Differences
This shows you the differences between two versions of the page.
| Both sides previous revisionPrevious revisionNext revision | Previous revision | ||
| home [2023-03-17] – [References] Martin Ziegler | home [2023-03-18] (current) – [Mission] Martin Ziegler | ||
|---|---|---|---|
| Line 2: | Line 2: | ||
| ===== Motivation ===== | ===== Motivation ===== | ||
| - | Digital Computers naturally process discrete data, such as bits or integers or strings or graphs. From bits to advanced data structures, from a first semi-conducting transistor to billions in wafer-scale integration, | + | Digital Computers naturally process discrete data, such as bits or integers or strings or graphs. From bits to advanced data structures, from a first semi-conducting transistor to billions in wafer-scale integration, |
| the success story of digital computing arguably is due to (1) hierarchical layers of abstraction and (2) the ultimate reliability of each layer for the next one to build on ― for processing discrete data. | the success story of digital computing arguably is due to (1) hierarchical layers of abstraction and (2) the ultimate reliability of each layer for the next one to build on ― for processing discrete data. | ||
| Line 17: | Line 17: | ||
| Deviations between mathematical structures and their hardware counterparts are common also in the discrete realm, | Deviations between mathematical structures and their hardware counterparts are common also in the discrete realm, | ||
| - | such as the “integer” wraparound 255+1=0 occurring in bytes that led to the ”Nuclear Gandhi'' | + | such as the “integer” wraparound 255+1=0 occurring in bytes that led to the // |
| - | Similarly, deviations between exact and approximate continuous data underlie infamous failures such as the Ariane 501 flight V88 or the Sleipner-A oil platform. | + | Similarly, deviations between exact and approximate continuous data underlie infamous failures such as the |
| - | Nowadays high-level programming languages (such as Java or Python) provide a user data type (called for example BigInt or mpz_t) that fully agrees with mathematical integers, simulated in software using a variable number of hardware bytes. | + | Nowadays high-level programming languages (such as Java or Python) provide a user data type (called for example |
| This additional layer of abstraction provides the reliability for advanced discrete data types (such as weighted or labelled graphs) to build on, as mentioned above. | This additional layer of abstraction provides the reliability for advanced discrete data types (such as weighted or labelled graphs) to build on, as mentioned above. | ||
| Line 27: | Line 27: | ||
| We develop Computer Science for continuous data, to catch up with the discrete case: from foundations via practical implementation to commercial applications. | We develop Computer Science for continuous data, to catch up with the discrete case: from foundations via practical implementation to commercial applications. | ||
| - | In fact some object-oriented software libraries, such as iRRAM or Core III or realLib or Ariadne or Aern, have long been providing general (=including all transcendental) real numbers as exact encapsulated user data type. | + | In fact some object-oriented software libraries, such as [[http:// |
| - | Technically they employ finite but variable precision approximations: | + | Technically they employ finite but variable precision approximations: |
| - | | + | |
| - | see also the preprint arXiv: | + | see also the preprint |
| * Thus reliably building on single real numbers leads to higher (but finite) dimensional data types, such as vectors or matrices. | * Thus reliably building on single real numbers leads to higher (but finite) dimensional data types, such as vectors or matrices. | ||
| - | Sewon Park has designed and analyzed and implemented a reliable variant of Gaussian Elimination, | + | [[http:// |
| - | Seokbin Lee has designed and analyzed and implemented a reliable variant of the Grassmannian, | + | [[https:// |
| - | * Infinite sequences of real numbers arise as elements of ℓp spaces; and as coefficients to analytic function germs. | + | * Infinite sequences of real numbers arise as elements of ℓ<sup>p</ |
| - | Holger Thies has implemented analytic functions for reliably solving ODEs and PDEs. | + | [[http:// |
| - | See the references below for this and more continuous data types on GitHub. | + | See the references below for this and more [[https:// |
| Like discrete data, processing continuous data on a digital computer eventually boils down to processing bits: finite sequences of bits in the discrete case, in the continuous case infinite sequences, approximated via finite initial segments. | Like discrete data, processing continuous data on a digital computer eventually boils down to processing bits: finite sequences of bits in the discrete case, in the continuous case infinite sequences, approximated via finite initial segments. | ||
| Coding theory of discrete data is well-established since Claude Shannon’s famous work. | Coding theory of discrete data is well-established since Claude Shannon’s famous work. | ||
| Encoding real numbers as infinite sequences of bits is non-trivial: | Encoding real numbers as infinite sequences of bits is non-trivial: | ||
| - | * Donghyun Lim in his MSc Thesis has investigated encoding more advanced (such as function) spaces; | + | * |
| - | see also the preprint arXiv: | + | see also the [[https:// |
| + | * [[https:// | ||
| + | |||
| + | * [[http:// | ||
| ===== References ===== | ===== References ===== | ||