Y. Crama, P. Hammer, and T. Ibaraki, "Cause--effect relationships and partially defined Boolean functions," Ann. Oper. Res., vol. 16, pp. 299--326, 1988.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....neural network architectures permit to construct a network in a straight forward manner for Boolean problems and can be trained iteratively. 2 The Approach It is relatively easy to construct higher order perceptrons or other types of neural networks for Boolean problems, as shown for example in [3], 6] 4] 16] 10] and [2] The networks produced by the algorithms presented in these papers differ mainly in properties like generalization performance, size, depth, weight range, etc. Unfortunately, these algorithms take advantage of the data being Boolean and are therefore not directly ....

....that the omitted data remain in the original data set used during the training phase and there is no reason why they should not be learnable by the constructed network. 2 fl Calculation of the Boolean Expression A for this approach suitable algorithm was initially developed by Y. Crama et al. [3] and further developed, as well as implemented by E. Mayoraz [9] This algorithm is in principle very similar to other algorithms for the calculation of Boolean expressions on the base of some Boolean vectors given. This program offers, besides being optimized in calculation time, some useful ....

Yves Crama, Peter L. Hammer, and Toshhihide Ibaraki. Cause-Effect Relationships and Partially Defined Boolean Functions. Technical Report # 39-88, Rutcor, Rutgers University, New Brunswick, N.J. 08903, USA, 1988.

....topological properties, such as having connected level sets, which has led to successful interior point methods for several problems, e.g. 12] 9] 13] 18] Guided by these results, we investigate the application of a similar approach for inductive inference. For related work, see e.g. 2] [5] and [25] Before we continue, we require some definitions. Consider the Boolean function F : f0; 1g n f0; 1g. An element of the domain of F is called a minterm of F . The set of minterms for which F evaluates to 1 (0) is called the on set (off set) An incompletely specified Boolean function ....

Y. Crama, P.L. Hammer, and T. Ibaraki. Cause-effect relationships and partially defined Boolean functions. Annals of O.R., 16:299--325, 1988.

....and false vectors v 2 f0; 1g n , respectively, where T F = It naturally generalizes a Boolean function, by allowing that the range function values on some input vectors are unknown. This concept has many applications, e.g. in circuit design, for representation of cause effect relationships [7], or in learning, to mention a few. A principal issue on pdBfs is the following: Given a pdBf (T ; F ) determine whether some f in a particular class of Boolean functions C exists such that T T (f) and F F (f) where T (f) and F (f) denote the sets of true and false vectors of f , ....

....that the extension problem for C 1 DL can be solved in linear time. This can be regarded as a positive result, since the extension problem for the renaming closures of classes that contain C 1 DL is mostly intractable, e.g. for C R Horn , C R pos , C R R 1 = CR 1 , C R 2M = C 2M [7, 6], or no linear time algorithms are known. We describe here an algorithm EXTENSION for the equivalent class CLR 1 , which uses Lemma 4.1 for a recursive extension test; it is similar to the more general algorithm described in [31] and also a relative of the algorithm total recall in [20] ....

Y. Crama, P. Hammer, and T. Ibaraki. Cause-Effect Relationships and Partially Defined Boolean Functions. Annals of Operations Research, 16:299--326, 1988.

....1; K, the set F Gamma1 (k) is called the class k. There is a wide variety of algorithms available in the literature to handle classification problems. They originate in different domains such as : statistics (e.g. Bayesian classifiers, see [DH73] logic (e.g. logical analysis of data [CHI88, BHI 96] neural networks (e.g. perceptron algorithm [Ros58] backpropagation [Wer74] artificial intelligence (e.g. decision trees [BOS84, Qui86] While some of them are suitable to learn polychotomies (multi layered perceptrons, decision trees) others are designed only for dichotomies ....

Yves Crama, Peter L. Hammer, and Toshihide Ibaraki. Cause-effect relationships and partially defined boolean functions. Annals of Operations Research, 16:299--326, 1988.

