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Track B: Automata, Logic, Semantics, and Theory of Programming

**Published in:** LIPIcs, Volume 229, 49th International Colloquium on Automata, Languages, and Programming (ICALP 2022)

The topological properties of a set have a strong impact on its computability properties. A striking illustration of this idea is given by spheres and closed manifolds: if a set X is homeomorphic to a sphere or a closed manifold, then any algorithm that semicomputes X in some sense can be converted into an algorithm that fully computes X. In other words, the topological properties of X enable one to derive full information about X from partial information about X. In that case, we say that X has computable type. Those results have been obtained by Miller, Iljazović, Sušić and others in the recent years. A similar notion of computable type was also defined for pairs (X,A) in order to cover more spaces, such as compact manifolds with boundary and finite graphs with endpoints.
We investigate the higher dimensional analog of graphs, namely the pairs (X,A) where X is a finite simplicial complex and A is a subcomplex of X. We give two topological characterizations of the pairs having computable type. The first one uses a global property of the pair, that we call the ε-surjection property. The second one uses a local property of neighborhoods of vertices, called the surjection property. We give a further characterization for 2-dimensional simplicial complexes, by identifying which local neighborhoods have the surjection property.
Using these characterizations, we give non-trivial applications to two famous sets: we prove that the dunce hat does not have computable type whereas Bing’s house does. Important concepts from topology, such as absolute neighborhood retracts and topological cones, play a key role in our proofs.

Djamel Eddine Amir and Mathieu Hoyrup. Computability of Finite Simplicial Complexes. In 49th International Colloquium on Automata, Languages, and Programming (ICALP 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 229, pp. 111:1-111:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{amir_et_al:LIPIcs.ICALP.2022.111, author = {Amir, Djamel Eddine and Hoyrup, Mathieu}, title = {{Computability of Finite Simplicial Complexes}}, booktitle = {49th International Colloquium on Automata, Languages, and Programming (ICALP 2022)}, pages = {111:1--111:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-235-8}, ISSN = {1868-8969}, year = {2022}, volume = {229}, editor = {Boja\'{n}czyk, Miko{\l}aj and Merelli, Emanuela and Woodruff, David P.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2022.111}, URN = {urn:nbn:de:0030-drops-164522}, doi = {10.4230/LIPIcs.ICALP.2022.111}, annote = {Keywords: Computable Type, Simplicial Complex, Surjection Property, Topological Cone, Absolute Neighborhood Retract, Dunce Hat, Bing’s House} }

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**Published in:** Dagstuhl Reports, Volume 11, Issue 10 (2022)

Computability and continuity are closely linked - in fact, continuity can be seen as computability relative to an arbitrary oracle. As such, concepts from topology and descriptive set theory feature heavily in the foundations of computable analysis. Conversely, techniques developed in computability theory can be fruitfully employed in topology and descriptive set theory, even if the desired results mention no computability at all. In this Dagstuhl Seminar, we brought together researchers from computable analysis, from classical computability theory, from descriptive set theory, formal topology, and other relevant areas. Our goals were to identify key open questions related to this interplay, to exploit synergies between the areas and to intensify collaboration between the relevant communities.

Mathieu Hoyrup, Arno Pauly, Victor Selivanov, and Mariya I. Soskova. Descriptive Set Theory and Computable Topology (Dagstuhl Seminar 21461). In Dagstuhl Reports, Volume 11, Issue 10, pp. 72-93, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@Article{hoyrup_et_al:DagRep.11.10.72, author = {Hoyrup, Mathieu and Pauly, Arno and Selivanov, Victor and Soskova, Mariya I.}, title = {{Descriptive Set Theory and Computable Topology (Dagstuhl Seminar 21461)}}, pages = {72--93}, journal = {Dagstuhl Reports}, ISSN = {2192-5283}, year = {2022}, volume = {11}, number = {10}, editor = {Hoyrup, Mathieu and Pauly, Arno and Selivanov, Victor and Soskova, Mariya I.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/DagRep.11.10.72}, URN = {urn:nbn:de:0030-drops-159293}, doi = {10.4230/DagRep.11.10.72}, annote = {Keywords: computable analysis, enumeration degrees, quasi-polish spaces, synthetic topology} }

Document

Track B: Automata, Logic, Semantics, and Theory of Programming

**Published in:** LIPIcs, Volume 168, 47th International Colloquium on Automata, Languages, and Programming (ICALP 2020)

This article is a study of descriptive complexity of subsets of represented spaces. Two competing measures of descriptive complexity are available. The first one is topological and measures how complex it is to obtain a set from open sets using boolean operations. The second one measures how complex it is to test membership in the set, and we call it symbolic complexity because it measures the complexity of the symbolic representation of the set. While topological and symbolic complexity are equivalent on countably-based spaces, they differ on more general spaces. Our investigation is aimed at explaining this difference and highly suggests that it is related to the well-known mismatch between topological and sequential aspects of topological spaces.

