Boolean equations

This means that Boolean equation 2.2-5 is the same as 2.2-7 and the third term in equation 2.2-5 is superfluous, Figure 2.2-2 interprets these equations as Venn diagrams. 

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... 

The Boolean equation for the probability of a chemical process system failure is R = A (B-i-C (D-rE (B-l-F G+C). Using Table 2.1-1, factor (he equation into a sum of products to get the mincut representation with each of the products representing an accident sequence. 

Boolean equations can be used to model any system the system s reliability is calculated by factoring the equations into cutsets and substituting the probabilities for component fai lure 1 lus can be done for either success or failure models. Working directly with equations is not e erv one s ctip of tea many individuals prefer graphical to mathematical methods. I hus, symbols and appearance of the methods differ but they must represent the same Boolean equation for them to be eqni valent. 

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... 

To determine the minimal cutsets of a fault tree, the tree is first translated into its equivalent Boolean equations and then either the "top-down" or "bottom-up" substitution method is used. The methods are straightforward and involve substituting and expanding Boolean expressions Two Boolean laws, the distributive law and the law of absorption (Table 2.1-1), are used to remove superfluous items. 

Consider the example fault tree (Figure 3.4,4-4) the Boolean equations taken, gate at a time,. ire T = EI E2, El = A+E3, E3 = B+C, E2 = C+E4, E4=A B. Start with the 7-cquation, substitute, and expand until the minimal cutset expression for the top event is obtaineil Subst ituting for El and E2 gives ... 

A fault tree is a graphical form of a Boolean equation, but the probability of the top event (and lesser events) can be found by substituting failure rates and probabilities for these iwo-staie events. The graphical fault tree is prepared for computer or manual evaluation by pruning" it of less significant events to focus on more significant events. Even pruned, the tree may be so large that it IS intractable and needs division into subtrees for separate evaluations. If this is done, care must be taken to insure that no information is lost such as interconnections between subtrees. 

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

It is illustrated in Section 3.4.4 by tracing the paths for leaking engine compression and applied to fault tree construction for the FFTF reactor Fullwood and Erdmann, 1974). The method involves writing Boolean equations for all paths whereby hazardous material may be released. It is primarily useful for enumerating release paths, but not for what started the release It was used to enumerate the possible paths for stealing nuclear bomb material from a facility. 

FTAP - Give it a fault tree or equivalent as linked Boolean equations, and it will find the cutsets (Section 2.2), but it does not numerically evaluate the probabilities. 

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... 

A fault tree is, itself, a Boolean equation relating basic events to the top event. The equation can be analyzed quantitatively or qualitatively by hand or by using computer code(s). If it is analyzed quantitatively, the probabilities or frequencies of the intermediate events and the top event are calculated. If it is analyzed qualitatively, a list of the failure combinations that can cause the top event is generated. These combinations are known as cut sets. A minimal cut set (MCS) is the smallest combination of basic events that, if they occur or exist simultaneously, cause the top event. These combinations are termed "minimal" because all of the basic events in a MCS must occur if the top event is to occur. Thus, a list of MCSs represents the known ways the top event can occur, stated in terms of equipment failures, human errors, and associated circumstances. 

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. 

F. M. Brown. Boolean Reasoning the Logic of Boolean Equations. Kluwer Academic Publishers, Boston, MA, USA, 1990. 

Screening of the final Boolean equation representing the Core Damage and the results of the importance analysis show clearly that the basic event 1IKPROBRKL, which represents the CCF of circuit breakers, is the most risk important contributor to the ACDF, which can be approximated using equation (5) as follows ... 

In Eq. (11), (B + Bi) is actually the approximation of(B+Bi-B-Bi)in the Boolean algebra by assuming that (B Bi) is negligible. Thus, true Boolean equation would be the following ... 

A description more suitable for formal verification is a combination of boolean equations and state registers (see the BCD recognizer example in section 4). To derive such a description from layout is not a trivial task, however, and is the topic of this section. A more extensive description of the algorithms described can be found elsewhere [3]. 

Using the top-down substitution method—actually writing Boolean equations from the top event down— we can write... 

Start at the top event and then create Boolean equations for each level or branch on the tree. Once the next couple of levels have been written, you can use the various Table 7.1 substitution laws. So, combining B and B2 and through Boolean manipulation,... 

Obviously, the bottom-up method of FTA is the exact opposite of what was just demonstrated. You start at the lowest level, substitute the Boolean equations, and solve for the top event. 

As stated earlier, the opposite of a fault tree is a success tree. In Boolean algebra, a success tree is the complement of a fault tree. The complement of a cut set is a path set. To solve a success tree, you have two options either draw a success tree from the start or draw a fault tree and then take the complement of the tree (along with the corresponding Boolean equations). 

The fault tree is drawn, and then the Boolean equations and minimal cut sets are derived for the top event. Probability estimates can be generated from hardware failure data, human error estimation, maintenance frequency, etc. Probability estimates are then assigned to the events. Be sure to take into consideration uncertainty limits to your failure data. Through the laws of probability, combine the... 

Logic (Gate) State Transitions, Boolean Equations, Tables Logic Gates, Latches Cells... 

IBM s ALERT system was one of the first high-level synthesis systems, and included scheduling, data path synthesis, and structural transformations. The output was a Register-Transfer level design, specified in terms of Boolean equations. 

Operations are mapped onto Boolean equations, and then onto combinational library modules. Variables are mapped onto registers only if they are declared as architectural registers, or if thqr are used in a loop otherwise, th r are implemented as wires. 

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