Set, Boolean

The Web-based graphical user interface permits a choice from numerous criteria and the performance of rapid searches. This service, based on the chemistry information toolkit CACTVS, provides complex Boolean searches. Flexible substructure searches have also been implemented. Users can conduct 3D pharmacophore queries in up to 25 conformations pre-calculated for each compound. Numerous output formats as well as 2D and 3D visuaHzation options are supplied. It is possible to export search results in various forms and with choices for data contents in the exported files, for structure sets ranging in size from a single compound to the entire database. Additional information and down-loadable files (in various formats) can be obtained from this service. 

Structural keys describe the chemical composition and structural motifs of molecules represented as a Boolean array. If a certain structural feature is present in a molecule or a substructure, a particular bit is set to 1 (true), otherwise to 0 (false). A bit in this array may encode a particular functional group (such as a carboxylic acid or an amidelinkage), a structural element (e.g., a substituted cyclohexane), or at least n occurrences of a particular element (e.g., a carbon atom). Alternatively, the structural key can be defined as an array of integers where the elements of this array contain the frequency of a specific feature in the molecule. 

Answers from aH of these searches contain CAS Registry Numbers. Answer sets may be combined, using the Boolean operators AND, OR, or NOT, with other answer sets or with text terms, such as names or molecular formulas. Any answer set also may be used to define subsets of the file for subsequent stmcture searching. Answer sets of up to 10,000 Registry Numbers from any type of search in this file may also be used as search terms in other files, such as the CA or CAOLD files (53). 

Union The union of two fuzzy sets A and B eorresponds to the Boolean OR funetion and is given by... 

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... 

Boolean equations show how component failures can fail a system. A minimal cut. he smallest combination of component failures that can fail a system. It is the set of non-sup us components, such as in the previous example, with the superfluous combination Y Z(X Y,. Z) e uded. If they all occurred they would cause the top event to occur. One-component minimtil cut s( if there are any, are single failures that cause system failure. Two-component minimal cutsets ai tairs of components, if they occur together cause system failure. Triple-components minimal Cl sts are sets of three components that, if they fail together cause system failure, and so on to hi er cutsets... 

A simple example of fault tree analysis applied to an internal combustion engine (Figure 3.4.4-2) is the Figure 3.4.4-3 fault tree diagram of how the undesired event "Low Cylinder Compression" may occur. The Boolean equation of this fault tree is in the caption of Figure 3.4.4-3. Let the occurrence of these events be represented by a 7, non-occurrence by 0, and consider that there may he a long history of occurrences with this engine. Several sets of occunrence.s (trials) are... 

VS alpha-XHi%, control. Ibuit-iree AND OR INHIBIT Minimal cutsets of up to 20 can be generated top-down successive Boolean substitution Path sets Cutsets can be automatically punched uds or online data lor ivITT or SUPERPOCUS IBM )C 760( Ava Arg(- c Center... 

ICSUP/ umeric Tnaiion. on/ Pandc. AND OR INHIBIT/ None computer memory is limiting factor. Minimal cuLscls up to order 10 an n icd Ton down Boolean substitutioii Minimal cutsets/path Can determine minimal sets of intermediate gates ... 

SETS is unique in that it performs symbolic manipulation of Boolean equations usi... 

Generating block files (i.e, a set of Boolean equations) for subsystems... 

Conducting common cause and dependent event analysis Dropping complemented events and performing the subsequent minimization Generating block files (i.e., a set of Boolean equations) for subsystems Eliminating mincutsets with mutually exclusive events... 

The algebra of sets-BooIean algebra-governs tlie way in wliich sets can be manipulated to form equivalent sets. The principal Boolean algebra laws used for tliis purpose are as follows. 

If the letter symbols for sets are replaced by numbers, tlie commutative and associative laws become familiar laws of aritlimetic. In Boolean algebra tlie first of tlie two distributive laws, Eq. (19.3.5), lias an analogous counterpart in arithmetic. Tlie second, Eq. (19.3.6), does not. In risk analysis. Boolean algebra is used to simplify e. pressions for complicated events. For example, consider tlie event... 

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. ... 

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 ... 

Since we will be dealing with finite graphs, we can analyze the behavior of random Boolean nets in the familiar fashion of looking at their attractor (or cycle) state structure. Specifically, we choose to look at (1) the number of attractor state cycles, (2) the average cyclic state length, (3) the sizes of the basins of attraction, (4) the stability of attractors with respect to minimal perturbations, and (4) the changes in the attractor states and basins of attraction induced by mutations in the lattice structure and/or the set of Boolean rules. 

Boolean OR operation will be performed if the synaptic weights between the second hidden layer and the output layer are equal to one and the output neuron s thresholds are set to 0.5 [lipp87]. 

HarM63a Harrison, M A. The number of transitivity sets of Boolean functions. SIAM J. 11 (1963) 886-878. 

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. 

Thus, propanol, C3H70H, has a membership of 1 in the three-carbon molecule class, while ethanol, C2H5OH, has a membership of 0 in the same class. As the membership in a crisp set must take one of only two possible values, Boolean (two-valued) logic can be used to manipulate crisp sets. If all the knowledge that we have can be described by placing objects in sets that are separated by crisp divisions, the sort of rule-based approach to the development of an expert system described in the previous chapter is appropriate. 

This statement might belong to the "True" set with a membership of, say, 0.9, and to the "False" set with a membership of 0.1. Statements that in Boolean logic are either true or false, but not both, can in fuzzy logic be simultaneously... 

Then, a set of cell types are defined each nonempty cell will contain a joint and between zero and five beams. The beams will extend in directions n, 3ir/4, n/2, n/4, and 0, labeled g0 through g4, respectively. Conversion from Boolean gene values to an integer is accomplished through the following equation ... 

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