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Smooth coalgebra: testing vector analysis

Published online by Cambridge University Press:  14 December 2015

DUSKO PAVLOVIC  and
BERTFRIED FAUSER
DUSKO PAVLOVIC
Affiliation:
Information and Computer Sciences Department, University of Hawaii, HI 96822 Honolulu, USA Email: [email protected]
BERTFRIED FAUSER
Affiliation:
Mathematisch- Naturwissenschaftliche Sektion, University of Konstanz, 78464 Konstanz, Germany Email: [email protected]
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Abstract

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Processes are often viewed as coalgebras, with the structure maps specifying the state transitions. In the simplest case, the state spaces are discrete, and the structure map simply takes each state to the next states. But the coalgebraic view is also quite effective for studying processes over structured state spaces, e.g. measurable, or continuous. In the present paper, we consider coalgebras over manifolds. This means that the captured processes evolve over state spaces that are not just continuous, but also locally homeomorphic to normed vector spaces, and thus carry a differential structure. Both dynamical systems and differential forms arise as coalgebras over such state spaces, for two different endofunctors over manifolds. A duality induced by these two endofunctors provides a formal underpinning for the informal geometric intuitions linking differential forms and dynamical systems in the various practical applications, e.g. in physics. This joint functorial reconstruction of tangent bundles and cotangent bundles uncovers the universal properties and a high-level view of these fundamental structures, which are implemented rather intricately in their standard form. The succinct coalgebraic presentation provides unexpected insights even about the situations as familiar as Newton's laws.

Type
Paper
Information
Mathematical Structures in Computer Science , Volume 27 , Special Issue 7: Special Issue: Coalgebraic Logic , October 2017 , pp. 1195 - 1235
Copyright
Copyright © Cambridge University Press 2015 

