E. Boros, T. Ibaraki and K. Makino, "Error-free and best-fit extensions of partially defined Boolean functions." Information and Computation 140 (1998) pp. 254-283.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....class of functions. For example, the class of monotone Boolean functions is often representative of real life phenomena and has been studied in the context of learning in such fields as medical diagnosis [5] manufacturing and reliability analysis [6] as well as signal processing [14] In [3], the consistency problem was considered for various classes of Boolean functions, such as positive (monotone) k DNF, h term DNF, self dual, and many others. Specifically, the question of the existence of polynomial time algorithms for various classes of functions was investigated. In this paper, ....

....Tampere, Finland (e mail: ilya cs.tut.fi) Publisher Item Identifier S 1083 4419(01)02508 0. functions is a Post class, since a superposition of monotone Boolean functions results in a monotone Boolean function. The consistency problem for M was shown to be polynomial time solvable in [3]. In fact, this was shown to be the case for all transitive classes. In this paper, the classes under consideration are F 8 , 3 , using the notation of Yablonsky [13] This and other notation is established in Section II. The above classes have practical relevance in the context of ....

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E. Boros, T. Ibaraki, and K. Makino, "Error-free and best-fit extensions of partially defined Boolean functions," Inform. Comput., vol. 140, pp. 254--283, 1998.

....related to such data sets include classification (i.e. identification of the type of a new observation not included in the data set) determination of characteristic properties of observations of the same type, analysis of the role of various attributes, etc. The logical analysis of data (LAD) [21, 13, 6, 5, 4, 7, 20, 8, 18] is a methodology addressing the above kinds of problems. The mathematical foundation of LAD is in discrete mathematics, with a special emphasis on the theory of Boolean functions. Patterns are the key building blocks in LAD [21, 13] as well as in many other rule induction algorithms (such as ....

E. Boros, T. Ibaraki, and K. Makino. "Error-free and best-fit extensions of partially defined Boolean functions", Information and Computation, 140 (2) (1998), 254--283.

....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 , respectively. Any such f is called an extension of f in C, and finding such an f is known as the extension problem [6, 27]. Since in general, a pdBf may have multiple extensions, it is sometimes desired to know all extensions, or to compute an extension of a certain quality (e.g. one described by a shortest formula, or having a smallest set T (f) The extension problem is closely related to problems in machine ....

....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] ....

E. Boros, T. Ibaraki, and K. Makino. Error-free and Best-fit Extensions of Partially Defined Boolean Functions. Information and Computation, 140:254--283, 1998.

....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 ....

E. Boros, T. Ibaraki, and K. Makino. Error-free and Best-fit Extensions of Partially Defined Boolean Functions. Information and Computation, 140:254--283, 1998.

....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 ....

E. Boros, T. Ibaraki, and K. Makino. Error-free and Best-fit Extensions of Partially Defined Boolean Functions. Information and Computation, 140:254--283, 1998. 37

....(resp. F (f) denotes the set of true (resp. false) vectors of f . This is known as the extension problem, and corresponds in a sense to the satisfiability problem of Boolean formulas. The extension problem and variants thereof have been investigated for a number of classes of Boolean functions [8, 6, 5, 28]. Among these classes are Horn functions, which are of central interest in many domains. A function is Horn if it can be represented by a DNF (disjunctive normal form) in which each term contains at most one negative literal. It is well known that the Horn functions f are those whose set F (f) of ....

....at most one negative literal. It is well known that the Horn functions f are those whose set F (f) of false vectors is closed under intersection (see Section 2) they play an important role in artificial intelligence, logical databases, and logic in computer science, cf. 16, 7, 21] As shown in [28, 6], a Horn extension of a pdBf can be found in polynomial time. In fact, a Horn extension for (T ; F ) exists precisely if the true vectors T are disjoint from the closure of the false vectors F under intersection. However, this characterization shows that the Horn extension problem is, in a sense, ....

[Article contains additional citation context not shown here]

E. Boros, T. Ibaraki and K. Makino, Error-free and best-fit extensions of partially defined Boolean functions, Information and Computation, 140 (1998) 254-283.

....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 , respectively. Any such f is called an extension of (T ; F ) in C, and finding such an f is known as the extension problem [6, 30]. Since in general, a pdBf may have multiple extensions, it is sometimes desired to know all extensions, or to compute an extension of a certain quality (e.g. one described by a shortest formula, or having a smallest set T (f) The extension problem is closely related to problems in machine ....

