2.1 Abduction (Hypothetical Reasoning)
According to the definition by Peirce, abduction is characterized as follows (Peirce, 1955):
Abduction is an operation for adopting an explanatory hypothesis, which is subject to certain conditions, and that in pure abduction, there can never be justification for accepting the hypothesis other than through interrogation.
Abduction is very powerful in the human reasoning. For the computation, abduction is usually used to find the reason (set of hypotheses) in a logical way to explain an observation. The original abduction is rather complicated reasoning system. For the computation a certain restriction such as selecting hypotheses from a hypothesis base is usually given. For instance, the inference mechanism of Theorist (Poole, Goebel, & Aleliunas, 1987) that explains an observation (O) by a consistent and minimal hypotheses set (h) selected from a set of hypotheses (H) is shown as followings.F ⊬ O. (O cannot be explained by only F.) (1)F
h
O. (O can be explained by F and h.) (2)F
h ⊬ 
. (F and h is consistent.) (3) Where F is a fact (background knowledge) and 
is an empty clause. A hypothesis set (h) is selected from a hypothesis base (h
H).
Thus, ``reason'' is usually selected from the knowledge (hypotheses) base. For instance, when Theorist is used for an LSI circuit design, F includes knowledge about the devices' function and their connections, and the knowledge of other rules. In addition, H includes candidate devices and their candidate connections. If the relation between input and output of the circuit is given as an observation O, Theorist computes the name of devices and their connections as hypotheses h. Therefore, usual abduction requires a perfect hypotheses base from which a consistent hypotheses set is selected to explain an observation. Here, ``perfect hypotheses base'' means the hypotheses base that contains all the necessary hypotheses.