Table Of ContentLecture Notes in Computer Science 6808
CommencedPublicationin1973
FoundingandFormerSeriesEditors:
GerhardGoos,JurisHartmanis,andJanvanLeeuwen
EditorialBoard
DavidHutchison
LancasterUniversity,UK
TakeoKanade
CarnegieMellonUniversity,Pittsburgh,PA,USA
JosefKittler
UniversityofSurrey,Guildford,UK
JonM.Kleinberg
CornellUniversity,Ithaca,NY,USA
AlfredKobsa
UniversityofCalifornia,Irvine,CA,USA
FriedemannMattern
ETHZurich,Switzerland
JohnC.Mitchell
StanfordUniversity,CA,USA
MoniNaor
WeizmannInstituteofScience,Rehovot,Israel
OscarNierstrasz
UniversityofBern,Switzerland
C.PanduRangan
IndianInstituteofTechnology,Madras,India
BernhardSteffen
TUDortmundUniversity,Germany
MadhuSudan
MicrosoftResearch,Cambridge,MA,USA
DemetriTerzopoulos
UniversityofCalifornia,LosAngeles,CA,USA
DougTygar
UniversityofCalifornia,Berkeley,CA,USA
GerhardWeikum
MaxPlanckInstituteforInformatics,Saarbruecken,Germany
Markus Holzer Martin Kutrib
Giovanni Pighizzini (Eds.)
Descriptional Complexity
of Formal Systems
13th International Workshop, DCFS 2011
Gießen/Limburg, Germany, July 25-27, 2011
Proceedings
1 3
VolumeEditors
MarkusHolzer
UniversitätGießen,InstitutfürInformatik
Arndtstraße2,35392Gießen,Germany
E-mail:[email protected]
MartinKutrib
UniversitätGießen,InstitutfürInformatik
Arndtstraße2,35392Gießen,Germany
E-mail:[email protected]
GiovanniPighizzini
UniversitàdegliStudidiMilano
DipartimentodiInformaticaeComunicazione
ViaComelico39,20135Milano,Italy
E-mail:[email protected]
ISSN0302-9743 e-ISSN1611-3349
ISBN978-3-642-22599-4 e-ISBN978-3-642-22600-7
DOI10.1007/978-3-642-22600-7
SpringerHeidelbergDordrechtLondonNewYork
LibraryofCongressControlNumber:2011931777
CRSubjectClassification(1998):F.1,D.2.4,F.3,F.4.2-3
LNCSSublibrary:SL1–TheoreticalComputerScienceandGeneralIssues
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Preface
The 13th International Workshop of Descriptional Complexity of Formal Sys-
tems(DCFS2011)wasorganizedbytheInstitutfu¨rInformatikoftheUniversita¨t
Giessen and took place in the vicinity of Giessen, in Limburg, Germany. It was
a three-day workshop starting July 25 and ending July 27, 2011. The city of
Limburg lies in the west of the province Hessen between the Taunus and the
Westerwald in the beautiful Lahn valley and it looks back on a history of more
than 1,100 years.
The DCFS workshop is the successor workshop and the merger of two re-
lated workshops, Descriptional Complexity of Automata, Grammars and Re-
lated Structures (DCAGRS) and Formal Descriptions and Software Reliability
(FDSR). The DCAGRS workshop took place in Magdeburg, Germany (1999),
London, Ontario, Canada (2000), and Vienna, Austria (2001), while the FDSR
workshoptook place inPaderborn,Germany (1998),BocaRaton,Florida,USA
(1999), and San Jose, California, USA (2000). The DCFS workshop has previ-
ouslybeenheldinLondon,Ontario,Canada(2002),Budapest,Hungary(2003),
London,Ontario,Canada (2004),Como,Italy (2005),Las Cruces,New Mexico,
USA (2006), Novy´ Smokovec, Slovakia (2007), Charlottetown, Prince Edward
Island, Canada (2008), Magdeburg, Germany (2009), and Saskatoon, Saskatch-
ewan, Canada (2010).
This volume contains the invited contributions and the accepted papers of
DCFS 2011. Special thanks go to the invited speakers:
– Jarkko Kari (Unversity of Turku, Finland)
– Friedrich Otto (Universita¨t Kassel, Germany)
– Stefan Schwoon (ENS de Cachan, France)
– Denis Th´erien (McGill University, Quebec, Canada)
for accepting our invitation and presenting their recent results at DCFS 2011.
