Quantifiers and Logical Reductions

Abstract

In Chapter 6 we have seen that least fixed-point logic FO(LFP) captures PTIME, but does not strongly capture PTIME. In fact, it is open until now, whether there is any “natural” logic strongly capturing PTIME. At a first glance the following idea might be successful: Consider, for example, FO(LFP). One could think of strengthening it by adding some PTIME concepts, thereby enlarging its expressive power just up to PTIME. In traditional model theory there is a well-established method to enrich logics by new concepts, namely by adjoining so-called Lindström quantifiers. We have already met such quantifiers, for example, the counting quantifiers (cf. section 2.4).