Continuity of operators on continuous and discrete time streams

Abstract We consider the semantics of networks processing streams of data from a complete metric space. We consider two types of data streams: those based on continuous time (used in networks of physical components and analog devices), and those based on discrete time (used in concurrent algorithms). The networks are both governed by global clocks and together model a huge range of systems. Previously, we have investigated these two types of networks separately. Here we combine their study in a unified theory of stream transformers, given as fixed points of equations. We begin to develop this theory by using the standard mathematical techniques of topology to prove certain computationally desirable properties of these semantic functions, notably continuity, which is significant for models of a physical system, according to Hadamard’s principle.

[1]  J. V. Tucker,et al.  Abstract versus concrete computation on metric partial algebras , 2001, TOCL.

[2]  J. V. Tucker,et al.  Unifying computers and dynamical systems using the theory of synchronous concurrent algorithms , 2009, Appl. Math. Comput..

[3]  George Finlay Simmons,et al.  Introduction to Topology and Modern Analysis , 1963 .

[4]  Ludomir Newelski,et al.  Logic Colloquium 2005 (Lecture Notes in Logic) , 2007 .

[5]  J. V. Tucker,et al.  Computable total functions on metric algebras, universal algebraic specifications and dynamical systems , 2005, J. Log. Algebraic Methods Program..

[6]  Wayne C. Myrvold,et al.  Computability in Quantum Mechanics , 1995 .

[7]  S. C. Kleene,et al.  Introduction to Metamathematics , 1952 .

[8]  F. Friedlander.  LA THEÓRIE DES ÉQUATIONS AUX DÉRIVÉES PARTIELLES , 1967 .

[9]  R. Courant,et al.  Methods of Mathematical Physics , 1962 .

[10]  Viggo Stoltenberg-Hansen,et al.  Streams, Stream Transformers and Domain Representations , 1998, Prospects for Hardware Foundations.

[11]  Eduardo D. Sontag,et al.  Mathematical Control Theory: Deterministic Finite Dimensional Systems , 1990 .

[12]  J. Hadamard,et al.  Lectures on Cauchy's Problem in Linear Partial Differential Equations , 1924 .

[13]  Min Chen,et al.  Constructive Volume Geometry , 2000, Comput. Graph. Forum.

[14]  Viggo Stoltenberg-Hansen,et al.  Concrete Models of Computation for Topological Algebras , 1999, Theor. Comput. Sci..

[15]  Kenneth Harold Anthony Johnson,et al.  The algebraic specification of spatial data types with applications to constructive volume geometry , 2007 .

[16]  J. V. Tucker,et al.  Computable Functions on Stream Algebras , 1995 .

[17]  Gianpiero Cattaneo,et al.  The Foundational Debate: Complexity and Constructivity in Mathematics and Physics , 1995 .

[18]  Robert Stephens,et al.  A survey of stream processing , 1997, Acta Informatica.

[19]  Stefan Friedrich,et al.  Topology , 2019, Arch. Formal Proofs.

[20]  Nick D. James,et al.  A Class of Contracting Stream Operators , 2013, Comput. J..

[21]  T. Hughes,et al.  Signals and systems , 2006, Genome Biology.

[22]  J. V. Tucker,et al.  Computability of analog networks , 2007, Theor. Comput. Sci..

[23]  Anil Nerode,et al.  Models for Hybrid Systems: Automata, Topologies, Controllability, Observability , 1992, Hybrid Systems.

[24]  James S. Small.  The Analogue Alternative: The Electronic Analogue Computer in Britain and the USA, 1930-1975 , 2001 .

[25]  J. V. Tucker,et al.  Computable functions and semicomputable sets on many-sorted algebras , 2001, Logic in Computer Science.

[26]  Viggo Stoltenberg-Hansen,et al.  Domain representations of partial functions, with applications to spatial objects and constructive volume geometry , 2002, Theor. Comput. Sci..

[27]  W. Rudin.  Principles of mathematical analysis , 1964 .

[28]  M. Beeson.  Foundations of Constructive Mathematics , 1985 .

[29]  Kellen Petersen August.  Real Analysis , 2009 .

[30]  Charles Care.  Technology for Modelling - Electrical Analogies, Engineering Practice, and the Development of Analogue Computing , 2010, History of Computing.