Fast algorithms for revision of some special propositional knowledge bases

In this paper,the computational complexity of propositional clause set counter-factuals is discussed.It is shown that the computational complexity of propositional clause set counterfactuals is at the second level of the polynomial hierarchy,and that the computational complexity of propositional Horn clause set counterfactuals is at the first level of the polynomial hierarchy.Furthermore,some polynomial algorithms are presented for some special propositional clauset set ,such as the unique satisfiable clause set and the clause set of which only one subset is minimally inconsistent with the input clause whose inconsistency check can be solved in polynomial time.     

一阶子句搜索方法

子句集的可满足性判定是自动证明领域的热点之一.提出了子句搜索方法判定命题子句集Φ的可满足性,该方法查找Φ中子句的一个公共不可扩展子句C,当且仅当找到C时Φ可满足,此时C中各文字的补构成一个模型.结合部分实例化方法将子句搜索方法提升至一阶.一阶子句搜索方法可以判定子句集的M可满足性,具备终止性、正确性和完备性,是一种判定子句集可满足性的有效方法.     

Arithmetical hierarchy and complexity of computation

Various sets of Turing machines naturally occuring in the theory of computational complexity are shown to be complete on the respective levels of the arithmetical hierarchy. Results saying that various assertions concerning computational complexity (e.g. some relativizations of the P = NP problem) are independent of formal systems like set theory are obtained as corollaries. Provable complexity classes are also investigated.     

A survey of one-way functions in complexity theory

In complexity theory a one-way function is defined to be a one-one, honest, function that is computable in polynomial time whose inverse is not computable in polynomial time. We examine relationships between the complexity of functional computational problems and ordinary set recognition problems. The complexity of inverting one-way functions follows from these relationships. Then we survey various forms of one-way functions that have arisen in relationship to some cryptographic investigations and in relationship to the isomorphism problem.The author acknowledges support by the National Science Foundation under Grant CCR-9002292.     

A uniform approach to obtain diagonal sets in complexity classes

A uniform method for constructing sets which diagonalize over certain complexity classes while preserving other complexity properties is given. We obtain some known results as well as some new ones as corollaries of our main theorem. The new results concern the complexity classes P, NP, co-NP, PSPACE, APT (almost polynomial time), R (random polynomial time), and the polynomial hierarchy.     

Some observations on NP real numbers and P-selective sets

We show that each standard left cut of a real number is a p-selective set. Classification results about NP real numbers and recursively enumerable real numbers follow from similar results about p-selective and semirecursive sets. In particular, it is proved that no left cut can be NP-hard unless the polynomial hierarchy collapses to ?₂^P. This result is surprising because it shows that the McLaughlin-Martin construction of a ?_tt-complete r.e. semirecursive set fails at the polynomial time complexity level.     

Robust Hierarchical Indexing based on Texture Features

In this paper, we present a hierarchical indexing method based on texture characterization for image retrieval. The novelty of our contribution is the hierarchical structure of the index: it exploits the multiresolution formulation of Wavelet Transforms to define a new set of approximated versions of the images for each level of resolution. On this set, the algorithm extracts significant signatures by means of statistical correlations; the experimental results and the analysis of computational complexity have proved that the algorithm presents the best performance at the highest level of the indexing hierarchy, where the computational complexity is the lowest. Our method has been evaluated by the following methodologies: (a) the study of the computational complexity for signature generation; (b) the comparison with analogous methods based on texture analysis by reporting the performance obtained on the same database (Brodatz); and (c) the evaluation of the robustness of the hierarchical indexing in different color spaces, by querying the database with different versions of the original images obtained by noise addition (gaussian and scanner acquisition noise and lossy compression distortion), brightness and contrast enhancement, color and scale adjustment and rotation. Even if our method is designed for texture databases, experiments show satisfactory results also on a real heterogeneous photographic database. This confirms the possibility of exploiting our method as a low computational complexity indexing tool based on texture characterization in a broader system for hierarchical content-based retrieval.     

Redundancy in logic II: 2CNF and Horn propositional formulae

We report results about the redundancy of formulae in 2CNF form. In particular, we give a slight improvement over the trivial redundancy algorithm and give some complexity results about some problems related to finding Irredundant Equivalent Subsets (i.e.s.) of 2CNF formulae. The problems of checking whether a 2CNF formula has a unique i.e.s. and checking whether a clause in is all its i.e.s.'s are polynomial. Checking whether a 2CNF formula has an i.e.s. of a given size and checking whether a clause is in some i.e.s.'s of a 2CNF formula are polynomial or NP-complete depending on whether the formula is cyclic. Some results about Horn formulae are also reported.     

Implicit complexity over an arbitrary structure: Quantifier alternations

We provide machine-independent characterizations of some complexity classes, over an arbitrary structure, in the model of computation proposed by L. Blum, M. Shub, and S. Smale. We show that the levels of the polynomial hierarchy correspond to safe recursion with predicative minimization and the levels of the digital polynomial hierarchy to safe recursion with digital predicative minimization. Also, we show that polynomial alternating time corresponds to safe recursion with predicative substitutions and that digital polynomial alternating time corresponds to safe recursion with digital predicative substitutions.     

