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Some recent developments on Shannon's General Purpose Analog Computer
Author(s) -
Silva Graça Daniel
Publication year - 2004
Publication title -
mathematical logic quarterly
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.473
H-Index - 28
eISSN - 1521-3870
pISSN - 0942-5616
DOI - 10.1002/malq.200310113
Subject(s) - computable analysis , computable number , computability , computable function , mathematics , riemann hypothesis , riemann zeta function , transcendental number , analog computer , computation , universality (dynamical systems) , algebra over a field , turing , transcendental function , class (philosophy) , turing machine , pure mathematics , discrete mathematics , computer science , algorithm , artificial intelligence , mathematical analysis , programming language , physics , quantum mechanics , electrical engineering , engineering
This paper revisits one of the first models of analog computation, the General Purpose Analog Computer (GPAC). In particular, we restrict our attention to the improved model presented in [11] and we show that it can be further refined. With this we prove the following: (i) the previous model can be simplified; (ii) it admits extensions having close connections with the class of smooth continuous time dynamical systems. As a consequence, we conclude that some of these extensions achieve Turing universality. Finally, it is shown that if we introduce a new notion of computability for the GPAC, based on ideas from computable analysis, then one can compute transcendentally transcendental functions such as the Gamma function or Riemann's Zeta function. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)
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