Abductive reasoning

Abduction, or abductive reasoning, is the process of reasoning to the best explanations. In other words, it is the reasoning process that starts from a set of facts and derives their most likely explanations. The term abduction is sometimes used to mean just the generation of hypotheses to explain observations or conclusions, but the former definition is more common both in philosophy and computing.

Deduction and abduction differ in the direction in which a rule like &ldquo;$$a$$ entails $$b$$&rdquo; is used for inference (see also logical reasoning for a comparison with induction):


 * Deduction : allows deriving $$b$$ as a consequence of $$a$$; in other words, deduction is the process of deriving the consequences of what is known;


 * Abduction : allows deriving $$a$$ as an explanation of $$b$$; abduction works in reverse to deduction, by allowing the precondition $$a$$ of &ldquo;$$a$$ entails $$b$$&rdquo; to be derived from the consequence $$b$$; in other words, abduction is the process of explaining what is known.

In rare occasions, the expression "explanatory conclusions" is used instead of "explanations" to name the result of the abductive process.

Applications in artificial intelligence include fault diagnosis, belief revision, and automated planning.

Logic-based abduction
In logic, abduction is done from a logical theory $$T$$ representing a domain and a set of observations $$O$$. Abduction is the process of deriving a set of explanations of $$O$$ according to $$T$$. For $$E$$ to be an explanation of $$O$$ according to $$T$$, it should satisfy two conditions:


 * $$O$$ follows from $$E$$ and $$T$$;


 * $$E$$ is consistent with $$T$$.

In formal logic, $$O$$ and $$E$$ are assumed to be sets of literals. The two conditions for $$E$$ being an explanation of $$O$$ according to theory $$T$$ are formalized as:


 * $$T \cup E \models O$$;
 * $$T \cup E$$ is consistent.

Among the possible explanations $$E$$ satisfying these two conditions, some other condition of minimality is usually imposed to avoid irrelevant facts (not contributing to the entailment of $$O$$) to be included in the explanations.

Set-cover abduction
A different formalization of abduction is based on inverting the function that calculates the visible effects of the hypotheses. Formally, we are given a set of hypotheses $$H$$ and a set of manifestations $$M$$; they are related by the domain knowledge, represented by a function $$e$$ that takes as an argument a set of hypotheses and gives as a result the corresponding set of manifestations. In other words, for every subset of the hypotheses $$H' \subseteq H$$, their effects are known to be $$e(H')$$.

Abduction is performed by finding a set $$H' \subseteq H$$ such that $$M \subseteq e(H')$$. In other words, abduction is performed by finding a set of hypotheses $$H'$$ such that their effects $$e(H')$$ include all observations $$M$$.

A common assumption is that the effects of the hypotheses are independent, that is, for every $$H' \subseteq H$$, it holds that $$e(H') = \bigcup_{h \in H'} e(\{h\})$$. If this condition is met, abduction can be seen as a form of set covering.

History of the concept
The philosopher Charles Peirce introduced abduction into modern logic. In his works before 1900, he mostly uses the term to mean the use of a known rule to explain an observation, e.g., &ldquo;if it rains the grass is wet&rdquo; is a known rule used to explain that the grass is wet.

He later used the term to mean creating new rules to explain new observations, emphasising that abduction is the only logical process that actually creates anything new. Namely, he described the process of science as a combination of abduction, deduction and implication, stressing that new knowledge is only created by abduction.

This is contrary to the common use of abduction in the social sciences and in artificial intelligence, where the old meaning is used. Contrary to this use, Peirce stated that the actual process of generating a new rule is not &ldquo;hampered&rdquo; by logic rules. Rather, he pointed out that humans have an innate ability to correctly do inference; possessing this ability is explained by the evolutionary advantage it gives. Pierce's second use of 'abduction' is most similar to induction.

Applications
Abduction has been applied in artificial intelligence for various tasks. The most direct application of abduction is that of automatically detect faults in systems: given a theory relating faults and manifestation and a set of manifestations (the visible effects of faults), abduction can be used to derive some set of faults that are likely to be the cause of the problem.

Abduction can also be used to model automated planning. Given a logical theory relating action occurrences with their effects (for example, a formula of the event calculus), the problem of finding a plan for reaching a state can be modeled as the problem of abducing a sequence of literals implying that the final state is the goal state.

Belief revision, the process of adapting beliefs in view of new information, is another field in which abduction has been applied. The main problem of belief revision is that the new information may be inconsistent with the corpus of beliefs, while the result of the incorporation cannot be inconsistent. This process can be done by the use of abduction: once an explanation for the observation has been found, integrating it does not generate inconsistency. This use of abduction is not straighforward, as adding propositional formulae to other propositional formulae can only make inconsistencies worst. Instead, abduction is done at the level of the ordering of preference of the possible worlds.