1,495 research outputs found
First-order queries on finite structures over the reals
We investigate properties of finite relational structures over the reals expressed by first-order sentences whose predicates are the relations of the structure plus arbitrary polynomial inequalities, and whose quantifiers can range over the whole set of reals. In constraint programming terminology, this corresponds to Boolean real polynomial constraint queries on finite structures. The fact that quantifiers range over all reals seems crucial; however, we observe that each sentence in the first-order theory of the reals can be evaluated by letting each quantifier range over only a finite set of real numbers without changing its truth value. Inspired by this observation, we then show that when all polynomials used are linear, each query can be expressed uniformly on all finite structures by a sentence of which the quantifiers range only over the finite domain of the structure. In other words, linear constraint programming on finite structures can be reduced to ordinary query evaluation as usual in finite model theory and databases. Moreover, if only "generic" queries are taken into consideration, we show that this can be reduced even further by proving that such queries can be expressed by sentences using as polynomial inequalities only those of the simple form x <y
Strong Turing Degrees for Additive BSS RAM's
For the additive real BSS machines using only constants 0 and 1 and order
tests we consider the corresponding Turing reducibility and characterize some
semi-decidable decision problems over the reals. In order to refine,
step-by-step, a linear hierarchy of Turing degrees with respect to this model,
we define several halting problems for classes of additive machines with
different abilities and construct further suitable decision problems. In the
construction we use methods of the classical recursion theory as well as
techniques for proving bounds resulting from algebraic properties. In this way
we extend a known hierarchy of problems below the halting problem for the
additive machines using only equality tests and we present a further
subhierarchy of semi-decidable problems between the halting problems for the
additive machines using only equality tests and using order tests,
respectively
Feedback computability on Cantor space
We introduce the notion of feedback computable functions from to
, extending feedback Turing computation in analogy with the standard
notion of computability for functions from to . We then
show that the feedback computable functions are precisely the effectively Borel
functions. With this as motivation we define the notion of a feedback
computable function on a structure, independent of any coding of the structure
as a real. We show that this notion is absolute, and as an example characterize
those functions that are computable from a Gandy ordinal with some finite
subset distinguished
Tarski's influence on computer science
The influence of Alfred Tarski on computer science was indirect but
significant in a number of directions and was in certain respects fundamental.
Here surveyed is the work of Tarski on the decision procedure for algebra and
geometry, the method of elimination of quantifiers, the semantics of formal
languages, modeltheoretic preservation theorems, and algebraic logic; various
connections of each with computer science are taken up
Two- versus three-dimensional connectivity testing of first-order queries to semi-algebraic sets
This paper addresses the question whether one can determine the connectivity
of a semi-algebraic set in three dimensions by testing the connectivity of a
finite number of two-dimensional ``samples'' of the set, where these samples
are defined by first-order queries. The question is answered negatively for two
classes of first-order queries: cartesian-product-free, and positive one-pass.Comment: corrected minor confusion in Proof of Theorem
The Integration of Connectionism and First-Order Knowledge Representation and Reasoning as a Challenge for Artificial Intelligence
Intelligent systems based on first-order logic on the one hand, and on
artificial neural networks (also called connectionist systems) on the other,
differ substantially. It would be very desirable to combine the robust neural
networking machinery with symbolic knowledge representation and reasoning
paradigms like logic programming in such a way that the strengths of either
paradigm will be retained. Current state-of-the-art research, however, fails by
far to achieve this ultimate goal. As one of the main obstacles to be overcome
we perceive the question how symbolic knowledge can be encoded by means of
connectionist systems: Satisfactory answers to this will naturally lead the way
to knowledge extraction algorithms and to integrated neural-symbolic systems.Comment: In Proceedings of INFORMATION'2004, Tokyo, Japan, to appear. 12 page
Definable Ellipsoid Method, Sums-of-Squares Proofs, and the Isomorphism Problem
The ellipsoid method is an algorithm that solves the (weak) feasibility and
linear optimization problems for convex sets by making oracle calls to their
(weak) separation problem. We observe that the previously known method for
showing that this reduction can be done in fixed-point logic with counting
(FPC) for linear and semidefinite programs applies to any family of explicitly
bounded convex sets. We use this observation to show that the exact feasibility
problem for semidefinite programs is expressible in the infinitary version of
FPC. As a corollary we get that, for the isomorphism problem, the
Lasserre/Sums-of-Squares semidefinite programming hierarchy of relaxations
collapses to the Sherali-Adams linear programming hierarchy, up to a small loss
in the degree
Definable ellipsoid method, sums-of-squares proofs, and the isomorphism problem
The ellipsoid method is an algorithm that solves the (weak) feasibility and linear optimization problems for convex sets by making oracle calls to their (weak) separation problem. We observe that the previously known method for showing that this reduction can be done in fixed-point logic with counting (FPC) for linear and semidefinite programs applies to any family of explicitly bounded convex sets. We use this observation to show that the exact feasibility problem for semidefinite programs is expressible in the infinitary version of FPC. As a corollary we get that, for the graph isomorphism problem, the Lasserre/Sums-of-Squares semidefinite programming hierarchy of relaxations collapses to the Sherali-Adams linear programming hierarchy, up to a small loss in the degree. © 2018 ACM.Peer ReviewedPostprint (author's final draft
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