Abstract
Rudimentary relations are those relations over natural integers that are defined by a first-order arithmetical formula, in which all quantifications are bounded by some variable. Paris and Wilkie conjectured that the class R of rudimentary relations is not closed under modular counting or constant-bounded recursion. A consequence would be the proper containment of R in LINSPACE. It is now known that the closure of R under counting modulo an unsolvable non-abelian finite group, or modulo the semigroup of all binary relations on a finite set, contains ALINTIME. We characterize here the semigroups for which the counting is reducible to modular counting, or contains counting modulo an unsolvable non-abelian finite group.
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