Variable, Boolean

A set G of logic gates is universal if an arbitrary ri-variable Boolean function T can be written as a composition of the logic gates in G a universal set of gates 9u- - dm is. sometimes also said to gorm a basis set for T. It is easy to show that the set consisting of the Boolean operators AND, OR and NOT, for example, is universal. ... 

Valence Shell Electron Pair Repulsion (VSEPR), 491 van der Waals interaction energy, 347, 796 van der Waals radius, 849 variable. Boolean, 382 variational function, 399 variational method, 838, 940... 

Table 2.1-1 compares the ordinary algebra of continuous variables with the Boolean algebra of 1 s and Os. This table uses the symbols and -h for the operations of intersection (AND) and union (OR) which mathematicians represent by n and u respectively. The symbols and -f which are the symbols of multiplication and addition, are used because of the similarity of their use to AND and OR in logic. 

Shannon s method, expands a Boolean function of n variables in minterms consisting of all combinations of occurrences and non-occurrences of the events of interest. Consider a function of n Boolean variables XJ which may be expanded about X, as shown in Equation 2.2-3 where f(l, Xj,..., XJ where 1 replaces X,. This says that a function of Boolean variables equals the function with a variable set to I plus the product of NOT the variable limes the function with the variable set to 0. By extending Equation 2.2-3, a Boolean function may be expanded about all of its... 

Just as there are universal computers that, given a particular input, can simulate any other com-puter, there are NP-complete problems that, with the appropriate input, are effectively equivalent to any NP-hard problem of a given size. For example, Boolean satisfiability -i.e. the problem of determining truth values of the variable s of a Boolean expression so that the expression is true -is known to be an NP-complete problem. See section 12.3.5.2... 

Consider an arbitrary set of Boolean input values, Xi,..., Xn- Using only the Boolean operators AND and NOT, we can write down an expre.ssion which takes the value 1 if and only if a particular configuration of O s and I s appears in the input. Suppose that n = 6 and we want to construct an expression that equals one if and only if xi = X3 = X4 = X5 = 1 and. X2 = xe = 0. A little thought will show that the required expression is given by joining the string of six Boolean variables with the AND operator, substituting NOT(xi) for all values x)i that are required to equal zero ... 

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 ... 

In order to write down the microscopic equations of motion more formally, we consider a size N x N 4-neighbor lattice with periodic boundary conditions. At each site (i, j) there are four cells, each of which is associated with one of the four neighbors of site (i,j). Each cell at time t can be in one of two states defined by a Boolean variable where d = 1,..., 4 labels, respectively, the east, north,... 

SleD53 Slepian, D. On the number of symmetry types of boolean functions of n variables. Canad. J. Math. 5 (1953) 185-193. 

The solution of Problem-1 requires extensive search over the set of potential sequences of operations. Prior work has tried either to identify all feasible operating sequences through explicit search techniques, or locate the optimum sequence (for single-objective problems) through the implicit enumeration of plans. The former have been used primarily to solve planning problems with Boolean or integer variables, whereas the latter have applied to problems with integer and continuous decisions. 

The observation will not be included in population analysis and denominator definitions. The response is "No" when the categorical variable is a Boolean variable. The observation will be included in population analysis and denominator definitions. 

Let us give the construction for block B from Example IV 4 which gave a division of the scheme of Example IV-3 into major blocks. The critical points are a and 8 and the new variables are u(21,a,x ) and u(21,6,x2). The Boolean expression which gives the conditions under which a computation goes from the entry... 

First let us extend the definition of recursion equation as we did the definition of WHILE scheme. Let a Boolean expression be any expression involving predicate terms P(t, ...,tm) where each is a terminal term (not necessarily a variable), and the connectives AND, OR, and NOT. We define a recursion expression inductively, by saying that first any term is a recursion expression, and then that any statement of the form IF Q THEN ELSE E2 is a recursion expression if Q is a Boolean predicate and E and E2 are recursion expressions. 

