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home [2023-03-17] – import from cs4contidat.eu 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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 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://irram.uni-trier.de/|iRRAM]] or [[https://cs.nyu.edu/exact/core_pages/intro.html|Core III]] or [[https://store.fmi.uni-sofia.bg/fmi/logic/theses/lambov/|realLib]] or [[http://ariadne-cps.org/|Ariadne]] or [[https://michalkonecny.github.io/aern/|Aern]], have long been providing general (=including all transcendental) real numbers as exact encapsulated user data type. 
-Technically they employ finite but variable precision approximations: much like BigInt, but with the added challenge of choosing said precision automatically and adaptively sufficient for the user program to appear as indistinguishable from exact reals. This requires a new (namely partial) semantics for real comparison: formalizing the folklore to “avoid” testing for equality, in terms of Kleene's ternary logic.+Technically they employ finite but variable precision approximations: much like ''BigInt'', but with the added challenge of choosing said precision automatically and adaptively sufficient for the user program to appear as indistinguishable from exact reals. This requires a new (namely partial) semantics for real comparison: formalizing the folklore to “avoid” testing for equality, in terms of [[https://en.wikipedia.org/wiki/Three-valued_logic|Kleene's ternary logic]].
  
-   *  Sewon Park in his PhD Thesis has extended that semantics to composite expressions, and further to command sequences aka programs, whose correctness can then be symbolically verified using an extension of Floyd-Hoare Logic; +   *  [[http://sewonpark.com/sewon_park|Sewon Park]] in [[http://realcomputation.asia/THESES/21Sewon.pdf|his PhD Thesis]] has extended that semantics to composite expressions, and further to command sequences aka programs, whose correctness can then be symbolically verified using an extension of [[https://en.wikipedia.org/wiki/Hoare_logic|Floyd-Hoare Logic]]; 
-see also the preprint arXiv:1608.05787.+see also the preprint [[http://arXiv.org/abs/1608.05787|arXiv:1608.05787]].
  
      * 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, in particular regarding pivot search. +[[http://sewonpark.com/sewon_park|Sewon Park]] has designed and analyzed and implemented a [[https://github.com/realcomputation/irramplus/tree/master/GAUSSELIM|reliable variant of Gaussian Elimination]], in particular regarding pivot search. 
-Seokbin Lee has designed and analyzed and implemented a 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. 
-Holger Thies has implemented 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]].
  
-See the references below for this and more continuous data types on GitHub.+See the references below for this and more [[https://github.com/realcomputation/irramplus|continuous data types on GitHub]].
  
 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: the binary expansion for example renders addition uncomputable. Encoding real numbers as infinite sequences of bits is non-trivial: the binary expansion for example renders addition uncomputable.
-  *  Donghyun Lim in his MSc Thesis has investigated encoding more advanced (such as function) spaces;+  *  [[https://www.donghyunlim.com/|Donghyun Lim]] in [[http://realcomputation.asia/THESES/19Donghyun.pdf|his MSc Thesis]] has investigated encoding more advanced (such as function) spaces;
  
-see also the 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 =====
-  * 30h online lecture series http://youtube.com/playlist?list=PLvcvykdwsGNE9HyG46aT6aqCvvJBZgKZh in YouTube channel+  * [[http://youtube.com/playlist?list=PLvcvykdwsGNE9HyG46aT6aqCvvJBZgKZh|30h online lecture series]] in [[https://youtube.com/channel/UCSehZy20EPRqgV0uiTzJkEA|YouTube channel]]
  
-  *  C++ classes for continuous data types: https://github.com/realcomputation, https://github.com/holgerthies+  *  C++ classes for continuous data types: [[https://github.com/realcomputation|RealComputation]], [[https://github.com/holgerthies|Holger Thies]]
  
-  * Vision paper https://doi.org/10.1007/978-3-031-14788-3_5+  * [[https://doi.org/10.1007/978-3-031-14788-3_5|Vision paper]]
  
-  * Recent publication characterizing the bit-complexity of Partial Differential Equations in terms of discrete complexity classes P/NP/#P/PSPACE/EXP: https://doi.org/10.1016/j.jco.2022.101727+  * [[https://doi.org/10.1016/j.jco.2022.101727|Recent publication characterizing the bit-complexity of Partial Differential Equations in terms of discrete complexity classes P/NP/#P/PSPACE/EXP]]
  
-  * Preprint on quantitative coding theory of continuous data: https://arxiv.org/abs/2002.04005+  * [[https://arxiv.org/abs/2002.04005|Preprint on quantitative coding theory of continuous data]]
  
-  * Preprint on semantics and formal verification of programs processing real numbers exactly with partial semantics of comparison: https://arxiv.org/abs/1608.05787+  * [[https://arxiv.org/abs/1608.05787|Preprint on semantics and formal verification of programs processing real numbers exactly with partial semantics of comparison]]
  
-  * Workshop on Computational Knot Theory organized by Alexander Stoimenow+  * [[http://stoimenov.net/stoimeno/homepage/ckt/index.html|Workshop on Computational Knot Theory]] organized by [[http://stoimenov.net/stoimeno/homepage/|Alexander Stoimenow]]
  
-  * Vision paper (in German) about Artificial Intelligence, written jointly with Andrea Reichenberger+  * [[https://www.fiff.de/publikationen/fiff-kommunikation/fk-2022/fk-2022-2/fk-2022-2-content/fk-2-22-p25.pdf|Vision paper (in German) about Artificial Intelligence]], written jointly with [[https://www.fernuni-hagen.de/philosophie/lg1/team/andrea.reichenberger.shtml|Andrea Reichenberger]]