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Multilinear representations of free pros. (English) Zbl 07273642
Summary: We describe a structure of pro on hypermatrices. This structure allows us to define multilinear representations of pros and in particular of free pros. As an example of applications, we investigate the relations of the representations of pros with the theory of automata.
MSC:
15A69 Multilinear algebra, tensor calculus
03D05 Automata and formal grammars in connection with logical questions
11B85 Automata sequences
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[1] Mac Lane, S., Categorical algebra, Bull Amer Math Soc, 71, 40-106 (1965) · Zbl 0161.01601
[2] Mac Lane, S., Natural associativity and commutativity, Rice Univ Studies, 49, 28-46 (1963) · Zbl 0244.18008
[3] Boardman, JM; Vogt, RM., Homotopy-everything H-spaces, Bull Amer Math Soc, 74, 1117-1122 (1968) · Zbl 0165.26204
[4] Boardman, JM; Vogt, RM., Homotopy invariant algebraic structures on topological spaces (1973), Berlin: Springer-Verlag
[5] Bultel, JP; Giraudo, S., Combinatorial Hopf algebras from pros, J Algebr Comb, 44, 2, 455-493 (2016) · Zbl 1351.05233
[6] Giraudo, S.Colored operads, series on colored operads, and combinatorial generating systems. Preprint arXiv:1605.04697v1. · Zbl 1412.68142
[7] Eckmann, B.; Hilton, PJ., Group-like structures in general categories. I. Multiplications and comultiplications, Math Ann, 145, 3, 227-255 (1962) · Zbl 0099.02101
[8] Chomsky, N., Three models for the description of language, IEEE Trans Inf Theory, 2, 113-124 (1956) · Zbl 0156.25401
[9] Chomsky, N, Schützenberger, MP.The algebraic theory of context free languages. In: Braffort P, Hirschberg D, editors. Computer programming and formal languages. Amsterdam: North Holland; 1963. p. 118-161. · Zbl 0148.00804
[10] Kleene, SC.Representation of events in nerve nets and finite automata. In: Shannon CE, McCarthy J, editors. Automata studies. Princeton: Princeton University Press; 1956. p. 3-41.
[11] McCulloch, WS; Pitt, E., A logical calculus of the ideas immanent in nervous activity, Bull Math Sci, 5, 541-544 (1943)
[12] Rabin, MO; Scott, D., Finite automata and their decision problems, IBM jour, Res Dev, 3, 2, 114-125 (1959)
[13] Schützenberger, MP., On the definition of a family of automata, Inf Control, 4, 245-270 (1961) · Zbl 0104.00702
[14] Fülöp, Z, Vogler, H.Weighted tree automata and tree transducers. In: Handbook of weighted automata. Droste-Kuich-Vogler, editor. Berlin: Springer; 2009. p. 313-404.
[15] Drewes, F.Lecture notes on tree automata. In: Chapter I bottom-up and top-down tree automata by Adam Sernheim. Frank Drewes, editor. Umea: Umea University. p. 1-7.
[16] Lodaya, K, Weil, P.A Kleene iteration for parallelism. International Conference on Foundations of Software Technology and Theoretical Computer Science; 1998. p. 355-366. · Zbl 0932.68060
[17] Lodaya, K.; Weil, P., Series-parallel languages and the bounded-width property, Theoret Comput Sci, 237, 1-2, 347-380 (2000) · Zbl 0939.68042
[18] Bedon, N.Logic and branching automata. Proceedings of MFCS 2013: 38nd International Symposium on Mathematical Foundations of Computer Science, Klosterneuburg, Austria. Krishnendu Chatterjee and Jiri Sgall Eds. vol. 8087 of Lecture Notes in Computer Science; 2013. p. 123-134. · Zbl 1398.68297
[19] Heindel, T.A Myhill-Nerode theorem beyond trees and forests via finite syntactic categories internal to monoids; 2017. Available from: http://www.informatik.uni-leipzig.de/∼heindel/MyhillNerodeFoSSaCS18.pdf
[20] Bossut, F.Rationalité et reconnaissabilité dans les graphes [PhD thesis]. Université Lille: Sciences et Technologies; 1986.
[21] Bossut, F.; Dauchet, M.; Warin, B., A Kleene theorem for a class of planar acyclic graphs, Inform Comput, 117, 2, 251-265 (1995) · Zbl 0826.68089
[22] Temperley, N.; Lieb, E., Relations between the ‘Percolation’ and ‘Colouring’ problem and other graph-theoretical problems associated with regular planar lattices: some exact results for the ‘Percolation’ problem, Proc R Soc Ser, A322, 251-280 (1971) · Zbl 0211.56703
[23] Ekert, A.Introduction to quantum computation. In: Fundamentals of quantum informations. Dieter Heiss, editor. Berlin: Springer-Verlag; 2002. · Zbl 1079.81510
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