150 research outputs found
An optimal construction of Hanf sentences
We give the first elementary construction of equivalent formulas in Hanf
normal form. The triply exponential upper bound is complemented by a matching
lower bound
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
Gaifman Normal Forms for Counting Extensions of First-Order Logic
We consider the extension of first-order logic FO by unary counting quantifiers and generalise the notion of Gaifman normal form from FO to this setting. For formulas that use only ultimately periodic counting quantifiers, we provide an algorithm that computes equivalent formulas in Gaifman normal form. We also show that this is not possible for formulas using at least one quantifier that is not ultimately periodic.
Now let d be a degree bound. We show that for any formula phi with arbitrary counting quantifiers, there is a formula gamma in Gaifman normal form that is equivalent to phi on all finite structures of degree <= d. If the quantifiers of phi are decidable (decidable in elementary time, ultimately periodic), gamma can be constructed effectively (in elementary time, in worst-case optimal 3-fold exponential time).
For the setting with unrestricted degree we show that by using our Gaifman normal form for formulas with only ultimately periodic counting quantifiers, a known fixed-parameter tractability result for FO on classes of structures of bounded local tree-width can be lifted to the extension of FO with ultimately periodic counting quantifiers (a logic equally expressive as FO+MOD, i.e., first-oder logic with modulo-counting quantifiers)
A presentation theorem for continuous logic and Metric Abstract Elementary Classes
We give a presentation theorem for continuous first-order logic and Metric
Abstract Elementary classes in terms of and Abstract
Elementary Classes, respectively. This presentation is accomplished by
analyzing dense subsets that are closed under functions. We extend this
correspondence to types and saturation
On Testability of First-Order Properties in Bounded-Degree Graphs and Connections to Proximity-Oblivious Testing
We study property testing of properties that are definable in first-order
logic (FO) in the bounded-degree graph and relational structure models. We show
that any FO property that is defined by a formula with quantifier prefix
is testable (i.e., testable with constant query
complexity), while there exists an FO property that is expressible by a formula
with quantifier prefix that is not testable. In the dense
graph model, a similar picture is long known (Alon, Fischer, Krivelevich,
Szegedy, Combinatorica 2000), despite the very different nature of the two
models. In particular, we obtain our lower bound by an FO formula that defines
a class of bounded-degree expanders, based on zig-zag products of graphs. We
expect this to be of independent interest.
We then use our class of FO definable bounded-degree expanders to answer a
long-standing open problem for proximity-oblivious testers (POTs). POTs are a
class of particularly simple testing algorithms, where a basic test is
performed a number of times that may depend on the proximity parameter, but the
basic test itself is independent of the proximity parameter. In their seminal
work, Goldreich and Ron [STOC 2009; SICOMP 2011] show that the graph properties
that are constant-query proximity-oblivious testable in the bounded-degree
model are precisely the properties that can be expressed as a generalised
subgraph freeness (GSF) property that satisfies the non-propagation condition.
It is left open whether the non-propagation condition is necessary. We give a
negative answer by showing that our property is a GSF property which is
propagating. Hence in particular, our property does not admit a POT. For this
result we establish a new connection between FO properties and GSF-local
properties via neighbourhood profiles.Comment: Preliminary version of this article appeared in SODA'21
(arXiv:2008.05800) and CCC'21 (arXiv:2105.08490
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