The classical Feferman–Vaught Theorem for First Order Logic explains how to compute the truth value of a first order sentence in a generalized product of first order structures by reducing this computation to the computation of truth values of other first order sentences in the factors and evaluation of a monadic second order sentence in the index structure. This technique was later extended by Läuchli, Shelah and Gurevich to monadic second order logic. The technique has wide applications in decidability and (...) definability theory. Here we give a unified presentation, including some new results, of how to use the Feferman–Vaught Theorem, and some new variations thereof, algorithmically in the case of Monadic Second Order Logic MSOL . We then extend the technique to graph polynomials where the range of the summation of the monomials is definable in MSOL . Here the Feferman–Vaught Theorem for these polynomials generalizes well known splitting theorems for graph polynomials. Again, these can be used algorithmically. Finally, we discuss extensions of MSOL for which the Feferman–Vaught Theorem holds as well. (shrink)

We show that the spectrum of a sentence ϕ in Counting Monadic Second Order Logic (CMSOL) using one binary relation symbol and finitely many unary relation symbols, is ultimately periodic, provided all the models of ϕ are of clique width at most k, for some fixed k. We prove a similar statement for arbitrary finite relational vocabularies τ and a variant of clique width for τ-structures. This includes the cases where the models of ϕ are of tree width at most (...) k. For the case of bounded tree-width, the ultimate periodicity is even proved for Guarded Second Order Logic GSOL. We also generalize this result to many-sorted spectra, which can be viewed as an analogue of Parikh's Theorem on context-free languages, and its analogues for context-free graph grammars due to Habel and Courcelle. Our work was inspired by Gurevich and Shelah (2003), who showed ultimate periodicity of the spectrum for sentences of Monadic Second Order Logic where only finitely many unary predicates and one unary function are allowed. This restriction implies that the models are all of tree width at most 2, and hence it follows from our result. (shrink)

Many logics considered in finite model theory have a natural notion of an arity. The purpose of this article is to study the hierarchies which are formed by the fragments of such logics whose formulae are of bounded arity.Based on a construction of finite graphs with a certain property of homogeneity, we develop a method that allows us to prove that the arity hierarchies are strict for several logics, including fixed-point logics, transitive closure logic and its deterministic version, variants of (...) the database language Datalog, and extensions of first-order logic by implicit definitions.Furthermore, we show that all our results already hold on the class of finite graphs. (shrink)

In 1952, Heinrich Scholz published a question in The Journal of Symbolic Logic asking for a characterization of spectra, i.e., sets of natural numbers that are the cardinalities of finite models of first order sentences. Günter Asser in turn asked whether the complement of a spectrum is always a spectrum. These innocent questions turned out to be seminal for the development of finite model theory and descriptive complexity. In this paper we survey developments over the last 50-odd years pertaining to (...) the spectrum problem. Our presentation follows conceptual developments rather than the chronological order. Originally a number theoretic problem, it has been approached by means of recursion theory, resource bounded complexity theory, classification by complexity of the defining sentences, and finally by means of structural graph theory. Although Scholz' question was answered in various ways, Asser's question remains open. (shrink)

The paper discusses the notion of finite model truth definitions (or FM-truth definitions), introduced by M. Mostowski as a finite model analogue of Tarski's classical notion of truth definition. We compare FM-truth definitions with Vardi's concept of the combined complexity of logics, noting an important difference: the difficulty of defining FM-truth for a logic ᵍ does not depend on the syntax of L, as long as it is decidable. It follows that for a natural ᵍ there exist FM-truth definitions whose (...) evaluation is much easier than the combined complexiy of ᵍ would suggest. We apply the general theory to give a complexity-theoretical characterization of the logics for which the $\Sigma_{m}^{d}$ classes (prenex classes of higher order logics) define FM-truth. For any $d \geq 2.m \geq 1$ we construct a family $\{[\Sigma_{m}^{d}]^{\leg\kappa}\}\kappa\in\omega$ of syntactically defined fragments of $\Sigma_{m}^{d}$ which satisfy this characterization. We also use the $[\Sigma_{m}^{d}]\leg\kappa$ classes to give a refinement of known results on the complexity classes captured by $\Sigma_{m}^{d}$ . We close with a few simple corollaries, one of which gives a sufficient condition for the existence, given a vocabulary $\sigma$ , of a fixed number $\kappa$ such that model checking for all first order sentences over $\sigma$ can be done in deterministic time $n^{\kappa}$. (shrink)