# Affine Parikh automata∗

Michaël Cadilhac; Alain Finkel; Pierre McKenzie

RAIRO - Theoretical Informatics and Applications (2012)

- Volume: 46, Issue: 4, page 511-545
- ISSN: 0988-3754

## Access Full Article

top## Abstract

top## How to cite

topCadilhac, Michaël, Finkel, Alain, and McKenzie, Pierre. "Affine Parikh automata∗." RAIRO - Theoretical Informatics and Applications 46.4 (2012): 511-545. <http://eudml.org/doc/277829>.

@article{Cadilhac2012,

abstract = {The Parikh finite word automaton (PA) was introduced and studied in 2003 by Klaedtke and
Rueß. Natural variants of the PA arise from viewing a PA equivalently as an automaton that
keeps a count of its transitions and semilinearly constrains their numbers. Here we adopt
this view and define the affine PA, that extends the PA by having each
transition induce an affine transformation on the PA registers, and the PA on
letters, that restricts the PA by forcing any two transitions on the same
letter to affect the registers equally. Then we report on the expressiveness, closure, and
decidability properties of such PA variants. We note that deterministic PA are strictly
weaker than deterministic reversal-bounded counter machines.},

author = {Cadilhac, Michaël, Finkel, Alain, McKenzie, Pierre},

journal = {RAIRO - Theoretical Informatics and Applications},

keywords = {Automata; semilinear sets; affine functions; counter machines; automata},

language = {eng},

month = {11},

number = {4},

pages = {511-545},

publisher = {EDP Sciences},

title = {Affine Parikh automata∗},

url = {http://eudml.org/doc/277829},

volume = {46},

year = {2012},

}

TY - JOUR

AU - Cadilhac, Michaël

AU - Finkel, Alain

AU - McKenzie, Pierre

TI - Affine Parikh automata∗

JO - RAIRO - Theoretical Informatics and Applications

DA - 2012/11//

PB - EDP Sciences

VL - 46

IS - 4

SP - 511

EP - 545

AB - The Parikh finite word automaton (PA) was introduced and studied in 2003 by Klaedtke and
Rueß. Natural variants of the PA arise from viewing a PA equivalently as an automaton that
keeps a count of its transitions and semilinearly constrains their numbers. Here we adopt
this view and define the affine PA, that extends the PA by having each
transition induce an affine transformation on the PA registers, and the PA on
letters, that restricts the PA by forcing any two transitions on the same
letter to affect the registers equally. Then we report on the expressiveness, closure, and
decidability properties of such PA variants. We note that deterministic PA are strictly
weaker than deterministic reversal-bounded counter machines.

LA - eng

KW - Automata; semilinear sets; affine functions; counter machines; automata

UR - http://eudml.org/doc/277829

ER -

## References

top- B.S. Baker and R.V. Book, Reversal-bounded multipushdown machines. J. Comput. Syst. Sci.8 (1974) 315–332.
- M. Blattner and M. Latteux, Parikh-bounded languages, in ICALP. Lect. Notes Comput. Sci.115 (1981) 316–323.
- R. Book, M. Nivat and M. Paterson, Reversal-bounded acceptors and intersections of linear languages. SIAM J. Comput.3 (1974) 283.
- F. Brandenburg, Analogies of PAL and COPY, in Fundamentals of Computation Theory. Lect. Notes in Comput. Sci.117 (1981) 61–70.
- E. Chiniforooshan, M. Daley, O.H. Ibarra, L. Kari and S. Seki, One-reversal counter machines and multihead automata : revisited, in Proc. of SOFSEM. ACM (2011) 166–177.
- H.B. Enderton, A Mathematical Introduction to Logic. Academic Press (1972).
- J. Ferrante and C. Rackoff, A decision procedure for the first order theory of real addition with order. SIAM J. Comput.4 (1975) 69–76.
- P. Ganty, R. Majumdar and B. Monmege, Bounded underapproximations. Form. Methods Syst. Des.40 (2012) 206–231.
- S. Ginsburg and E.H. Spanier, Semigroups, Presburger formulas and languages. Pacific J. Math.16 (1966) 285–296.
- S. Ginsburg and E. Spanier, Finite-turn pushdown automata. SIAM J. Control Optim.4 (1966) 429.
- S.A. Greibach, A note on undecidable properties of formal languages. Math. Syst. Theor.2 (1968) 1–6.
- O.H. Ibarra, Reversal-bounded multicounter machines and their decision problems. J. ACM25 (1978) 116–133.
- O.H. Ibarra and J. Su, A technique for proving decidability of containment and equivalence of linear constraint queries. J. Comput. Syst. Sci.59 (1999) 1–28.
- O.H. Ibarra, J. Su, Z. Dang, T. Bultan and R.A. Kemmerer, Counter machines and verification problems. Theor. Comput. Sci.289 (2002) 165–189.
- W. Karianto, Parikh automata with pushdown stack. Diploma thesis, RWTH Aachen (2004).
- F. Klaedtke and H. Rueß, Parikh automata and monadic second-order logics with linear cardinality constraints. Universität Freiburg, Tech. Rep. 177 (2002).
- F. Klaedtke and H. Rueß, Monadic second-order logics with cardinalities, in Proc. of ICALP. Lect. Notes Comput. Sci.2719 (2003) 681–696.
- S.Y. Kuroda, Classes of languages and linear bounded automata. Inform. Control7 (1964) 207–223.
- M. Latteux, Mots infinis et langages commutatifs. RAIRO Inform. Théor. Appl.12 (1978) 185–192.
- D.R. Mazur, Combinatorics : A Guided Tour. Mathematical Association of Mathematics (2010).
- P. McKenzie, M. Thomas and H. Vollmer, Extensional uniformity for boolean circuits. SIAM J. Comput.39 (2010) 3186–3206.
- H. Seidl, T. Schwentick and A. Muscholl, Numerical document queries, in Principles of Database Systems. ACM, San Diego, CA, USA (2003) 155–166.
- H. Straubing, Finite Automata, Formal Logic, and Circuit Complexity. Birkhäuser, Boston (1994).
- P. Tesson and D. Thérien, Logic meets algebra : the case of regular languages. Log. Meth. Comput. Sci.3 (2007) 1–37.
- L.P.D. van den Dries, Tame Topology and O-minimal Structures. Cambridge Univ. Press (1998).
- P. Wolper and B. Boigelot, An automata-theoretic approach to Presburger arithmetic constraints, in Static Analysis (SAS’95). Lect. Notes Comput. Sci.983 (1995) 21–32.
- S.D. Zilio and D. Lugiez, Xml schema, tree logic and sheaves automata, in Rewriting Techniques and Applications (2003) 246–263.

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.