S. Rudeanu. Boolean Functions and Equations. North Holland, 1974.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....query was introduced. Let B = B; 0; 1; be a Boolean algebra, where 0; 1 2 B are two distinguished elements of B, is a unary operation, and ; are two binary associative, commutative, and idempotent operation that satisfy the usual axioms of Boolean algebras (see, for example [7]) Here 0 and 1 are the least and the largest element of the algebra, respectively. We de ne x x if b = 1 x if b = 0; for x 2 B and b 2 f0; 1g. It is a well known fact that a Boolean algebra B = B; 0; 1; de nes a Boolean ring structure, B = B; 0; 1; where plays ....

S. Rudeanu, Boolean Functions and Equation, North-Holland, Amsterdam, 1974. 10

....null or undefined values. Let B = B; 0; 1; be a Boolean algebra, where 0; 1 2 B are two distinguished elements of B, is a unary operation, and ; are two binary associative, commutative, and idempotent operation that satisfy the usual axioms of Boolean algebras (see, for example [6, 2]) Here 0 and 1 are the least and the largest element of the algebra, respectively. We define x b x if b = 1 x if b = 0; for x 2 B and b 2 f0; 1g. It is a well known fact (see, for instance [6] that a Boolean algebra B = B; 0; 1; defines a Boolean ring structure, B = B; 0; ....

....and idempotent operation that satisfy the usual axioms of Boolean algebras (see, for example [6, 2] Here 0 and 1 are the least and the largest element of the algebra, respectively. We define x b x if b = 1 x if b = 0; for x 2 B and b 2 f0; 1g. It is a well known fact (see, for instance [6]) that a Boolean algebra B = B; 0; 1; defines a Boolean ring structure, B = B; 0; 1; where plays the role of the multiplication, and the role of addition, where x y = x y) x y) for x; y 2 B. This ring is unitary, commutative, and has characteristic 2 (since ....

S. Rudeanu. Boolean Functions and Equation. NorthHolland, Amsterdam, 1974.

....of variables, and addition of ine#ective variables to functions that already have ine#ective variables. Familiarity with the basic standard terminology of first order logic, universal algebra, lattice theory, and the theory of Boolean functions is assumed. For definitions and background see [1, 4, 5, 7, 8, 10, 11, 13, 14]. The HSP Theorem in particular is proved and discussed in [1, 4, 5, 7, 10] 2. Lattice and algebra terminology Recall that a Boolean lattice is a bounded distributive lattice in which every element has a complement. A Boolean sublattice is not just a sublattice that is a Boolean lattice, it is ....

S. Rudeanu, Boolean Functions and Equations, North-Holland, 1974.

....for C = C ) for C = CHorn (and for all related classes, such as k quasi Horn and k quasi reverse Horn for any fixed k) and for C = C k DNF with k fixed. To discuss properties of very robust extensions, we shall recall the consensus method and some of its properties (see e.g. [17, 18]) Given two terms t = V j#P x j V j#N x j and t # = V j#P # x j V j#N # x j , we say that they are in conflict at variable x j if j # (P #N # )#(N #P # ) i.e. if x j appears in one and x j appears in the other) If t and t # are in conflict at exactly one of the variables, then ....

S. Rudeanu. Boolean Functions and Equations. North-Holland, Amsterdam, 1974.

....k quasi reverse Horn for any fixed k) and for C = C k DNF with k fixed. Let us recall that a class C is minor closed, if f # C and g # f imply g # C. To discuss properties of very robust extensions, we shall further recall the consensus method and some of its properties (see e.g. [14, 15]) Given two terms t = # j#P x j # j#N x j and t # = # j#P # x j # j#N # x j , we say that they are in conflict at variable x j if j # (P #N # ) # (N #P # ) i.e. if x j appears in one and x j appears in the other) If t and t # are in conflict at exactly one of the variables, ....

S. Rudeanu. Boolean Functions and Equations. North-Holland, Amsterdam, 1974.

....for C = C ) for C = CHorn (and for all related classes, such as k quasi Horn and k quasi reverse Horn for any fixed k) and for C = C k DNF with k fixed. To discuss properties of very robust extensions, we shall recall the consensus method and some of its properties (see e.g. [18, 19]) Given two terms t = # j#P x j # j#N x j and t # = # j#P # x j # j#N # x j , we say that they are in conflict at variable x j if j # (P #N # )#(N #P # ) i.e. if x j appears in one and x j appears in the other) If t and t # are in conflict at exactly one of the variables, ....

S. Rudeanu. Boolean Functions and Equations. North-Holland, Amsterdam, 1974. 27

....we get an even smaller number of representable n ary operations: 2 2 n . For readers convenience, all the proofs are placed at the end of the paper) Comment. The fact that not every operation can be represented is known; moreover, there exist conditions (see, e.g. Ch. 13, Section 4, of [6]) which are necessary and sufficient for the existence of a representable operation f for which f(a k1 ; a kn ) b k for known values a ki and b k , 1 k K. The standard two valued logical algebra B (2) f0; 1g describes the standard (two valued) logic. When a Boolean algebra is ....

....form, i.e. to the form C 1 C 2 : C k , where each conjunction C i has the form x 1 1 : x n n a 1 : n ; with i 2 f ; Gammag, x def = x, x Gamma def = x 0 , and a 1 : n a constant. This representation is known in Boolean algebra theory; see, e.g. [6]. Different expressions of this type can be described by assigning, to each of 2 n combinations ( 1 ; n ) 2 f ; Gammag n of s and Gamma s, an element a 1 : n 2 B. Different assignments lead to different functions: indeed, if we fix the n ary operation f , i.e. if we fix ....

S. Rudeanu, Boolean functions and equations, North Holland, Amsterdam, 1974.

.... research in unification theory since, unlike theories such as associativity commutativity, the theory of Boolean algebras is unitary even for unification with constants (where the terms to be unified may contain additional free constant symbols) In addition, well known results from mathematics [2, 12, 16] can be used to compute the most general unifier of a given (solvable) unification problem. General Boolean unification (where the terms to be unified may contain additional free function symbols) is still finitary, but no longer unitary [17] From a practical point of view, a Prolog system ....

Sergiu Rudeanu. Boolean Functions and Equations. North-Holland, 1974.

....sub problems, namely determining the dependent state variables, and finding the functions which express the dependencies. The following theorem provides the theoretical means to solve both problems; it is a reformulation of a general theorem by Schr der to solve a boolean equation in one unknown [10]. In this theorem, R(V) v and R(V) v respectively denote the positive and negative cofactor of the expression R(V) with respect to a variable v. Theorem 1 A state variable v is functionally dependent in a function R : B n # B iff: R(V) v i # R(V) v i # 0 . The set of solutions for the ....

S. Rudeanu, "Boolean Functions and Equations," NorthHolland Publishing, Amsterdam, 1974.

....the theoretical means to determine whether a variable is functional 83 Sequential Verification ly dependent in a given set, and if it is dependent, to determine a valid dependency function. It is a reformulation of a general theorem by Schr der to solve a Boolean equation in one unknown [Rud74]. THEOREM 5.2 A variable r is functionally dependent in a function Q : S # B iff: Q r Q r # 0 . Each function j r : S # B with j r r # j r r that satisfies the following condition is a valid dependency function: Q r # j r ) # (j r # Q r #Q r ) In words, a variable is ....

S. Rudeanu, "Boolean Functions and Equations", North-Holland Publishing, Amsterdam, 1974.

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S. Rudeanu. Boolean Functions and Equations. North Holland, 1974.

No context found.

Rudeanu, S., (1974), "Boolean functions and equations", North-Holland, NY.

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