## Coinductive Counting With Weighted Automata (2002)

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Citations: | 4 - 0 self |

### BibTeX

@MISC{Rutten02coinductivecounting,

author = {J. J. M. M. Rutten},

title = {Coinductive Counting With Weighted Automata},

year = {2002}

}

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### Abstract

A general methodology is developed to compute the solution of a wide variety of basic counting problems in a uniform way: (1) the objects to be counted are enumerated by means of an infinite weighted automaton; (2) the automaton is reduced by means of the quantitative notion of stream bisimulation; (3) the reduced automaton is used to compute an expression (in terms of stream constants and operators) that represents the stream of all counts.

### Citations

365 |
Combinatorial enumeration
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- 1983
(Show Context)
Citation Context ...zations (addressing the foundational question: what is a combinatorial structure? How do we “specify” it? What is the relation of such specifications to counting?) being introduced by Goulden-Jackson =-=[GJ83]-=-, Flajolet-Sedgewick [FS93, FS01], Joyal [BLL98], Stanley [Sta97, Sta99], and several others. Here we add one more formal system to the list, called the method of coinductive counting. From the enumer... |

323 | Universal coalgebra: A theory of systems - Rutten - 1996 |

244 | A tutorial on (co)algebras and (co)induction - Jacobs, Rutten - 1996 |

183 |
Combinatorial Species and Tree-Like Structures, volume 67 of Encyclopedia of Mathematics and its Applications
- Bergereon, Labelle, et al.
- 1998
(Show Context)
Citation Context ...what is a combinatorial structure? How do we “specify” it? What is the relation of such specifications to counting?) being introduced by Goulden-Jackson [GJ83], Flajolet-Sedgewick [FS93, FS01], Joyal =-=[BLL98]-=-, Stanley [Sta97, Sta99], and several others. Here we add one more formal system to the list, called the method of coinductive counting. From the enumerative point of view, it makes it possible to der... |

171 |
Rational Series and Their Languages
- Berstel, Reutenauer
- 1988
(Show Context)
Citation Context ...(u =1) �� u �� �� ··· Stirling numbers (2nd) 3 �� 1 �� �� ··· Bell numbers (u =1) 3 Figure 1: Representations of special numbers 28ssimpler and more uniform. A basic reference on weighted automata is =-=[BR88]-=-. Most of the Sections 2 through 4 can be fairly straightforwardly generalised from streams to so-called multivariate streams or, more generally, to formal power series in many non-commutative variabl... |

146 | Combinatorial aspects of continued fractions
- Flajolet
- 1980
(Show Context)
Citation Context ...s might be used, though, at the point where our methods stops: many of the obtained stream expressions are suited for further analytical treatment. The use of continued fractions has been inspired by =-=[Fla80]-=-; see also [GJ83, Chapter 5.2]. The formal treatment of such continued fractions seems somewhat easier in the present setting of coinductive stream calculus. Also, some of the combinatorial interpreta... |

58 |
ECO: a methodology for the enumeration of combinatorial objects
- Barcucci, Lungo, et al.
- 1999
(Show Context)
Citation Context ...erties expressed as a kind of domain equation (such as T =1+T × X × T for binary trees), as present in for instance [FS93] and also [BLL98]. Also the work of the Florence school of Pinzani and others =-=[BLPP99]-=- is related to this point. (ii) The issue of minimization of weighted automata has of course only been touched upon. In the presented examples, there usually was an obvious minimized candidate, but a ... |

54 | Behavioural Differential Equations: a Coinductive Calculus
- Rutten
- 2003
(Show Context)
Citation Context ...f related (and future) work. 2 Basic facts from stream calculus We present the basic facts of a coinductive calculus of streams, by repeating parts of earlier work on formal power series and streams (=-=[Rut00a]-=- and [Rut01]). Since the concepts of bisimulation and coinduction may be new to some readers, a number of examples will be treated as well. The set of all streams is defined by IR ω = {σ | σ : {0, 1, ... |

29 |
Analitic combinatorics: functional equations, rational and algebraic functions
- Flajolet, Sedgewick
(Show Context)
Citation Context ...s worthwhile to develop also the multivariate case in some detail. (iv) In one case (Section 8), we have distinguished between rational and algebraic streams. Notably in the work of Flajolet, such as =-=[FS01]-=-, much more has already been said about the classification of streams (there rather: generating functions) in analytical terms. Looking at the various weighted automata that we have encountered sofar,... |

23 | Concrete mathematics, second edition - Graham, Knuth, et al. - 1994 |

20 | Escard#o, Calculus in coinductive form
- Pavlovic, H
- 1998
(Show Context)
Citation Context ...1+X ) • Forafunctionf :IR→ IR that is analytical in 0, the stream T (f) of Taylor coefficients of f(x) can be defined, coinductively, by the following behavioural differential equation over IR ω (cf. =-=[PE98]-=-): behavioural differential equation initial value name T (f) ′ = T (df /dx) T (f)(0) = f(0) Taylor series (Here df /dx denotes the analytical derivative of the function f(x).) One can readily check t... |

20 |
Elements of stream calculus (an extensive exercise in coinduction
- Rutten
(Show Context)
Citation Context ...re queens at level k. Thuswehave translated the original counting problem into a question about streams and their representation by automata, thereby entering the coinductive world of stream calculus =-=[Rut01]-=-. A crucial ingredient of stream calculus is the notion of stream bisimulation, with the help of which the above automaton can be simplified by identifying all equivalent states as follows. Every dron... |

17 |
The average case analysis of algorithms: Counting and generating functions
- Flajolet, Sedgewick
(Show Context)
Citation Context ...ly related to an alternative approach to counting, which is based on structural properties expressed as a kind of domain equation (such as T =1+T × X × T for binary trees), as present in for instance =-=[FS93]-=- and also [BLL98]. Also the work of the Florence school of Pinzani and others [BLPP99] is related to this point. (ii) The issue of minimization of weighted automata has of course only been touched upo... |

7 | power series, and coinduction: Taking input derivatives seriously (extended abstract - Automata |

5 | Rational series and their - Berstel, Reutenauer |

4 | Behavioural di®erential equations: a coinductive calculus of streams, automata, and power series - Rutten |