The Specker-Blatter Theorem Revisited: Generating Functions for Definable Classes of Structures

In this paper we study the generating function of classes of graphs and hypergraphs. For a class of labeled graphs C we denote by fC(n) the number ofstructures of size n. For C de nable in Monadic Second Order Logic with unary and binary relation symbols only, E.Specker and C. Blatter showed in 1981 that for every m 2 N, fC(n) satisfies a linear recurrence relation fC(n) = Pdm j=1 a (m) j fC(n; j)� over Zm, and hence is ultimately periodic for each m. To show this they introduced what we call the Specker-index of C and rst showed the theorem to hold for any C of nite Specker-index, and then showed that every C definable in Monadic Second Order Logic is indeed of finite Specker-index. E. Fischer showed in 2002 that the Specker-Blatter Theorem does not hold for quaternary relations. In this paper we show how the Specker-Blatter Theorem is related to Schützenberger's Theorem and the Myhill-Nerode criterion for the characterization of regular languages, and discuss in detail how the behavior of this generating function depends on the choice of constant and relation symbols allowed in the definition of C. Among the main results we havethe following: -- We consider n-ary relations of degree at most d, where each element a is related to at most d other elements by any of the relations. We show that the Specker-Blatter Theorem holds for those, irrespective ofthe arity of the relations involved. -- Every C de nable in Monadic Second Order Logic with (modular) Counting (CMSOL) is of finite Specker-index. This covers many new cases, for which such a recurrence relation was not known before. -- There are continuum many C of finite Specker-index. Hence, contrary to the Myhill-Nerode characterization of regular languages, the recognizable classes of graphs cannot be characterized by the finiteness of the Specker-index