....MODEL is somewhat related to the extension problem for double Horn functions [16] where the extension problem is to establish a Boolean function f that is consistent with a given partially defined Boolean function (pdBf) T ; F ) i.e. f(v) 1 (resp. 0) holds for all v 2 T (resp. v 2 F ) [4, 11], and a double Horn extension f is a natural restriction of Horn function. This relationship is by the kind of efficient algorithms for solving the extension problem and problem MODEL, which results from related inherent subproblems. However, no deep semantical relation exists. Further operations ....

Y. Crama, P. Hammer, and T. Ibaraki. Cause-Effect Relationships and Partially Defined Boolean Functions. Annals of Operations Research, 16:299--326, 1988.

....we present the results of a series of computational experiments which show the high degree of robustness of spanned patterns. 1 Introduction Logical Analysis of Data (LAD) is a method based on combinatorics, optimization, and Boolean logic, for data analysis. LAD was first introduced in [10] [8] as a method for the analysis of binary data, and extended later in [6] to the analysis of datasets having numerical independent variables and binary outcomes (positive and negative observations) LAD produces highly accurate, completely reproducible, and robust classification models with high ....

....N A N A pid 1000 N A N A N A # Spanned Patterns Sensitivity of SPIC to discretization. We shall show next that the input consensus algorithm has another important quality, namely, it has a very low sensitivity to the increase of the discretization grid resolution. Originally, LAD was proposed [8] as a method for the analysis of binary (0 1) data. In [6] it was shown that the applicability of LAD can be extended to the analysis of problems with numerical data, by replacing each numerical variable by several binary ones. Also in [6] it was shown that the choice of a support set using a ....

Crama, Y., Hammer, P.L., and Ibaraki, T. Cause-Effect Relationships and Partially Defined Boolean Functions. Annals of Operations Research 16 (1988), 299-326.

....of study of classical disciplines, like statistics, and newer ones, like machine learning and data mining. Numerous methods have been developed to address this type of problems, and substantial success with their use is reported in the literature. The Logical Analysis of Data (LAD) introduced in [8], 9] is a combinatorial approach to the analysis of datasets consisting of positive and negative observations, each of which is represented as a vector of n (usually real) attribute values. It has been established in previous studies ( 7] that LAD provides a competitive classification tool ....

....The relative positive (respectively, negative) coverage of a term C is defined as ) T C COV (respectively ) F C COV ) LAD was built around two central concepts: positive or negative) patterns and (positive or negative) theories. Following the terminology of [8], for any number c (0,1] a term C will be called a positive c pattern of (T,F) if 4 1. COV (C) and 2. COV (C) c COV (C) and c will be called the homogeneity of C. Note that condition 1 is equivalent to the condition that C(Z) 1 for at least one vector Z T. It should be ....

Y. Crama, P.L. Hammer, and T. Ibaraki. Cause-Effect Relationships and Partially Defined Boolean Functions. Annals of Operations Research 16 (1988) 299-326.

....called Logical Analysis of Data (LAD) is based on combinatorics, optimization and the theory of Boolean functions, and was adapted to handle the type of inseparable data which are typical for medical applications. LAD was introduced in 1986 ( 9] the first paper on the topic appeared in 1988 [6], and since then numerous papers (e.g. 1] 2] 7] 8] have been devoted to various mathematical, computational and applied aspects of this method. The objective of this study was the construction of a model for distinguishing groups of patients at high and at low mortality risk. Risk ....

....having their resting heart rate of at least 97 beats minute, i.e. those satisfying the conditions = 97, RETSTHR 0 GENDER (2) includes 5 positive observations from the CCF dataset and no negative observations, thus representing a positive pattern. Positive and negative patterns, introduced in [6], represent in fact one of the fundamental tools used in LAD, and were seen in [2] to give a strong insight into the structure of numerous practical problems of data analysis. Positive and negative patterns, in their original pure form, represent homogeneous intervals in R , containing only ....

