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**Published in:** LIPIcs, Volume 309, 15th International Conference on Interactive Theorem Proving (ITP 2024)

In exact real computation real numbers are manipulated exactly without round-off errors, making it well-suited for high precision verified computation. In recent work we propose an axiomatic formalization of exact real computation in the Coq theorem prover. The formalization admits an extended extraction mechanism that lets us extract computational content from constructive parts of proofs to efficient programs built on top of AERN, a Haskell library for exact real computation.
Many processes in science and engineering are modeled by ordinary differential equations (ODEs), and often safety-critical applications depend on computing their solutions correctly. The primary goal of the current work is to extend our framework to spaces of functions and to support computation of solutions to ODEs and other essential operators.
In numerical mathematics, the most common way to represent continuous functions is to use polynomial approximations. This can be modeled by so-called Taylor models, that encode a function as a polynomial and a rigorous error-bound over some domain. We define types of classical functions that do not hold any computational content and formalize Taylor models to computationally approximate those classical functions. Classical functions are defined in a way to admit classical principles in their constructions and verification. We define various basic operations on Taylor models and verify their correctness based on the classical functions that they approximate. We then shift our interest to analytic functions as a generalization of Taylor models where polynomials are replaced by infinite power series. We use the formalization to develop a theory of non-linear polynomial ODEs. From the proofs we can extract certified exact real computation programs that compute solutions of ODEs on some time interval up to any precision.

Sewon Park and Holger Thies. A Coq Formalization of Taylor Models and Power Series for Solving Ordinary Differential Equations. In 15th International Conference on Interactive Theorem Proving (ITP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 309, pp. 30:1-30:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{park_et_al:LIPIcs.ITP.2024.30, author = {Park, Sewon and Thies, Holger}, title = {{A Coq Formalization of Taylor Models and Power Series for Solving Ordinary Differential Equations}}, booktitle = {15th International Conference on Interactive Theorem Proving (ITP 2024)}, pages = {30:1--30:19}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-337-9}, ISSN = {1868-8969}, year = {2024}, volume = {309}, editor = {Bertot, Yves and Kutsia, Temur and Norrish, Michael}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITP.2024.30}, URN = {urn:nbn:de:0030-drops-207581}, doi = {10.4230/LIPIcs.ITP.2024.30}, annote = {Keywords: Exact real computation, Taylor models, Analytic functions, Computable analysis, Program extraction} }

Document

**Published in:** LIPIcs, Volume 272, 48th International Symposium on Mathematical Foundations of Computer Science (MFCS 2023)

We propose a framework for certified computation on hyperspaces by formalizing various higher-order data types and operations in a constructive dependent type theory. Our approach builds on our previous work on axiomatization of exact real computation where we formalize nondeterministic first-order partial computations over real and complex numbers. Based on the axiomatization, we first define open, closed, compact and overt subsets in an abstract topological way that allows short and elegant proofs with computational content coinciding with standard definitions in computable analysis. From these proofs we extract programs for testing inclusion, overlapping of sets, et cetera.
To improve extracted programs, our framework specializes the Euclidean space ℝ^m making use of metric properties. To define interesting operations over hyperspaces of Euclidean space, we introduce a nondeterministic version of a continuity principle valid under the standard type-2 realizability interpretation. Instead of choosing one of the usual formulations, we define it in a way similar to an interval extension operator, which often is already available in exact real computation software.
We prove that the operations on subsets preserve the encoding, and thereby define a small calculus to built new subsets from given ones, including limits of converging sequences with regards to the Hausdorff metric. From the proofs, we extract programs that generate drawings of subsets of ℝ^m with any given precision efficiently. As an application we provide a function that constructs fractals, such as the Sierpinski triangle, from iterated function systems using the limit operation, resulting in certified programs that errorlessly draw such fractals up to any desired resolution.

