Infinitary Logic and Inductive Definability Over Finite Structures

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November 1991

Department of Computer & Information Science

Inﬁnitary Logic and Inductive Deﬁnability Over Finite Structures

Anuj Dawar

University of Pennsylvania

Steven Lindell

University of Pennsylvania

Scott Weinstein

University of Pennsylvania

Follow this and additional works at: https://repository.upenn.edu/cis_reports

Recommended Citation

Anuj Dawar, Steven Lindell, and Scott Weinstein, "Inﬁnitary Logic and Inductive Deﬁnability Over Finite Structures", . November 1991.

University of Pennsylvania Department of Computer and Information Science Technical Report No. MS-CIS-91-97.

This paper is posted at ScholarlyCommons. https://repository.upenn.edu/cis_reports/365

For more information, please contact [email protected].

Inﬁnitary Logic and Inductive Deﬁnability Over Finite Structures

Abstract

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, L
(see, for instance, [KV90]). We present a treatment of

∞ω

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 L recursively enumerable class.
, raising the question of whether the latter fragment is a

∞ω

Comments

University of Pennsylvania Department of Computer and Information Science Technical Report No. MS- CIS-91-97.

This technical report is available at ScholarlyCommons: https://repository.upenn.edu/cis_reports/365

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