Distances and limits on Herbrand interpretations
A notion of distances between Herbrand interpretations enables us to measure how good a certain program, learned from examples, approximates some target program. The distance introduced in  has the disadvantage that it does not fit the notion of “identification in the limit”. We use a distance defined by a level mapping  to overcome this problem, and study in particular the mapping TII induced by a definite program 11 on the metric space. Continuity of TII holds under certain conditions, and we give a concrete level mapping that satisfies these conditions, based on . This allows us to prove the existence of fixed points without using the Banach Fixed Point Theorem.
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