Michal Konečný, Sewon Park, and Holger Thies. Formalizing Hyperspaces for Extracting Efficient Exact Real Computation. In 48th International Symposium on Mathematical Foundations of Computer Science (MFCS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 272, pp. 59:1-59:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{konecny_et_al:LIPIcs.MFCS.2023.59, author = {Kone\v{c}n\'{y}, Michal and Park, Sewon and Thies, Holger}, title = {{Formalizing Hyperspaces for Extracting Efficient Exact Real Computation}}, booktitle = {48th International Symposium on Mathematical Foundations of Computer Science (MFCS 2023)}, pages = {59:1--59:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-292-1}, ISSN = {1868-8969}, year = {2023}, volume = {272}, editor = {Leroux, J\'{e}r\^{o}me and Lombardy, Sylvain and Peleg, David}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2023.59}, URN = {urn:nbn:de:0030-drops-185935}, doi = {10.4230/LIPIcs.MFCS.2023.59}, annote = {Keywords: Computable analysis, type theory, program extraction} }

Document

**Published in:** LIPIcs, Volume 182, 40th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2020)

We use ideas from computable analysis to formalize exact real number computation in the Coq proof assistant. Our formalization is built on top of the Incone library, a Coq library for computable analysis. We use the theoretical framework that computable analysis provides to systematically generate target specifications for real number algorithms. First we give very simple algorithms that fulfill these specifications based on rational approximations. To provide more efficient algorithms, we develop alternate representations that utilize an existing formalization of floating-point algorithms and interval arithmetic in combination with methods used by software packages for exact real arithmetic that focus on execution speed. We also define a general framework to define real number algorithms independently of their concrete encoding and to prove them correct. Algorithms verified in our framework can be extracted to Haskell programs for efficient computation. The performance of the extracted code is comparable to programs produced using non-verified software packages. This is without the need to optimize the extracted code by hand.
As an example, we formalize an algorithm for the square root function based on the Heron method. The algorithm is parametric in the implementation of the real number datatype, not referring to any details of its implementation. Thus the same verified algorithm can be used with different real number representations. Since Boolean valued comparisons of real numbers are not decidable, our algorithms use basic operations that take values in the Kleeneans and Sierpinski space. We develop some of the theory of these spaces. To capture the semantics of non-sequential operations, such as the "parallel or", we use multivalued functions.

Michal Konečný, Florian Steinberg, and Holger Thies. Computable Analysis for Verified Exact Real Computation. In 40th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 182, pp. 50:1-50:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{konecny_et_al:LIPIcs.FSTTCS.2020.50, author = {Kone\v{c}n\'{y}, Michal and Steinberg, Florian and Thies, Holger}, title = {{Computable Analysis for Verified Exact Real Computation}}, booktitle = {40th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2020)}, pages = {50:1--50:18}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-174-0}, ISSN = {1868-8969}, year = {2020}, volume = {182}, editor = {Saxena, Nitin and Simon, Sunil}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2020.50}, URN = {urn:nbn:de:0030-drops-132914}, doi = {10.4230/LIPIcs.FSTTCS.2020.50}, annote = {Keywords: Computable Analysis, exact real computation, formal proofs, proof assistant, Coq} }

Document

**Published in:** LIPIcs, Volume 170, 45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020)

We investigate a variant of the fuel-based approach to modeling diverging computation in type theories and use it to abstractly capture the essence of oracle Turing machines. The resulting objects we call continuous machines. We prove that it is possible to translate back and forth between such machines and names in the standard function encoding used in computable analysis. Put differently, among the operators on Baire space, exactly the partial continuous ones are implementable by continuous machines and the data that such a machine provides is a description of the operator as a sequentially realizable functional. Continuous machines are naturally formulated in type theories and we have formalized our findings in Coq as part of Incone, a Coq library for computable analysis.
The correctness proofs use a classical meta-theory with countable choice. Along the way we formally prove some known results such as the existence of a self-modulating modulus of continuity for partial continuous operators on Baire space. To illustrate their versatility we use continuous machines to specify some algorithms that operate on objects that cannot be fully described by finite means, such as real numbers and functions. We present particularly simple algorithms for finding the multiplicative inverse of a real number and for composition of partial continuous operators on Baire space. Some of the simplicity is achieved by utilizing the fact that continuous machines are compatible with multivalued semantics.

