# 6 Meta-Logical Layer¶

The meta-logical layer is populated with constructs like common cause groups, delete terms, recovery rules, and exchange events that are used to give flavors to fault trees. This chapter reviews all these constructs.

## 6.1 Common Cause Groups¶

### 6.1.1 Description¶

From a theoretical view point, one of the basic assumptions of the fault tree technique is that occurrences of basic events are independent from a statistical viewpoint. However, most of the PSA models include, to a large extent, so-called common cause groups. Common cause groups are groups of basic events whose failure are not independent from a statistical view point. They may occur either independently or dependently due to a common cause failure. So far, existing tools embed three models for common cause failures (CCF): the beta-factor model, the Multiple Greek letters (MGL) model, and the alpha-factor model. Alpha-factor and MGL models differ only in the way the factors for each level (2 components fail, 3 components fail, etc.) are given. The Model Exchange Format proposes the three mentioned models plus a fourth one, so-called phi-factor, which is a more direct way to set factors.

Beta-Factor

The $$\beta$$-factor model assumes that if a common cause occurs, then all components of the group fail simultaneously. Components can fail independently. Multiple independent failures are neglected. The $$\beta$$-factor model assumes, moreover, that all the components of the group have the same probability distribution. It is characterized by this probability distribution and the conditional probability $$\beta$$ that all components fail, given that one component failed.

Let $$BE_1, BE_2, \ldots, BE_n$$ be the $$n$$ basic events of a common cause group with a probability distribution $$Q$$ and a beta-factor $$\beta$$. Applying the beta-factor model on the fault tree consists in following operations.

1. Create new basic events $$BE_{CCF_i}$$ for each $$BE_i$$ to represent the independent occurrence of $$BE_i$$ and $$BE_{CCF_i}$$ to represent the occurrence of all $$BE_i$$ together.
2. Substitute a gate $$G_i = BE_{CCF_i} \lor BE_i$$ for each basic event $$BE_i$$.
3. Associate the probability distribution (e.g., $$\beta \times Q$$) with the event $$BE_{CCF_i}$$.
Multiple Greek Letters

The Multiple Greek Letters (MGL) model generalizes the beta-factor model. It considers the cases where sub-groups of $$1, 2, \ldots, n-1$$ components of the group fail together. This model is characterized by the probability distribution of failure of the components, and $$n-1$$ factors $$\rho_2, \ldots, \rho_n$$, $$\rho_k$$ denotes the conditional probability that $$k$$ components of the group fail given that $$k-1$$ failed.

Let $$BE_1, BE_2, \ldots, BE_n$$ be the $$n$$ basic events of a common cause group with a probability distribution $$Q$$ and factors $$\rho_2, \ldots, \rho_n$$. Applying the MGL model on the fault tree consists in following operations.

1. Create a basic event for each combination of basic events of the group (there are $$2^n-1$$ such combinations).
2. Transform each basic event $$BE_i$$ into an OR-gate $$G_i$$ over all newly created event basic events that represent a group that contains $$BE_i$$.
3. Associate the following probability distribution with each newly created basic event representing a group of $$k$$ components (with $$\rho_{n+1} = 0$$).
$Q_k = \frac{1}{\binom{n-1}{k-1}} \times \left(\prod_{i=2}^{k}\rho_i \right) \times (1 - \rho_{k+1}) \times Q$

For instance, for a group of 4 basic events: A, B, C, and D, the basic event A is transformed into a gate $$G_A = A \lor AB \lor AC \lor AD \lor ABC \lor ABD \lor ACD \lor ABDC$$, and the $$Q_k$$‘s are as follows.

$\begin{split}Q_1& = (1 - \rho_2) \times Q\\ Q_2& = \tfrac{1}{3} \times \rho_2 \times (1 - \rho_3) \times Q\\ Q_3& = \tfrac{1}{3} \times \rho_2 \times \rho_3 \times (1 - \rho_4) \times Q\\ Q_4& = \rho_2 \times \rho_3 \times \rho_4 \times Q\end{split}$

Note that if $$\rho_k = 0$$, then $$Q_k, Q_{k+1}, \ldots$$ are null as well. In such a case it is not necessary to create the groups with k elements or more.

