4,401 research outputs found
Tameness in least fixed-point logic and McColm's conjecture
We investigate four model-theoretic tameness properties in the context of
least fixed-point logic over a family of finite structures. We find that each
of these properties depends only on the elementary (i.e., first-order) limit
theory, and we completely determine the valid entailments among them. In
contrast to the context of first-order logic on arbitrary structures, the order
property and independence property are equivalent in this setting. McColm
conjectured that least fixed-point definability collapses to first-order
definability exactly when proficiency fails. McColm's conjecture is known to be
false in general. However, we show that McColm's conjecture is true for any
family of finite structures whose limit theory is model-theoretically tame
Fixed-parameter tractability, definability, and model checking
In this article, we study parameterized complexity theory from the
perspective of logic, or more specifically, descriptive complexity theory.
We propose to consider parameterized model-checking problems for various
fragments of first-order logic as generic parameterized problems and show how
this approach can be useful in studying both fixed-parameter tractability and
intractability. For example, we establish the equivalence between the
model-checking for existential first-order logic, the homomorphism problem for
relational structures, and the substructure isomorphism problem. Our main
tractability result shows that model-checking for first-order formulas is
fixed-parameter tractable when restricted to a class of input structures with
an excluded minor. On the intractability side, for every t >= 0 we prove an
equivalence between model-checking for first-order formulas with t quantifier
alternations and the parameterized halting problem for alternating Turing
machines with t alternations. We discuss the close connection between this
alternation hierarchy and Downey and Fellows' W-hierarchy.
On a more abstract level, we consider two forms of definability, called Fagin
definability and slicewise definability, that are appropriate for describing
parameterized problems. We give a characterization of the class FPT of all
fixed-parameter tractable problems in terms of slicewise definability in finite
variable least fixed-point logic, which is reminiscent of the Immerman-Vardi
Theorem characterizing the class PTIME in terms of definability in least
fixed-point logic.Comment: To appear in SIAM Journal on Computin
Defining Recursive Predicates in Graph Orders
We study the first order theory of structures over graphs i.e. structures of
the form () where is the set of all
(isomorphism types of) finite undirected graphs and some vocabulary. We
define the notion of a recursive predicate over graphs using Turing Machine
recognizable string encodings of graphs. We also define the notion of an
arithmetical relation over graphs using a total order on the set
such that () is isomorphic to
().
We introduce the notion of a \textit{capable} structure over graphs, which is
one satisfying the conditions : (1) definability of arithmetic, (2)
definability of cardinality of a graph, and (3) definability of two particular
graph predicates related to vertex labellings of graphs. We then show any
capable structure can define every arithmetical predicate over graphs. As a
corollary, any capable structure also defines every recursive graph relation.
We identify capable structures which are expansions of graph orders, which are
structures of the form () where is a partial order. We
show that the subgraph order i.e. (), induced subgraph
order with one constant i.e. () and an expansion
of the minor order for counting edges i.e. ()
are capable structures. In the course of the proof, we show the definability of
several natural graph theoretic predicates in the subgraph order which may be
of independent interest. We discuss the implications of our results and
connections to Descriptive Complexity
Tameness in least fixed-point logic and McColm's conjecture
We investigate four model-theoretic tameness properties in the context of
least fixed-point logic over a family of finite structures. We find that each
of these properties depends only on the elementary (i.e., first-order) limit
theory, and we completely determine the valid entailments among them. In
contrast to the context of first-order logic on arbitrary structures, the order
property and independence property are equivalent in this setting. McColm
conjectured that least fixed-point definability collapses to first-order
definability exactly when proficiency fails. McColm's conjecture is known to be
false in general. However, we show that McColm's conjecture is true for any
family of finite structures whose limit theory is model-theoretically tame
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
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
- β¦