Mathieu Hoyrup. Descriptive Complexity on Non-Polish Spaces II. In 47th International Colloquium on Automata, Languages, and Programming (ICALP 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 168, pp. 132:1-132:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{hoyrup:LIPIcs.ICALP.2020.132, author = {Hoyrup, Mathieu}, title = {{Descriptive Complexity on Non-Polish Spaces II}}, booktitle = {47th International Colloquium on Automata, Languages, and Programming (ICALP 2020)}, pages = {132:1--132:17}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-138-2}, ISSN = {1868-8969}, year = {2020}, volume = {168}, editor = {Czumaj, Artur and Dawar, Anuj and Merelli, Emanuela}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2020.132}, URN = {urn:nbn:de:0030-drops-125395}, doi = {10.4230/LIPIcs.ICALP.2020.132}, annote = {Keywords: Represented space, Computable analysis, Descriptive set theory, Scott topology} }

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**Published in:** LIPIcs, Volume 154, 37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020)

Represented spaces are the spaces on which computations can be performed. We investigate the descriptive complexity of sets in represented spaces. We prove that the standard representation of a countably-based space preserves the effective descriptive complexity of sets. We prove that some results from descriptive set theory on Polish spaces extend to arbitrary countably-based spaces. We study the larger class of coPolish spaces, showing that their representation does not always preserve the complexity of sets, and we relate this mismatch with the sequential aspects of the space. We study in particular the space of polynomials.

Antonin Callard and Mathieu Hoyrup. Descriptive Complexity on Non-Polish Spaces. In 37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 154, pp. 8:1-8:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{callard_et_al:LIPIcs.STACS.2020.8, author = {Callard, Antonin and Hoyrup, Mathieu}, title = {{Descriptive Complexity on Non-Polish Spaces}}, booktitle = {37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020)}, pages = {8:1--8:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-140-5}, ISSN = {1868-8969}, year = {2020}, volume = {154}, editor = {Paul, Christophe and Bl\"{a}ser, Markus}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2020.8}, URN = {urn:nbn:de:0030-drops-118694}, doi = {10.4230/LIPIcs.STACS.2020.8}, annote = {Keywords: Represented space, Computable analysis, Descriptive set theory, CoPolish spaces} }

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**Published in:** LIPIcs, Volume 138, 44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019)

We introduce the notion of a semicomputable point in R^n, defined as a point having left-c.e. projections. We study the range of such a point, which is the set of directions on which its projections are left-c.e., and is a convex cone. We provide a thorough study of these notions, proving along the way new results on the computability of convex sets. We prove realization results, by identifying computability properties of convex cones that make them ranges of semicomputable points. We give two applications of the theory. The first one provides a better understanding of the Solovay derivatives. The second one is the investigation of left-c.e. quadratic polynomials. We show that this is, in fact, a particular case of the general theory of semicomputable points.

Mathieu Hoyrup and Donald M. Stull. Semicomputable Points in Euclidean Spaces. In 44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 138, pp. 48:1-48:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{hoyrup_et_al:LIPIcs.MFCS.2019.48, author = {Hoyrup, Mathieu and Stull, Donald M.}, title = {{Semicomputable Points in Euclidean Spaces}}, booktitle = {44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019)}, pages = {48:1--48:13}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-117-7}, ISSN = {1868-8969}, year = {2019}, volume = {138}, editor = {Rossmanith, Peter and Heggernes, Pinar and Katoen, Joost-Pieter}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2019.48}, URN = {urn:nbn:de:0030-drops-109928}, doi = {10.4230/LIPIcs.MFCS.2019.48}, annote = {Keywords: Semicomputable point, Left-c.e. real, Convex cone, Solovay reducibility, Genericity} }

Document

**Published in:** LIPIcs, Volume 107, 45th International Colloquium on Automata, Languages, and Programming (ICALP 2018)

Computability and semicomputability of compact subsets of the Euclidean spaces are important notions, that have been investigated for many classes of sets including fractals (Julia sets, Mandelbrot set) and objects with geometrical or topological constraints (embedding of a sphere). In this paper we investigate one of the simplest classes, namely the filled triangles in the plane. We study the properties of the parameters of semicomputable triangles, such as the coordinates of their vertices. This problem is surprisingly rich. We introduce and develop a notion of semicomputability of points of the plane which is a generalization in dimension 2 of the left-c.e. and right-c.e. numbers. We relate this notion to Solovay reducibility. We show that semicomputable triangles admit no finite parametrization, for some notion of parametrization.