....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 ....

E. Boros, T. Ibaraki, and K. Makino. Error-free and Best-fit Extensions of Partially Defined Boolean Functions. Information and Computation, 140:254--283, 1998.

..... 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) ....

....f by exchanging (or) and Delta (and) as well as the constants 0 and 1. It is easy to see that (f g) d = f d g d , and so on. A function f is called dual minor if f f d , dual major if f f d , dual comparable if f f d or f f d , and self dual if f d = f . It is known [5] that a function f is dual minor (resp. dual major, self dual) if and only if at most (resp. at least, exactly) one of f(a) 1 and f(a) 1 holds for every a 2 IB n . An assignment A of binary values 0 or 1 to k variables x i 1 ; x i 2 ; x i k is called a k assignment, and is ....

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E. Boros, T. Ibaraki and K. Makino, Error-free and best-fit extensions of a partially defined Boolean function, RUTCOR Research Report RRR 14-95, Rutgers University, 1995. RRR 6-96 Page 57

....E(T, F ) #= #. It may not be evident, however, to find out if a given pdBf has an extension belonging to a particular class C of Boolean functions, or not. This problem has been studied in various fields such as learning theory, knowledge discovery, data mining and logical analysis of data [1, 5, 6, 10, 11, 13]. In practical cases, the fact that C #E(T,F ) # might be due to some classification errors in the input. To correct this type of errors, provided that they are not in a large number, one can consider the optimization problem of finding the largest subsets T # # T and F # # F for which ....

....errors in the input. To correct this type of errors, provided that they are not in a large number, one can consider the optimization problem of finding the largest subsets T # # T and F # # F for which E(T # , F # ) # C #= # holds. These problems have extensively been studied (e.g. in [10, 13]) for a large variety of classes. In this paper we shall consider another type of error in the input, the case in which some data vectors are incomplete in the sense that some of their components are not available at the time of reading the input. Such missing information may either be due to ....

[Article contains additional citation context not shown here]

E. Boros, T. Ibaraki, and K. Makino. Error-free and best-fit extensions of partially defined boolean functions. Information and Computation, 140:254--283, 1998.

....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) ....

....In this case, however, even the problem of deciding whether there is an extension or not for a given pdBf (T; F ) may not be trivial, depending on the class of functions at hand. This problem and the problem of finding an extension with minimum number of errors are extensively discussed in [4] for such classes as positive (i.e. monotone) functions, Horn functions, functions with k DNF and or h term DNF, threshold functions, regular functions, RRR 7 96 Page 13 read once functions, self dual functions, decomposable functions and so on. This direction is further pursued in [5] to ....

E. Boros, T. Ibaraki and K. Makino, Error-free and best-fit extensions of a partially defined Boolean function, RUTCOR Research Report RRR 14-95, Rutgers University, 1995.

....of 4 the sets T and F is a necessary and su#cient condition for the existence of an extension. It may not be evident, however, how to find out whether a given pdBf has a extension that belongs to a class of functions C. Therefore, we have extensively studied the following problems in [11]. As noted in Introduction, the first problem is called the consistency problem in learning theory, and some results obtained therein (e.g. 31] overlap with those in [11] Problem EXTENSION(C) Input: a pdBf (T, F ) where T, F # IB n . Question: Is there an extension f # C of (T , F ) ....

....has a extension that belongs to a class of functions C. Therefore, we have extensively studied the following problems in [11] As noted in Introduction, the first problem is called the consistency problem in learning theory, and some results obtained therein (e.g. 31] overlap with those in [11]. Problem EXTENSION(C) Input: a pdBf (T, F ) where T, F # IB n . Question: Is there an extension f # C of (T , F ) Problem BEST FIT(C) Input: a pdBf (T, F ) where T, F # IB n , and a weight function w : T # F ## IR (nonnegative reals) Output: Subsets T # and F # such that T ....

[Article contains additional citation context not shown here]

E. Boros, T. Ibaraki and K. Makino, Error-free and best-fit extensions of a partially defined Boolean function, Information and Computation, 140 (1998) 254-283. 34

....E(T , F ) #= #. It may not be evident, however, to find out if a given pdBf has an extension belonging to a particular class C of Boolean functions, or not. This problem has been studied in various fields such as learning theory, knowledge discovery, data mining and logical analysis of data [1, 4 6, 8, 9, 11, 13]. # This research was partially supported by ONR (Grant N00014 92 J 1375) DARPA (Contract Number N66001 97 C 8537) and the Scientific Grants in Aid by the Ministry of Education, Science, Sports and Culture of Japan. The visit of the first author to Kyoto University was made possible by the ....