The state of the automaton at time t can be completely determined by the boolean variable n, r,t), which is equal to 1(0) if a particle is present (absent) on site r with velocity c,. From this it follows that the local microscopic density p and flow velocity u at site r are given by... 

As Boolean variable names that can be bound before the pre/post specification section define normal and exception in terms of the success and failure indicators that the operation will use. They are treated as special names, as opposed to names introduced locally within a let.. ., because their binding must be shared across all specifications of that action. 

As Boolean variables that can be used within a postcondition. For example ... 

The rows of the occurrence matrix correspond to the balance equations and the columns to the process variables, both measured and unmeasured. An element of the matrix Ojj is a Boolean 1 or 0, that is,... 

Since the systems of equations to be considered are quite large, it is necessary to use some compact method to represent the information flow among them. A very convenient technique is to relate the system equations to a digraph and its associated Boolean matrix, which represent the structure of the information flow in the system. The Boolean matrix to be used is called the occurrence matrix (HI, S3), and is defind as follows (1) each row of the occurrence matrix corresponds to a system equation, and each column corresponds to a system variable (2) an element of the matrix, s -, is either a Boolean 1 or 0 according to the rule... 

Once an output set has been established, the direction of information flow is fixed in the system of equations, and they can be represented either by a linear diagraph or its associated Boolean adjacency matrix. For our purposes it is more convenient to work with the Boolean adjacency matrix, which can be obtained directly from the occurrence matrix and output set as follows. First, assign numbers to the equations that correspond to the rows, and numbers to the variables that correspond to the columns of the occurrence matrix as in Fig. 5. Then pick an output set by the methods described in Section III. For the first equation, and the number of the column containing its output variable. The information flow transmitted by the variable designated by the column number goes to all other equations that have nonzero... 

C IF THE VARIABLE CAN BE AN ITERATE THE BOOLEAN UNION OF THE COLUMN C ANO THE LAST LOOP COLUMN IS PLACED IN THE LAST LOOP COLUMN. 

Most systems involve several interconnected feedback loops. Such systems cannot be analyzed seriously without a proper formalism, but their detailed description using differential equations is often too heavy. For these reasons we (as many others before) turned to a logical (or Boolean) description, that is, a description in which variables and functions can take only a limited number of values, typically two (1 and 0). Section II is an updated description of a logical method ( kinetic logic ) whose essential aspects were first presented by Thomas and Thomas and Van Ham.2 A less detailed version of this part can be found in Thomas.3 The present paper puts special emphasis on the fact that for each system the Boolean trajectories and final states can be obtained analytically (i.e.,... 

The very nature of the Boolean variables used has varied considerably. For instance, in his initial papers Glass1 2 associated with each element, r/of the system a Boolean variable whose value is 1 where dxjdt > 0 and 0 where dXjldt < 0. Subsequently, instead of using Boolean variables whose value was determined by the value of the time derivative of the corresponding continuous variable, Glass22 shifted to Boolean variables x, associated with the concentrations of the elements and such that jc, = 1 if x, > 0, and x, = 0 if x, < 0, (like our variables a,). 

As remarked in Section II.E an essential feature of the asynchronous treatment is the fact that certain Boolean states have two or more possible next states. Note that this has no philosophical pretentions at the level of the problem of determinism a Boolean state may have more than one possible next state depending on the time delays, essentially in the same way as, in the continuous description, different values of the parameters may result in different trajectories. But, in addition, it must be realized that a Boolean state is not punctual it would correspond, in the continuous description, to a whole domain in the space of variables. More specifically, in a continuous description, one can cut the n-dimensional space of variables into 2" boxes, each of which corresponds to a state of the Boolean description. In the continuous space of variables a state is entirely defined by its coordinates in the Boolean description, it is defined in a broad way by a Boolean number and in more detail by data about the carryover of the time delays. Finally, the simplicity of the Boolean treatment permits to include in an especially easy way the possibility that the values of the time delays fluctuate with time (of course, this has more to do with the problem of determinism). 

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