ThepapersweresubmittedtoDCFS2011byatotalof54authorsfrom16differ-
entcountries,fromallovertheworld,Canada,CzechRepublic,Finland,France,
Germany, Hungary, India, Italy, Republic of Korea,Latvia, Malaysia,Portugal,
Romania, Slovakia, Spain, and USA. From these submissions, on the basis of
threerefereereportseach,theProgramCommitteeselected21papers—thesub-
missionandrefereeingprocesswassupportedbytheEasyChairconferenceman-
agementsystem.We warmlythank the members ofthe ProgramCommittee for
their excellent work in making this selection. Moreover,we also thank the addi-
tional external reviewers for their careful evaluation. All these efforts were the
basis for the success of the workshop. We are indebted to Alfred Hofmann and
AnnaKramer,fromSpringer,fortheirefficientcollaborationinmakingthisvol-
ume available before the conference. Their timely instructions were very helpful
to our preparation of this volume.
VI Preface
WearegratefultotheOrganizingCommitteeconsistingofSusanneGretschel,
MarkusHolzer(Co-chair),SebastianJakobi,MartinKutrib(Co-chair),Andreas
Malcher,KatjaMeckel,HeinzRu¨beling,andMatthiasWendlandt(Co-chair)for
their support of the sessions,the excursion and the other accompanying events.
Thanks also go to the staff of the Dom Hotel in Limburg, where the conference
tookplace,andalltheotherhelpinghandsthatwereworkinginthebackground
for the success of this workshop.
Finally, we would like to thank all the participants for attending the DCFS
workshop.We hopethatthis year’sworkshopstimulatednew investigationsand
scientific co-operations in the field of descriptional complexity, as in previous
years. Looking forward to DCFS 2012 in Porto, Portugal.
July 2011 Markus Holzer
Martin Kutrib
Giovanni Pighizzini
Organization
DCFS2011wasorganizedbytheInstitutfu¨rInformatikoftheUniversita¨tGiessen,
Germany.TheconferencetookplaceattheDomHotelinLimburg,Germany.
Program Committee
Jean-Marc Champarnaud Universit´e de Rouen, France
Erzs´ebet Csuhaj-Varju´ MTA SZTAKI, Hungary
Zoltan E´sik University of Szeged, Hungary
Markus Holzer Universit¨at Giessen, Germany (Co-chair)
Galina Jira´skov´a Slovak Academy of Sciences, Slovakia
Martin Kutrib Universita¨t Giessen, Germany (Co-chair)
Carlos Mart´ın-Vide Roviri i Virgili University, Spain
Tom´aˇs Masopust Czech Academy of Sciences, Czech Republic;
Centrum Wiskunde & Infromatica,
The Netherlands
Ian McQuillan University of Saskatoon, Canada
Carlo Mereghetti Universita` degli Studi di Milano, Italy
Victor Mitrana Universitatea din Bucure¸sti, Romania
Alexander Okhotin University of Turku, Finland
Giovanni Pighizzini Universita` degli Studi di Milano, Italy (Co-chair)
Bala Ravikumar Sonoma State University, USA
Rog´erio Reis Universidade do Porto, Portugal
Kai Salomaa Queen’s University, Canada
Bianca Truthe Universita¨t Magdeburg, Germany
External Referees
Alberto Bertoni Christof Lo¨ding Rama Raghavan
Sabine Broda Andreas Malcher Shinnosuke Seki
Flavio D’Alessandro Florin Manea Ralf Stiebe
Mike Domaratzki Wim Martens Maurice H. ter Beek
Stefan Gulan Giancarlo Mauri Sandor Vagvolgyi
Yo-Sub Han Katja Meckel Lynette Van Zijl
Szabolcs Ivan Nelma Moreira Gyo¨rgy Vaszil
Sebastian Jakobi Zoltan L. Nemeth Claudio Zandron
Tomasz Jurdzinski Dana Pardubska
Lakshmanan Kuppusamy Xiaoxue Piao
Sponsoring Institutions
Universita¨t Giessen
Table of Contents
Invited Papers
Linear Algebra Based Bounds for One-Dimensional Cellular
Automata....................................................... 1
Jarkko Kari
On Restarting Automata with Window Size One..................... 8
Friedrich Otto
Construction and SAT-Based Verification of Contextual Unfoldings..... 34
Stefan Schwoon and C´esar Rodr´ıguez
The Power of Diversity ........................................... 43
Denis Th´erien
Regular Papers
Decidability and Shortest Strings in Formal Languages ............... 55
Levent Alpoge, Thomas Ang, Luke Schaeffer, and Jeffrey Shallit
On the Degree of Team Cooperation in CD Grammar Systems......... 68
Fernando Arroyo, Juan Castellanos, and Victor Mitrana
