a spate of knowledge representation and reasoning forms.The widespread use of such extended programs requires the definition of a correct top-down querying mechanism, much as for Prolog wrt. normal programs. One purpose of this paper is to present and exploit a SLDNF-like derivation procedure, SLX, for programs with explicit negation under well-founded semantics (WFSX) and prove its soundness and completeness. (Its soundness wrt. the answer-sets semantics is also shown.) Our choice ofWFSX as the base semantics is justi-fied by the structural properties it enjoys, which are paramount for top-down query evaluation.Of course, introducing explicit negation requires dealing with contradiction. Consequently, we allow for contradiction to appear, and show moreover how it can be removed by freely changing the truth-values of some subset of a set of predefined revisable literals. To achieve this, we introduce a paraconsistent version ofWFSX, WFSX _p , that allows contradictions and for which our SLX top-down procedure is proven correct as well.This procedure can be used to detect the existence of pairs of complementary literals inWESX _p simply by detecting the violation of integrity rulesf L, -L introduced for eachL in the language of the program. Furthermore, integrity constraints of a more general form are allowed, whose violation can likewise be detected by SLX.Removal of contradiction or integrity violation is accomplished by a variant of the SLX procedure that collects, in a formula, the alternative combinations of revisable literals' truth-values that ensure the said removal. The formulas, after simplification, can then be satisfied by a number of truth-values changes in the revisable, among true, false, and undefined. A notion of minimal change is defined as well that establishes a closeness relation between a program and its revisions. Forthwith, the changes can be enforced by introducing or deleting program rules for the revisable literals.To illustrate the usefulness and originality of our framework, we applied it to obtain a novel logic programming approach, and results, in declarative debugging and model-based diagnosis problems.     

Autoepistemic logic of first order and its expressive power

We study the expressive power of first-order autoepistemic logic. We argue that full introspection of rational agents should be carried out by minimizing positive introspection and maximizing negative introspection. Based on full introspection, we propose the maximal well-founded semantics that characterizes autoepistemic reasoning processes of rational agents, and show that breadth of the semantics covers all theories in autoepistemic logic of first order, Moore's AE logic, and Reiter's default logic. Our study demonstrates that the autoepistemic logic of first order is a very powerful framework for nonmonotonic reasoning, logic programming, deductive databases, and knowledge representation.This research is partially supported by NSERC grant OGP42193.     

一种新的辩论推理模式及其应用

文中定义了一个新的辩论推理模式,建立了一个形式化的知识表示框架,并把它应用于研究扩展逻辑程序类的说明语义,结果表明,新语义克服了择优语义的不足。作者还根据上述研究结果实现了逻辑程序设计风格下的知识框架。     

N-SHOQ(D): 描述逻辑SHOQ(D)的一个非单调扩展

描述逻辑SHOQ(D)给出了Web本体语言DAML+OIL的语义,但SHOQ(D)只能处理严格成立的完备 知识,不能处理在实际情况中经常出现的不完备知识.对描述逻辑SHOQ(D)进行扩展,提出了 能够处理不完备知识的非单调描述逻辑N-SHOQ(D).给出了N-SHOQ(D)的语法和语义,定义了N -SHOQ(D)中的蕴涵推理关系,研究了N-SHOQ(D)所具有的性质. N-SHOQ(D)为扩展DAML+OIL语 言到能够处理不完备知识的情形提供了语义支持.     

A proof procedure for the logic of hereditary Harrop formulas

A proof procedure is presented for a class of formulas in intuitionistic logic. These formulas are the so-calledgoal formulas in the theory of hereditary Harrop formulas. Proof search in intuitionistic logic is complicated by the nonexistence of a Herbrand-like theorem for this logic: formulas cannot in general be preprocessed into a form such as the clausal form, and the construction of a proof is often sensitive to the order in which the connectives and quantifiers are analyzed. An interesting aspect of the formulas we consider here is that this analysis can be carried out in a relatively controlled manner in their context. In particular, the task of finding a proof can be reduced to one of demonstrating that a formula follows from a set of assumptions, with the next step in this process being determined by the structure of the conclusion formula. An acceptable implementation of this observation must utilize unification. However, since our formulas may contain universal and existential quantifiers in mixed order, care must be exercised to ensure the correctness of unification. One way of realizing this requirement involves labelling constants and variables and then using these labels to constrain unification. This form of unification is presented and used in a proof procedure for goal formulas in a first-order version of hereditary Harrop formulas. Modifications to this procedure for the relevant formulas in a higher-order logic are also described. The proof procedure that we present has a practical value in that it provides the basis for an implementation of the logic programming language Prolog.     