References

Abraham, R., Marsden, J. and Ratiu, T. (1988). Manifolds, Tensor Analysis, and Applications. Applied Mathematical Sciences, volume 75, Springer-Verlag, New York.CrossRefGoogle Scholar
Aczel, P. (1988). Non-Well-Founded Sets. Number 14 in Lecture Notes. Center for the Study of Language and Information, Stanford University.Google Scholar
Artin, M., Grothendieck, A. and Verdier, J. (1972). Théorie des topos et cohomologie étalé des schémas, Lecture Notes in Mathematics, volume 269, 270, 305, Springer-Verlag.Google Scholar
Barr, M. (1979). *-Autonomous Categories, Lecture Notes in Mathematics, 752, Springer-Verlag.CrossRefGoogle Scholar
Barr, M. (1996). *-autonomous categories, revisited. Journal of Pure and Applied Algebra 111 (1) 1–20.Google Scholar
Barr, M. (2006). The chu construction: history of an idea. Theory and Application of Categories 17 (1) 10–16.Google Scholar
Bernardo, M., De Nicola, R. and Loreti, M. (2012). Revisiting trace and testing equivalences for nondeterministic and probabilistic processes. In: Birkedal, L. (ed.) Proceedings of FOSSACS 2012. Lecture Notes in Computer Science 7213, Springer, Berlin, 195–209.Google Scholar
Blute, R., Cockett, J.R.B. and Seely, R.A.G. (2006). Differential categories. Mathematical Structures in Computer Science 16 (6) 1049–1083.CrossRefGoogle Scholar
Blute, R., Cockett, J.R.B. and Seely, R.A.G. (2009). Cartesian differential categories. Theory and Applications of Categories 22 (23) 622–672.Google Scholar
Caenepeel, S., Militaru, G. and Zhu, S. (2002). Frobenius and Separable Functors for Generalized Module Categories and Nonlinear Equations, Springer-Verlag.CrossRefGoogle Scholar
Cockett, J.R.B. and Cruttwell, G.S.H. (2014). Differential structure, tangent structure, and SDG. Applied Categorical Structures 22 (2) 331–417.Google Scholar
Cockett, J.R.B. and Seely, R.A.G. (2011). The Faà di Bruno construction. Theory and Application of Categories 25 (15) 393–425.Google Scholar
De Nicola, R. and Hennessy, M. (1984). Testing equivalences for processes. Theoretical Computer Science 34 (1–2) 83–133.CrossRefGoogle Scholar
Dold, A. (1995). Lectures on Algebraic Topology, Classics in Mathematics, Springer, Berlin, Heidelberg.Google Scholar
Frölicher, A. and Kriegl, A. (1988). Linear Spaces and Differentiation Theory, Pure and Applied Mathematics, Wiley.Google Scholar
Grothendieck, A. (1973). Topological Vector Spaces, Notes on mathematics and its applications, Gordon and Breach.Google Scholar
Haghverdi, E., Tabuada, P. and Pappas, G.J. (2005). Bisimulation relations for dynamical, control, and hybrid systems. Theoretical Computer Science 342 (2–3) 229–261.Google Scholar
Harel, D., Tiuryn, J. and Kozen, D. (2000). Dynamic Logic, MIT Press, Cambridge, MA, USA.CrossRefGoogle Scholar
Hasuo, I., Jacobs, B. and Sokolova, A. (2007). Generic trace semantics via coinduction. Logical Methods in Computer Science 3 (4).Google Scholar
Jackson, A. (2004). Comme Appelé du Néant - as if summoned from the void: The life of Alexandre Grothendieck. Notices of the AMS 51 (4,10) 1038–1056, 1196–1212.Google Scholar
Jacobs, B. and Sokolova, A. (2010). Exemplaric expressivity of modal logics. Journal of Logic and Computation 20 (5) 1041–1068.Google Scholar
Jubin, B. t. M. (2012). The Tangent Functor Monad and Foliations. PhD thesis, University of Berkeley, Berkeley. arxiv:1401.0940.Google Scholar
Kapulkin, K., Kurz, A. and Velebil, J. (2012). Expressiveness of positive coalgebraic logic. In: Bolander, T., Braüner, T., Ghilardi, S. and Moss, L. S. (eds.) Advances in Modal Logic, College Publications 368–385.Google Scholar
Klin, B. (2007a). Bialgebraic operational semantics and modal logic. In: LICS, IEEE Computer Society 336–345.Google Scholar
Klin, B. (2007b). Coalgebraic modal logic beyond sets. Electronic Notes in Theoretical Computer Science 173 177–201.Google Scholar
Klin, B. (2009). Bialgebraic methods and modal logic in structural operational semantics. Information and Computation 207 (2) 237–257.Google Scholar
Kock, A. (2006). Synthetic Differential Geometry , volume 333 of London Mathematical Society Lecture Note Series. Cambridge University Press.Google Scholar
Kolar, I., Michor, P. and Slovak, J. (1993). Natural Operations in Differential Geometry, Springer-Verlag.CrossRefGoogle Scholar
Kupke, C., Kurz, A. and Pattinson, D. (2004). Algebraic semantics for coalgebraic logics. Electronic Notes in Theoretical Computer Science 106 219–241.CrossRefGoogle Scholar
Kupke, C., Kurz, A. and Pattinson, D. (2005). Ultrafilter extensions for coalgebras. In: et al, J. L. F. (eds.) CALCO. Lecture Notes in Computer Science 3629, Springer, Berlin, 263–277.Google Scholar
Kurz, A. and Rosický, J. (2012). Strongly complete logics for coalgebras. Logical Methods in Computer Science 8 (3) 1–32.Google Scholar
Lawvere, F.W. (1980). Toward the description in a smooth topos of the dynamically possible motions and deformations of a continuous body. Cahiers de Topologie et Gomtrie Diffrentielle Catgoriques 21 (4) 377–392.Google Scholar
