75 research outputs found
Topological Complexity with Continuous Operations
AbstractThe topological complexity of algorithms is studied in a general context in the first part and for zero-finding in the second part. In the first part thelevel of discontinuityof a functionfis introduced and it is proved that it is a lower bound for the total number of comparisons plus 1 in any algorithm computingfthat uses only continuous operations and comparisons. This lower bound is proved to be sharp if arbitrary continuous operations are allowed. Then there exists even a balanced optimal computation tree forf. In the second part we use these results in order to determine the topological complexity of zero-finding for continuous functionsfon the unit interval withf(0) ·f(1) < 0. It is proved that roughly log2log2ϵ−1comparisons are optimal during a computation in order to approximate a zero up to ϵ. This is true regardless of whether one allows arbitrary continuous operations or just function evaluations, the arithmetic operations {+, −, *, /}, and the absolute value. It is true also for the subclass of nondecreasing functions. But for the subclass of increasing functions the topological complexity drops to zero even for the smaller class of operations
Shifts with decidable language and non-computable entropy
Automata, Logic and Semantic
08271 Abstracts Collection -- Topological and Game-Theoretic Aspects of Infinite Computations
From June 29, 2008, to July 4, 2008, the Dagstuhl Seminar 08271 ``Topological and Game-Theoretic Aspects of Infinite Computations\u27\u27 was held in the International Conference and Research Center (IBFI), Schloss Dagstuhl. During the seminar, many participants presented their current research, and ongoing work and open problems were discussed. Abstracts of the presentations given during the seminar as well as abstracts of seminar results and ideas are put together in this paper. The first section describes the seminar topics and goals in general. Links to extended abstracts or full papers are provided, if available
Computation with Advice
Computation with advice is suggested as generalization of both computation
with discrete advice and Type-2 Nondeterminism. Several embodiments of the
generic concept are discussed, and the close connection to Weihrauch
reducibility is pointed out. As a novel concept, computability with random
advice is studied; which corresponds to correct solutions being guessable with
positive probability. In the framework of computation with advice, it is
possible to define computational complexity for certain concepts of
hypercomputation. Finally, some examples are given which illuminate the
interplay of uniform and non-uniform techniques in order to investigate both
computability with advice and the Weihrauch lattice
Donagi-Markman cubic for the generalised Hitchin system
Donagi and Markman (1993) have shown that the infinitesimal period map for an algebraic completely integrable Hamiltonian system (ACIHS) is encoded in a section of the third symmetric power of the cotangent bundle to the base of the system. For the ordinary Hitchin system the cubic is given by a formula of Balduzzi and Pantev. We show that the Balduzzi\u2013Pantev formula holds on maximal rank symplectic leaves of the G-generalised Hitchin system
Turing machines on represented sets, a model of computation for Analysis
We introduce a new type of generalized Turing machines (GTMs), which are
intended as a tool for the mathematician who studies computability in Analysis.
In a single tape cell a GTM can store a symbol, a real number, a continuous
real function or a probability measure, for example. The model is based on TTE,
the representation approach for computable analysis. As a main result we prove
that the functions that are computable via given representations are closed
under GTM programming. This generalizes the well known fact that these
functions are closed under composition. The theorem allows to speak about
objects themselves instead of names in algorithms and proofs. By using GTMs for
specifying algorithms, many proofs become more rigorous and also simpler and
more transparent since the GTM model is very simple and allows to apply
well-known techniques from Turing machine theory. We also show how finite or
infinite sequences as names can be replaced by sets (generalized
representations) on which computability is already defined via representations.
This allows further simplification of proofs. All of this is done for
multi-functions, which are essential in Computable Analysis, and
multi-representations, which often allow more elegant formulations. As a
byproduct we show that the computable functions on finite and infinite
sequences of symbols are closed under programming with GTMs. We conclude with
examples of application
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