Theory Curbing
Circumscription does not always correctly handle disjunctive information. Ray Reiter provided the following example: a coin is tossed over a checkboard, and the result is that the coin is either on a black area, or on a white area, or both. However, there are a large number of other possible places where the coin is not supposed to be on; for example, it is implicit that the coin is not on the floor, or on the refrigerator, or on the moon surface. Circumscription can therefore be used to minimize the extension of predicate, so that is false even if this is not explicitly stated.
On the other hand, the minimization of the predicate leads to the wrong result that the coin is either on a black area or on a white area, but not both. This is because the models in which is true only on and only on have a minimal extension of, while the model in which the extension of is composed of both pairs is not minimal.
Theory curbing is a solution proposed by Thomas Eiter, Georg Gottlob, and Yuri Gurevich. The idea is that the model that circumscription fails to select, the one in which both and are true, is a model of the formula that is greater (w.r.t. the extension of ) than both the two models that are selected. More specifically, among the models of the formula, the excluded model is the least upper bound of the two selected models. Theory curbing selects such least upper bounds models in addition to the ones selected by circumscription. This inclusion is done until the set of models is closed, in the sense that it includes all least upper bounds of all sets of models it contains.
Read more about this topic: Circumscription (logic)
Famous quotes containing the words theory and/or curbing:
“Everything to which we concede existence is a posit from the standpoint of a description of the theory-building process, and simultaneously real from the standpoint of the theory that is being built. Nor let us look down on the standpoint of the theory as make-believe; for we can never do better than occupy the standpoint of some theory or other, the best we can muster at the time.”
—Willard Van Orman Quine (b. 1908)
“Imagine believing in the control of inflation by curbing the money supply! That is like deciding to stop your dog fouling the sidewalk by plugging up its rear end. It is highly unlikely to succeed, but if it does it kills the hound.”
—Michael D. Stephens. “On Sinai, There’s No Economics,” New York Times (Nov. 13, 1981)