Editing Abductive reasoning (section)

===Logic-based abduction===
In [[logic]], [[explanation]] is accomplished through the use of a [[logical theory]] <math>T</math> representing a [[domain of discourse|domain]] and a set of observations <math>O</math>. Abduction is the process of deriving a set of explanations of <math>O</math> according to <math>T</math> and picking out one of those explanations. For <math>E</math> to be an explanation of <math>O</math> according to <math>T</math>, it should satisfy two conditions:

* <math>O</math> follows from <math>E</math> and <math>T</math>;
* <math>E</math> is [[Consistent|consistent]] with <math>T</math>.

In formal logic, <math>O</math> and <math>E</math> are assumed to be sets of [[Literal (mathematical logic)|literal]]s. The two conditions for <math>E</math> being an explanation of <math>O</math> according to theory <math>T</math> are formalized as:

:<math>T \cup E \models O;</math>
:<math>T \cup E</math> is consistent.

Among the possible explanations <math>E</math> satisfying these two conditions, some other condition of minimality is usually imposed to avoid irrelevant facts (not contributing to the entailment of <math>O</math>) being included in the explanations. Abduction is then the process that picks out some member of <math>E</math>. Criteria for picking out a member representing "the best" explanation include the [[simplicity]], the [[prior probability]], or the explanatory power of the explanation.

A [[Proof theory|proof-theoretical]] abduction method for [[first-order logic|first-order]] classical logic based on the [[sequent calculus]] and a dual one, based on semantic tableaux ([[analytic tableaux]]) have been proposed.<ref>Cialdea Mayer, Marta and Pirri, Fiora (1993) "First order abduction via tableau and sequent calculi" Logic Jnl IGPL 1993 1: 99–117; {{doi|10.1093/jigpal/1.1.99}}. Oxford Journals</ref> The methods are sound and complete and work for full first-order logic, without requiring any preliminary reduction of formulae into normal forms. These methods have also been extended to [[modal logic]].<ref>Cialdea Mayer, Marta and Pirri, Fiora (1993) "Propositional abduction in modal logic" Logic Jnl IGPL 1995 3(6) 907–919; {{doi|10.1093/jigpal/3.6.907}}. Oxford Journals</ref>

[[Abductive logic programming]] is a computational framework that extends normal [[logic programming]] with abduction. It separates the theory <math>T</math> into two components, one of which is a normal logic program, used to generate <math>E</math> by means of [[backward reasoning]], the other of which is a set of integrity constraints, used to filter the set of candidate explanations.