A Variational Model for Capturing Illusory Contours Using Curvature

Illusory contours, such as the classical Kanizsa triangle and square [9], are intrinsic phenomena in human vision. These contours are not completely defined by real object boundaries, but also include illusory boundaries which are not explicitly present in the images. Therefore, the major computational challenge of capturing illusory contours is to complete the illusory boundaries. In this paper, we propose a level set based variational model to capture a typical class of illusory contours such as Kanizsa triangle. Our model completes missing boundaries in a smooth way via Euler’s elastica, and also preserves corners by incorporating curvature information of object boundaries. Our model can capture illusory contours regardless of whether the missing boundaries are straight lines or curves. We compare the choice of the second order Euler’s elastica used in our model and that of the first order Euler’s elastica developed in Nitzberg-Mumford-Shiota’s work on the problem of segmentation with depth [15, 16]. We also prove that with the incorporation of curvature information of objects boundaries our model can preserve corners as completely as one wants. Finally we present the numerical results by applying our model on some standard illusory contours. This work has been supported by ONR contract N00014-03-1-0888, NSF contract DMS-9973341 and NIH contract P20 MH65166. Wei Zhu received the B.S. degree in Mathematics from Tsinghua University in 1994, the M.S. degree in Mathematics from Peking University in 1999, and the Ph.D. degree in Applied Mathematics from UCLA in 2004. He is currently a Postdoc at Courant Institute, New York University. His research interests include mathematical problems in image processing and visual neuroscience. Tony Chan received the B.S. degree in engineering and the M.S. degree in aerospace engineering, both in 1973, from the California Institute of Technology, Pasadena, and the Ph.D. degree in computer science from Stanford University, Stanford, CA, in 1978. Heis currently the Dean of the Division of Physical Science and College of Letters and Science, UCLA, where he has been a Professor in the Department of Mathematics since 1986. His research interests include PDE methods for image processing, multigrid, and domain decomposition algorithms, iterative methods, Krylov subspace methods, and parallel algorithms.     

基于关键迹和ASP的CSP模型检测

模型检测是通信顺序进程(communicating sequential processes,简称CSP)形式化验证的重要手段.当前, CSP模型检测方法基于操作语义,需将进程转化为迁移系统,进而提取语义模型,但转化过程较为复杂;待验证性质采用CSP语言进行描述,虽然有利于精炼检测(refinement checking),但描述能力较弱,通用性不强.鉴于此,提出了一种新的CSP指称语义模型——关键迹模型(critical-trace model)及基于该指称语义模型的CSP模型检测方法,并证明了其验证的可靠性,避免了上述问题.关键迹模型采用递归策略计算,待验证性质采用线性时态逻辑(linear temporal logic,简称LTL)描述.基于回答集程序设计(answer set programming,简称ASP)实现了关键迹模型的自动生成及LTL的自动验证,并开发了一个CSP模型检测原型系统——T_ASP.实验结果表明:与类似系统相比,该系统的描述能力更强,验证结果的准确性更高,且可同时验证多条性质,在性质不满足时还可提供多条反例.     

The density of truth in monadic fragments of some intermediate logics

This paper is an attempt to count the proportion of tautologies of some intermediate logics among all formulas. Our interest concentrates especially on Medvedev’s logic and its fragment over language with one propositional variable.     

A generalization of Cobham’s theorem to automata over real numbers

This article studies the expressive power of finite-state automata recognizing sets of real numbers encoded positionally. It is known that the sets that are definable in the first-order additive theory of real and integer variables 〈R,Z,+,<〉$〈\mathbb{R},\mathbb{Z},+,<〉$ can all be recognized by weak deterministic Büchi automata, regardless of the encoding base r>1$r>1$. In this article, we prove the reciprocal property, i.e., a subset of R$\mathbb{R}$ that is recognizable by weak deterministic automata in every base r>1$r>1$ is necessarily definable in 〈R,Z,+,<〉$〈\mathbb{R},\mathbb{Z},+,<〉$. This result generalizes to real numbers the well-known Cobham’s theorem on the finite-state recognizability of sets of integers. Our proof gives interesting insight into the internal structure of automata recognizing sets of real numbers, which may lead to efficient data structures for handling these sets.     

Average Case Analysis of Gosper’s Algorithm for a Class of Urn Model Inputs

In this paper we perform an asymptotic average case analysis of some of the most important steps of Gosper’s algorithm for indefinite summation of hypergeometric terms. The space of input functions of the algorithm is described in terms of urn models, and the analysis is performed by using specialized probabilistic transform techniques. We analyze two different types of urn model classes: one in which the input functions are assumed to be rational, and another for which a certain function of two inputs is assumed to be rational. The first set of results shows that the asymptotic complexity of the algorithm is the same within each of the two classes. The second set of results indicates that the complexity of the algorithm scales differently for the two classes of models: one can observe the logarithmic versus square root type of difference that is also present in other combinatorial models.