The Size-CostofBooleanOperationson ConstantHeight Deterministic
Pushdown Automata ............................................. 80
Zuzana Bedn´arov´a, Viliam Geffert, Carlo Mereghetti, and
Beatrice Palano
Syntactic Complexity of Prefix-, Suffix-, and Bifix-Free Regular
Languages ...................................................... 93
Janusz Brzozowski, Baiyu Li, and Yuli Ye
Geometrical Regular Languages and Linear Diophantine Equations..... 107
Jean-Marc Champarnaud, Jean-Philippe Dubernard,
Franck Guingne, and Hadrien Jeanne
On the Number of Components and Clusters of Non-returning Parallel
Communicating Grammar Systems................................. 121
Erzs´ebet Csuhaj-Varju´ and Gyo¨rgy Vaszil
On Contextual Grammars with Subregular Selection Languages........ 135
Ju¨rgen Dassow, Florin Manea, and Bianca Truthe
X Table of Contents
Remarks on Separating Words..................................... 147
Erik D. Demaine, Sarah Eisenstat, Jeffrey Shallit, and
David A. Wilson
State Complexity of Four Combined Operations Composed of Union,
Intersection, Star and Reversal .................................... 158
Yuan Gao and Sheng Yu
k-Local Internal Contextual Grammars ............................. 172
Radu Gramatovici and Florin Manea
On Synchronized Multitape and Multihead Automata ................ 184
Oscar H. Ibarra and Nicholas Q. Tran
State Complexity of Projected Languages ........................... 198
Galina Jira´skova´ and Tom´aˇs Masopust
Note on Reversal of Binary Regular Languages ...................... 212
Galina Jira´skova´ and Juraj Sˇebej
State Complexity of Operations on Two-Way Deterministic Finite
Automata over a Unary Alphabet.................................. 222
Michal Kunc and Alexander Okhotin
Kleene Theorems for Product Systems.............................. 235
Kamal Lodaya, Madhavan Mukund, and Ramchandra Phawade
Descriptional Complexity of Two-Way Pushdown Automata with
Restricted Head Reversals......................................... 248
Andreas Malcher, Carlo Mereghetti, and Beatrice Palano
State Trade-Offs in Unranked Tree Automata........................ 261
Xiaoxue Piao and Kai Salomaa
A ΣP ∪ΠP Lower Bound Using Mobile Membranes.................. 275
2 2
Shankara Narayanan Krishna and Gabriel Ciobanu
Language Classes Generated by Tree Controlled Grammars with
Bounded Nonterminal Complexity ................................. 289
Sherzod Turaev, Ju¨rgen Dassow, and Mohd Hasan Selamat
Transition Function Complexity of Finite Automata.................. 301
Ma¯ris Valdats
Complexity of Nondeterministic Multitape Computations Based on
Crossing Sequences............................................... 314
Jiˇr´ı Wiedermann
Author Index.................................................. 329
Linear Algebra Based Bounds for
One-Dimensional Cellular Automata
Jarkko Kari(cid:2)
Department of Mathematics, Universityof Turku
FI-20014 Turku,Finland
[email protected]
Abstract. Onepossiblecomplexitymeasureforacellularautomatonis
the size of its neighborhood. If a cellular automaton is reversible with
a small neighborhood, the inverse automaton may need a much larger
neighborhood. Our interest is to find good upper bounds for the size of
this inverse neighborhood. It turns out that a linear algebra approach
providesbetterboundsthananyknowncombinatorialmethods.Wealso
consider cellular automata that are not surjective. In this case there
must exist so-called orphans, finite patterns without a pre-image. The
length of the shortest orphan measures the degree of non-surjectiveness
of the map. Again, a linear algebra approach provides better bounds
on this length than known combinatorial methods. We also use linear
algebra to bound the minimum lengths of any diamond and any word
with a non-balanced number of pre-images. These both exist when the
cellularautomatoninquestionisnotsurjective.Allourresultsdealwith
one-dimensional cellular automata. Undecidability results imply that in
higher dimensional cases no computable upper bound exists for any of
theconsidered quantities.
