FTA: Preprocessing Techniques

In order to optimize the analysis and deal with the exponential complexity, fault tree preprocessing (or transformation) is attempted before initiating other fault tree analysis algorithms.

The preprocessing algorithms deal with transformations of propositional directed acyclic graphs ([PDAG]) to reduce the complexity and to divide-and-conquer the problem. There are many proposed algorithms, and successful application of preprocessing techniques helps reduce substantially the analysis complexity of large graphs in some cases. However, the ordering of preprocessing algorithms is not always clear due to their side-effects, and the performance gain is not guaranteed (may even be negative). Some preprocessing techniques may only work for certain structures or particular setups in the graph.

Constant Propagation

House or external events are treated as Boolean constants and propagated according to the Boolean logic before other more complex and expensive preprocessing steps [NR99] [Rau03]. This procedure prunes the PDAG. Null and unity branches are removed from the PDAG, leaving only variables and gates.

Gate Normalization

The PDAG is simplified to contain only AND and OR gates by rewriting complex gates like ATLEAST and XOR with AND and OR gates [Nie94] [Rau03]. After this operation, the graph is in normal form.

Complement Propagation

Complements or negations of gates are pushed down to leaves (variables) according to the De Morgan’s law [Rau03]. This procedure transforms the graph into negation normal form ([NNF]) if the graph is normal before the propagation.

Gate Coalescing

Gates of the same type or logic are coalesced according to the rules of the Boolean algebra [Nie94] [Rau03]. For example, AND gate parent and AND gate child are joined into a new gate, or the arguments of the child are added to the parent gate. This operation attempts to reduce the number of gates to expand later. However, this operation may complicate other preprocessing steps that may try to find modules or propagate failure in the PDAG.

Module Detection

Modules are defined as gates or group of nodes whose sub-graph does not have common nodes with the rest of the graph. Modules are detected and analyzed as separate and independent PDAGs [DR96]. If a module appears in the final products, then the products are populated with the sum of products of the module. This operation guarantees that the final, joint sum is minimal, and no expensive check for minimality is needed. However, most complex fault trees do not contain big modules in their original Boolean formula.

Multiple Definition Detection

Gates with the same logic and arguments can be considered to be multiply defined. Any gate from this group of multiply defined gates can represent all of them in the graph, reducing the size of the graph [Nie94] [NR99]. The result of this preprocessing technique can help other preprocessing algorithms that work with common nodes or detect independent sub-graphs.

The Shannon Decomposition for Common Nodes

Application of the Shannon decomposition for particular setups with an AND/OR gate with common descendant nodes in the gate’s sub-graph.

\[ \begin{align}\begin{aligned}x \& f(x, y) = x \& f(1, y)\\x \| f(x, y) = x \| f(0, y)\end{aligned}\end{align} \]

This technique is also called Constant Propagation [NR99] [Rau03], but the actual constant propagation is only the last part of the procedure; though, it is the main benefit of this preprocessing technique. Some ancestors of the common node in the sub-graph may need to be cloned, which increases the size of the graph, if the ancestors are common nodes themselves and linked to other parts of the whole graph. The application of this technique may be limited due to performance and memory considerations for complex graphs with many common nodes.

Distributivity Detection

This is a formula rewriting technique that detects common arguments in particular setups corresponding to the distributivity of AND and OR operators [Nie94].

\[ \begin{align}\begin{aligned}(x \| y) \& (x \| z) = x \| (y \& z)\\(x \& y) \| (x \& z) = x \& (y \| z)\end{aligned}\end{align} \]

This technique helps reduce the number of common nodes; however, it gets trickier to find the most optimal rewriting with more complex setups where more than one rewriting is possible.

\[(x \| y) \& (x \| z) \& (y \| z)\]

Merging Common Arguments

Common arguments of gates with the same logic can be merged into a new gate with the same logic as the group [NR99]. This new gate can replace the common arguments in the set of arguments of gates in the group. Successful application of this technique helps reduce the complexity of the BDD construction from the graph. Moreover, by reducing the number of common nodes, this technique may help isolate the common nodes into modules.

Boolean Optimization

This optimization technique detects redundancies in the graph by propagating failures of common nodes and noting the failure destinations [Nie94]. The redundant occurrences of common nodes are minimized by directly transferring the common node and its failure logic to the failure destinations.

The generalization of this technique comes from the observations of special cases for the Shannon decomposition. Given a Boolean formula \(f(x, y)\), the following cases are the special cases of its Shannon decomposition:

  1. If f(x, y) = 1/True/Failure assuming x = 1/True/Failure:

    \[f(x, y) = x \| f(0, y)\]
  2. If f(x, y) = 0/False/Success assuming x = 1/True/Failure:

    \[f(x, y) = \overline{x} \& f(0, y)\]
  3. If f(x, y) = 1/True/Failure assuming x = 0/False/Success:

    \[f(x, y) = \overline{x} \| f(1, y)\]
  4. If f(x, y) = 0/False/Success assuming x = 0/False/Success:

    \[f(x, y) = x \& f(1, y)\]

There may be many setups that satisfy these special cases in a PDAG, but only few transformations are beneficial. Transformations with disjunctions of the formula (cases 1 and 3) are the most desirable for analysis because the final result of the analysis is the disjunction of products.

The main optimization criterion for transformations is to decrease the complexity or multiplicity of the graph. That is, the transformation must yield fewer destinations than its original multiplicity. This kind of successful transformations may help other preprocessing techniques achieve better results with the simpler graph as well.