1,135 research outputs found
Logics of Finite Hankel Rank
We discuss the Feferman-Vaught Theorem in the setting of abstract model
theory for finite structures. We look at sum-like and product-like binary
operations on finite structures and their Hankel matrices. We show the
connection between Hankel matrices and the Feferman-Vaught Theorem. The largest
logic known to satisfy a Feferman-Vaught Theorem for product-like operations is
CFOL, first order logic with modular counting quantifiers. For sum-like
operations it is CMSOL, the corresponding monadic second order logic. We
discuss whether there are maximal logics satisfying Feferman-Vaught Theorems
for finite structures.Comment: Appeared in YuriFest 2015, held in honor of Yuri Gurevich's 75th
birthday. The final publication is available at Springer via
http://dx.doi.org/10.1007/978-3-319-23534-9_1
Connection Matrices and the Definability of Graph Parameters
In this paper we extend and prove in detail the Finite Rank Theorem for
connection matrices of graph parameters definable in Monadic Second Order Logic
with counting (CMSOL) from B. Godlin, T. Kotek and J.A. Makowsky (2008) and
J.A. Makowsky (2009). We demonstrate its vast applicability in simplifying
known and new non-definability results of graph properties and finding new
non-definability results for graph parameters. We also prove a Feferman-Vaught
Theorem for the logic CFOL, First Order Logic with the modular counting
quantifiers
Preservation and decomposition theorems for bounded degree structures
We provide elementary algorithms for two preservation theorems for
first-order sentences (FO) on the class \^ad of all finite structures of degree
at most d: For each FO-sentence that is preserved under extensions
(homomorphisms) on \^ad, a \^ad-equivalent existential (existential-positive)
FO-sentence can be constructed in 5-fold (4-fold) exponential time. This is
complemented by lower bounds showing that a 3-fold exponential blow-up of the
computed existential (existential-positive) sentence is unavoidable. Both
algorithms can be extended (while maintaining the upper and lower bounds on
their time complexity) to input first-order sentences with modulo m counting
quantifiers (FO+MODm). Furthermore, we show that for an input FO-formula, a
\^ad-equivalent Feferman-Vaught decomposition can be computed in 3-fold
exponential time. We also provide a matching lower bound.Comment: 42 pages and 3 figures. This is the full version of: Frederik
Harwath, Lucas Heimberg, and Nicole Schweikardt. Preservation and
decomposition theorems for bounded degree structures. In Joint Meeting of the
23rd EACSL Annual Conference on Computer Science Logic (CSL) and the 29th
Annual ACM/IEEE Symposium on Logic in Computer Science (LICS), CSL-LICS'14,
pages 49:1-49:10. ACM, 201
First-order modal logic in the necessary framework of objects
I consider the first-order modal logic which counts as valid those sentences which are true on every interpretation of the non-logical constants. Based on the assumptions that it is necessary what individuals there are and that it is necessary which propositions are necessary, Timothy Williamson has tentatively suggested an argument for the claim that this logic is determined by a possible world structure consisting of an infinite set of individuals and an infinite set of worlds. He notes that only the cardinalities of these sets matters, and that not all pairs of infinite sets determine the same logic. I use so-called two-cardinal theorems from model theory to investigate the space of logics and consequence relations determined by pairs of infinite sets, and show how to eliminate the assumption that worlds are individuals from Williamson’s argument
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