Alpha-Factor

The alpha-factor model is the same as the MGL model except in the way the factors are given. Here $$n$$ factors $$\alpha_1, \ldots, \alpha_n$$ are given. $$\alpha_k$$ represents the fraction of the total failure probability due to common cause failures that impact exactly $$k$$ components. The distribution associated with a group of size $$k$$ is as follows:

$Q_k = \frac{k}{\binom{n-1}{k-1}} \times \frac{\alpha_k}{\sum_{i=1}^{n}i\cdot\alpha_i} \times Q$
Phi-Factor

The phi-factor model is the same as MGL and alpha-factor models except that factors for each level are given directly.

Indeed, the sum of the $$\phi_i$$‘s should equal 1.

### 6.1.2 XML representation¶

The RNC schema for the XML description of Common Cause Failure Groups is given in Listing 6.1. Note that the number of factors depends on the model. Tools are in charge of checking that there is the good number of factors. Note also that each created basic event is associated with a factor that depends on the model and the level of the basic event. The sum of the factors associated with basic events of a member of the CCF group should be equal to 1; although, this is not strictly required by the Model Exchange Format.

Listing 6.1 The RNC schema for the XML representation of CCF-groups
CCF-group-definition =
element define-CCF-group {
name,
attribute model { CCF-model },
label?,
attributes?,
members,
distribution,
factors
}

members = element members { basic-event+ }

factors =
element factors { factor+ }
| factor

factor =
element factor {
attribute level { xsd:positiveInteger }?,
expression
}

distribution = element distribution { expression }

CCF-model = "beta-factor" | "MGL" | "alpha-factor" | "phi-factor"


#### 6.1.2.1 Example¶

Here follows a declaration of a CCF-group with four elements under the MGL model.

<define-CCF-group name="pumps" model="MGL">
<members>
<basic-event name="pumpA"/>
<basic-event name="pumpB"/>
<basic-event name="pumpC"/>
<basic-event name="pumpD"/>
</members>
<factors>
<factor level="2">
<float value="0.10"/>
</factor>
<factor level="3">
<float value="0.20"/>
</factor>
<factor level="4">
<float value="0.30"/>
</factor>
</factors>
<distribution>
<exponential>
<parameter name="lambda"/>
<system-mission-time/>
</exponential>
</distribution>
</define-CCF-group>


## 6.2 Delete Terms, Recovery Rules, and Exchange Events¶

### 6.2.1 Description¶

Delete Terms

Delete Terms are groups of pairwise exclusive basic events, used to model impossible configurations. A typical example is the case where:

• The basic event a can only occur when the equipment A is in maintenance.
• The basic event b can only occur when the equipment B is in maintenance.
• Equipment A and B are redundant and cannot be simultaneously in maintenance.

In most of the tools, delete terms are considered as a post-processing mechanism: minimal cut sets containing two basic events of a delete terms are discarded. In order to speed-up calculations, some tools use basic events to discard minimal cut sets on the fly, during their generation.

Delete Terms can be handled in several ways. Let $$G = \{e_1, e_2, e_3\}$$ be a Delete Term (group).

• A first way to handle $$G$$, is to use it to post-process minimal cut sets, or to discard them on the fly during their generation. If a minimal cut set contains at least two of the elements of $$G$$, it is discarded.

• A global constraint $$C_G = \lnot \binom{3}{2}(e_1, e_2, e_3)$$ is introduced, and each top event (or event tree sequences) “top” is rewritten as $$top \land C_G$$.

