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20cs700 [2020-09-03] – [Synopsis] Martin Ziegler | 20cs700 [2020-09-17] – [Synopsis] Martin Ziegler | ||
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==== Synopsis ==== | ==== Synopsis ==== | ||
- | 0. Introduction & Motivation ({{20cs700_.pdf|pdf}}, | + | **0. Introduction & Motivation** ({{20cs700_.pdf|pdf}}, |
" | " | ||
Abstract Data Types / Structures in Computer Science and Mathematics, | Abstract Data Types / Structures in Computer Science and Mathematics, | ||
IEEE 754 / Floatingpoint Discretization Properties and Reliability, | IEEE 754 / Floatingpoint Discretization Properties and Reliability, | ||
Numerical Folklore and Myths \\ | Numerical Folklore and Myths \\ | ||
- | I. Recap on discrete Theory of Computation ({{20cs700a.pdf|pdf}}, | + | **I. Recap on discrete Theory of Computation** ({{20cs700a.pdf|pdf}}, |
- | Un-/ | + | Un-/ |
- | II. Computability theory over the Reals: \\ | + | **II. Computability theory over the Reals**: \\ |
- | real numbers: binary vs. approximation | + | a) |
- | sequences, limits, rate of convergence | + | b) Computing Real Sequences: semi-decidability / strong undecidability |
- | qualitative stability: computability | + | c) Computing Real Functions: closure under composition |
- | real arithmetic, join, maximum, integral | + | d) Advanced Un/ |
- | root finding, argmax, derivative | + | e) Multi-Functions & Enrichment: generalized restriction, |
- | uncomputable | + | f) Computing Real Operators: encoding continuous functions, encoding compact subsets |
- | analytic functions, discrete enrichment | + | **III. Computability on T0 spaces**: \\ |
- | multivaluedness, computability | + | |
- | III. Computability on separable metric | + | |
C[0;1] and computable Weierstrass Theorem | C[0;1] and computable Weierstrass Theorem | ||
encoding compact Euclidean subsets | encoding compact Euclidean subsets | ||
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Main Theorem: continuity = oracle computability | Main Theorem: continuity = oracle computability | ||
Henkin-continuity of multivalued ' | Henkin-continuity of multivalued ' | ||
- | IV. Complexity theory over the Reals: \\ | + | **IV. Complexity theory over the Reals**: \\ |
quantitative stability and polynomial-time computability | quantitative stability and polynomial-time computability | ||
maximizing smooth polynomial-time functions is NP-' | maximizing smooth polynomial-time functions is NP-' | ||
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fixed-parametrized complexity and Gevrey' | fixed-parametrized complexity and Gevrey' | ||
average-case complexity of real functions \\ | average-case complexity of real functions \\ | ||
- | V. Imperative programming over the Reals: \\ | + | **V. Complexity Theory on Compact Metric Spaces** \\ |
+ | **VI. Imperative programming over the Reals**: \\ | ||
Semantics of tests | Semantics of tests | ||
Example: Gaussian Elimination | Example: Gaussian Elimination | ||
Verification in Floyd-Hoare Logic | Verification in Floyd-Hoare Logic | ||
Practical implementation in iRRAM \\ | Practical implementation in iRRAM \\ | ||
- | VI. Uniform complexity of operators in Analysis: \\ | + | **VII. Uniform complexity of operators in Analysis**: \\ |
TTE and computational complexity | TTE and computational complexity | ||
encoding metric spaces of polynomial entropy | encoding metric spaces of polynomial entropy | ||
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2nd-order polynomial complexity theory | 2nd-order polynomial complexity theory | ||
Weihrauch reductions | Weihrauch reductions | ||
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==== Test questions for self-assessment ==== | ==== Test questions for self-assessment ==== | ||
Mathematical background: | Mathematical background: |