Binary Boolean variable

It should be noted that problem (GDP) can be reformulated as an MINLP problem by replacing the Boolean variables with binary variables yj,, ... 

Semantics are given to the graph structure by associating nodes with function decompositions. In the following, a binary decomposition that can be described in terms of equation (1), will be called a Kronecker decomposition. Rmctions diow xi) and dhigh xi) only depend on the boolean variable and (or) on constant values. Operations -I- and represent addition and multiplication in the domain of function /. 

The matrix condition IOC, stated by Theorem 2.2 or by Remark 2.6, is called the intrinsic order criterion, because it is independent of the basic probabilities p, and it only depends on the relative positions of the Os and Is in the binary n-tuples u, v. Theorem 2.2 naturally leads to the following partial order relation on the set 0,1 " (3). The so-called intrinsic order will be denoted by , and we shall write u > V (u < v)to indicate that u is intrinsically greater (less) than or equal to v. The partially ordered set (from now on, poset, for short) ( 0,1 ", on n Boolean variables will be denoted by / . 

The independent boolean variables in conformal coordinate scales have exactly the same order as in the boolean function argument list, as depicted in Fig. 1.46. Conformal assignment of the independent variables to the K-map coordinate scales makes the catenated position coordinates for a minterm (or maxterm) identical to the minterm (or maxterm) number. Utilization of this identity eliminates the need for the placement of minterm identification numbers in each square or for a separate position identification table. This significantly decreases the time required to construct K-maps and makes their construction less error prone. The minterm number, given by the catenation of the vertical and horizontal coordinate numbers, is obvious if the binary or octal number system is used. 

In Boolean algebra, binary states 1 and 0 are used to represent the two states of each event (i.e. occurrence and non-occurrence). Any event has an associated Boolean variable. Events A and B can be described as follows using Boolean algebra ... 

Boolean Network with connectivity k- or N, )-net - generalizes the basic binary k = 2) CA model by evolving each site variable Xi 0,1 of according to a randomly selected Boolean function of k inputs ... 

The logical relations can advantageously be represented if Boolean or binary variables are used for describing component and system states [30], i.e. 

Boolean or binary variables and the corresponding functions only adopt two values 0 or 1. 

Many operations on Boolean functions can be implemented by simple graph algorithms that work recursively on their BDD representation in a conventional depth-first fashion. For example if f and g are Boolean functions represented by BDDs, if x, is one of their variables and if op is a generic binary operator, we express f op g as ... 

Contrary to Boolean logic where variables can have only binary states, in fuzzy logic aU variables may have any values between zero and one. The fuzzy logic consists of the same basic A - AND, V-OR, and NOT operators... 

Boolean algebra A branch of mathematics describing the behavior of linear functions of variables which are binary in nature on or off, open or closed, true or false. All coherent fault trees can be converted into an equivalent set of Boolean equations. 

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