Brown, B. and Chalmers, M. Tourism and mobile technology, Proc of ECSCW 2003. Kluwer academic publishers. pp 335-354.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....positive cofactor) is = 021 3 545454 5 A 545454 ;1 . The cofactor of with respect to variable 1: the negative cofactor) is = B 021 3 54 4545 C# 454545 1 6 . The Boole s expansion of a function over a variable (also called Shannon s expansion) is given as follows [22] [23] Let D)E . Then, 021 3 1F9 545454 1 454545 1 G6 H1: F = I 1 J = Let and be two functions with support variables 1: L NM 454545 8 O be an arbitrary binary operator, representing a Boolean function of two arguments. The orthonormal ....

F. Brown, Boolean Reasoning, Kluwer Academic Publishers, 1990.

....formulas. These are sufficient to provide equality, dis equality and strict containment, as x = y , x y y x; x ae y , x y y 6 x; x 6= y , x Delta y x Delta y 6 ; Although systems of positive Boolean constraints have been extensively studied since Boole [2] see for example [3] or [11] the extension to negative constraints has not, to our knowledge, been addressed. This may be because in the case of twovalued Boolean algebras, negative constraints add no power since the constraint x 6 y is equivalent to x = 1 y = For more general Boolean algebras, however, systems ....

....and their duals: They allow us to replace semantic inequality by syllogistic inequality, which can be checked syntactically: Theorem 4. 10 (Blake) For all Boolean formulas f and sum of products formulas g: g f , g BCF (f) There exist a number of algorithms to compute BCF (f ) One method [3] first converts f to an arbitrary sum of products formula and then repeatedly forms the consensus of two terms in f and simplifies by absorption until a fixpoint is reached. Forming consensus means rewriting according to the rule: x Delta p x Delta q = x Delta p x Delta q p Delta q: ....

F.M. Brown. Boolean Reasoning, The Logic of Boolean Equations. Kluwer Academic Publishers, Norwell, Massachusetts, USA, 1990.

....the function and its cofactors. For example, if jf Delta f 0 DC j jgj no matching is obviously possible. The interested reader is referred to [39] for details. Cell based logic optimization 25 Boolean unification. Boolean unification is the process of finding a solution of a Boolean equation [8]. A method for finding Boolean matching with don t care conditions based on Boolean unification was proposed by Chen [12] A matching is searched for by solving a Boolean equation in which the unknowns are the variable matching functions representing input assignments. Note that these functions ....

F. Brown. Boolean reasoning. Kluwer Academic Publishers, 1990.

....failure if and only if a faulty predicate P 0 evaluates to a different value than the correct predicate P . That is, where : P , P 0 ) or P Phi P 0 , where Phi is exclusive or. This is simply the boolean difference of P with respect to P 0 [15] 16] also called the boolean derivative [17] [18] or predicate difference [19] when P contains expressions rather than strictly boolean terms. To determine, for example, the conditions under which a variable negation fault for variable v will be detected, we simply compute P Phi P v v ; where P x e is predicate P with all free ....

F.M. Brown. Boolean Reasoning. Kluwer Academic Publishers, 1990.

..... A set D of prime implicates is a covering of A iff for every clause c such that A c there is c 0 2 D such that c 0 c. Fact 3 If D is a covering set of prime implicates of A then D A and atm(D) atm(A#) Another name for a covering set of prime implicates is Blake canonical form 6 [3]. Note that there may be several sets of prime implicates covering a given formula. Fact 3 can also be used to get rid of redundant atoms (this has been proposed by [9] Nevertheless, the number of prime implicates can be exponential in the length of the original formula. Therefore the algorithm ....

F. M. Brown, Boolean Reasoning, Kluwer Academic Publishers (1990).

.... x i (the positive cofactor) is f x i = f(x 1 ; 1; xn ) The cofactor of f with respect to variable x i (the negative cofactor) is f x i = f(x 1 ; 0; xn ) The Boole s expansion of a function over a variable (also called Shannon s expansion) is given as follows [22] [23] Let f : B n B. Then, f(x 1 ; x 2 ; x i ; xn ) x i f x i x i f x i (1) Let f and g be two functions with support variables fx i ; i = 1; 2; ng. Let be an arbitrary binary operator, representing a Boolean function of two arguments. The orthonormal ....

