From Parikh’s Theorem to Many-Sorted Spectra

We discuss a generalization of Parikh’s Theorem for contextfree languages to classes of many-sorted relational structures which are both definable in Monadic Second Order Logic and which are of bounded patch-width. Patch-width is a generalization of both tree-width and clique-width. This gives a powerful unifying tool to prove that certain classes of graphs are of unbounded width. For R. Parikh, at the occasion of his 70th birthday 1 Generalizing Parikh’s Theorem R. Parikh’s celebrated theorem, first proved in [Par66], counts the number of occurrences of letters in words of a context-free languages L over an alphabet of k letters. For a given word w, the numbers of these occurrences is denoted by a vector n(w) ∈ N, and the theorem states Theorem 1 (Parikh 1966). For a context-free language L, the set Par(L) = {n(w) ∈ N : w ∈ L} is semi-linear. A set X ⊆ N is linear in N iff there is vector ā ∈ N and a matrix M ∈ N such that X = Aā,M̄ = {b̄ ∈ N s : there is ū ∈ N with b̄ = ā +M · ū}. Singletons are linear sets with M = 0. If M 6= 0 the series is nontrivial. X ⊆ N is semi-linear in N iff X is a finite union of linear sets Ai ⊆ N. For s = 1 the semi-linear sets are exactly the ultimately periodic sets. The terminology is from [Par66], and has since become standard terminology in formal language theory. Several alternative proofs of Parikh’s Theorem have appeared since. D. Pilling [Pil73] put it into a more algebraic form, and more recently, L. Aceto, Z. Esik and A. Ingolfsdottir [AÉI02] showed that it depends only on a few equational properties of least pre-fixed points. B. Courcelle [Cou95]. has generalized Theorem 1 further to certain graph grammars, and relational structures which are ⋆ Partially supported by the Israel Science Foundation for the project ”Model Theoretic Interpretations of Counting Functions” (2007-2010) and the Grant for Promotion of Research by the Technion–Israel Institute of Technology. definable in Monadic Second Order Logic (MSOL) in labeled trees, counting not only letter occurrences but the size of MSOL-definable subsets. In [FM04] a similar generalization was proven, inspired by a theorem due to Gurevich and Shelah [GS03] on spectra of MSOL-sentences with one unary function symbol and a finite number of unary relation symbols. The results of [FM04] are formulated in a model theoretic framework rather than using the language of graph grammars. But it turned out that some of their result could have been obtained also using the techniques of [Cou95]. In this paper we explain discuss these generalizations of Parikh’s Theorem without detailed proofs, but with the emphasis on the concepts involved and on applications. Like in the well known characterization of regular languages using Monadic Second Order Logic (MSOL) we also use MSOL, and an extension thereof, CMSOL, which allows for modular counting quantifiers. However, the languages in Parikh’s Theorem are replaced by arbitrary finite relational structures which are of bounded width. The most general notion of width we shall use is patch-width, which was first introduced in [FM04]. It generalizes both tree-width and clique-width. The detailed discussion of these notions of width is given in Section 4. Finally, like in Courcelle’s generalization of Parikh’s Theorem, rather than counting occurrences of letters, we count cardinalities of CMSOL-definable unary relations. We shall first explain this for the case where these relations are given by unary predicates, where we speak of many-sorted spectra. The applications consist mostly in proving that certain classes of graphs and relational structures have unbounded tree-width, clique-width or patch-width. These are given in Section 5. We assume the reader is familiar with the basics of Monadic Second Order Logic MSOL, cf. [Cou92,EF95]. For convenience, we collect the basics in Section 2. Otherwise this paper is rather self-contained. 2 Monadic Second Order Logic and its Extension First Order Logic FOL restricts quantification to elements of the structure, Monadic Second Order Logic MSOL also allows for quantification over subsets, but not over binary relations,, or relations of arity r ≥ 2. The logic MSOL can be extended by modular counting quantifiers Ck,m, where Ck,mx φ(x) is interpreted as “there are, modulo m, exactly k elements satisfying φ(x)”. We denote the extension of MSOL obtained by adding, for all k,m ∈ N the quantifiers Ck,m, by CMSOL. Over structures which have a linear order the quantifiers Ck,mx φ(x) can be eliminated without loss of expressive power. Typical graph theoretic concepts expressible in FOL are the presence or absence (up to isomorphism) of a fixed (induced) subgraph H , and fixed lower or upper bounds on the degree of the vertices (hence also r-regularity). Typical graph theoretic concepts expressible in MSOL but not in FOL are connectivity, k-connectivity, reachability, k-colorability (of the vertices), and the presence or absence of a fixed (topological) minor. The latter includes planarity, and more generally, graphs of a fixed genus g. Typical graph theoretic concepts expressible