A Robust Class of Linear Recurrence Sequences

We introduce a subclass of linear recurrence sequences which we call poly-rational sequences because they are denoted by rational expressions closed under sum and product. We show that this class is robust by giving several characterisations: polynomially ambiguous weighted automata, copyless cost-register automata, rational formal series, and linear recurrence sequences whose eigenvalues are roots of rational numbers.

[1]  Joël Ouaknine,et al.  On the Positivity Problem for Simple Linear Recurrence Sequences, , 2013, ICALP.

[2]  Marcel Paul Schützenberger,et al.  On the Definition of a Family of Automata , 1961, Inf. Control..

[3]  Rajeev Alur,et al.  Regular Functions and Cost Register Automata , 2013, 2013 28th Annual ACM/IEEE Symposium on Logic in Computer Science.

[4]  Terence Tao.  Structure and Randomness: Pages from Year One of a Mathematical Blog , 2008 .

[5]  Michaël Cadilhac,et al.  Weak Cost Register Automata are Still Powerful , 2018, DLT.

[6]  Joël Ouaknine,et al.  Ultimate Positivity is Decidable for Simple Linear Recurrence Sequences , 2013, ICALP.

[7]  Cristian Riveros,et al.  Copyless Cost-Register Automata: Structure, Expressiveness, and Closure Properties , 2016, STACS.

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

[9]  Cristian Riveros,et al.  Maximal Partition Logic: Towards a Logical Characterization of Copyless Cost Register Automata , 2015, CSL.

[10]  Stephan Kreutzer,et al.  Quantitative Monadic Second-Order Logic , 2013, 2013 28th Annual ACM/IEEE Symposium on Logic in Computer Science.

[11]  Ronald L. Graham,et al.  Concrete mathematics - a foundation for computer science (2. ed.) , 1994 .

[12]  James Worrell,et al.  Probabilistic Automata of Bounded Ambiguity , 2020, CONCUR.

[13]  Mireille Bousquet-Mélou,et al.  Algebraic Generating Functions in Enumerative Combinatorics and Context-Free Languages , 2005, STACS.

[14]  Arnaldo Vieira Moura,et al.  On the Termination of Linear and Affine Programs over the Integers , 2014, ArXiv.

[15]  S. Akshay,et al.  Complexity of Restricted Variants of Skolem and Related Problems , 2017, MFCS.

[16]  Ines Klimann,et al.  Deciding unambiguity and sequentiality from a finitely ambiguous max-plus automaton , 2004, Theor. Comput. Sci..

[17]  Cristian Riveros,et al.  Pumping Lemmas for Weighted Automata , 2018, STACS.

[18]  Joël Ouaknine,et al.  On linear recurrence sequences and loop termination , 2015, SIGL.

[19]  Daniel Kirsten,et al.  Deciding Unambiguity and Sequentiality of Polynomially Ambiguous Min-Plus Automata , 2009, STACS.