F. Brown, Boolean Reasoning, Kluwer Academic Publishers, 1990.

.... table (see [1] We report results of experiments on three medical data sets, concerning lymphography, breast cancer and primary tumor (see [8] We compare standard methods for extracting laws from decision tables (see e.g. 17] 1] based on rough set (see [13] and boolean reasoning (see [2]) with the method based on algorithms calculating relevant decision rules for new objects. We also compare the results of computer experiments on those data sets obtained by applying our system based on rough set methods with the results on the same data sets obtained with help of several ....

.... of other classification systems (see also [1] for more complete discussion) 2 Classification Algorithms based on Decision Rules We assume that the reader is familiar with basic notions of the rough set theory (see [13] and methods of decision rules generation based on boolean reasoning (see [2]) In particular by A = U; A [ fdg) we denote the decision table and by RUL(A) we denote the set of all optimal basic decision rules of A (i.e. decision rule with minimal number of descriptors in predecessor and only one decision descriptor in successor see [13] 1] The cardinality of the ....

Brown, E., M.: Boolean reasoning. Kluwer Academic Publishers, Dordrecht (1990)

....tools, which are powerful enough to deal with the extraction of optimal (H; approximate decision reducts from real life data. Besides, rough set foundations enable us to understand better the nature of complexity of the above problem by its relationship to (approximate) boolean reasoning (cf. [2, 18, 19]) For a brief illustration, let us mention about yet another approach to approximation of the notion of a decision reduct. By analogy with discernibility characteristics developed for classical reducts ( 18, 19] we can say that a given subset of attributes approximately defines decision ....

.... approximate decision reducts to operations over boolean functions encoding the required discernibility of pairs w 1 ; w 2 2 V U A satisfying (16) Complexity of this problem can be thus treated as comparable to that of the problem of finding (approximate) prime implicants for boolean functions ([2, 18, 19]) 4 Approximate Independence The property of defining decision by a given subset of conditional attributes is a special case of conditional independence statement. Definition 7 Let A = U; A) and mutually disjoint subsets X;Y; Z A be given. We say that X is conditionally independent on Z ....

Brown, E.M.: Boolean Reasoning. Kluwer Academic Publishers, Dordrecht (1990).

....Boolean formulas. These are sufficient to provide equality, dis equality and strict containment, as x = y , x y y x; x ae y , x y y 6 x; x 6= y , x Delta y x Delta y 6 ; Although systems of positive Boolean constraints have been extensively studied since Boole [2] see for example [3] or [11] the extension to negative constraints has not, to our knowledge, been addressed. This may be because in the case of twovalued Boolean algebras, negative constraints add no power since the constraint x 6 y is equivalent to x = 1 y = For more general Boolean algebras, however, ....

....and their duals: They allow us to replace semantic inequality by syllogistic inequality, which can be checked syntactically: Theorem 4. 10 (Blake) For all Boolean formulas f and sum of products formulas g: g f , g BCF (f) There exist a number of algorithms to compute BCF (f ) One method [3] first converts f to an arbitrary sum of products formula and then repeatedly forms the consensus of two terms in f and simplifies by absorption until a fixpoint is reached. Forming consensus means rewriting according to the rule: x Delta p x Delta q = x Delta p x Delta q p Delta q: ....

F.M. Brown. Boolean Reasoning, The Logic of Boolean Equations. Kluwer Academic Publishers, Norwell, Massachusetts, USA, 1990.

