Algorithmic uses of the Feferman–Vaught Theorem

Abstract

The classical Feferman-Vaught Theorem for First Order Logic explains how to compute the truth value of a ÿrst order sentence in a generalized product of ÿrst order structures by reducing this computation to the computation of truth values of other ÿrst order sentences in the factors and evaluation of a monadic second order sentence in the index structure. This technique was later extended by L auchli, Shelah and Gurevich to monadic second order logic. The technique has wide applications in decidability and deÿnability theory.Here we give a uniÿed presentation, including some new results, of how to use the FefermanVaught 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 deÿnable 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. (J.A. Makowsky).1 At the occasion of his 75th birthday. It is also a tribute to Feferman's work in model theory, which went seemingly unnoticed at the celebrations of his 70th birthday. Feferman returned several times to topics related to the preservation of truth under various model theoretic constructions, so in [54][55][56][57][58]60]. The FefermanVaught Theorem can also be viewed as falling into this category.

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“…Ehrenfeucht used it in [6] and our proofs follow his work. The method was developed and used by Fefeman-Vaught [7], Shelah [22] and others (see e.g., surveys [11,23,20] This theorem justifies the notation τ 1 + τ 2 for the k-type of an m-chain which is the sum of two m-chains of k-types τ 1 and τ 2 .…”

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“…This had already been investigated by Atserias, Dawar and Kolaitis [2] in which one of the main results is that HPT holds for every class of finite structures of bounded treewidth; another result is that HPT holds for the class of all planar graphs. At the Tarski Centenary Conference held in Warsaw in 2001, Johann A. Makowsky presented a survey of applications of another kind of preservation result that goes under the heading of the Feferman-Vaught Theorem [31]. What Vaught and I had shown in our joint 1959 paper [16] was that for a great variety of sum and product operations O on structures M i (i ∈ I), the first-order properties of…”

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“…An outline of such a study was presented in [Mak06]. In [Mak04] the second author has introduced the MSOL-definable and the SOL-definable graph polynomials, the class of graph polynomials where the range of summation is definable in (monadic) second order logic. He has verified that all the examples of graph polynomials discussed in the literature, with the exception of the weighted graph polynomial of [NW99], are actually SOL-polynomials over some expansions (by adding order relations) of the graph, cf.…”

“…The notion of graph polynomials definable in Monadic Second Order Logic, MSOL, was introduced in [Mak04]. It was shown that the Tutte polynomial and its generalization, as well as the matching polynomial, the cover polynomial and the various interlace polynomials fall into this category.…”