## A characterization of alternating log time by first order functional programs (2006)

Venue: | In LPAR 2006, volume 4246 of LNAI |

Citations: | 7 - 5 self |

### BibTeX

@INPROCEEDINGS{Bonfante06acharacterization,

author = {Guillaume Bonfante and Jean-yves Marion and Romain Péchoux},

title = {A characterization of alternating log time by first order functional programs},

booktitle = {In LPAR 2006, volume 4246 of LNAI},

year = {2006},

pages = {90--104},

publisher = {Springer}

}

### OpenURL

### Abstract

Abstract. We a give an intrinsic characterization of the class of functions which are computable in NC 1 that is by a uniform, logarithmic depth and polynomial size family circuit. Recall that the class of functions in ALogTime, that is in logarithmic time on an Alternating Turing Machine, is NC 1. Our characterization is in terms of first order functional programming languages. We define measure-tools called Supinterpretations, which allow to give space and time bounds and allow also to capture a lot of program schemas. This study is part of a research on static analysis in order to predict program resources. It is related to the notion of Quasi-interpretations and belongs to the implicit computational complexity line of research. 1

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Citation Context ...esource verification of bytecode programs is obtained by compiling first order functional and reactive programs. See for example [3,2,17]. 3. There are heuristics to determine program complexity. See =-=[1, 10]-=- 1.1 Backgrounds on ALogTime and NC 1 We write log(n) to mean ⌈log 2 (n + 1)⌉. Recall that the floor function ⌊x⌋ is the greatest integer ≤ x, and the ceiling function ⌈x⌉ is least integer ≥ x. We ref... |

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Citation Context ...amic programming [28] and deal with non-terminating programs [29]. 2. Resource verification of bytecode programs is obtained by compiling first order functional and reactive programs. See for example =-=[3,2,17]-=-. 3. There are heuristics to determine program complexity. See [1, 10] 1.1 Backgrounds on ALogTime and NC 1 We write log(n) to mean ⌈log 2 (n + 1)⌉. Recall that the floor function ⌊x⌋ is the greatest ... |

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Citation Context ... in [29]. The main features of interpretation methods for proving complexity bounds are the following. 1. The analysis include broad classes of algorithms, like greedy algorithms, dynamic programming =-=[28]-=- and deal with non-terminating programs [29]. 2. Resource verification of bytecode programs is obtained by compiling first order functional and reactive programs. See for example [3,2,17]. 3. There ar... |

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Citation Context ...amic programming [28] and deal with non-terminating programs [29]. 2. Resource verification of bytecode programs is obtained by compiling first order functional and reactive programs. See for example =-=[3,2,17]-=-. 3. There are heuristics to determine program complexity. See [1, 10] 1.1 Backgrounds on ALogTime and NC 1 We write log(n) to mean ⌈log 2 (n + 1)⌉. Recall that the floor function ⌊x⌋ is the greatest ... |

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Citation Context ...or proving complexity. The first method concerns Quasi-interpretations, which is surveyed in [9]. The second method, which concerns this paper, is the sup-interpretation method, that we introduced in =-=[29]-=-. The main features of interpretation methods for proving complexity bounds are the following. 1. The analysis include broad classes of algorithms, like greedy algorithms, dynamic programming [28] and... |

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Citation Context ...r order functional programming. Unlike the two former approaches and the next one, the third one considers imperative programming language and is developed by Kristiansen-Jones [22], Niggl-Wunderlich =-=[31]-=-, and Marion-Moyen [30]. Lastly, the fourth approach is the one on which we focus in this paper. It concerns term rewriting systems and interpretation methods for proving complexity bounds. This metho... |

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Citation Context ...based on finite global functions. These results are clearly a guideline for us. However, there are only a few characterizations of ALogTime from which a resource static analysis is conceivable. Bloch =-=[8]-=- gives a characterization of ALogTime using a divide and conquer ramified recursion schema. Leivant and Marion [27] propose another characterization based on linear ramified recursion with substitutio... |

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Citation Context ...acterizations of ALogTime from which a resource static analysis is conceivable. Bloch [8] gives a characterization of ALogTime using a divide and conquer ramified recursion schema. Leivant and Marion =-=[27]-=- propose another characterization based on linear ramified recursion with substitutions. It is also worth mentioning [25,19] which capture NC. These purely syntactic characterizations capture a few al... |

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Citation Context ...esource verification of bytecode programs is obtained by compiling first order functional and reactive programs. See for example [3,2,17]. 3. There are heuristics to determine program complexity. See =-=[1, 10]-=- 1.1 Backgrounds on ALogTime and NC 1 We write log(n) to mean ⌈log 2 (n + 1)⌉. Recall that the floor function ⌊x⌋ is the greatest integer ≤ x, and the ceiling function ⌈x⌉ is least integer ≥ x. We ref... |

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Citation Context ...llowing Cook [16], we say that a function F : {0, 1} ∗ → {0, 1} ∗s4 There are various characterizations of ALogTime, which are surveyed in [13] based on bounded recursion schema. Compton and Laflamme =-=[15]-=- give a characterization of ALogTime based on finite global functions. These results are clearly a guideline for us. However, there are only a few characterizations of ALogTime from which a resource s... |

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Citation Context ...ramming. Unlike the two former approaches and the next one, the third one considers imperative programming language and is developed by Kristiansen-Jones [22], Niggl-Wunderlich [31], and Marion-Moyen =-=[30]-=-. Lastly, the fourth approach is the one on which we focus in this paper. It concerns term rewriting systems and interpretation methods for proving complexity bounds. This method consists in giving an... |

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Citation Context ...amic programming [28] and deal with non-terminating programs [29]. 2. Resource verification of bytecode programs is obtained by compiling first order functional and reactive programs. See for example =-=[3,2,17]-=-. 3. There are heuristics to determine program complexity. See [1, 10] 1.1 Backgrounds on ALogTime and NC 1 We write log(n) to mean ⌈log 2 (n + 1)⌉. Recall that the floor function ⌊x⌋ is the greatest ... |

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Citation Context ...gTime using a divide and conquer ramified recursion schema. Leivant and Marion [27] propose another characterization based on linear ramified recursion with substitutions. It is also worth mentioning =-=[25,19]-=- which capture NC. These purely syntactic characterizations capture a few algorithmic patterns. On the contrary, this work tries to delineate a broad class of algorithms. Parallel algorithms are diffi... |

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Citation Context ...ta flow in order to measure the program complexity. We have developed two kinds of interpretation methods for proving complexity. The first method concerns Quasi-interpretations, which is surveyed in =-=[9]-=-. The second method, which concerns this paper, is the sup-interpretation method, that we introduced in [29]. The main features of interpretation methods for proving complexity bounds are the followin... |

1 |
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Citation Context ...gTime using a divide and conquer ramified recursion schema. Leivant and Marion [27] propose another characterization based on linear ramified recursion with substitutions. It is also worth mentioning =-=[25,19]-=- which capture NC. These purely syntactic characterizations capture a few algorithmic patterns. On the contrary, this work tries to delineate a broad class of algorithms. Parallel algorithms are diffi... |