Abstract
There is a need for a language able to reconcile the recent upsurge of interest in quantitative methods in the software sciences with logic and set theory that have been used for so many years in capturing the qualitative aspects of the same body of knowledge. Such a lingua franca should be typed, polymorphic, diagrammatic, calculational and easy to blend with traditional notation.
This paper puts forward typed linear algebra (LA) as a candidate notation for such a role. Typed LA emerges from regarding matrices as morphisms of suitable categories whereby traditional linear algebra is equipped with a type system.
In this paper we show typed LA at work in describing weighted (probabilistic) automata. Some attention is paid to the interface between the index-free language of matrix combinators and the corresponding index-wise notation, so as to blend with traditional set theoretic notation.
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References
Abadir, K., Magnus, J.: Matrix algebra. Econometric exercises, vol. 1. Cambridge University Press (2005)
Andova, S., McIver, A., D’Argenio, P.R., Cuijpers, P.J.L., Markovski, J., Morgan, C., Núñez, M. (eds.): Proceedings First Workshop on Quantitative Formal Methods: Theory and Applications. EPTCS, vol. 13 (2009)
Backhouse, R., Michaelis, D.: Exercises in Quantifier Manipulation. In: Uustalu, T. (ed.) MPC 2006. LNCS, vol. 4014, pp. 69–81. Springer, Heidelberg (2006)
Backhouse, R.: Mathematics of Program Construction, 608 pages. Univ. of Nottingham (2004), draft of book in preparation
Barbosa, L.: Towards a Calculus of State-based Software Components. Journal of Universal Computer Science 9(8), 891–909 (2003)
Bird, R., de Moor, O.: Algebra of Programming. Series in Computer Science. Prentice-Hall International (1997)
Bloom, S., Sabadini, N., Walters, R.: Matrices, machines and behaviors. Applied Categorical Structures 4(4), 343–360 (1996)
Bonchi, F., Bonsangue, M., Boreale, M., Rutten, J., Silva, A.: A coalgebraic perspective on linear weighted automata. Information and Computation 211, 77–105 (2012)
Buchholz, P.: Bisimulation relations for weighted automata. Theoretical Computer Science 393(1-3), 109–123 (2008)
Droste, M., Gastin, P.: Weighted automata and weighted logics. In: Kuich, W., Vogler, H., Droste, M. (eds.) Handbook of Weighted Automata. EATCS Monographs in Theoretical Computer Science, ch. 5, pp. 175–211. Springer (2009)
Erwig, M., Kollmansberger, S.: Functional pearls: Probabilistic functional programming in Haskell. J. Funct. Program. 16, 21–34 (2006)
Freyd, P., Scedrov, A.: Categories, Allegories, Mathematical Library, vol. 39. North-Holland (1990)
Gibbons, J., Hinze, R.: Just do it: simple monadic equational reasoning. In: Proceedings of the 16th ACM SIGPLAN International Conference on Functional Programming, ICFP 2011, pp. 2–14. ACM, New York (2011)
Hehner, E.: A probability perspective. Formal Aspects of Computing 23, 391–419 (2011)
Larsen, K., Skou, A.: Bisimulation through probabilistic testing. Inf. Comput. 94(1), 1–28 (1991)
Macedo, H.D., Oliveira, J.N.: Matrices As Arrows! A Biproduct Approach to Typed Linear Algebra. In: Bolduc, C., Desharnais, J., Ktari, B. (eds.) MPC 2010. LNCS, vol. 6120, pp. 271–287. Springer, Heidelberg (2010)
Macedo, H.D., Oliveira, J.N.: Do the middle letters of “OLAP” stand for linear algebra (“LA”)? Technical Report TR-HASLab:04:2011, INESC TEC and University of Minho, Gualtar Campus, Braga (2011a)
Macedo, H.D., Oliveira, J.N.: Towards Linear Algebras of Components. In: Barbosa, L.S. (ed.) FACS 2010. LNCS, vol. 6921, pp. 300–303. Springer, Heidelberg (2010)
Macedo, H.D., Oliveira, J.N.: Typing linear algebra: A biproduct-oriented approach (2011) (accepted for publication in SCP)
MacLane, S.: Categories for the Working Mathematician. Springer, New-York (1971)
MacLane, S., Birkhoff, G.: Algebra. AMS Chelsea (1999)
McIver, A., Morgan, C.: Abstraction, Refinement and Proof For Probabilistic Systems. Monographs in Computer Science. Springer (2005)
Oliveira, J.: Towards a linear algebra of programming. Accepted for publication in Formal Aspects of Computing (2012)
Schmidt, G.: Relational Mathematics. Encyclopedia of Mathematics and its Applications, vol. 132. Cambridge University Press (November 2010)
Sernadas, A., Ramos, J., Mateus, P.: Linear algebra techniques for deciding the correctness of probabilistic programs with bounded resources. Tech. rep., SQIG - IT and IST - TU Lisbon, 1049-001 Lisboa, Portugal (2008), short paper presented at LPAR 2008, Doha, Qatar, November 22-27
Sokolova, A.: Coalgebraic Analysis of Probabilistic Systems. Ph.D. dissertation, Tech. Univ. Eindhoven, Eindhoven, The Netherlands (2005)
SQIG-Group: LAP: Linear algebra of bounded resources programs, iT & Tech. Univ. Lisbon (2011), http://sqig.math.ist.utl.pt/work/LAP
Trčka, N.: Strong, weak and branching bisimulation for transition systems and Markov reward chains: A unifying matrix approach. In: [2], pp. 55–65
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Oliveira, J.N. (2012). Typed Linear Algebra for Weigthed (Probabilistic) Automata. In: Moreira, N., Reis, R. (eds) Implementation and Application of Automata. CIAA 2012. Lecture Notes in Computer Science, vol 7381. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-31606-7_5
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