2,545 research outputs found
A note on first-order spectra with binary relations
The spectrum of a first-order sentence is the set of the cardinalities of its
finite models. In this paper, we consider the spectra of sentences over binary
relations that use at least three variables. We show that for every such
sentence , there is a sentence that uses the same number of
variables, but only one symmetric binary relation, such that its spectrum is
linearly proportional to the spectrum of . Moreover, the models of
are all bipartite graphs. As a corollary, we obtain that to settle
Asser's conjecture, i.e., whether the class of spectra is closed under
complement, it is sufficient to consider only sentences using only three
variables whose models are restricted to undirected bipartite graphs
Model Checking Lower Bounds for Simple Graphs
A well-known result by Frick and Grohe shows that deciding FO logic on trees
involves a parameter dependence that is a tower of exponentials. Though this
lower bound is tight for Courcelle's theorem, it has been evaded by a series of
recent meta-theorems for other graph classes. Here we provide some additional
non-elementary lower bound results, which are in some senses stronger. Our goal
is to explain common traits in these recent meta-theorems and identify barriers
to further progress. More specifically, first, we show that on the class of
threshold graphs, and therefore also on any union and complement-closed class,
there is no model-checking algorithm with elementary parameter dependence even
for FO logic. Second, we show that there is no model-checking algorithm with
elementary parameter dependence for MSO logic even restricted to paths (or
equivalently to unary strings), unless E=NE. As a corollary, we resolve an open
problem on the complexity of MSO model-checking on graphs of bounded max-leaf
number. Finally, we look at MSO on the class of colored trees of depth d. We
show that, assuming the ETH, for every fixed d>=1 at least d+1 levels of
exponentiation are necessary for this problem, thus showing that the (d+1)-fold
exponential algorithm recently given by Gajarsk\`{y} and Hlin\u{e}n\`{y} is
essentially optimal
Local Sentences and Mahlo Cardinals
Local sentences were introduced by J.-P. Ressayre who proved certain
remarkable stretching theorems establishing the equivalence between the
existence of finite models for these sentences and the existence of some
infinite well ordered models. Two of these stretching theorems were only proved
under certain large cardinal axioms but the question of their exact
(consistency) strength was left open in [O. Finkel and J.-P. Ressayre,
Stretchings, Journal of Symbolic Logic, Volume 61 (2), 1996, p. 563-585 ].
Here, we solve this problem, using a combinatorial result of J. H. Schmerl. In
fact, we show that the stretching principles are equivalent to the existence of
n-Mahlo cardinals for appropriate integers n. This is done by proving first
that for each integer n, there is a local sentence phi_n which has well ordered
models of order type alpha, for every infinite ordinal alpha > omega which is
not an n-Mahlo cardinal
Data Mining the SDSS SkyServer Database
An earlier paper (Szalay et. al. "Designing and Mining MultiTerabyte
Astronomy Archives: The Sloan Digital Sky Survey," ACM SIGMOD 2000) described
the Sloan Digital Sky Survey's (SDSS) data management needs by defining twenty
database queries and twelve data visualization tasks that a good data
management system should support. We built a database and interfaces to support
both the query load and also a website for ad-hoc access. This paper reports on
the database design, describes the data loading pipeline, and reports on the
query implementation and performance. The queries typically translated to a
single SQL statement. Most queries run in less than 20 seconds, allowing
scientists to interactively explore the database. This paper is an in-depth
tour of those queries. Readers should first have studied the companion overview
paper Szalay et. al. "The SDSS SkyServer, Public Access to the Sloan Digital
Sky Server Data" ACM SIGMOND 2002.Comment: 40 pages, Original source is at
http://research.microsoft.com/~gray/Papers/MSR_TR_O2_01_20_queries.do
A note on the expressive power of linear orders
This article shows that there exist two particular linear orders such that
first-order logic with these two linear orders has the same expressive power as
first-order logic with the Bit-predicate FO(Bit). As a corollary we obtain that
there also exists a built-in permutation such that first-order logic with a
linear order and this permutation is as expressive as FO(Bit)
Lower Complexity Bounds for Lifted Inference
One of the big challenges in the development of probabilistic relational (or
probabilistic logical) modeling and learning frameworks is the design of
inference techniques that operate on the level of the abstract model
representation language, rather than on the level of ground, propositional
instances of the model. Numerous approaches for such "lifted inference"
techniques have been proposed. While it has been demonstrated that these
techniques will lead to significantly more efficient inference on some specific
models, there are only very recent and still quite restricted results that show
the feasibility of lifted inference on certain syntactically defined classes of
models. Lower complexity bounds that imply some limitations for the feasibility
of lifted inference on more expressive model classes were established early on
in (Jaeger 2000). However, it is not immediate that these results also apply to
the type of modeling languages that currently receive the most attention, i.e.,
weighted, quantifier-free formulas. In this paper we extend these earlier
results, and show that under the assumption that NETIME =/= ETIME, there is no
polynomial lifted inference algorithm for knowledge bases of weighted,
quantifier- and function-free formulas. Further strengthening earlier results,
this is also shown to hold for approximate inference, and for knowledge bases
not containing the equality predicate.Comment: To appear in Theory and Practice of Logic Programming (TPLP
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