4,039 research outputs found
Computing with Classical Real Numbers
There are two incompatible Coq libraries that have a theory of the real
numbers; the Coq standard library gives an axiomatic treatment of classical
real numbers, while the CoRN library from Nijmegen defines constructively valid
real numbers. Unfortunately, this means results about one structure cannot
easily be used in the other structure. We present a way interfacing these two
libraries by showing that their real number structures are isomorphic assuming
the classical axioms already present in the standard library reals. This allows
us to use O'Connor's decision procedure for solving ground inequalities present
in CoRN to solve inequalities about the reals from the Coq standard library,
and it allows theorems from the Coq standard library to apply to problem about
the CoRN reals
Independence, Relative Randomness, and PA Degrees
We study pairs of reals that are mutually Martin-L\"{o}f random with respect
to a common, not necessarily computable probability measure. We show that a
generalized version of van Lambalgen's Theorem holds for non-computable
probability measures, too. We study, for a given real , the
\emph{independence spectrum} of , the set of all so that there exists a
probability measure so that and is
-random. We prove that if is r.e., then no set
is in the independence spectrum of . We obtain applications of this fact to
PA degrees. In particular, we show that if is r.e.\ and is of PA degree
so that , then
Comparing hierarchies of total functionals
In this paper we consider two hierarchies of hereditarily total and
continuous functionals over the reals based on one extensional and one
intensional representation of real numbers, and we discuss under which
asumptions these hierarchies coincide. This coincidense problem is equivalent
to a statement about the topology of the Kleene-Kreisel continuous functionals.
As a tool of independent interest, we show that the Kleene-Kreisel functionals
may be embedded into both these hierarchies.Comment: 28 page
On the Polytope Escape Problem for Continuous Linear Dynamical Systems
The Polyhedral Escape Problem for continuous linear dynamical systems
consists of deciding, given an affine function and a convex polyhedron ,
whether, for some initial point in , the
trajectory of the unique solution to the differential equation
,
, is entirely contained in .
We show that this problem is decidable, by reducing it in polynomial time to
the decision version of linear programming with real algebraic coefficients,
thus placing it in , which lies between NP and PSPACE. Our
algorithm makes use of spectral techniques and relies among others on tools
from Diophantine approximation.Comment: Accepted to HSCC 201
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
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