消除子句冗余的算法及复杂性

本文讨论了知识子句表示的冗余消除，着重讨论了子句中冗余文字的消除的民政部压缩是消除子句文字冗余的一种重要类型除对这种类型的问题的可判定性以及复杂性结果进行讨论外，本文还给出了压缩问题的一个子问题的多项式算法。     

Locally Determined Logic Programs and Recursive Stable Models

In general, the set of stable models of a recursive logic program can be quite complex. For example, it follows from results of Marek, Nerode, and Remmel [Ann. Pure and Appl. Logic (1992)] that there exists finite predicate logic programs and recursive propositional logic programs which have stable models but no hyperarithmetic stable models. In this paper, we shall define several conditions which ensure that recursive logic program P has a stable model which is of low complexity, e.g., a recursive stable model, a polynomial time stable model, or a stable model which lies in a low level of the polynomial time hierarchy.     

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.     

Complexity parameters for first order classes

We study several complexity parameters for first order formulas and their suitability for first order learning models. We show that the standard notion of size is not captured by sets of parameters that are used in the literature and thus they cannot give a complete characterization in terms of learnability with polynomial resources. We then identify an alternative notion of size and a simple set of parameters that are useful for first order Horn Expressions. These parameters are the number of clauses in the expression, the maximum number of distinct terms in a clause, and the maximum number of literals in a clause. Matching lower bounds derived using the Vapnik Chervonenkis dimension complete the picture showing that these parameters are indeed crucial. This work has been partly supported by NSF Grant IIS-0099446. A preliminary version of this paper appeared in the proceeding of the conference on Inductive Logic Programming 2003. Most of this work was done while M.A. was at Tufts University. Editors: Tamás Horváth and Akihiro Yamamoto     

Exact satisfiability,a natural extension of set partition,and its average case behavior

The problem of determining whether a Boolean formula in conjunctive normal form is satisfiable in such a way that in each clause exactly one literal is set true, and all the other literals are set false is called the exact satisfiability problem. The exact satisfiability problem is well known to be NP-complete [5] and it contains the well known set partitioning problem as a special case. We study here the average time complexity of a simple backtracking strategy for solving the exact satisfiability problem under two probability models, the constant density model and the constant degree model. For both models we present results sharply separating classes of instances solvable in low degree polynomial time in the average from classes for which superpolynomial or exponential time is needed in the average.     

A linear programming approach to reasoning about probabilities

We continue our study, initiated in [9], of the following computational problem proposed by Nilsson: Several clauses (Boolean functions of several variables) are given, and for each clause the probability that the clause is true is specified. We are asked whether these probabilities are consistent. They are if there is a probability distribution on the truth assignments such that the probability of each clause is the measure of its satisfying set of assignments. Since this is a generalization of the satisfiability problem of predicate calculus, it is immediately NP-hard. In [9] we showed certain restricted cases of the problem to be NP-complete, and used the Ellipsoid Algorithm to show that a certain special case is in P. In this paper we use the Simplex method, column generation techniques, and variable-depth local search to derive an effective heuristic for the general problem. Experiments show that our heuristic performs successfully on instances with many dozens of variables and clauses. We also prove several interesting complexity results that answer open questions in [9] and motivate our approach.     

子空间搜索广义主成分分析

GPCA(Generalized Principal Component Analysis)是近几年提出的一种数据聚类和降维方法,它通过将样本聚类为不同的子空间得到样本的低维表达.GPCA方法已经被应用于图像分割、图像聚类等问题.原有的GPCA算法具有指数计算复杂度,很难应用于高维数据的实际处理.文中针对此问题,提出了基于子空间搜索的SGPCA算法,将聚类问题分解为单个平面的单个垂直向量的搜索问题,对不同子空间分别搜索,从而实现多项式复杂度算法.实验表明,新方法不仅计算复杂度低,而且对噪声的鲁棒性也更强.     

命题逻辑的子句集中文字的分类

检测和消除命题逻辑公式中的冗余文字,是人工智能领域广泛研究的基本问题。针对命题逻辑的子句集中子句的划分,结合冗余子句和冗余文字的概念,将命题逻辑的子句集中的文字分为必需文字、有用文字和无用文字3类,并分别给出其定义。讨论3种文字与无冗余等价子集的性质,给出其等价子集的等价描述方法。得到题逻辑的子句集中必需文字、有用文字和无用文字的判定方法,借助子句集的可满足性得到3种文字与子句集的可满足性的等价条件。上述结果对命题逻辑中文字属性的判断提供了多种可选择方法,同时为命题逻辑公式的化简奠定了理论基础。     

Extending stratified datalog to capture complexity classes ranging from {\cal P} to {\cal QH}

This paper presents a unified solution to the problem of extending stratified DATALOG to express database complexity classes ranging from is the query hierarchy containing the decision problems that can be solved in polynomial time by a deterministic Turing machine using a constant number of calls to an -oracle. The solution is based on (i) stratified negation as the core of a simple, declarative semantics for negation, (ii) the use of a “choice” construct to capture the nondeterminism of stable models in a disciplined fashion, (iii) the ability to bind a query to the lowest complexity level that includes the problem at hand, and (iv) a general algorithm that adapts its behavior to the desired level of complexity required by the query so that exponential time computation is only required for hard problems.