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home [2023-03-17] – [Mission] Martin Zieglerhome [2023-03-18] (current) – [Mission] Martin Ziegler
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 ===== 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, from individual Boolean connectives to entire CPU circuits, from kB to TB memories, from 10^2 to 10^9 instructions per second, from assembly code to high-level programming languages: \\+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, from individual Boolean connectives to entire CPU circuits, from kB to TB memories, from 10<sup>2</sup> to 10<sup>9</sup> instructions per second, from assembly code to high-level programming languages: \\
 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.
  
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 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'' programming bug.+such as the “integer” wraparound 255+1=0 occurring in bytes that led to the //[[https://en.wikipedia.org/wiki/Nuclear_Gandhi|Nuclear Gandhi]]// programming bug.
  
-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  [[https://en.wikipedia.org/wiki/Ariane_flight_V88|Ariane 501 flight V88]] or the [[https://en.wikipedia.org/wiki/Sleipner_A|Sleipner-A oil platform]].
  
-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 ''BigInt'' or ''mpz_t'') that fully agrees with mathematical integers, simulated in software using a variable number of hardware bytes.
 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.
  
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 [[https://slee3379.math.gatech.edu/|Seokbin Lee]] has designed and analyzed and implemented a [[https://github.com/realcomputation/irramplus/tree/master/GRASSMANN|reliable variant of the Grassmannian]], i.e., the orthomodular lattice of subspaces of some fixed finite-dimensional Euclidean or unitary vector space. [[https://slee3379.math.gatech.edu/|Seokbin Lee]] has designed and analyzed and implemented a [[https://github.com/realcomputation/irramplus/tree/master/GRASSMANN|reliable variant of the Grassmannian]], i.e., the orthomodular lattice of subspaces of some fixed finite-dimensional Euclidean or unitary vector space.
  
-  * 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</sup> spaces; and as coefficients to analytic function germs.
 [[http://www.holgerthies.com/|Holger Thies]] has implemented [[https://github.com/holgerthies|analytic functions for reliably solving ODEs and PDEs]]. [[http://www.holgerthies.com/|Holger Thies]] has implemented [[https://github.com/holgerthies|analytic functions for reliably solving ODEs and PDEs]].
  
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 see also the [[https://arxiv.org/abs/2002.04005|preprint arXiv:2002.04005]]. see also the [[https://arxiv.org/abs/2002.04005|preprint arXiv:2002.04005]].
 +  *  [[https://www.comp.nus.edu.sg/programmes/pg/phdcs/directory/|Ivan Koswara]] and [[http://www.lix.polytechnique.fr/Labo/Gleb.POGUDIN/|Gleb Pogudin]] and [[https://www.researchgate.net/profile/Svetlana-Selivanova|Svetlana Selivanova]] have [[http://cs4contidat.eu/yjcom101727.pdf|related the bit-complexity intrinsic to approximate solutions of linear partial differential equations]] to discrete complexity classes #P and PSPACE.
 +
 +  * [[http://informatik.uni-trier.de/~brausse/personal/index.xhtml|Franz Brauße]] and [[https://www.maastrichtuniversity.nl/pieter.collins|Pieter Collins]] envision a Computer <del>Algebra</del>//Analysis// System
  
 ===== References ===== ===== References =====