Michal Konečný, Florian Steinberg, and Holger Thies. Continuous and Monotone Machines. In 45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 170, pp. 56:1-56:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{konecny_et_al:LIPIcs.MFCS.2020.56, author = {Kone\v{c}n\'{y}, Michal and Steinberg, Florian and Thies, Holger}, title = {{Continuous and Monotone Machines}}, booktitle = {45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020)}, pages = {56:1--56:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-159-7}, ISSN = {1868-8969}, year = {2020}, volume = {170}, editor = {Esparza, Javier and Kr\'{a}l', Daniel}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2020.56}, URN = {urn:nbn:de:0030-drops-127230}, doi = {10.4230/LIPIcs.MFCS.2020.56}, annote = {Keywords: Computable Analysis, exact real computation, formal proofs, proof assistant, Coq} }

Document

**Published in:** LIPIcs, Volume 141, 10th International Conference on Interactive Theorem Proving (ITP 2019)

We give a number of formal proofs of theorems from the field of computable analysis. Many of our results specify executable algorithms that work on infinite inputs by means of operating on finite approximations and are proven correct in the sense of computable analysis. The development is done in the proof assistant Coq and heavily relies on the Incone library for information theoretic continuity. This library is developed by one of the authors and the results of this paper extend the library. While full executability in a formal development of mathematical statements about real numbers and the like is not a feature that is unique to the Incone library, its original contribution is to adhere to the conventions of computable analysis to provide a general purpose interface for algorithmic reasoning on continuous structures. The paper includes a brief description of the most important concepts of Incone and its sub libraries mf and Metric.
The results that provide complete computational content include that the algebraic operations and the efficient limit operator on the reals are computable, that the countably infinite product of a space with itself is isomorphic to a space of functions, compatibility of the enumeration representation of subsets of natural numbers with the abstract definition of the space of open subsets of the natural numbers, and that continuous realizability implies sequential continuity. We also describe many non-computational results that support the correctness of definitions from the library. These include that the information theoretic notion of continuity used in the library is equivalent to the metric notion of continuity on Baire space, a complete comparison of the different concepts of continuity that arise from metric and represented space structures and the discontinuity of the unrestricted limit operator on the real numbers and the task of selecting an element of a closed subset of the natural numbers.

Florian Steinberg, Laurent Théry, and Holger Thies. Quantitative Continuity and Computable Analysis in Coq. In 10th International Conference on Interactive Theorem Proving (ITP 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 141, pp. 28:1-28:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{steinberg_et_al:LIPIcs.ITP.2019.28, author = {Steinberg, Florian and Th\'{e}ry, Laurent and Thies, Holger}, title = {{Quantitative Continuity and Computable Analysis in Coq}}, booktitle = {10th International Conference on Interactive Theorem Proving (ITP 2019)}, pages = {28:1--28:21}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-122-1}, ISSN = {1868-8969}, year = {2019}, volume = {141}, editor = {Harrison, John and O'Leary, John and Tolmach, Andrew}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITP.2019.28}, URN = {urn:nbn:de:0030-drops-110830}, doi = {10.4230/LIPIcs.ITP.2019.28}, annote = {Keywords: computable analysis, Coq, contionuous functionals, discontinuity, closed choice on the naturals} }

Document

**Published in:** LIPIcs, Volume 117, 43rd International Symposium on Mathematical Foundations of Computer Science (MFCS 2018)

We apply average-case complexity theory to physical problems modeled by continuous-time dynamical systems. The computational complexity when simulating such systems for a bounded time-frame mainly stems from trajectories coming close to complex singularities of the system. We show that if for most initial values the trajectories do not come close to singularities the simulation can be done in polynomial time on average. For Hamiltonian systems we relate this to the volume of "almost singularities" in phase space and give some general criteria to show that a Hamiltonian system can be simulated efficiently on average. As an application we show that the planar circular-restricted three-body problem is average-case polynomial-time computable.

Akitoshi Kawamura, Holger Thies, and Martin Ziegler. Average-Case Polynomial-Time Computability of Hamiltonian Dynamics. In 43rd International Symposium on Mathematical Foundations of Computer Science (MFCS 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 117, pp. 30:1-30:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{kawamura_et_al:LIPIcs.MFCS.2018.30, author = {Kawamura, Akitoshi and Thies, Holger and Ziegler, Martin}, title = {{Average-Case Polynomial-Time Computability of Hamiltonian Dynamics}}, booktitle = {43rd International Symposium on Mathematical Foundations of Computer Science (MFCS 2018)}, pages = {30:1--30:17}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-086-6}, ISSN = {1868-8969}, year = {2018}, volume = {117}, editor = {Potapov, Igor and Spirakis, Paul and Worrell, James}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2018.30}, URN = {urn:nbn:de:0030-drops-96125}, doi = {10.4230/LIPIcs.MFCS.2018.30}, annote = {Keywords: Computable Analysis, Real computation, Dynamical systems, Average-case complexity, Computation in physics} }

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