Lawvere, W.F. (1975). Continuously variable sets: Algebraic geometry = geometric logic. In: Logic Colloquium '73 (Bristol, 1973). Studies in Logic and the Foundations of Mathematics 80, North-Holland, Amsterdam 135–156.Google Scholar
Lawvere, W.F. (1986). Categories of spaces may not be generalized spaces as exemplified by directed graphs. Revista Colombiana de Matematicas 20 179–186.Google Scholar
Leray, J. (1945). Sur la forme des espaces topologiques et sur les pointes fixes des représentations. Journal de Mathématiques Pures et Appliquées 9 95–249.Google Scholar
Mac Lane, S. and Moerdijk, I. (1992). Sheaves in Geometry and Logic: A First Introduction to Topos Theory, Universitext. Springer-Verlag, New York.Google Scholar
Mackey, G. (1945). On infinite dimensional vector spaces. Transactions of the American Mathematical Society 51 (57) 155–207.CrossRefGoogle Scholar
Mesablishvili, B. and Wisbauer, R. (2010). Galois functors and entwining structures. J. Algebra 324 464–506.CrossRefGoogle Scholar
Mesablishvili, B. and Wisbauer, R. (2011). Bimonads and Hopf monads on categories. Journal of K-Theory 7 (2) 349–388.Google Scholar
Milner, R. (1989). Communication and Concurrency, International Series in Computer Science, Prentice Hall, London.Google Scholar
Modungo, M. and Stefani, G. (1978). Some results on second tangent and cotangent spaces. Quaderni dell'Instituto di Matematica dell' Universitá di Lecce Q. 16 1–23.Google Scholar
Moerdijk, I. (2002). Monads on tensor categories. Journal of Pure and Applied Algebra 168 (2–3) 189–208.Google Scholar
Moore, E. (1956). Gedanken experiments on sequential machines. In: Shannon, C. E. and McCarthy, J. (eds.) Automata Studies, Annals of Mathematics Studies, Princeton, Princeton University Press 129–153.Google Scholar
Pavlovic, D. (1995). Convenient categories of processes and simulations I: Modulo strong bisimilarity. In: Pitt, D., Rydeheard, D.E. and Johnstone, P. (eds.) Category Theory and Computer Science '95. Lecture Notes in Computer Science 953, Springer-Verlag, Berlin, 3–24.Google Scholar
Pavlovic, D. (1997). Chu i: Cofree equivalences, dualities and *-autonomous categories. Mathematical Structures in Computer Science 7 (2) 49–73.Google Scholar
Pavlović, D. (2001). Towards semantics of self-adaptive software. In Robertson, P., Shrobe, H. and Laddaga, R. (eds.) Proceedings of the Workshop on Self-Adaptive Software. Lecture Notes in Computer Science 1936, Springer-Verlag, Berlin, 50–64.Google Scholar
Pavlovic, D. (2013). Monoidal computer I: Basic computability by string diagrams. Information and Computation 226 94–116.Google Scholar
Pavlović, D. and Escardó, M. (1998). Calculus in coinductive form. In: Pratt, V. (eds.) Proceedings 13th Annual IEEE Symposium on Logic in Computer Science, IEEE Computer Society 408–417.Google Scholar
Pavlovic, D., Mislove, M. and Worrell, J. (2006). Testing semantics: Connecting processes and process logic. In: Johnson, M. and Vene, V. (eds.) AMAST 2006. Springer-Verlag Lecture Notes in Computer Science 4019 308–322 Berlin. (The version with the Appendix is available from dusko.org).Google Scholar
Pavlovic, D., Pepper, P. and Smith, D.R. (2008). Evolving specification engineering. In: Meseguer, J. and Rosu, G. (eds.) Proceedings of AMAST 2008. Springer Verlag Lecture Notes in Computer Science 5140 299–314.Google Scholar
Pavlovic, D., Pepper, P. and Smith, D.R. (2010). Formal derivation of concurrent garbage collectors. In: Desharnais, J. (ed.) Proceedings of MPC 2010. Springer Verlag Lecture Notes in Computer Science 6120 353–376. full version arxiv.org:1006.4342.Google Scholar
Pavlovic, D. and Smith, D.R. (2001). Composition and refinement of behavioral specifications. In: The 16th International Conference on Automated Software Engineering, IEEE.Google Scholar
Pavlovic, D. and Smith, D. R. (2002). Guarded transitions in evolving specifications. In: Kirchner, H. and Ringeissen, C. (eds.) Proceedings of AMAST 2002. Springer Verlag Lecture Notes in Computer Science 2422 411–425.CrossRefGoogle Scholar
Pratt, V.R. (1976). Semantical consideration on Floyd-Hoare logic. In: Proceeding of the 17th Annual Symposium on Foundations of Computer Science, IEEE 109–121.Google Scholar
Rosický, J. (1984). Abstract tangent functors. Diagrammes 12 (3) JR1–JR11.Google Scholar
Rutten, J. (2000). Universal coalgebra: A theory of systems. Theoretical Computer Science 249 3–80.Google Scholar
Schwartz, L. (1950). Théorie des Distributions, Hermann, Paris.Google Scholar
Tennison, B. (1975). Sheaf Theory, Cambridge Monographs on Physics, Cambridge University Press.Google Scholar
Turi, D. and Plotkin, G.D. (1997). Towards a mathematical operational semantics. In: LICS, IEEE Computer Society, 280–291.Google Scholar
van Glabbeek, R.J. (1990). The linear time-branching time spectrum (extended abstract). In: Baeten, J.C.M. and Klop, J.W. (eds.) Proceedings of CONCUR '90. Springer Lecture Notes in Computer Science 458 278–297.Google Scholar
van Glabbeek, R.J. (1993). The linear time - branching time spectrum II. In: Best, E. (eds.) Proceedings of CONCUR '93. Springer Lecture Notes in Computer Science 715.Google Scholar