Boolean function

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... 

Jisjoinl. Another reason for representing a Boolean function in minterms is their... 

Input/Output statements—Two boolean functions that are useful in input processing are EOF, which is TRUE if the pointer is currently at the end of the input file and FALSE otherwise, and EOLN, which is TRUE if the pointer is at the end of the current input line and FALSE otherwise. 

Non-homogeneous CA. These are CA in which the state-transition rules are allowed to vary from cell to cell. The simplest such example is one where there are only two different rules randomly distributed throughout the lattice. Kauffman [kauff84] has studied the otlier extreme in whidi tlie lattice is randomly populated with all possible Boolean functions of k inputs. 

Non-Homogeneous CA a characteristic feature of all CA rules defined so far has been that of homogeneity - each cell of the system evolves according to the same rule 0. Hartman and Vichniac [hartSfi] were the first to systematically study a class of inhomogeneous CA (INCA), in which the state-transition rules are allowed to vary from cell to cell. The simplest such example is one where there are only two different 0 s, which are randomly distributed throughout the lattice. Kauffman has studied the other extreme in which the lattice is randomly populated with all 2 possible boolean functions of k inputs. The results of such studies, as well as the relationship with the dynamics of random, mappings, are covered in detail in chapter 8.3. 

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. ... 

In order to construct an expression representing an arbitrary Boolean function J-, we must therefore j)erform two simple steps (1) construct expressions of the above form for each occurrence of a 1 in J- s truth table, and (2) join each such expression by the OR operator. Since each term is explicitly constructed so as to pick out a particular input configuration yielding the value 1 under J-, no more than one such term in the resulting OR expansion can yield the value 1 for a given input. All other configurations are identically zero. 

This model can be simplified by reducing it to a single parameter. Vichniac, et.al. [vich86b] for example, consider the behavior along the line p = 1 — Pi- They point out that this reduction affords pi and p2 a simple interpretation in terms of the familiar Boolean functions AND and XOR (exclusive OR). In particular, for the line p2 = 1 — pi, we can equate pi = p(XOR) and P2 = 1 — Pi = p(AND), so that... 

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 ... 

Since there are two choices for every combination of states of the k inputs at each site, is randomly selected from among 2 possible Boolean functions of k... 

Table 8.9 State cycle length and the number of state cycles for random boolean nets of size N and connectivity k a = pn — 1/2, where Pk is the mean internal homogeneity of all Boolean functions on K inputs (see text).

In this case, each site can have many identical outputs but receives only a single input. There are four possible Boolean functions with one input two yield fixed values of 0 or 1, independent of input (these two static functions, and Fi are always among the 2 possible Boolean functions), the third inverts the input T = —) and the fourth is the identity = +). We will discuss behavior arising only ft om the latter two active functions. Exact results for the analytically tractable case of allowing a distribution of all four Boolean functions have been derived by Flyvbjerg and Kjaer [flyvb88]. 

Figure 8.16 shows a portion of typical N, l)-net. The arrows indicate inputs to given sites and + and represent the two active Boolean functions at the sites toward which the arrows are pointing. Note that the structure degenerates into subnets of dynamically independent loops with outwmdly radiating tails . An input signal cannot enter a loop since the site by which it would enter the loop... 

Fig. 8.16 Typical random Boolean network with connectivity k == 1. Arrows indicate inputs to given sites and represent the Boolean functions at the sites toward which the arrows are pointing.

Attractor states arc tyirically stable with respect to 80 - 95% of the possible minimal perturbations, and mutations (by which sites are deleted or Boolean functions of single sites are altered) affect the dynamical behavior only slightly. 

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. 

Internal Homogeneity Clusters Walker and Ashby [walker66] first showed that increasing the sameness of the entries or the internal homogeneity - of a Boolean function s rule table tends to decrease the cycle length. 

Since, in this case, the state of each site depends oii the states of all other sites, including itself, the specification of each Boolean function requires that all... 

The problem was first addressed by Cover ([cover64], [cover65]), who found a lower bound on the minimum number of synaptic weights needed for an ADALINE network (see section 10.5.1) to realize any Boolean function. More recently. Cover s early work has been extended by Baum ([baumSSa], [baum89]). who has found a... 

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

HarM64 Harrison, M. A. On the classification of Boolean functions by the general linear and affine groups. SIAM J. 12 (1964) 285-299. 

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

Then the assignQualified invariant corresponds to the following Boolean function, which should evaluate to true upon completion of any external operation invocation. Note that the code form is the same as that of the type model invariant, with each reference to an attribute in the type model expanded to its corresponding representation in the implementation. 

Well-written postconditions can be used as the basis for verification and testing. For this purpose, we should write the postconditions in a more precise style as test (Boolean) functions. You can use the Boolean expression part of your favorite programming language we will use a general syntax from UML called Object Constraint Language (OCL). It translates readily to most programming languages but is more convenient for specification. 

So we generalize our model into a framework by creating a template package, as shown in Figure 9.3. (We will later drop the framework stereotype. We ve taken the opportunity to add more details, particularly about Resources not being double-booked. (Timelnterval will have to be defined somewhere we ve assumed it has a Boolean function noOverlap that compares two Timelntervals.)... 

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