Is Nonmonotonic Reasoning Always Harder

Abstract. Although it has been shown that non-monotonic reasoning
1 Introduction
In the last two decades, many non-monotonic reasoning principles have been introduced to overcome some de ciencies of classical logic when applied to everyday reasoning. This reasoning is characterized by drawing conclusions even in under-speci ed situations. Rules of thumb like \Professors are good teachers" are used with the proviso that, for few instances, exceptions are possible. The worst-case complexities of various (propositional) non-monotonic reasoning systems have been thoroughly investigated (e.g. 3, 13, 15, 16, 18], an overview is given in 8]). It turned out that, basically, the usual reasoning tasks in the context of non-monotonic reasoning are at the second level of the polynomial hierarchy. In other words, unless the hierarchy collapses, non-monotonic reasoning is strictly harder than classical (monotonic) reasoning.2 This somewhat negative result seems to contradict the intuition that common-sense reasoning should actually be easier because of its non-monotonic character, and not more di cult (otherwise we would hardly use it the way we obviously do). As a case in point, in 6, 7] it is shown that (propositional) non-monotonic reasoning systems allow a \super-compact" representation of knowledge as compared with (monotonic) classical logic.
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The authors would like to thank Georg Gottlob and Thomas Eiter for their useful comments and constructive criticisms on an earlier version of this paper. The rst author was partly supported by the Christian Doppler Labor fur Expertensysteme, Wien, Austria. 2 The seemingly single exception of this pattern is the modal logic S4. In 21] it is shown that non-monotonic S4 is at the second level of the polynomial hierarchy, whereas it is well-known that monotonic S4 is PSPACE-complete.
If rst-order non-monotonic systems are considered, other complexity measures have to be used due to the undecidability of rst-order logic. One suitable measure is to determine \how undecidable" a problem is (similar to propositional problems, where one determines \how decidable" a problem is). For circumscription this has been carried out in 20], where it is shown that, e.g., the problem of deciding whether a given rst-order formula has a countably in nite minimal 1 model is 2 -complete. Hence, model checking for circumscription is at the second level of the analytical hierarchy and is thus strictly harder than ordinary model checking. Another measure of complexity is the minimal proof length of a theorem (the length of a proof is the number of symbols occurring in it). Assume we have theorems H and H 0 , where H 0 is a formula obtained from H by an application of a non-monotonic rule schema (e.g., completion or circumscription). Then the minimal proof lengths of these two formulae (in the same calculus) can be compared. We use a cut-free sequent calculus here for the comparison, because it is a prototype of many other calculi improved for proof search (like, e.g., semantic tableaux or various connection calculi). We will show that Clark's completion lifted to full rst-order logic (i.e., circumscription) can tremendously shorten proofs. More precisely, we will show that there is an in nite sequence of formulae H1 ; H2 ; : : : with the following two properties: 1. The minimal proof length of Hk in a cut-free sequent calculus is non-elementary in k, i.e., the proof length of Hk is of the order s(k ? 1), where s(0) = 1 and s(n + 1) = 2s(n) for all n 2 IN0 . 0 2. The minimal proof length of Hk is exponential in k. 0 In particular, each Hk is of the form Tk ! Fk , and each Hk is of the form CIRC(Tk ; q) ! Fk , where CIRC(Tk ; q) is the circumscription of the predicate q in the formula Tk . Thus, our result states that showing ` Tk ! Fk is much LK harder than showing ` CIRC(Tk ; q) ! Fk , where \ ` " denotes the derivability LK LK in the cut-free sequent calculus LK. The reason for this extreme decrease of complexity is the possibility to simulate certain instances of the cut rule by the formula introduced by circumscription. Moreover, since for rst-order cut-free sequent calculi the size of the search space is elementarily related to the minimal proof length if, e.g., breadth- rst search is assumed, a non-elementary decrease of the search space is also achieved. In addition to this result for LK, we also show a similar result for Herbrand complexity 1], which is a more \calculusindependent" complexity measure. A motivation of our method can be given as follows. Usually, non-monotonic techniques are applied in case a classical proof cannot be found. Although this is a reasonable procedure in decidable systems, it is not appropriate for undecidable systems like rst-order logic. Indeed, if we integrate non-monotonic rules into rst-order theorem provers, we have to invoke non-monotonic mechanisms after a certain amount of time, whenever the goal formula has not been proven classically up to this point. Accordingly, it may happen that a formula is provable both classically and with the help of non-monotonic rules. Our result shows therefore that, in certain cases, the theorem prover may easier nd a proof because the presence of non-monotonic rules yields a much smaller search space.
Is Non-Monotonic Reasoning Always Harder?
Uwe Eglyபைடு நூலகம்and Hans Tompits?
Technische Universitat Wien Abt. Wissensbasierte Systeme 184/3 Treitlstra e 3, A{1040 Wien, Austria e-mail: uwe,tompits]@kr.tuwien.ac.at is presumably harder than classical reasoning, there are cases where a non-monotonic treatment actually simpli es matters. Indeed, one of the reasons for considering non-monotonic systems is the hope of speeding up reasoning, and not to slow it down. In this paper, we consider proof lengths in a cut-free sequent calculus, and we show that the application of circumscription (or completion) to certain rst-order formulae leads to a non-elementary speed-up of proof length. This is possible because the introduction of the completion formula can simulate the cut rule.