## Productivity of Stream Definitions (2008)

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Citations: | 14 - 3 self |

### BibTeX

@MISC{Endrullis08productivityof,

author = {Jörg Endrullis and Clemens Grabmayer and Dimitri Hendriks and Ariya Isihara and Jan Willem Klop},

title = {Productivity of Stream Definitions},

year = {2008}

}

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### Abstract

We give an algorithm for deciding productivity of a large and natural class of recursive stream definitions. A stream definition is called ‘productive’ if it can be evaluated continually in such a way that a uniquely determined stream in constructor normal form is obtained as the limit. Whereas productivity is undecidable for stream definitions in general, we show that it can be decided for ‘pure’ stream definitions. For every pure stream definition the process of its evaluation can be modelled by the dataflow of abstract stream elements, called ‘pebbles’, in a finite ‘pebbleflow net(work)’. And the production of a pebbleflow net associated with a pure stream definition, that is, the amount of pebbles the net is able to produce at its output port, can be calculated by reducing nets to trivial nets.

### Citations

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(Show Context)
Citation Context ...ly capture the consumption/production behaviour of a large class of stream functions. Related Work. It is well-known that networks are devices for computing least fixed points of systems of equations =-=[7]-=-. The notion of ‘productivity’ (sometimes also referred to as ‘liveness’) was first mentioned by Dijkstra [3]. Since then several papers [17,13,2,6,15,1] have been devoted to criteria ensuring product... |

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(Show Context)
Citation Context ...ements (data) in an SCS in favour of occurrences of the symbol •, which we call ‘pebble’. Thus, a stream term d : s is translated to [d : s] = •([s]). Pebbleflow nets are inspired by interaction nets =-=[11]-=-, and could be implemented in the framework of interaction nets with little effort. We give an operational description of pebbleflow nets and define a production preserving translation of pure SCSs in... |

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- 1996
(Show Context)
Citation Context ...m ≥ n.∆ν(m) = ∆ν(m + p). This class of moduli is effectively closed under composition, and allows to calculate fixed points of unary functions. Rational production moduli generalise those employed by =-=[17,6,2,15]-=-, and enable us to precisely capture the consumption/production behaviour of a large class of stream functions. Related Work. It is well-known that networks are devices for computing least fixed point... |

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(Show Context)
Citation Context ...calculus, [14]). All of these methods pertain to finite data only. In the last two decades interest has grown towards infinite data, as witnessed by the application of type theory to infinite objects =-=[2]-=-, and the emergence of coalgebraic techniques for infinite data types like streams [12]. While termination cannot be expected when infinite data are processed, infinitary notions of ⋆ This research ha... |

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Citation Context ...0.502. Preprint submitted to Theoretical Computer Science 22 January 2008stermination become relevant. For example, in frameworks for the manipulation of infinite objects such as infinitary rewriting =-=[8]-=- and infinitary λ-calculus [9], basic notions are the properties WN ∞ and SN ∞ of infinitary weak and strong normalisation [10], and UN ∞ of uniqueness of (infinitary) normal forms. In the functional ... |

54 | Behavioural Differential Equations: a Coinductive Calculus
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(Show Context)
Citation Context ...ades interest has grown towards infinite data, as witnessed by the application of type theory to infinite objects [2], and the emergence of coalgebraic techniques for infinite data types like streams =-=[12]-=-. While termination cannot be expected when infinite data are processed, infinitary notions of ⋆ This research has been partially funded by the Netherlands Organisation for Scientific Research (NWO) u... |

48 |
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Citation Context ...ction In functional programming, term rewriting and λ-calculus, there is a wide arsenal of methods for proving termination such as recursive path orders, dependency pairs (for term rewriting systems, =-=[16]-=-) and the method of computability (for λ-calculus, [14]). All of these methods pertain to finite data only. In the last two decades interest has grown towards infinite data, as witnessed by the applic... |

