Logical expressions

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 ... 

A reasonable question is what is the significance of superfluence in the logical expression As logic, there is nothing wrong, the overstatement is extraneous. But when... 

IF(e)st—Logical if, where e is a logical expression (see Table 1-26 for relational and logical operators) and st is any executable statement except DO, IF, ELSEIF, ENDIF, or END. For example,... 

Otherwise. The Kronecker delta symbol is a particular case of a LKD, where in the logical expression L there is involved an equivalence symbol. 

The NSS concept corresponds to an operator attached to an arbitrary number of nested sums. In other words, a NSS represents a set of summation symbols where the number of them can be variable. In a general notation one can write a NSS as E (j=i,f,s,L) where the meaning of this convention corresponds to perform all the sums involved in the generation of all the possible values of the index vector j under the fulfillment of the set of logical expressions collected in the components of the vector L. The elements of the vector j have the following limits ... 

The logical Kronecker delta, which appears when integration is performed over the coordinates of the remaining n-r electrons, can be easily substituted by the equivalent logical expression ... 

While ANNs cannot do anything that would be impossible to accomplish using an alternative algorithmic method, they can execute tasks that would otherwise be very difficult, such as forming a model from sensory data or from data that is extracted from a continuous industrial process for which no comprehensive theoretical model exists. Their principal limitation is that the numerical model that they create, while open to inspection, is difficult to interpret. This is in marked contrast to the model used by expert systems, in which the knowledge of the system is expressed in readily interpreted statements of fact and logical expressions. Nevertheless, the power of the ANN is... 

Semantic uncertainty is the type of uncertainty for which we shall need fuzzy logic. Expressed by phrases such as "acidic" or "much weaker," this is imprecision in the description of an event, state, or object rather than its measurement. Fuzzy logic offers a way to make credible deductions from uncertain statements. We shall illustrate this with a simple example. 

Then a proposition is any logical expression that is based on a set of clauses Pi which are related via the aforementioned logical operators. 

The propositional logic expressions can be expressed in an equivalent mathematical representation by associating a binary variable yi with each clause P. The clause P being true, corresponds to yi = 1, while the clause P, being false, corresponds to yt = 0. Note that (-iPj) is represented by (1 — 3/i). Examples of basic equivalence relations between propositions and linear constraints in the binary variables include the following (Williams (1988)) ... 

Remark 2 Note that using the above basic equivalence relations we can systematically convert any arbitrary propositional logic expression into a set of linear equality and inequality constraints. The basic idea in an approach that obtains such an equivalence is to reduce the logical expression into its equivalent conjunctive normal form which has the form ... 

Clocksin and Mellish (1981) proposed a systematic procedure that results in a conjunctive normal form of a propositional logic expression. This procedure involves the following three steps ... 

Illustration 7.4.1 This example is taken from Raman and Grossmann (1991), who first introduced the propositional logic expressions equivalence in chemical engineering. It involves the proposition ... 

It should be noted that when several m input streams with flowrates F are directed to the same process unit i, then there are two basic alternatives of writing the logical expressions. Alternative 1 is of the form ... 

This logical expression can be written in the following equivalent mathematical form by introducing a 0-1 variable yi which takes the value of 1 if f (x) < 0 and 0 if f2(x) < 0 ... 

Illustration 7.4.3 The second example involves the logical expression ... 

A logical Kronecker delta S( L]) is a symbolic factor which is unity when the logical expression L is true and zero otherwise. ... 

In LCVD, the simplest parameter that can be correlated with the flow rate of monomer is the polymer deposition rate, which is generally and most logically expressed by (mass)/(area)(time). As long as the dependence of polymer deposition rate on monomer flow rate is sought for a given monomer only, the monomer flow rate given by seem can be used without difficulty when such a correlation is extended to different monomers and the polymer deposition characteristics are compared, however, the flow rate based on cubic centimeters per minute cannot be used because the mass of a mole of gas depends on the molecular weight of the monomer. The polymer deposition rates of various monomers should be compared on the basis of the mass flow rate otherwise, polymer deposition rates are not directly proportional to the polymerization rates. 

IF(NOT(ISERROR(logical expression)),valueJf true,value if false) returns value if true if logicaLexpression does not return an error value. 

Now, change the formula to the logical expression =(YOG=96). Once again, highlight the formula and use function key F9 to display the result in the formula bar (only the first portion of the array is shown) ... 

Earlier we learned that FALSE can be represented by zero and TRUE by any non-zero value. If a logical expression is included in an arithmetic operation, FALSE becomes zero and TRUE becomes 1. Thus we can convert the array of TRUE and FALSE values to I s and O s by the expression =(YOG=96) 1. Again, use function key F9 to display the array of values ... 

Notice that multiplying the two logical expressions has the effect of converting the result to a number, making it unnecessary to multiply by 1. 

You can use other data types as Boolean variables. When a variable is used in a logical expression, zero is converted to False while any non-zero value is converted to True. Thus the expression... 

If LogicalExpression Then statement Else statement or, as illustrated in Figure 14-8, you can employ several logical expressions. 

The preceding expressions must evaluate to True or False that is, they must be logical expressions. The logical operators are almost always used in combination with If statements. 

You can also use the To keyword in expression, e.g.. Case "A" To "M". Expression can also be a logical expression. Use Case Else (not required) to handle all cases not covered by the preceding Case statements. 

With the basic equivalent relations given in Table I (e.g., see Williams, 1988), one can systematically model an arbitrary propositional logic expression that is given in terms of OR, AND, IMPLICATION operators, as a set of linear equality and inequality constraints. One approach is to systematically convert the logical expression into its equivalent conjunctive normal form representation, which involves the application of pure logical operations (Raman and Gross-mann, 1991). The conjunctive normal form is a conjunction of clauses, gi gj A A gj. Hence, for the conjunctive normal form to be true, each clause g, must be true independent of the others. Also, since a clause g, is just a disjunction of literals, Pj V V " V t can be expressed in the linear mathematical form as the inequality. 

The procedure to convert a logical expression into its corresponding conjunctive normal form was formalized by Clocksin and Mellish (1981). The systematic procedure consists of applying the following three steps to each logical proposition ... 

As an example, consider the logic condition if the absorber is selected or the membrane separator is selected, then do not use cryogenic separation. Assigning the Boolean literals to each action — select absorber, = select membrane separator, Pqs = select cryogenic separation, the logic expression is given by... 

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