Mathieu Hoyrup, Diego Nava Saucedo, and Don M. Stull. Semicomputable Geometry. In 45th International Colloquium on Automata, Languages, and Programming (ICALP 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 107, pp. 129:1-129:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{hoyrup_et_al:LIPIcs.ICALP.2018.129, author = {Hoyrup, Mathieu and Nava Saucedo, Diego and Stull, Don M.}, title = {{Semicomputable Geometry}}, booktitle = {45th International Colloquium on Automata, Languages, and Programming (ICALP 2018)}, pages = {129:1--129:13}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-076-7}, ISSN = {1868-8969}, year = {2018}, volume = {107}, editor = {Chatzigiannakis, Ioannis and Kaklamanis, Christos and Marx, D\'{a}niel and Sannella, Donald}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2018.129}, URN = {urn:nbn:de:0030-drops-91336}, doi = {10.4230/LIPIcs.ICALP.2018.129}, annote = {Keywords: Computable set, Semicomputable set, Solovay reducibility, Left-ce real, Genericity} }

Document

**Published in:** LIPIcs, Volume 55, 43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016)

What can be decided or semidecided about a primitive recursive function, given a definition of that function by primitive recursion? What about subrecursive classes other than primitive recursive functions? We provide a complete and explicit characterization of the decidable and semidecidable properties. This characterization uses a variant of Kolmogorov complexity where only programs in a subrecursive programming language are allowed. More precisely, we prove that all the decidable and semidecidable properties can be obtained as combinations of two classes of basic decidable properties: (i) the function takes some particular values on a finite set of inputs, and (ii) every finite part of the function can be compressed to some extent.

Mathieu Hoyrup. The Decidable Properties of Subrecursive Functions. In 43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 55, pp. 108:1-108:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{hoyrup:LIPIcs.ICALP.2016.108, author = {Hoyrup, Mathieu}, title = {{The Decidable Properties of Subrecursive Functions}}, booktitle = {43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016)}, pages = {108:1--108:13}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-013-2}, ISSN = {1868-8969}, year = {2016}, volume = {55}, editor = {Chatzigiannakis, Ioannis and Mitzenmacher, Michael and Rabani, Yuval and Sangiorgi, Davide}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2016.108}, URN = {urn:nbn:de:0030-drops-62435}, doi = {10.4230/LIPIcs.ICALP.2016.108}, annote = {Keywords: Rice theorem, subrecursive class, decidable property, Kolmogorov complexity, compressibility} }

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**Published in:** LIPIcs, Volume 30, 32nd International Symposium on Theoretical Aspects of Computer Science (STACS 2015)

In computability theory and computable analysis, finite programs can compute infinite objects. Presenting a computable object via any program for it, provides at least as much information as presenting the object itself, written on an infinite tape. What additional information do programs provide? We characterize this additional information to be any upper bound on the Kolmogorov complexity of the object. Hence we identify the exact relationship between Markov-computability and Type-2-computability. We then use this relationship to obtain several results characterizing the computational and topological structure of Markov-semidecidable sets.

Mathieu Hoyrup and Cristóbal Rojas. On the Information Carried by Programs about the Objects They Compute. In 32nd International Symposium on Theoretical Aspects of Computer Science (STACS 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 30, pp. 447-459, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)

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@InProceedings{hoyrup_et_al:LIPIcs.STACS.2015.447, author = {Hoyrup, Mathieu and Rojas, Crist\'{o}bal}, title = {{On the Information Carried by Programs about the Objects They Compute}}, booktitle = {32nd International Symposium on Theoretical Aspects of Computer Science (STACS 2015)}, pages = {447--459}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-78-1}, ISSN = {1868-8969}, year = {2015}, volume = {30}, editor = {Mayr, Ernst W. and Ollinger, Nicolas}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2015.447}, URN = {urn:nbn:de:0030-drops-49337}, doi = {10.4230/LIPIcs.STACS.2015.447}, annote = {Keywords: Markov-computable, representation, Kolmogorov complexity, Ershov topology} }

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**Published in:** LIPIcs, Volume 25, 31st International Symposium on Theoretical Aspects of Computer Science (STACS 2014)

The strong relationship between topology and computations has played a central role in the development of several branches of theoretical computer science: foundations of functional programming, computational geometry, computability theory, computable analysis. Often it happens that a given function is not computable simply because it is not continuous. In many cases, the function can moreover be proved to be non-computable in the stronger sense that it does not preserve computability: it maps a computable input to a non-computable output. To date, there is no connection between topology and this kind of non-computability, apart from Pour-El and Richards "First Main Theorem", applicable to linear operators on Banach spaces only.
In the present paper, we establish such a connection. We identify the discontinuity notion, for the inverse of a computable function, that implies non-preservation of computability. Our result is applicable to a wide range of functions, it unifies many existing ad hoc constructions explaining at the same time what makes these constructions possible in particular contexts, sheds light on the relationship between topology and computability and most importantly allows us to solve open problems. In particular it enables us to answer the following open question in the negative: if the sum of two shift-invariant ergodic measures is computable, must these measures be computable as well? We also investigate how generic a point with computable image can be. To this end we introduce a notion of genericity of a point w.r.t. a function, which enables us to unify several finite injury constructions from computability theory.