....errors in the input. To correct this type of errors, provided that they are not in a large number, one can consider the optimization problem of finding the largest subsets T # # T and F # # F for which E(T # , F # ) # C #= # holds. These problems have extensively been studied (e.g. in [8, 11]) for a large variety of classes. In this paper we shall consider another type of errors in the input, the case in which some data vectors are incomplete in the sense that some of their components are not available at the time of reading the input. Such missing information may either be due to ....

E. Boros, T. Ibaraki, and K. Makino. Error-free and best-fit extensions of partially defined Boolean functions. Information and Computation, 140:254--283, 1998.

....robust, depending upon how to deal with missing bits. We then provide polynomial time algorithms or prove their NP hardness for the problems under various restrictions. 1 Introduction The 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) ....

....In this case, however, even the problem of deciding whether there is an extension or not for a given pdBf (T , F ) may not be trivial, depending on the class of functions at hand. This problem and the problem of finding an extension with minimum number of errors are extensively discussed in [4] for such classes as positive (i.e. monotone) functions, Horn functions, functions with k DNF and or h term DNF, threshold functions, regular functions, read once functions, self dual functions, decomposable functions and so on. This direction is further pursued in [5] to consider the problems ....

E. Boros, T. Ibaraki and K. Makino, Error-free and best-fit extensions of a partially defined Boolean function, RUTCOR Research Report RRR 14-95, Rutgers University, 1995.

....E(T , F ) #= #. It may not be evident, however, to find out if a given pdBf has an extension belonging to a particular class C of Boolean functions, or not. This problem has been studied in various fields such as learning theory, knowledge discovery, data mining and logical analysis of data [1, 5, 6, 10, 11, 13, 16]. In practical cases, the fact that C # E(T, F ) # might be due to some classification errors in the input. To correct this type of errors, provided that they are not in a large number, one can consider the optimization problem of finding the largest subsets T # # T and F # # F for ....

....errors in the input. To correct this type of errors, provided that they are not in a large number, one can consider the optimization problem of finding the largest subsets T # # T and F # # F for which E(T # , F # ) # C #= # holds. These problems have extensively been studied (e.g. in [10, 13]) for a large variety of classes. In this paper we shall consider another type of error in the input, the case in which some data vectors are incomplete in the sense that some of their components are not available at the time of reading the input. Such missing information may either be due to ....

[Article contains additional citation context not shown here]

E. Boros, T. Ibaraki, and K. Makino. Error-free and best-fit extensions of partially defined Boolean functions. Information and Computation, 140:254--283, 1998.

....h term ) and EXTENSION(C h term ) are NP complete. # In the remaining of this subsection, we describe a theorem (without proof) which shows that, for some special cases, problems BEST FIT(C h term ) and BEST FIT(C h term ) can be solved in polynomial time. Its proof is found in [9]. Theorem 11 If h # T c for a given pdBf (T , F ) and c is fixed, then BEST FIT(C h term ) and BEST FIT(C h term ) can be solved in polynomial time. # 13 4.3 h term k DNF functions A DNF #(x) m # i=1 # j#P i x j # j#N i x j is an h term k DNF if m # h and P i # N ....

....C TH the class of threshold functions, and by C 2M the class of 2monotonic positive functions. It is easy to see that EXTENSION(C TH ) can be computed in polynomial time by using linear programming, and it is the folklore that BEST FIT(C TH ) is NP hard, where the proofs can also be found in [2, 9]. Theorem 18 EXTENSION(C TH ) can be solved in polynomial time, but BEST FIT(C TH ) is NP hard. # It is shown in Theorem 2 that EXTENSION for the class of regular function C = i.e. 2 monotonic positive function with a fixed order (10) can be solved in polynomial time. However, we have a ....

Boros, E., Ibaraki, T., and Makino, K. (1995), Error-free and best-fit extensions of a partially defined Boolean function, RUTCOR Research Report RRR 14-95, Rutgers University.