• As for Common Causes Groups, the $$e_i$$‘s are locally rewritten in as gates:

• $$e_1$$ is rewritten as a gate $$ge_1 = e_1 \land \lnot e_2 \land \lnot e_3$$
• $$e_2$$ is rewritten as a gate $$ge_2 = e_2 \land \lnot e_1 \land \lnot e_3$$
• $$e_3$$ is rewritten as a gate $$ge_3 = e_3 \land \lnot e_1 \land \lnot e_2$$
Recovery Rules

Recovery Rules are an extension of Delete Terms. A Recovery Rule is a couple $$(H, e)$$, where $$H$$ is a set of basic events, and $$e$$ is a (fake) basic event. It is used to post-process minimal cut sets: if a minimal cut set $$C$$ contains $$H$$, the $$e$$ is added to $$C$$. Recovery Rules are used to model actions taken in some specific configurations to mitigate the risk (hence their name).

Here several remarks can be made.

• It is possible to mimic Delete Terms by means of recovery rules. To do so, it suffices to assign the basic event e to the value “false” or the probability 0.0.
• As for Delete Terms, it is possible to give purely logical interpretation to Recovery Rules. The idea is to add a global constraint $$H \Rightarrow e$$, i.e., $$\lnot H \lor e$$, for each Recovery Rule $$(H, e)$$.
• Another definition of Recovery Rules as a post-processing is that the event $$e$$ is substituted for subset $$H$$ in the minimal cut set. This definition, however, has the major drawback by being impossible to interpret with a Boolean logic. No Boolean formula can withdraw events from a configuration.
Exchange Events
Exchange Events are very similar to Recovery Rules. An Exchange Event (Rule) is a triple $$(H, e, e')$$, where $$H$$ is a set of basic events, and $$e$$ and $$e'$$ are two basic events. Considered as a post-processing of minimal cut sets, such a rule is interpreted as follows. If the minimal cut set contains both the set $$H$$ and the basic event $$e$$, then the basic event $$e'$$ is substituted for $$e$$ in the cut set. For the same reason as above, Exchange Events cannot be interpreted with a Boolean logic.

### 6.2.2 All Extra-Logical Constructs in One: the Notion of Substitution¶

Constructs that cannot be interpreted with a Boolean logic should be avoided for at least two reasons. First, models containing such constructs are not declarative. Second, and more importantly, they tighten assessment tools to one specific type of algorithms. The second interpretation of Recovery Rules and Exchange Events tighten the models to be assessed by means of the minimal cut sets approach.

Nevertheless, Recovery Rules and Exchange Events are useful and broadly used in practice. Fortunately, Exchange Events (considered as a post processing mechanism) can be avoided in many cases by using the instructions that give flavors to fault trees while walking along event tree sequences: in a given sequence, one may decide to substitute the event $$e'$$ for the event $$e$$ (or the parameter $$p'$$ for the parameter $$p$$) in the Fault Trees collected so far. This mechanism is perfectly acceptable because it applies while creating the Boolean formula to be assessed.

It is not yet possible to decide whether Recovery Rules (under the second interpretation) and Exchange Events can be replaced by purely declarative constructs or by instructions of event trees. This has to be checked on real-life models. To represent Delete Term, Recovery Rules and Exchange Events, the Model Exchange Format introduces a unique construct: the notion of substitution.

A substitution is a triple $$(H, S, t)$$ where:

• $$H$$, the hypothesis, is a (simple) Boolean formula built over basic events.
• $$S$$, the source, is also a possibly empty set of basic events.
• $$t$$, the target, is either a basic event or a constant.

Let $$C$$ be a minimal cut set, i.e., a set of basic events. The substitution $$(H, S, t)$$ is applicable on $$C$$ if $$C$$ satisfies $$H$$ (i.e., if $$H$$ is true when $$C$$ is realized). The application of $$(H, S, t)$$ on $$C$$ consists in removing from $$C$$ all the basic events of $$S$$ and in adding to $$C$$ the target $$t$$.

Note that if t is the constant “true”, adding t to $$C$$ is equivalent to adding nothing. If $$t$$ is the constant “false”, adding $$t$$ to $$C$$ is equivalent to discard $$C$$.

