The Microcosm Principle and Concurrency in Coalgebra

Coalgebras are categorical presentations of state-based systems. In investigating parallel composition of coalgebras (realizing concurrency), we observe that the same algebraic theory is interpreted in two different domains in a nested manner, namely: in the category of coalgebras, and in the final coalgebra as an object in it. This phenomenon is what Baez and Dolan have called the microcosm principle, a prototypical example of which is "a monoid in a monoidal category." In this paper we obtain a formalization of the microcosm principle in which such a nested model is expressed categorically as a suitable lax natural transformation. An application of this account is a general compositionality result which supports modular verification of complex systems.

[1]  Peter Dybjer,et al.  Extracting a Proof of Coherence for Monoidal Categories from a Proof of Normalization for Monoids , 1995, TYPES.

[2]  Jan Friso Groote,et al.  A syntactic commutativity format for SOS , 2005, Inf. Process. Lett..

[3]  F. Bartels,et al.  On Generalised Coinduction and Probabilistic Specification Formats , 2004 .

[4]  Andrew M. Pitts,et al.  Category Theory and Computer Science , 1987, Lecture Notes in Computer Science.

[5]  C. A. R. Hoare,et al.  Communicating sequential processes , 1978, CACM.

[6]  Ross Street,et al.  Fibrations and Yoneda's lemma in a 2-category , 1974 .

[7]  John Power,et al.  Discrete Lawvere theories and computational effects , 2006, Theor. Comput. Sci..

[8]  Ichiro Hasuo,et al.  Generic Forward and Backward Simulations , 2006, CONCUR.

[9]  Ichiro Hasuo,et al.  Context-Free Languages via Coalgebraic Trace Semantics , 2005, CALCO.

[10]  M. Barr,et al.  Toposes, Triples and Theories , 1984 .

[11]  John Power,et al.  A Coalgebraic Foundation for Linear Time Semantics , 1999, CTCS.

[12]  Robin Milner,et al.  Communication and concurrency , 1989, PHI Series in computer science.

[13]  Donald Sannella,et al.  Horizontal Composability Revisited , 2006, Essays Dedicated to Joseph A. Goguen.

[14]  Bart Jacobs.  Trace Semantics for Coalgebras , 2004, CMCS.

[15]  Ana Sokolova,et al.  Generic Trace Semantics via Coinduction , 2007, Log. Methods Comput. Sci..

[16]  F. W. Lawvere,et al.  Some algebraic problems in the context of functorial semantics of algebraic theories , 1968 .

[17]  Bernhard Steffen,et al.  Reactive, Generative and Stratified Models of Probabilistic Processes , 1995, Inf. Comput..

[18]  G. M. Kelly.  Elementary observations on 2-categorical limits , 1989, Bulletin of the Australian Mathematical Society.

[19]  Jan A. Bergstra,et al.  Algebra of Communicating Processes with Abstraction , 1985, Theor. Comput. Sci..

[20]  Bart Jacobs,et al.  Structural Induction and Coinduction in a Fibrational Setting , 1998, Inf. Comput..

[21]  Bartek Klin,et al.  From Bialgebraic Semantics to Congruence Formats , 2005, SOS@CONCUR.

[22]  Bart Jacobs,et al.  Categorical Logic and Type Theory , 2001, Studies in logic and the foundations of mathematics.

[23]  Rocco De Nicola,et al.  Testing Equivalences for Processes , 1984, Theor. Comput. Sci..

[24]  Ana Sokolova,et al.  Generic Trace Theory , 2006, CMCS.

[25]  Christel Baier,et al.  CONCUR 2006 - Concurrency Theory, 17th International Conference, CONCUR 2006, Bonn, Germany, August 27-30, 2006, Proceedings , 2006, CONCUR.

[26]  A. Kock.  Monads on symmetric monoidal closed categories , 1970 .

[27]  Bartek Klin,et al.  Bialgebraic Operational Semantics and Modal Logic , 2007, 22nd Annual IEEE Symposium on Logic in Computer Science (LICS 2007).

[28]  H. Keisler,et al.  Handbook of mathematical logic , 1977 .

[29]  F. Bartels.  On generalised coinduction and probabilistic specification formats : Distributive laws in coalgebraic modelling , 2004 .

[30]  James Stuart Tanton,et al.  Encyclopedia of Mathematics , 2005 .

[31]  Eugenio Moggi,et al.  Notions of Computation and Monads , 1991, Inf. Comput..

[32]  James Worrell,et al.  An axiomatics for categories of transition systems as coalgebras , 1998, Proceedings. Thirteenth Annual IEEE Symposium on Logic in Computer Science (Cat. No.98CB36226).

[33]  S. Lack,et al.  The formal theory of monads II , 2002 .

[34]  Jan J. M. M. Rutten,et al.  Universal coalgebra: a theory of systems , 2000, Theor. Comput. Sci..

[35]  John Power,et al.  Coalgebraic semantics for timed processes , 2006, Inf. Comput..

[36]  Gonzalo E. Reyes,et al.  Doctrines in Categorical Logic , 1977 .

[37]  John Power,et al.  Lawvere theories enriched over a general base , 2009 .

[38]  James Dolan,et al.  Higher-Dimensional Algebra III: n-Categories and the Algebra of Opetopes , 1997 .

[39]  Sally Popkorn,et al.  A Handbook of Categorical Algebra , 2009 .

[40]  Gordon D. Plotkin,et al.  Towards a mathematical operational semantics , 1997, Proceedings of Twelfth Annual IEEE Symposium on Logic in Computer Science.

[41]  Bart Jacobs,et al.  A Bialgebraic Review of Deterministic Automata, Regular Expressions and Languages , 2006, Essays Dedicated to Joseph A. Goguen.