....Horn, threshold, linear and quadratic functions (see Section 2. 1 for their definitions) The class of decomposable Boolean functions and decomposable extensions of pdBfs were studied in [5] The complexity of finding extensions of a pdBf in various classes of functions was considered in [7], and extended in [8] to the more general cases in which some bits in the data vectors may be missing. These problems were shown to be polynomially solvable for all of the above six classes, as will be recalled in Section 2.1. Several classes of Boolean functions which may be particularly relevant ....

....functions, and the class CQUAD of quadratic (i.e. 2 DNF) functions. A partially defined Boolean function (pdBf, for short) is defined by a pair (T; F ) where T; F f0; 1g n . A function f is an extension of pdBf (T; F ) if T T (f) and F F (f) hold. The following six lemmas, proved in [7, 10], give the basis for our discussion. Lemma 2.1 A pdBf (T; F ) has an extension in CALL if and only if T F = 2 Lemma 2.2 A pdBf (T; F ) has an extension in C P if and only if there is no a 2 T and b 2 F such that a b. 2 Lemma 2.3 A pdBf (T; F ) has an extension in CHORN if and only if there ....

E. Boros, T. Ibaraki and K. Makino, Error-free and best-fit extensions of partially defined Boolean functions, Technical Report RRR 14-95, RUTCOR, Rutgers University, 1995.

....and is therefore replaced by a more realistic search for extensions that would approximate f as closely as possible according to various criteria. Such criteria may include membership in specific classes of Boolean functions, simplicity considerations, or other optimality criteria (see [6, 1]) A common special class of Boolean functions frequently used for choosing an extension is the class of threshold (or linearly separable) functions in which the classification is decided by whether a weighted sum of the attributes does or does not exceed a certain threshold. Frequently, the quest ....

Endre Boros, Toshihide Ibaraki, and Kazuhisa Makino. Error-free and best-fit extensions of partially defined Boolean functions. RUTCOR Research Report, RRR 14-95, RUTCOR -- Rutgers University's Center for Operations Research, April 1995.

....total Boolean function from a particular class of Boolean functions; this is known as the extension problem, and corresponds in a sense to the satisfiability problem of Boolean formulas. The extension problem and variants thereof have been investigated for a number of classes of Boolean functions [5, 3, 2]. Among these classes are Horn functions; a function is Horn, if it can be represented by a DNF in which each term contains at most one negative literal. Denote for any Boolean function f by T (f) and F (f) the sets of its true and false vectors of f , i.e. T (f) fv j f(v) 1g and F (f) fw j ....

....by Cl (S) resp. Cl (S) the closure of S under intersection resp. union, i.e. conjunction resp. disjunction of vectors) It is well known that the Horn functions are those functions f whose set F (f) of false vectors is closed under intersection, i.e. satisfying F (f) Cl (F (f ) As shown in [13, 3], a Horn extension of a pdBF (T ; F ) can be found in polynomial time (if any exists) In fact, a Horn extension for (T ; F ) exists precisely if T is disjoint from Cl (F ) However, this characterization shows that the Horn extension problem is, in a sense, unbalanced in (T ; F ) From a ....

E. Boros, T. Ibaraki and K. Makino, Error-free and best-fit extensions of partially defined Boolean functions, RUTCOR Research Report RRR 14-95, Rutgers University, 1995.

....and is therefore replaced by a more realistic search for extensions that would approximate f as closely as possible according to various criteria. Such criteria may include membership in specific classes of Boolean functions, simplicity considerations, or other optimality criteria (see [6, 1]) A common special class of Boolean functions frequently used for choosing an extension is the class of threshold (or linearly separable) functions in which the classification is decided by whether a weighted sum of the attributes does or does not exceed a certain threshold. RRR 22 96 Page 3 ....

Endre Boros, Toshihide Ibaraki, and Kazuhisa Makino. Error-free and best-fit extensions of partially defined Boolean functions. RUTCOR Research Report, RRR 14-95, RUTCOR -- Rutgers University's Center for Operations Research, April 1995.

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E. Boros, T. Ibaraki and K. Makino, "Error-free and best-fit extensions of partially defined Boolean functions." Information and Computation 140 (1998) pp. 254-283.

No context found.

E. Boros, T. Ibaraki, and K. Makino, "Error-free and best-fit extensions of partially defined Boolean functions," Inform. Computat., vol. 140, pp. 254--283, 1998.

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E. Boros, T. Ibaraki, and K. Makino, "Error-free and best-fit extensions of partially defined Boolean functions," Inform. Computat., vol. 140, pp. 254--283, 1998.

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