Infinitary Logic and Inductive Difinability Over Finite Structures MS-CIS-91-97 LOGIC & COMPUTATION 44

The extensions of first-order logic with a least fixed point operators (FO + LFP) and with a partial fixed point operator (FO + PFP) are known to capture the complexity classes P and PSPACE respectively in the presence of an ordering relation over finite structures. Recently, Abiteboul and Vianu [AV91b] investigated the relation of these two logics in the absence of an ordering, using a mchine model of generic computation. In particular, they showed that the two languages have equivalent expressive power if and only if P = PSPACE. These languages can also be seen as fragments of an infinitary logic where each formula has a bounded number of variables, Lω∞ω (see, for instance, [KV90]). We present a treatment of the results in [AV91b] from this point of view. In particular, we show that we can write a formula of FO + LFP and P from ordered structures to classes of structures where every element is definable. We also settle a conjecture mentioned in [AV91b] by showing that FO + LFP in properly contained in the polynomial time computable fragment of Lω∞ω, raising the question of whether the latter fragment is a recursively enumerable class. Comments University of Pennsylvania Department of Computer and Information Science Technical Report No. MSCIS-91-97. This technical report is available at ScholarlyCommons: http://repository.upenn.edu/cis_reports/365 Infinitary Logic and Inductive Difinability Over Finite Structures MS-CIS-91-97 LOGIC & COMPUTATION 44 Anuj Dawar St even Lindell Scott Weinstein Department of Computer and Information Science School of Engineering and Applied Science University of Pennsylvania Philadelphia, PA 19104-6389