[Article contains additional citation context not shown here]

Y. Crama, P.L. Hammer, T. Ibaraki, Cause-Effect Relationships and Partially Defined Boolean Functions, Annals of Operations Research. 16 (1988) 299-326

....MODEL is somewhat related to the extension problem for double Horn functions [16] where the extension problem is to establish a Boolean function f that is consistent with a given partially defined Boolean function (pdBf) T ; F ) i.e. f(v) 1 (resp. 0) holds for all v 2 T (resp. v 2 F ) [4, 11], and a double Horn extension f is a natural restriction of Horn function. This relationship comes from the similarity between two efficient algorithms for solving the extension problem and problem MODEL. However, no deep semantical relation is known. Further operations in combining theories ....

Y. Crama, P. Hammer, and T. Ibaraki. Cause-Effect Relationships and Partially Defined Boolean Functions. Annals of Operations Research, 16:299--326, 1988.

....if and only if there is such an algorithm for dualizing a positive function. Keywords: Boolean functions, Horn formulas, satisfiability, partially defined Boolean functions, characteristic models, polynomial algorithms 1 Introduction The concept of a partially defined Boolean function (pdBf) [5, 8] is a natural generalization of the familiar concept of a Boolean function, by allowing that the function values on some input vectors are unknown. Those pdBfs have many applications, in particular in computer science and knowledge engineering. The major part of this research was conducted ....

....the inputs on which the circuit must output 1 and the inputs on which it must output 0; the output on the remaining inputs remains unspecified and is considered as don t care . Another application of pdBfs is with the representation of incomplete information about cause effect relationships [8]. e.g. the effect of a number of facts (e.g. a patient is male, is a smoker etc. on a specific disease (e.g. cancer) can be modeled as a Boolean function f(x 1 ; x 2 ; x n ) where the arguments x i represent presence of the facts, and the value of f tells whether the disease is ....

[Article contains additional citation context not shown here]

Y. Crama, P. L. Hammer and T. Ibaraki, Cause-effect relationships and partially defined Boolean functions, Annals of Operations Research, 16 (1988) 299-326.

.... data, Boolean functions, decomposable functions, computational learning theory, random graphs, probabilistic analysis 1 Introduction Extracting knowledge from given data has been studied in such fields as knowl edge engineering, data mining, artificial intelligence and database theory (e.g. [4, 6, 8]) Logical analysis of data (LAD) is one of the methodologies for knowledge discovery. LAD is based on Boolean logic, that is, a given data set is repre sented as a pair of set T of true vectors (examples that cause the phenomenon to occur) and set F of false vectors (examples not causing the ....

....in the class of all Boolean functions. It may not be trivial, however, to judge whether a given pdBf has an extension in a certain subclass C of Boolean functions, such as the class of positive functions, the class of k DNF functions (DNF functions with at most k literals in each term) and so on [6]. The problem to find an extension in a subclass C of a given pdBf (T, F) is called the consistency problem in computational learning theory [2] For a subset C 5: let 0,1 denote the vector space defined by an attribute set . Given a pair of subsets o, 1 C , a function f is called ....

Y. Crama, P.L. Hammer and T. Ibaraki, Cause-effect relationships and partially defined Boolean functions, Annals of Operations Research, 16 (1988) 299-325.

....question of LAD is to learn as much as possible about the structure of the sets W and W . In RRR 59 2001 PAGE 16 particular, LAD can be used for classification , i.e. for establishing the positive or negative nature of points not included in W . The original version of LAD ( 10] [5]) dealt with the case of binary (0 1) data. In a subsequent development ( 2] 3] the methodology was extended to the case of numerical data, based on a transformation which converted each real valued variable into several binary ones. This binarization process transforms a problem with ....

Y. Crama, P.L. Hammer, T. Ibaraki, Cause-effect relationships and partially defined Boolean functions, Annals of Operations Research, 16 (1988) 299-326

....and false vectors v 2 f0; 1g n , respectively, where T F = It naturally generalizes a Boolean function, by allowing that the range function values on some input vectors are unknown. This concept has many applications, e.g. in circuit design, for representation of cause effect relationships [7], or in learning, to mention a few. A principal issue on pdBfs is the following: Given a pdBf (T ; F ) determine whether some f in a particular class of Boolean functions C exists such that T T (f) and F F (f) where T (f) and F (f) denote the sets of true and false vectors of f , ....

