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Within the field of Inductive Logic Programming (ILP) some recent work has also focussed on the problem of regression. FORS (Karalic, 1995, Karalic & Bratko, 1997) induces a model that has the form of a Prolog program. This program consists of clauses of the form f(Y,X1,?,Xa) that represent the obtained regression model. The system is able to receive, as input, samples of the unknown regression function and background knowledge, which constitutes a major advantage. FORS uses a covering algorithm at the top level that keeps generating new clauses followed by the removal of the cases covered until a termination criterion is met. The clause generation step follows a general to specific search that keeps adding new literals to the clause. FORS uses several pre-pruning mechanisms for controlling clause generation. Among them is the use of Minimum Description Length (MDL) (Rissanen,1982).
FFOIL (Quinlan, 1996) is another example of an ILP system able to deal with regression. FFOIL is a derivation of the FOIL system (Quinlan, 1990). FFOIL follows a similar covering algorithm as FOIL?s. It starts with an empty program and keeps adding clauses until all cases are covered. Each added clause starts with an empty body and literals are appended as a form of specialising the clause. A function with k arguments (the input variables) is represented by a k+1-ary relation where one argument holds the function outcome. The result obtained by FFOIL consists of a Prolog program with clauses of this relation.
First order approaches to regression have a much larger search space than their propositional counterparts. This fact has two contradictory effects. While being able to find solutions not available to propositional systems, this increased expressiveness has a strong impact in the computational complexity of these systems. This makes them hardly applicable to extremely large domains that are found in some applications (like in a typical Data Mining situation). However, computation power grows at a very fast rate and ILP may well become the major trend in Machine Learning approaches to regression .
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