Mathieu Hoyrup. Irreversible computable functions. In 31st International Symposium on Theoretical Aspects of Computer Science (STACS 2014). Leibniz International Proceedings in Informatics (LIPIcs), Volume 25, pp. 362-373, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2014)

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@InProceedings{hoyrup:LIPIcs.STACS.2014.362, author = {Hoyrup, Mathieu}, title = {{Irreversible computable functions}}, booktitle = {31st International Symposium on Theoretical Aspects of Computer Science (STACS 2014)}, pages = {362--373}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-65-1}, ISSN = {1868-8969}, year = {2014}, volume = {25}, editor = {Mayr, Ernst W. and Portier, Natacha}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2014.362}, URN = {urn:nbn:de:0030-drops-44712}, doi = {10.4230/LIPIcs.STACS.2014.362}, annote = {Keywords: Computability theory, computable analysis, finite injury, generic set} }

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**Published in:** LIPIcs, Volume 14, 29th International Symposium on Theoretical Aspects of Computer Science (STACS 2012)

Let m be a computable ergodic shift-invariant measure over the set of infinite binary sequences. Providing a constructive proof of Shannon-McMillan-Breiman theorem, V'yugin proved that if x is a Martin-Löf random binary sequence w.r.t. m then its strong effective dimension Dim(x) equals the entropy of m. Whether its effective dimension dim(x) also equals the entropy was left as an open problem. In this paper we settle this problem, providing a positive answer. A key step in the proof consists in extending recent results on Birkhoff's ergodic theorem for Martin-Löf random sequences. At the same time, we present extensions of some previous results.
As pointed out by a referee the main result can also be derived from results by Hochman [Upcrossing inequalities for stationary sequences and applications. The Annals of Probability, 37(6):2135--2149, 2009], using rather different considerations.

Mathieu Hoyrup. The dimension of ergodic random sequences. In 29th International Symposium on Theoretical Aspects of Computer Science (STACS 2012). Leibniz International Proceedings in Informatics (LIPIcs), Volume 14, pp. 567-576, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2012)

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@InProceedings{hoyrup:LIPIcs.STACS.2012.567, author = {Hoyrup, Mathieu}, title = {{The dimension of ergodic random sequences}}, booktitle = {29th International Symposium on Theoretical Aspects of Computer Science (STACS 2012)}, pages = {567--576}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-35-4}, ISSN = {1868-8969}, year = {2012}, volume = {14}, editor = {D\"{u}rr, Christoph and Wilke, Thomas}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2012.567}, URN = {urn:nbn:de:0030-drops-33917}, doi = {10.4230/LIPIcs.STACS.2012.567}, annote = {Keywords: Shannon-McMillan-Breiman theorem, Martin-L\"{o}f random sequence, effective Hausdorff dimension, compression rate, entropy} }

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**Published in:** LIPIcs, Volume 3, 26th International Symposium on Theoretical Aspects of Computer Science (2009)

We extend the notion of randomness (in the version introduced by Schnorr) to computable Probability Spaces and compare it to a \emph{dynamical} notion of randomness: typicality. Roughly, a point is \emph{typical} for some dynamic, if it follows the statistical behavior of the system (Birkhoff's pointwise ergodic theorem). We prove that a point is Schnorr random if and only if it is typical for every \emph{mixing} computable dynamics. To prove the result we develop some tools for the theory of computable probability spaces (for example, morphisms) that are expected to have other applications.

Peter Gacs, Mathieu Hoyrup, and Cristobal Rojas. Randomness on Computable Probability Spaces - A Dynamical Point of View. In 26th International Symposium on Theoretical Aspects of Computer Science. Leibniz International Proceedings in Informatics (LIPIcs), Volume 3, pp. 469-480, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2009)

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@InProceedings{gacs_et_al:LIPIcs.STACS.2009.1828, author = {Gacs, Peter and Hoyrup, Mathieu and Rojas, Cristobal}, title = {{Randomness on Computable Probability Spaces - A Dynamical Point of View}}, booktitle = {26th International Symposium on Theoretical Aspects of Computer Science}, pages = {469--480}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-09-5}, ISSN = {1868-8969}, year = {2009}, volume = {3}, editor = {Albers, Susanne and Marion, Jean-Yves}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2009.1828}, URN = {urn:nbn:de:0030-drops-18280}, doi = {10.4230/LIPIcs.STACS.2009.1828}, annote = {Keywords: Schnorr randomness, Birkhoff's ergodic theorem, Computable measures} }

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