....showing that the extension problem for C 1 DL can be solved in linear time. This can be regarded as a positive result, since the extension problem for the renaming closures of classes that contain C 1 DL is mostly intractable, e.g. for C R Horn , C R pos , C R R 1 = CR 1 , C R 2M = C 2M [7, 6], or no linear time algorithms are known. We describe here an algorithm EXTENSION (see Figure 1) which outputs a 1 decision list for an extension of a given pdBf (T ; F ) It uses Lemma 4.1 for the equivalent class CLR 1 for a recursive extension test. The algorithm is similar to the more ....

Y. Crama, P. Hammer, and T. Ibaraki. Cause-Effect Relationships and Partially Defined Boolean Functions. Annals of Operations Research, 16:299--326, 1988.

..... 42 7 Dual comparable functions 44 8 Threshold functions 48 9 Decomposable functions 50 10 Conclusion 56 References 56 Summary tables of results 57 Page 2 RRR 6 96 1 Introduction Knowledge acquisition in the form of Boolean logic has been intensively studied in the recent research (e.g. [3, 5, 8, 12, 18]) given a set of data, represented as a set T of binary true n vectors (or positive examples ) and a set F of false n vectors (or negative examples ) establish a Boolean function (extension) f in a specified class C, such that f is true (resp. false) in every given true (resp. false) ....

Y. Crama, P. L. Hammer and T. Ibaraki, Cause-effect relationships and partially defined boolean functions, Annals of Operations Research 16 (1988) 299-326.

....and Culture of Japan. The third author is supported by Research Fellowships of the Japan Society for the Promotion of Science for Young Scientists. RRR 7 96 Page 1 1 Introduction Knowledge acquisition in the form of Boolean logic has been intensively studied in the recent research (e.g. [4, 6, 10]) given a set of data, represented as a set T of binary true n vectors (or positive examples ) and a set F of false n vectors (or negative examples ) establish a Boolean function (extension) f , such that f is true (resp. false) in every given true (resp. false) vector; i.e. T T (f) ....

Y. Crama, P. L. Hammer and T. Ibaraki, Cause-effect relationships and partially defined boolean functions, Annals of Operations Research 16 (1988) 299-326.

....the boolean function that predicts, on the basis of the test results, whether a substance is carcinogenic. In a third class of applications we want to explain an effect. A physician, for instance, may want to determine which combinations of foods cause his patient s suspected food allergy [5]. He asks the patient to record, each day, whether he eats certain foods, and whether an allergic reaction develops. The object is to find the boolean function that relates the occurrence of a reaction to which foods were eaten. 1.2 Advantages of Boolean Regression We propose a regression based ....

....x. It may be useful, however, to find a relatively simple logical formula that agrees with f on all observed x s. One may wish the formula to be a disjunction of as few terms as possible, or to involve as few x j s as possible. Techniques for obtaining such a formula are discussed in detail in [5]. 5 Positive Functions 5.1 Bayesian Estimation It is frequently the case that one has to find a good boolean regression in the presence of some a priori structural information on the function f , e.g. f must be a monotone boolean function, or more general, there is known a partial order OE on ....

Crama, Y., P.L. Hammer and T. Ibaraki, Cause-effect relationships and partially defined Boolean functions, Annals of Operations Research 16 (1988) 299-325.

....tax auditing, medicine, and real estate valuation. In particular, we display a special class of problems for which the best mono tone prediction can be found in polynomial time. 1 Introduction The problem of establishing cause effect relationship based on incomplete observations was studied in [4]. In this paper we address the problem of finding a good approximation of an unknown discrete function on the basis of a set of observations, which is incomplete in two senses. We observe the values of only GSIA Working Paper 1991 12. The first and second authors gratefully acknowledge the partial ....

Crama, Y., P.L. Hammer and T. Ibaraki, "Cause-effect relationships and partially defined Boolean functions", Annals of Operations Research 16 (1988) 299-325.

