AND function, Boolean

From equation (10.5) the Boolean OR function becomes the fuzzy max operation, and from equation (10.6) the Boolean AND function becomes the fuzzy min operation. Hence equation (10.22) can be written as... 

The Boolean AND function. A single straight line can separate the open and filled points. 

Programmable Logic Controllers The programmable logic controller (PLC) originated as a solid-state replacement for the hardwired relay control panel and was first used in the automotive indus-tiy for discrete manufacturing control. Today, PLCs are used to implement Boolean logic functions, timers, counters, and some math functions and PID control. PLCs are often used with on/off process control valves. 

Forcing Structures Consider the Boolean OR function. Note that the value of cti or CT2 is fixed as 1 whenever either a or C2 is equal to 1. Kauffman calls any function with the property that at least one value of at least one of its inputs fixes its output, a canalizing function [kauff84]. The Boolean functions OR and AND , for example, are both canalizing functions, but the EXCLUSIVE OR function is not. 

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.,... 

Molecules can display characteristics of electronic components such as the conduction of energy or electrons and are capable of simple Boolean logic functions. 

The package implementation provides the graph types BDD, FDD, OKFDD, MTBDD, EVBDD, BMD, BMD, p BMD, HDD, K BMD and ZDD in order to represent boolean and integer functions as well as sets. Applications of the presented method exist virtually in any approach based on decision diagrams. The paper investigates manipulation of bit-level and word-level functions as well as bit selection from arithmetic expressions. 

Theoretical results support the conjecture, that there exist no single graph type that can represent boolean and arithmetic functions with the same efficiency. Therefore, it is important to provide a framework for the integration of multiple graph types. 

For several basic forms of component linkage, the Boolean system function, the mathematical system function, and the probability are shown in Table 4.2. The assigned graphic presentation as it is used in the case of fault trees is completed (see Section 4.2.2). 

Table 4.2 Boolean system functions, mathematical system functions, and occur rence probabilities for the system behavior for several baac forms of component linkage. The graphic symbols correspond to [4-10)...

To the AND or OR linkages of component states considered in Section 4.2.1 and represented in Table 4.2, specific structures are assigned in the decision tables. AND means that the undesired system incident will occur only when all components under observation have failed. If, in the example, five components are again used as point of departure, then the compressed decision table according to Table 4.7(a) will correspond to the state equation represented as the Boolean system function... 

The multiplication sign, , is used for the and function, sometimes called conjunction. This function obeys the conventional arithmetical rules for products. Thus if the binary values are taken arithmetically as one and zero, 11 = 1, and 1-0 = 0. But, in Boolean notation, 1 -b 1 = 1, which is not correct by arithmetical notation. 

We computed the TE between every pair of RSs. To assess the significance of these values, we compared them with TE computed over the same time series in which the observations are randomly permuted. Such a time series has the same statistical properties of the original one except for the causal relations produced by the boolean update functions. For each time series, we generated 50 random shuffling and computed the TE between the RSs identified. These values were then used to compute a p-value for assessing the significance of the TE values... 

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