A Final Coalgebra for k-regular Sequences

We study k-regular sequences from a coalgebraic perspective. Building on the observation that the set of streams over a semiring S can be turned into a final coalgebra, we obtain characterizations of k-regular sequences in terms of finite weighted automata, finite systems of behavioral differential equations, and recognizable power series. The latter characterization is obtained via an isomorphism of final coalgebras based on the k-adic numeration system.

[1]  M. Douglas McIlroy,et al.  The music of streams , 2001, Inf. Process. Lett..

[2]  Jeffrey Shallit,et al.  Automatic Sequences by Jean-Paul Allouche , 2003 .

[3]  Jeffrey Shallit,et al.  The ring of k-regular sequences, II , 2003, Theor. Comput. Sci..

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

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

[6]  Yossi Moshe,et al.  On some questions regarding k -regular and k -context-free sequences , 2008, Theor. Comput. Sci..

[7]  Robert L. Constable,et al.  Infinite Objects in Type Theory , 1986, LICS.

[8]  Ronald L. Graham,et al.  Concrete mathematics - a foundation for computer science , 1991 .

[9]  Jan J. M. M. Rutten,et al.  On the Final Coalgebra of Automatic Sequences , 2012, Logic and Program Semantics.

[10]  Marcello M. Bonsangue,et al.  Defining Context-Free Power Series Coalgebraically , 2012, CMCS.

[11]  Sanjay V. Rajopadhye,et al.  Verification of Systolic Arrays: A Stream Function Approach , 1986, ICPP.

[12]  Zoltán Ésik,et al.  The Category of Simulations for Weighted Tree Automata , 2011, Int. J. Found. Comput. Sci..

[13]  Joost Winter.  QStream: A Suite of Streams , 2013, CALCO.

[14]  Alexandra Silva,et al.  Generalizing determinization from automata to coalgebras , 2013, Log. Methods Comput. Sci..

[15]  Jan J. M. M. Rutten,et al.  Behavioural differential equations: a coinductive calculus of streams, automata, and power series , 2003, Theor. Comput. Sci..

[16]  Ralf Hinze,et al.  Concrete stream calculus: An extended study , 2010, Journal of Functional Programming.

[17]  C. Reutenauer,et al.  Noncommutative Rational Series with Applications , 2010 .

[18]  J. Shallit,et al.  Automatic Sequences: Contents , 2003 .

[19]  Jason P. Bell.  On the values attained by a k-regular sequence , 2005, Adv. Appl. Math..

[20]  Michael Hauhs,et al.  Algebrais-Coalgebraic recursion theory of histroy-dependent dynamical system models , 2014 .

[21]  Jean Berstel.  Review of "Automatic sequences: theory, applications, generalizations" by Jean-Paul Allouche and Jeffrey Shallit. Cambridge University Press. , 2004, SIGA.

[22]  Ralf Stephan.  Divide-and-conquer generating functions. Part I. Elementary sequences , 2003 .

[23]  Jeffrey Shallit,et al.  Automatic Sequences: Theory, Applications, Generalizations , 2003 .

[24]  Jeffrey Shallit,et al.  The Ring of k-Regular Sequences , 1990, Theor. Comput. Sci..

[25]  Lawrence S. Moss,et al.  Automatic Sequences and Zip-Specifications , 2012, 2012 27th Annual IEEE Symposium on Logic in Computer Science.

[26]  Ronald L. Graham,et al.  Concrete Mathematics, a Foundation for Computer Science , 1991, The Mathematical Gazette.