## How to compose Presburger-Accelerations: Applications to Broadcast Protocols (2002)

Venue: | IN PROC. 22ND CONF. FOUND. OF SOFTWARE TECHNOLOGY AND THEOR. COMP. SCI. (FST&TCS'2002), KANPUR |

Citations: | 48 - 17 self |

### BibTeX

@INPROCEEDINGS{Finkel02howto,

author = {Alain Finkel and J. Leroux},

title = {How to compose Presburger-Accelerations: Applications to Broadcast Protocols},

booktitle = {IN PROC. 22ND CONF. FOUND. OF SOFTWARE TECHNOLOGY AND THEOR. COMP. SCI. (FST&TCS'2002), KANPUR},

year = {2002},

pages = {145--156},

publisher = {Springer}

}

### Years of Citing Articles

### OpenURL

### Abstract

Finite linear systems are finite sets of linear functions whose guards are de fined by Presburger formulas, and whose the squares matrice associated generate a finite multiplicative monoid. We prove that for finite linear systems, the accelerations of sequences of transitions always produce an effective Presburger-definable relation. We then show how to choose the good sequences of length n whose number is polynomial in n although the total number of cycles of length n is exponential in n. We implement these theoretical results in the tool FAST [FAS] (Fast Acceleration of Symbolic Transition systems). FAST computes in few seconds the minimal deterministic finite automata that represent the reachability sets of 8 well-known broadcast protocols.

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Citation Context ... reachability problem is undecidable for linear systems even of dimension 3 (by using a Post problem reduction [FB95]). We call Presburger Model Checking the Regular Model Checking [BJNT00], [FO97b], =-=[BF00-=-], [BGP97] in which Presburger formulas (instead of regular languages) are used as a symbolic representation of innite set of states. Presburger formulas enjoy good properties because, as regular sets... |

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Citation Context ... Accelerations are also called meta-transitions in [BW94], [ABJN99], [Boi98] or exact widening in theseld of abstract interpretation. Boigelot's model and results B. Boigelot and P. Wolper in [BW94], =-=[WB98-=-] and [Boi98] consider a unique linear function f whose guard is of the form Ax b. They gave two decidable technical conditions [B1] and [B2] over a linear function f (called here a Boigelot's functi... |

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Citation Context ...cation. Let us recall that verication of safety properties can be reduced to reachability of a given state from a set of initial states. For some particular models like lossy channel systems [AJ96], [=-=ABJ98]-=- automata with stacks [FWW97], [BEM97], reactive FIFO automata [SFRC99],sat counters automata ([CJ98] and reset/transfer 2-counters automata [FS00b], [FS00a], it is possible to test reachability from ... |

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Citation Context ...cceleration set. We will compute and use accelerations of dierent sequences, to speed up the reachability set construction and help its termination. Accelerations are also called meta-transitions in [=-=BW94]-=-, [ABJN99], [Boi98] or exact widening in theseld of abstract interpretation. Boigelot's model and results B. Boigelot and P. Wolper in [BW94], [WB98] and [Boi98] consider a unique linear function f wh... |

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Citation Context ...me what we called the Presburger Model Checking, we can cite [BF00] and [BGP97] which both also use the tool MONA [MON]. For FIFO channels systems, acceleration of loops have been recently studied in =-=[FPS00]-=- with SLRE, in [BGWW97] with QDD and in [BH99] with CQDD; for lossy FIFO channels systems, acceleration of loops and SRE have been studied in [ABJ98] and [AAB99]. Plan of the paper Section 2 and 3 int... |

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Citation Context ... models like lossy channel systems [AJ96], [ABJ98] automata with stacks [FWW97], [BEM97], reactive FIFO automata [SFRC99],sat counters automata ([CJ98] and reset/transfer 2-counters automata [FS00b], =-=[FS00a], it-=- is possible to test reachability from a regular set of initial states because the reachability set (or the set of all predecessors) is regular and is eectively computable. But, these \decidable"... |

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Citation Context ...of loops have been recently studied in [FPS00] with SLRE, in [BGWW97] with QDD and in [BH99] with CQDD; for lossy FIFO channels systems, acceleration of loops and SRE have been studied in [ABJ98] and =-=[AAB99]-=-. Plan of the paper Section 2 and 3 intuitively introduce, on a simple example, acceleration of cycles. Then, after recalling the basic notions (labeled transition systems, matrices and 3 functions, P... |

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Citation Context ... 2-counters automata and to FIFO channel systems. [AAB00] guesses the result of the innite iteration of a cycle, in a hybrid system, and verify whether the guess is correct or not. [Rev90], [FO97b], [=-=FO97a]-=- and [BF99] automatically accelerate a given non-elementary cycle. [PS00] attacks the acceleration techniques using formulas in a variant of WS1S for the symbolic representation of sets of states: the... |

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Citation Context ... that the reachability problem is undecidable for linear systems even of dimension 3 (by using a Post problem reduction [FB95]). We call Presburger Model Checking the Regular Model Checking [BJNT00], =-=[FO97b-=-], [BF00], [BGP97] in which Presburger formulas (instead of regular languages) are used as a symbolic representation of innite set of states. Presburger formulas enjoy good properties because, as regu... |

2 |
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Citation Context ...uced to reachability of a given state from a set of initial states. For some particular models like lossy channel systems [AJ96], [ABJ98] automata with stacks [FWW97], [BEM97], reactive FIFO automata =-=[SFRC99]-=-,sat counters automata ([CJ98] and reset/transfer 2-counters automata [FS00b], [FS00a], it is possible to test reachability from a regular set of initial states because the reachability set (or the se... |

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Citation Context ...reover as all broadcast protocols are parameterized Transfer/Reset/Inhibitors Petri Nets, we can prove that Transfer/Reset/Inhibitors Petri Nets and all broadcast protocols [EN98], [EFM99], [Del00a], =-=[Del00b]-=- aresnite linear systems. Proposition 1. Transfer/Reset/Inhibitor Petri Nets [DFS98] and all broadcast protocols [EN98], [EFM99], [Del00a], [Del00b] aresnite linear systems. 7 Proof. All these classes... |

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Citation Context ...are Presburger-denable; such systems are now called linear systems. It is clear that the reachability problem is undecidable for linear systems even of dimension 3 (by using a Post problem reduction [=-=FB95]-=-). We call Presburger Model Checking the Regular Model Checking [BJNT00], [FO97b], [BF00], [BGP97] in which Presburger formulas (instead of regular languages) are used as a symbolic representation of ... |

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Citation Context ... exists a word such that M = M . As f (S 0 ) = Reach(L; S 0 ; ) is Presburger-denable for every Presburger-denable set S 0 , then by using [Boi98], the monoidsissnite. The Burnside theorem [MZ75], [Jac78], [MS77] implies that M L issnite. ut 5 How tosnd out the good accelerations ? For asnite linear system L = (; m; f ), the number of words with a length less than or equal to an integer ... |

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Citation Context ... integers satisfying x+y = z. On the other hand, actions are relations (they are not restricted to be functions). Hence, their model is not comparable withsnite linear systems. A. Finkel and G. Sutre =-=[Sut0-=-0], [FS00b], [FS00a] study 2-counters automata with reset/transfer/+1/-1 actions and guards of the form x i c and x i = 0 (zero test). They prove that cycle-acceleration leads to the eective computat... |