About the decision of reachability for register machines

Véronique Cortier

RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications (2002)

  • Volume: 36, Issue: 4, page 341-358
  • ISSN: 0988-3754

Abstract

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We study the decidability of the following problem: given p affine functions f 1 , ... , f p over ℕ k and two vectors v 1 , v 2 ∈ ℕ k , is v 2 reachable from v 1 by successive iterations of f 1 , ... , f p (in this given order)? We show that this question is decidable for p = 1 , 2 and undecidable for some fixed p .

How to cite

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Cortier, Véronique. "About the decision of reachability for register machines." RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications 36.4 (2002): 341-358. <http://eudml.org/doc/244872>.

@article{Cortier2002,
abstract = {We study the decidability of the following problem: given $p$ affine functions $f_1,\ldots , f_p$ over $\mathbb \{N\}^k$ and two vectors $v_1, v_2\in \mathbb \{N\}^k$, is $v_2$ reachable from $v_1$ by successive iterations of $f_1,\ldots , f_p$ (in this given order)? We show that this question is decidable for $p = 1, 2$ and undecidable for some fixed $p$.},
author = {Cortier, Véronique},
journal = {RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications},
keywords = {verification; infinite state systems; decidability},
language = {eng},
number = {4},
pages = {341-358},
publisher = {EDP-Sciences},
title = {About the decision of reachability for register machines},
url = {http://eudml.org/doc/244872},
volume = {36},
year = {2002},
}

TY - JOUR
AU - Cortier, Véronique
TI - About the decision of reachability for register machines
JO - RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications
PY - 2002
PB - EDP-Sciences
VL - 36
IS - 4
SP - 341
EP - 358
AB - We study the decidability of the following problem: given $p$ affine functions $f_1,\ldots , f_p$ over $\mathbb {N}^k$ and two vectors $v_1, v_2\in \mathbb {N}^k$, is $v_2$ reachable from $v_1$ by successive iterations of $f_1,\ldots , f_p$ (in this given order)? We show that this question is decidable for $p = 1, 2$ and undecidable for some fixed $p$.
LA - eng
KW - verification; infinite state systems; decidability
UR - http://eudml.org/doc/244872
ER -

References

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  1. [1] E. Asarin, G. Schneider and S. Yovine, Towards computing phase portraits of polygonal differential inclusions, in HSCC’2002, Hybrid Systems: Computation and Control. Stanford, USA, Lecture Notes in Comput. Sci. 2289 (2002) 49-61. Zbl1054.93030
  2. [2] B. Boigelot, Symbolic Methods for Exploring Infinite Sate Spaces, Ph.D. Thesis. Université de Liège (1998) 225. 
  3. [3] H. Comon and Y. Jurski, Multiple counters automata, safety analysis and Presburger arithmetic, in Proc. Computer Aided Verification. Springer Verlag, Lecture Notes in Comput. Sci. 1427 (1998) 268-279. MR1729041
  4. [4] V. Cortier, Vérification de systèmes à compteurs (in French), Master’s Thesis. Université Paris 7 (1999) http://www.lsv.ens-cachan.fr/ ∼ cortier/memoiredea.ps 
  5. [5] L.E. Dickson, Finiteness of the odd perfect and primitive abundant numbers with n distinct prime factors. Amer. J. Math. 35 (1913) 413-422. Zbl44.0220.02MR1506194JFM44.0220.02
  6. [6] C. Dufourd, A. Finkel and Ph. Schnoebelen, Between decidability and undecidability, in Proc. ICALP 1998. Springer-Verlag, Lecture Notes in Comput. Sci. 1448 (1998) 103-115. Zbl0909.68124MR1683488
  7. [7] A. Finkel, P. McKenzie and C. Picaronny, A well-structured framework for analysing Petri nets extensions. Technical Report, Research Report LSV-99-2 (1999) http://www.lsv.ens-cachan.fr/Publis/RAPPORTS_LSV/rr-lsv-1999-2.rr.ps Zbl1101.68696
  8. [8] M.Yu. Matijacevitch, M. David and J. Robinson, Hilbert’s Tenth Problem, Chapter 3 (1976). Zbl0346.02026
  9. [9] C. Rorres and H. Anton, Applications of Linear Algebra, Chapters 9 and 13 (1979). Zbl0439.15001

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