....The observations fall in two classes: positive ones (e.g. healthy patients, potenitial buyers, oil rich areas, etc. and negative ones (e.g. sick patients, etc. The goal of data analysis is to arrive to logical explanations distinguishing positive and negative observations. It was shown in [10, 14] that in the case of binary data this goal can be achived by the use of partialy defined Boolean functions. The usefulness of logical or Boolean techniques is an important ingredient of many approaches in this field [1, 9, 11, 16, 21, 22] The classification power of the Boolean based ....

....or to find an extension having a special functional form (e.g. positive) or both. The problem of checking whether a pdBf (i.e. a binary data set partitioned into two classes) has an extension in a specified class C has been studied in relation to the detection of cause effect relationships [10, 14], learning and identification of a Boolean function [1, 4, 23] structural analysis of binary data [5] and others. Beside studying extensions in the class of all Boolean functions, we shall pay special attention to the important classes of monotone, Horn, threshold, linear and quadratic ....

[Article contains additional citation context not shown here]

Y. Crama, P. L. Hammer and T. Ibaraki, Cause-effect relationships and partially defined boolean functions, Annals of Operations Research, 16 (1988) 299-326.

....applications. Given a set of true points and a set of false points, the central question of logical analysis of data (LAD) is the study of those Boolean functions (called extensions of data sets) whose values agree with those of the given points. The basic concepts of LAD are introduced in [9], and an implementation of LAD is described in [6] A typical data set will usually have exponentially many extensions. In the absence of any additional information about the properties of the data set, the choice of an extension would be totally arbitrary, and therefore would risk to omit the ....

Y. Crama, P.L. Hammer, and T. Ibaraki. Cause-effect relationships and partially defined Boolean functions, Annals of Operations Research, 16 (1988), 299--326.

.... Bidual Horn Functions (extended abstract) Thomas Eiter y Toshihide Ibaraki z Kazuhisa Makino z 1 Introduction A partially defined Boolean function (pdBF) is a natural generalization of the familiar concept of Boolean function, by allowing that the function value on some vectors is unknown [5]. Such functions have many applications in computer science, including representation of incomplete information about cause effect relationships [5] switching circuit design, and machine learning. Formally, a pdBF is a pair (T ; F ) of sets T and F of true and false vectors v 2 f0; 1g n , ....

.... (pdBF) is a natural generalization of the familiar concept of Boolean function, by allowing that the function value on some vectors is unknown [5] Such functions have many applications in computer science, including representation of incomplete information about cause effect relationships [5], switching circuit design, and machine learning. Formally, a pdBF is a pair (T ; F ) of sets T and F of true and false vectors v 2 f0; 1g n , respectively, where T F = Clearly, each pdBF can be completed to a total Boolean function (which has T [ F = f0; 1g n ) by arbitrarily assigning ....

[Article contains additional citation context not shown here]

Y. Crama, P. L. Hammer and T. Ibaraki, Cause-effect relationships and partially defined boolean functions, Annals of Operations Research, 16 (1988) 299-326.

No context found.

Y. Crama, P. Hammer, and T. Ibaraki, "Cause--effect relationships and partially defined Boolean functions," Ann. Oper. Res., vol. 16, pp. 299--326, 1988.

No context found.

Y. Crama, P. Hammer, and T. Ibaraki, "Cause--effect relationships and partially defined Boolean functions," Ann. Oper. Res., vol. 16, pp. 299--326, 1988.

No context found.

Crama,Y., Hammer,P.L. and Ibaraki,T. 1988. Cause-Effect Relationships and Partially Defined Boolean Functions. Annals of Operations Research, 16: 299--326.

No context found.

Boros,E., Hammer,P.L. and Ibaraki,T. (1988) Cause-Effect Relationships and Partially Defined Boolean Functions. Annals of Operations Research, 16, 299--326.

Online articles have much greater impact   More about CiteSeer.IST   Add search form to your site   Submit documents   Feedback  

CiteSeer.IST - Copyright Penn State and NEC