41,858 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
Topological Birkhoff
One of the most fundamental mathematical contributions of Garrett Birkhoff is
the HSP theorem, which implies that a finite algebra B satisfies all equations
that hold in a finite algebra A of the same signature if and only if B is a
homomorphic image of a subalgebra of a finite power of A. On the other hand, if
A is infinite, then in general one needs to take an infinite power in order to
obtain a representation of B in terms of A, even if B is finite.
We show that by considering the natural topology on the functions of A and B
in addition to the equations that hold between them, one can do with finite
powers even for many interesting infinite algebras A. More precisely, we prove
that if A and B are at most countable algebras which are oligomorphic, then the
mapping which sends each function from A to the corresponding function in B
preserves equations and is continuous if and only if B is a homomorphic image
of a subalgebra of a finite power of A.
Our result has the following consequences in model theory and in theoretical
computer science: two \omega-categorical structures are primitive positive
bi-interpretable if and only if their topological polymorphism clones are
isomorphic. In particular, the complexity of the constraint satisfaction
problem of an \omega-categorical structure only depends on its topological
polymorphism clone.Comment: 21 page
Total Representations
Almost all representations considered in computable analysis are partial. We
provide arguments in favor of total representations (by elements of the Baire
space). Total representations make the well known analogy between numberings
and representations closer, unify some terminology, simplify some technical
details, suggest interesting open questions and new invariants of topological
spaces relevant to computable analysis.Comment: 30 page
Topological complexity of motion planning and Massey products
We employ Massey products to find sharper lower bounds for the Schwarz genus
of a fibration than those previously known. In particular we give examples of
non-formal spaces for which the topological complexity \TC(X) (defined to
be the genus of the free path fibration on ) is greater than the
zero-divisors cup-length plus one.Comment: 11 pages; minor revisions and 1 added reference; to appear in the
Proceedings of the M. M. Postnikov Memorial Conferenc
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