This notion of substitution generalizes the notions of Delete Terms, Recovery Rules, and Exchange Events:

• Let $$D = \{e_1, e_2, \ldots, e_n\}$$ be a group of pairwise exclusive events (a Delete Term). Then $$D$$ is represented as the substitution $$(\binom{n}{2}(e_1, e_2, \ldots, e_n), \varnothing, \text{false})$$.
• Let $$(H, e)$$ be a Recovery Rule, under the first interpretation, where $$H = \{e_1, e_2, \ldots, e_n\}$$. Then, $$(H, e)$$ is represented by the substitution $$(e_1 \land e_2 \land \ldots \land e_n, \varnothing, e)$$.
• Let $$(H, e)$$ be a Recovery Rule, under the second interpretation, where $$H = \{e_1, e_2, \ldots, e_n\}$$. Then $$(H, e)$$ is represented by the substitution $$(e_1 \land e_2 \land \ldots \land e_n, H, e)$$.
• Finally, let $$(H, e, e')$$ be an Exchange Event Rule, where $$H = \{e_1, e_2, \ldots, e_n\}$$. Then $$(H, e, e')$$ is represented by the substitution $$(e_1 \land e_2 \land \ldots \land e_n \land e, {e}, e')$$.

Note that a substitution $$(H, \varnothing, t)$$ can always be interpreted as the global constraint $$H \Rightarrow t$$.

### 6.2.3 XML Representation¶

The RNC schema for the XML description of substitutions is given in Listing 6.2. The optional attribute “type” is used to help tools that implement “traditional” substitutions.

Listing 6.2 The RNC schema for the XML representation of exclusive-groups
substitution-definition =
element define-substitution {
name?,
attribute type { xsd:string }?,
label?,
attributes?,
element hypothesis { formula },
element source { basic-event+ }?,
element target { basic-event+ | Boolean-constant }
}


#### 6.2.3.1 Example¶

Assume that Basic Events “failure-pump-A”, “failure-pump-B”, and “failure-pump-C” are pairwise exclusive (they form a delete term) because they can only occur when, respectively, equipment A, B, and C are under maintenance and only one equipment can be in maintenance at once. The representation of such a delete term is as follows.

<define-substitution name="pumps" type="delete-terms">
<hypothesis>
<atleast min="2">
<basic-event name="failure-pump-A"/>
<basic-event name="failure-pump-B"/>
<basic-event name="failure-pump-C"/>
</atleast>
</hypothesis>
<target>
<constant value="false"/>
</target>
</define-substitution>


#### 6.2.3.2 Example¶

Assume that if the valve V is broken, and an overpressure is detected in pipe P, then a mitigating action A is performed. This is a typical Recovery Rule (under the first interpretation), where the hypothesis is the conjunction of Basic Events “valve-V-broken” and “overpressure-pipe-P”, and the added Basic Event is “failure-action-A”. It is encoded as follows.

<define-substitution name="mitigation" type="recovery-rule">
<hypothesis>
<and>
<basic-event name="valve-V-broken"/>
<basic-event name="overpressure-pipe-P"/>
</and>
</hypothesis>
<target>
<basic-event name="failure-action-A"/>
</target>
</define-substitution>


#### 6.2.3.3 Example¶

Assume that if magnitude of the earthquake is 5, 6 or 7, the size of a leak of a given pipe P gets large, while it is small for magnitudes below 5. We can use an exchange event rule to model this situation.

<define-substitution name="magnitude-impact" type="exchange-event">
<hypothesis>
<or>
<basic-event name="magnitude-5"/>
<basic-event name="magnitude-6"/>
<basic-event name="magnitude-7"/>
</or>
</hypothesis>
<source>
<basic-event name="small-leak-pipe-P"/>
</source>
<target>
<basic-event name="large-leak-pipe-P"/>
</target>
</define-substitution>