....point of view. Below, we present a 2 Recall that ff[p ] resp. ff[p ] is the formula obtained from ff by replacing all occurrences of p by (resp. generally more efficient method based on the notion of the Blake canonical form of a formula. Our discussion follows Brown [2]. We start with preliminary terminology. A term is either , or a conjunction of literals in which no atom appears more than once. A formula is said to be in disjunctive normal form (DNF, for short) if it is a disjunction of different terms. 3 It is wellknown that each formula can be ....

....Denote the resulting formula by fi. 2) Repeat as long as possible: If fi contains a pair t 1 and t 2 of terms whose consensus exists and no term of fi is a subformula of c(t 1 ; t 2 ) then fi : fi c(t 1 ; t 2 ) 3) Take ABS(fi) This is BCF (ff) The following results can be found in Brown [2]. Theorem 1 (1) Formulas ff and BCF (ff) are equivalent. 2) All atoms occurring in BCF (ff) are non redundant for ff. 3 In the logical literature DNF is often defined as a disjunction of terms where a term is understood as either , or or any conjunction of literals. Note, however, that we ....

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F. M. Brown. Boolean Reasoning. Kluwer Academic Publishers, 1990.

....is based on the boolean algebra of boolean functions. Let B be a boolean algebra comprising k elements 1 and Fn(B) be the set of n variable boolean functions on B. The algebraic system (Fn (B) Delta, 0, 1) is a boolean algebra in which 0 is the zero function and 1 is the onefunction [20]. The boolean functions such as s i (X) and f i (X) can be considered as elements in Fn (B) Definition 3 Let g(X) h(X) 2 Fn (B) the inclusion relation is defined as follows: g(X) h(X)ifandonlyifg(X)h(X) 0: In this paper, we are especially interested in sets of boolean functions expressed ....

....h(X) is defined as [g(X) h(X) ff(X)jg(X) f(X) h(X)g; where g(X) is the lower bound and h(X) is the upper bound. An interval [g(X) h(X) is nonempty if and only if the condition g(X) h(X) is satisfied. Property 1 A function f(X) 2 [g(X) h(X) if and only if f(X)g(X) f(X)h(X) 0 [20]: Property 1 will be used in developing our gate correction procedure to identify whether a function belongs to a given interval. III. Search for Error Location When a gate level implementation N has been shown to be incorrect, the first step is to search for an error location. If the circuit ....

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F. M. Brown, Boolean Reasoning. Kluwer Academic Publishers, 1990.

....this subject. 2 Background 2.1 Boolean Functions, Operators and Relations In this section we briefly review some basic concepts and terminology used throughout the discussion. We assume that the reader is familiar with Boolean algebras and BDD based Boolean function manipulation (please refer to [2, 3, 4] for details) We denote vectors and matrices in bold, i.e. x = x 1 ; x 2 ; xn ] T . We use the symbols: 8 x f = f j x Delta f j x 0 and 9 x f = f j x f j x 0 to designate the consensus (or universal quantification) and the smoothing (or existential quantification) of Boolean ....

F. Brown. Boolean Reasoning. Kluwer Academic Publishers, 1990.

....in numerous application areas. In this section we introduce our terminology and review properties of positive Boolean constraints that we shall make use of in the sequel. We assume that the reader has an elementary knowledge of Boolean algebras and Boolean equations. Suitable references are [3] and [13] A Boolean formula is a variable, a constant ; or 1, the complement of a formula, a disjunction of formulas, or a conjunction of formulas. A formula is atomic if it is a variable or a constant. A literal is an atomic formula or its complement. A term is a conjunction of literals. A ....

F. Brown. Boolean Reasoning, The Logic of Boolean Equations. Kluwer Academic Publishers, Norwell, Massachusetts, USA, 1990.

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Brown, B. and Chalmers, M. Tourism and mobile technology, Proc of ECSCW 2003. Kluwer academic publishers. pp 335-354.

No context found.

F. M. Brown. Boolean Reasoning.Kluwer Academic Publishers, 1990.

No context found.

F. Brown. Boolean reasoning. Kluwer Academic Publishers, 1990.

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Brown E.M.: Boolean reasoning. Kluwer Academic Publishers, Dordrecht (1990)

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