The Specker Blatter Theorem Revisited Generating Functions for De nable 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 of structures 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 that for every m N fC n satis es a linear recurrence relation fC n Pdm j 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 de nable in Monadic Second Order Logic is indeed of nite Specker index E Fischer showed in that the Specker Blatter Theorem does not hold for quaternary relations In this paper we show how the Specker Blatter Theorem is related to Sch utzenberger 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 de nition of C Among the main results we have the 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 of the arity of the relations involved Every C de nable in Monadic Second Order Logic with modular Counting CMSOL is of nite Specker index This covers many new cases for which such a recurrence relation was not known before There are continuum many C of nite Specker index Hence contrary to the Myhill Nerode char acterization of regular languages the recognizable classes of graphs cannot be characterized by the niteness of the Specker index Introduction and main results Counting objects of a speci ed kind belongs to the oldest activities in mathematics In particular counting the number of labeled or unlabeled graphs satisfying a given property is a classic un dertaking in combinatorial theory The rst deep results for counting unlabeled graphs are due to J H Red eld and to G Polya but were only popularized after F Harary E M Palmer and R C Read uni ed these early results as witnessed in the still enjoyable HP It is unfortunate that a remarkable theorem by E Specker and C Blatter on counting la beled graphs and more generally labeled binary relational structures rst announced in cf BS BS Spe has not found the attention it deserves both for the beauty of the result and the ingenuity in its proof E Specker and C Blatter look at the function fC n which counts the number of labeled relational structures of size n with k relations R Rk which belong to a class C We shall call this function the density function for C It is required that C be de nable in Monadic Second Order Logic and that the relations are all unary or binary relations The theorem says that under these hypotheses the function fC n satis es a linear recurrence relation modulo m for every m Z Special cases of this theorem have been studied extensively cf HP Ges Wil and the references therein However the possibility of using a formal logical classi cation as a means to collect many special cases seems to have mostly escaped notice in this case In the present paper we shall discuss both the Specker Blatter theorem and its variations and limits of generalizabilty We rst set up our framework of logic For the reader not familiar with logic we recommend consulting EF We also give numerous examples in Appendix E which in turn provide combinatorial corollaries to the Specker Blatter Theorem Proving directly the linear recurrence relations over every modulus m for all the given examples would have been a nearly impossible undertaking We should also note that counting structures up to isomorphism is a very di erent task cf HP From Proposition below one can easily deduce that the Specker Blatter Theorem does not hold in this setting Counting labeled structures Let R fR R g be a set of relation symbols where each Ri is of arity i Let C be a class of relational R structures For an R structure A with universe A we denote the interpretation of Ri by Ri A We denote by fC n the number of structures in C over the labeled set An f ng i e fC n j f R An R An hAn R An R An i Cg j The notion of R isomorphism is the expected one Two structures A B are isomorphic if there is a bijection between their respective universes which preserves relations in both directions Proviso When we speak of a class of structures C we always assume that C is closed under R isomorphisms However we count two isomorphic but di erently labeled structures as two di erent members of C