31 |
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- 1981
(Show Context)
Citation Context ...m ≥ n.∆ν(m) = ∆ν(m + p). This class of moduli is effectively closed under composition, and allows to calculate fixed points of unary functions. Rational production moduli generalise those employed by =-=[17,6,2,15]-=-, and enable us to precisely capture the consumption/production behaviour of a large class of stream functions. Related Work. It is well-known that networks are devices for computing least fixed point... |

30 |
Vries. Infinitary lambda calculus
- Kennaway, Klop, et al.
- 1997
(Show Context)
Citation Context ...are processed, infinitary notions of termination become relevant. For example, in formal frameworks for the manipulation of infinite objects such as infinitary rewriting [7] and infinitary λ-calculus =-=[8]-=-, basic notions are the properties WN∞ of infinitary weak normalisation and SN∞ of infinitary strong normalisation [9]. In the functional programming literature the notion of ‘productivity’ has arisen... |

27 |
On the Productivity of Recursive List Definitions
- Sijtsma
- 1989
(Show Context)
Citation Context ... normalisation [10], and UN ∞ of uniqueness of (infinitary) normal forms. In the functional programming literature the notion of ‘productivity’ has arisen, initially in the pioneering work of Sijtsma =-=[13]-=-, as a natural strengthening of what in our setting are the properties WN ∞ and UN ∞ . A stream definition is called productive if not only can the definition be evaluated continually to build up a un... |

21 | Term Rewriting Systems, volume 55 of Cambridge Tracts - Terese - 2003 |

15 |
A Term Calculus for (Co-)Recursive Definitions on Streamlike Data Structures
- Buchholz
- 2005
(Show Context)
Citation Context ...ion/production steps during the evaluation of the stream specification in a quantitatively precise manner. This paper is a revised and extended version of the paper [5] presented at FCT’07. We follow =-=[13,1]-=- in describing the quantitative input/output behaviour of a stream function f by a ‘modulus of production’ νf : (N) r → N with the property that the first νf(n1, . . . , nr) elements of f(t1, . . . , ... |

15 | Data-Oblivious Stream Productivity - Endrullis, Grabmayer, et al. - 2008 |

12 | A Finite Semantics of Simply-Typed Lambda Terms For Infinite Runs of Automata - Aehlig - 2007 |

9 |
On the Productivity of Recursive Definitions
- Dijkstra
- 1980
(Show Context)
Citation Context ...l-known that networks are devices for computing least fixed points of systems of equations [7]. The notion of ‘productivity’ (sometimes also referred to as ‘liveness’) was first mentioned by Dijkstra =-=[3]-=-. Since then several papers [17,13,2,6,15,1] have been devoted to criteria ensuring productivity. The common essence of these ap2sproaches is a quantitative analysis. In [17], Wadge uses dataflow netw... |

8 | Infinitary Normalization
- Klop, Vrijer
- 2005
(Show Context)
Citation Context ...rks for the manipulation of infinite objects such as infinitary rewriting [8] and infinitary λ-calculus [9], basic notions are the properties WN ∞ and SN ∞ of infinitary weak and strong normalisation =-=[10]-=-, and UN ∞ of uniqueness of (infinitary) normal forms. In the functional programming literature the notion of ‘productivity’ has arisen, initially in the pioneering work of Sijtsma [13], as a natural ... |

5 | Applications of Infinitary Lambda Calculus - Barendregt, Klop |

4 | Ensuring the Productivity of Infinite Structures
- Telford, Turner
- 1997
(Show Context)
Citation Context ...m ≥ n.∆ν(m) = ∆ν(m + p). This class of moduli is effectively closed under composition, and allows to calculate fixed points of unary functions. Rational production moduli generalise those employed by =-=[17,6,2,15]-=-, and enable us to precisely capture the consumption/production behaviour of a large class of stream functions. Related Work. It is well-known that networks are devices for computing least fixed point... |