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32
Weihrauch degrees, omniscience principles and weak computability
, 2009
"... In this paper we study a reducibility that has been introduced by Klaus Weihrauch or, more precisely, a natural extension of this reducibility for multi-valued functions on represented spaces. We call the corresponding equivalence classes Weihrauch degrees and we show that the corresponding partia ..."
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Cited by 28 (5 self)
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In this paper we study a reducibility that has been introduced by Klaus Weihrauch or, more precisely, a natural extension of this reducibility for multi-valued functions on represented spaces. We call the corresponding equivalence classes Weihrauch degrees and we show that the corresponding partial order induces a lower semi-lattice with the disjoint union of multi-valued functions as greatest lower bound operation. We show that parallelization is a closure operator for this semilattice and the parallelized Weihrauch degrees even form a lattice with the product of multi-valued functions as greatest lower bound operation. We show that the Medvedev lattice and hence the Turing upper semi-lattice can both be embedded into the parallelized Weihrauch lattice in a natural way. The importance of Weihrauch degrees is based on the fact that multi-valued functions on represented spaces can be considered as realizers of mathematical theorems in a very natural way and studying the Weihrauch reductions between theorems in this sense means
Effective Choice and Boundedness Principles in Computable Analysis
, 2009
"... In this paper we study a new approach to classify mathematical theorems according to their computational content. Basically, we are asking the question which theorems can be continuously or computably transferred into each other? For this purpose theorems are considered via their realizers which a ..."
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Cited by 24 (6 self)
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In this paper we study a new approach to classify mathematical theorems according to their computational content. Basically, we are asking the question which theorems can be continuously or computably transferred into each other? For this purpose theorems are considered via their realizers which are operations with certain input and output data. The technical tool to express continuous or computable relations between such operations is Weihrauch reducibility and the partially ordered degree structure induced by it. We have identified certain choice principles on closed sets which are cornerstones among Weihrauch degrees and it turns out that certain core theorems in analysis can be classified naturally in this structure. In particular, we study theorems such as the Intermediate Value Theorem, the Baire Category Theorem, the Banach Inverse Mapping Theorem, the Closed Graph Theorem and the Uniform Boundedness Theorem. Well-known omniscience principles from constructive mathematics such as LPO and LLPO can also naturally be considered as Weihrauch degrees and they play an important role in our classification. Our classification scheme does not require any particular logical framework or axiomatic setting, but it can be carried out in the framework of classical mathematics using tools of topology, computability theory and computable analysis. Finally, we present a number of metatheorems that allow to derive upper bounds for the classification of the Weihrauch degree of many theorems and we discuss the Brouwer Fixed Point Theorem as an example.
On the (semi)lattices induced by continuous reducibilities
, 2009
"... Continuous reducibilities are a proven tool in computable analysis, and have applications in other fields such as constructive mathematics or reverse mathematics. We study the order-theoretic properties of several variants of the two most important definitions, and especially introduce suprema for t ..."
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Cited by 14 (8 self)
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Continuous reducibilities are a proven tool in computable analysis, and have applications in other fields such as constructive mathematics or reverse mathematics. We study the order-theoretic properties of several variants of the two most important definitions, and especially introduce suprema for them. The suprema are shown to commutate with several characteristic numbers.
A.: How much incomputable is the separable Hahn-Banach Theorem
- Conference on Computability and Complexity in Analysis. Number 348 in Informatik Berichte, FernUniversität Hagen (2008) 101 – 117
"... Abstract. We determine the computational complexity of the Hahn-Banach Extension Theorem. To do so, we investigate some basic connections between reverse mathematics and computable analysis. In particular, we use Weak König’s Lemma within the framework of computable analysis to classify incomputable ..."
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Cited by 13 (3 self)
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Abstract. We determine the computational complexity of the Hahn-Banach Extension Theorem. To do so, we investigate some basic connections between reverse mathematics and computable analysis. In particular, we use Weak König’s Lemma within the framework of computable analysis to classify incomputable functions of low complexity. By defining the multi-valued function Sep and a natural notion of reducibility for multi-valued functions, we obtain a computational counterpart of the subsystem of second order arithmetic WKL0. We study analogies and differences between WKL0 and the class of Sep-computable multi-valued functions. Extending work of Brattka, we show that a natural multi-valued function associated with the Hahn-Banach Extension Theorem is Sep-complete. 1.
The Bolzano-Weierstrass theorem is the jump of weak Kőnig’s lemma
- Annals of Pure and Applied Logic 163 (2012
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Decomposing Borel functions using the Shore-Slaman join theorem. submitted
, 2013
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Singular Coverings and Non-Uniform Notions of Closed Set Computability
, 2007
"... Abstract. The empty set of course contains no computable point. On the other hand, surprising results due to Zaslavskiĭ, Tseĭtin, Kreisel, and Lacombe assert the existence of non-empty co-r.e. closed sets devoid of computable points: sets which are even ‘large ’ in the sense of positive Lebesgue mea ..."
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Cited by 5 (0 self)
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Abstract. The empty set of course contains no computable point. On the other hand, surprising results due to Zaslavskiĭ, Tseĭtin, Kreisel, and Lacombe assert the existence of non-empty co-r.e. closed sets devoid of computable points: sets which are even ‘large ’ in the sense of positive Lebesgue measure. We observe that a certain size is in fact necessary: every non-empty co-r.e. closed real set without computable points has continuum cardinality. This initiates a comparison of different notions of computability for closed real subsets non-uniformly like, e.g., for sets of fixed cardinality or sets containing a (not necessarily effectively findable) computable point. By relativization we obtain a bounded recursive rational sequence of which every accumulation point is not even computable with support of a Halting oracle. Finally the question is treated whether compact sets have co-r.e. closed connected components; and every star-shaped co-r.e. closed set is asserted to contain a computable point. 1
Real Computation with Least Discrete Advice: A Complexity Theory of Nonuniform Computability
, 2009
"... It is folklore particularly in numerical and computer sciences that, instead of solving some general problem f: A → B, additional structural information about the input x ∈ A (that is any kind of promise that x belongs to a certain subset A ′ ⊆ A) should be taken advantage of. Some examples from ..."
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Cited by 4 (0 self)
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It is folklore particularly in numerical and computer sciences that, instead of solving some general problem f: A → B, additional structural information about the input x ∈ A (that is any kind of promise that x belongs to a certain subset A ′ ⊆ A) should be taken advantage of. Some examples from real number computation show that such discrete advice can even make the difference between computability and uncomputability. We turn this into a both topological and combinatorial complexity theory of information, investigating for several practical problems how much advice is necessary and sufficient to render them computable. Specifically, finding a nontrivial solution to a homogeneous linear equation A · x = 0 for a given singular real n × n-matrix A is possible when knowing rank(A) ∈ {0, 1,..., n−1}; and we show this to be best possible. Similarly, diagonalizing (i.e. finding a basis of eigenvectors of) a given real symmetric n × n-matrix A is possible when knowing the number of distinct eigenvalues: an integer between 1 and n (the latter corresponding to the nondegenerate case). And again we show that n–fold (i.e. roughly log n bits of) additional information is indeed necessary in order to render this problem (continuous and) computable; whereas finding some single eigenvector of A requires and suffices with Θ(log n)–fold advice.
