## 3 Integrating Plausible Rules with Ontologies
### Plausible Logic  
As a non-monotonic formalism, plausible logic (PL) deals with the problem of deriving conclusions in case of incomplete or uncertain information. Plausible logic is considered an improvement of defeasible logic [20,21]. A clause in PL \(\lor a_{1},a_{2},...a_{n}\) is the disjunction of positive or negative atoms \(a_{i}\). If both an atom and its negation appear, the clause is a tautology. A _contingent_ clause is a clause which is neither empty nor a tautology [20]. **Definition 1**: _A plausible description of a situation is a tuple \(PD=(Ax,R_{p},R_{d},\succ)\), where \(Ax\) is a set of contingent clauses, called axioms, characterising the aspects of the situation that are certain, \(R_{p}\) is a set of plausible rules, \(R_{d}\) is a set of defeater rules, and \(\succ\) is a priority relation on \(R_{p}\cup R_{d}\)._  
A plausible theory is computed from a plausible description by deriving the set \(R_{s}\) of strict rules from the definite facts \(Ax\). Thus, a plausible knowledge base consists of strict rules (\(\rightarrow\)), plausible rules (\(\Rightarrow\)), defeater (warning) rules (\(\rightarrow\)), and a priority relation on the rules (\(\succ\)). Strict rules are rules in the classical sense (that is, whenever the premises are indisputable), then so is the conclusion. An atomic fact is represented by a strict rule with an empty antecedent. The plausible rule \(a_{i}\Rightarrow c\) means that if all the antecedents \(a_{i}\) are proved and all the evidence against the consequent \(c\) has been defeated, then \(c\) can be deduced. The plausible conclusion \(c\) can be defeated by contrary evidence. The only use of defeaters is to prevent some conclusions, as in \"if the buyer is a regular one and he has a short delay for paying, we might not ask for penalties\". This rule does not provide sufficient evidence to support a \"non-penalty\" conclusion, but it is strong enough to prevent the derivation of the penalty consequent. The priority relation \(\succ\) allows the representation of preferences among non-strict rules. Decisive plausible logic consists of a plausible theory and a proof function \(P(\lambda f,)\) that, given a proof algorithm \(\lambda\) and a formula \(f\) in conjunctive normal form, returns \(+1\) if \(\lambda f\) was proved, \(-1\) if there is no proof for \(\lambda f\), or 0 when \(\lambda f\) is undecidable due to looping. Plausible Logic has five proof algorithms \(\{\mu,\alpha,\pi,\beta,\gamma\}\), in which one is monotonic and four are non-monotonic: \(\mu\) monotonic, strict, like classical logic; \(\alpha=\beta\wedge\pi\); \(\pi\) plausible, propagating-ambiguity; \(\beta\) plausible, blocking-ambiguity; and \(\gamma=\pi\vee\beta\). **Example 1**: _Consider the following rules for triggering an alarm system in case the temperature rises above 0 \({}^{\circ}C\) in a refrigerator room._  
\(r_{1}:sensor(x),measures(x,y),y>4\to highTemperatureAlarm.\)  
\(r_{2}:sensor(x),measures(x,y),y>0\Rightarrow highTemperatureAlarm.\)  
\(r_{3}:inside(x)\Rightarrow\
eg highTemperatureAlarm.\)  
\(r_{4}:opendoor\rightarrow\
eg highTemperatureAlarm.\)  
\(r_{4}>r_{3}\)The rule \(r_{1}\) is a strict one: whenever the temperature \(y\) measured by the sensor \(x\) is above 4 \({}^{\circ}\)C the alarm starts automatically. The rule \(r_{2}\) is a plausible one: if the measured temperature is above 0 \({}^{\circ}\)C, there is a reason to support the decision to trigger the alarm. The algorithm analyses if there are other reasons supporting the opposite conclusion or trying to defeat it. Rule \(r_{3}\) is such a rule. If both rules can be fired, we have an ambiguity. In case of the propagating-ambiguity strategy, both consequences are derived by the system. In case of the blocking-ambiguity strategy, no conclusion would be valid. Note that the rule \(r_{4}\) is a defeater that, in case of activation, blocks the derivation of the conclusion \(\
eg highTemperatureAlarm\). If active, \(r_{4}\) being stronger than \(r_{3}\) is able to block the corresponding decision.