13 research outputs found
A Generalization of the {\L}o\'s-Tarski Preservation Theorem over Classes of Finite Structures
We investigate a generalization of the {\L}o\'s-Tarski preservation theorem
via the semantic notion of \emph{preservation under substructures modulo
-sized cores}. It was shown earlier that over arbitrary structures, this
semantic notion for first-order logic corresponds to definability by
sentences. In this paper, we identify two properties of
classes of finite structures that ensure the above correspondence. The first is
based on well-quasi-ordering under the embedding relation. The second is a
logic-based combinatorial property that strictly generalizes the first. We show
that starting with classes satisfying any of these properties, the classes
obtained by applying operations like disjoint union, cartesian and tensor
products, or by forming words and trees over the classes, inherit the same
property. As a fallout, we obtain interesting classes of structures over which
an effective version of the {\L}o\'s-Tarski theorem holds.Comment: 28 pages, 1 figur
Extension Preservation in the Finite and Prefix Classes of First Order Logic
It is well known that the classic ?o?-Tarski preservation theorem fails in the finite: there are first-order definable classes of finite structures closed under extensions which are not definable (in the finite) in the existential fragment of first-order logic. We strengthen this by constructing for every n, first-order definable classes of finite structures closed under extensions which are not definable with n quantifier alternations. The classes we construct are definable in the extension of Datalog with negation and indeed in the existential fragment of transitive-closure logic. This answers negatively an open question posed by Rosen and Weinstein
MSO Undecidability for some Hereditary Classes of Unbounded Clique-Width
Seese's conjecture for finite graphs states that monadic second-order logic
(MSO) is undecidable on all graph classes of unbounded clique-width. We show
that to establish this it would suffice to show that grids of unbounded size
can be interpreted in two families of graph classes: minimal hereditary classes
of unbounded clique-width; and antichains of unbounded clique-width under the
induced subgraph relation. We explore a number of known examples of the former
category and establish that grids of unbounded size can indeed be interpreted
in them.Comment: 23 page