Boolean logic

Boolean equations represent combinational logic. Boolean equations are best represented using continuous assignment statements. Here is an example of a Gray code to binary code convertor using boolean equations. 

In this section we will summarize some of the most useful mathematical operations available in Excel. This section is merely for your information, just to give you an idea of what is available it is certainly not meant to be memorized. There are many more functions, not listed here, that are mainly used in connection with statistics, with logic (Boolean algebra), with business and database applications, with the manipulation of text strings, and with conversions between binary, octal, decimal and hexadecimal notation. 

Let us give a simple example of a Boolean network, taken from [8], shown in fig. 13.4. Part (a) of that figure shows the connectivity of the network part (b) gives the logical (Boolean) rules and part (c) the state transition table which results from the network structure and the logical rules. For example, consider the second line in the transition table, and we reason from the primed letters to the unprimed If A = 0, then B = 0 if B = 0, then A or C = 1 if C = 0, then A and B = 0. The task of reverse engineering is to find the network and rules that produce the observed transitions as displayed in fig. 13.4. 

Fig. 13.4 A simple Boolean network, (a) Wiring diagram, (b) Logical Boolean rules, (c) State transition table which defines the network. The input column corresponds to time t, the output column to time t +. (Figure and caption taken from [8], with permission.)...

The While loop continues as long as the logical (Boolean) statement is True the last statement in the loop range is the Wend. 

There are many different types of expressions in VBA. Among these are numerical, logical (Boolean), and character string expressions. 

The Boolean search operators are used to specify Logical relationships among the terms being searched. 

Conventional computers initially were not conceived to handle vague data. Human reasoning, however, uses vague information and uncertainty to come to a decision. In the mid-1960 this discrepancy led to the conception of fuzzy theory [14]. In fuzzy logic the strict scheme of Boolean logic, which has only two statements true and false), is extended to handle information about partial truth, i.e., truth values between "absolutely true" and absolutely false". It thus gives a mathematical representation of uncertainty and vagueness and provides a tool to treat them. 

Many expert systems also apply fnt zy logic (see Section 9.6) instead of strict Boolean logic and arc thus able to cope with rules which arc imprecise or incomplete. 

Fuzzy logic extends the Boolean logic so as to handle information about truth values which are between absolutely true and "absolutely false . 

Coordinate Indexing and Boolean Logic. Three methods of indexing have been prominent in the chemical Hterature in recent times. The first, articulated indexing, has been used in printed Chemicaly hstracts subject indexes from their earliest days until well into the 1990s. A number of important concepts are identified as permissible index entries, including specific compounds, material types, reactions, and processes. One or more modifying statements foUow each basic index entry. Thus, eg. 

In the earliest days of on-line databases, all three indexing types coUapsed into the third. Using older manual tools, it was difficult to coordinate more than two or three concepts, but the computer made that easier. Each concept in a search can be represented by a string of synonyms or alternatives, and searching can be done for two such parameters or more. Thus, Boolean logic expressions can easily be constmcted as follows ... 

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. 

The logical structure of a fault tree can be described in terms of boolean algebraic equations. Some specific prerequisites to the application of this methodology are as follows. 

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. 

The term "min" is readily understood when Boolean algebraic reduction is applied to the logic equation, because terms like A + A C collapse to A, thereby affording considerable simplification. Tlie reason for this is if A causes failure, then additional failures are superfluous. 

Logical—Data type w ith only two possible values True (represented as 1) and False (represented as 0), also referred to as Boolean. 

Real—Predefined scalar type, decimal, or scientific notation may be used. Boolean—Predefined logical type, ordered so that false < true. 

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. ... 

The first observation consists merely of an alternative but natural interpretation of the presence or absence of balls the movement of balls is equated with the communication of binary signals. Once this interpretation is made, the second observation also becomes a natural one wherever balls collide, either among themselves or with some collection of rigid mirrors, the effect of the collision may be viewed as a Boolean logic gate. 

Computer science and algebra The symbolic system of mathematical logic called Boolean algebra represents relationships between entities either ideas or objects. George Boole of England formulated the basic rules of the system in 1847. The Boolean algebra eventually became a cornerstone of computer science. 

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