4,039 research outputs found

    Computing with Classical Real Numbers

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    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

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    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 AA, the \emph{independence spectrum} of AA, the set of all BB so that there exists a probability measure μ\mu so that μ{A,B}=0\mu\{A,B\} = 0 and (A,B)(A,B) is μ×μ\mu\times\mu-random. We prove that if AA is r.e., then no Δ20\Delta^0_2 set is in the independence spectrum of AA. We obtain applications of this fact to PA degrees. In particular, we show that if AA is r.e.\ and PP is of PA degree so that P̸≥TAP \not\geq_{T} A, then A⊕P≥T0′A \oplus P \geq_{T} 0'

    Comparing hierarchies of total functionals

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    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

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    The Polyhedral Escape Problem for continuous linear dynamical systems consists of deciding, given an affine function f:Rd→Rdf: \mathbb{R}^{d} \rightarrow \mathbb{R}^{d} and a convex polyhedron P⊆Rd\mathcal{P} \subseteq \mathbb{R}^{d}, whether, for some initial point x0\boldsymbol{x}_{0} in P\mathcal{P}, the trajectory of the unique solution to the differential equation x˙(t)=f(x(t))\dot{\boldsymbol{x}}(t)=f(\boldsymbol{x}(t)), x(0)=x0\boldsymbol{x}(0)=\boldsymbol{x}_{0}, is entirely contained in P\mathcal{P}. 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 ∃R\exists \mathbb{R}, 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

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    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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