Choiceless Polynomial Time, Counting and the Cai--Fürer--Immerman Graphs

Authors:
Anuj Dawar;David Richerby;Benjamin Rossman
Affiliations:
University of Cambridge Computer Laboratory, William Gates Building, J.J. Thomson Avenue, Cambridge, CB3 0FD, United Kingdom;University of Cambridge Computer Laboratory, William Gates Building, J.J. Thomson Avenue, Cambridge, CB3 0FD, United Kingdom;University of Cambridge Computer Laboratory, William Gates Building, J.J. Thomson Avenue, Cambridge, CB3 0FD, United Kingdom
Venue:
Electronic Notes in Theoretical Computer Science (ENTCS)
Year:
2006

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Abstract

We consider Choiceless Polynomial Time (C@?PT), a language introduced by Blass, Gurevich and Shelah, and show that it can express a query originally constructed by Cai, Furer and Immerman to separate fixed-point logic with counting (IFP + C) from P. This settles a question posed by Blass et al. The program we present uses sets of unbounded finite rank: we demonstrate that this is necessary by showing that the query cannot be computed by any program that has a constant bound on the rank of sets used, even in C@?PT(Card), an extension of C@?PT with counting.