A one-dimensional cellular automaton (CA) over a finite alphabet A is a trans-
formation F : AZ −→ AZ that is defined by a local update rule f : Am −→ A
applied uniformly across the cellular space Z. Bi-infinite sequences c ∈ AZ are
configurations.Foreverycell i∈Z,thestate c(i)∈Awillbedenotedbyci.The
local rule is applied at all cells simultaneously on the pattern around the cell to
get the state of the cell in the next configuration: For all c∈AZ and i∈Z
F(c)i =f(ci+k,ci+k+1,...,ci+k+m−1). (1)
Here, k ∈Z is a constant offset and m is the range of the neighborhood {k,k+
1,...,k+m−1} of the CA.
Thecaseofthesmallestnon-trivialrangem=2istermedtheradius-1 neigh-
2
borhood. Any neighborhood range m can be simulated by a radius-1 neighbor-
2
hoodbyblockingsegmentsofm−1cellsinto“supercells”thattaketheirvalues
over the alphabet Am−1. Therefore we mostly consider the radius-1 case.
2
Cellularautomataaremuchstudiedcomplexsystemsandmodelsofmassively
parallelcomputation.Viewingthemasdiscretedynamicalsystemsofteninvolves
(cid:2) Research supported by theAcademy of Finland Grant 131558.
M.Holzer,M.Kutrib,andG.Pighizzini(Eds.):DCFS2011,LNCS6808,pp.1–7,2011.
(cid:2)c Springer-VerlagBerlinHeidelberg2011
2 J. Kari
considering a natural compact topology on the configuration space AZ. The
topology is defined by a subbase consisting of sets Sa,i of configurations that
assignafixedstatea∈Ainafixedcelli∈Z.Cellularautomatatransformations
are continuous under this topology. Also the converse is true: cellular automata
maps are precisely those transformations AZ −→ AZ that are continuous and
that commute with translations [1].
Cellular automaton F : AZ −→ AZ is called reversible if it is bijective and
the inverse function is also a CA. A compactness argument directly implies
that for bijective CA the inverse function is automatically a CA. It is also easy
to see that an injective CA function is automatically surjective, so reversibility,
bijectivityandinjectivityareequivalentconceptsonCA.Ifacellularautomaton
is not surjective then there exist configurations without a pre-image, known as
Garden-Of-Eden configurations.
An application of a range m local CA rule on a finite word of length l yields
– by applying (1) inthe obviousway – a wordof lengthl−m+1.Compactness
of AZ implies that in non-surjective CA there must exist finite words without
a pre-image, so that an occurrence of such a word in a configuration forces the
configuration to be a Garden-Of-Eden. We call these words orphans. One can
also prove that in surjective one-dimensional CA all finite words have exactly
|A|m−1 pre-images.Inmeasuretheoreticterms this balance propertystates that
theuniformBorelmeasureispreservedbyallsurjectiveCA.Weseethatallnon-
surjective CA have unbalanced words whose number of pre-images is different
from the average |A|m−1. Clearly, there is a word of length l with too many
pre-images if and only if there is another one with too few pre-images.
A pair of configurations c,e ∈ AZ is called asymptotic if the set diff(c,e) =
{i ∈ Z | ci (cid:5)= ei} of cells where they differ, is finite. A CA is pre-injective if no
twodistinctasymptoticconfigurationscanhavethe sameimage.The celebrated
Garden-Of-Eden-theorem byMooreandMyhillstatesthatacellularautomaton
is surjective if and only it is pre-injective [6,7]. In particular, a non-surjective,
radius-1 cellularautomatonhasadiamond:apairaubandavbofdistinctwords
2
of equal length (u (cid:5)= v and |u| = |v|) that begin and end in identical states
a,b ∈ A, and that are mapped to the same word by the CA. We define the
length of the diamond to be the common length of the words u and v.
These fundamental concepts and results can be extended to higher dimen-
sionalCA, where the cellular space Z is replacedby Zd, for dimension d. See [3]
for more basic facts about cellular automata.
The topologicalapproachoutlinedaboveprovidesthe existence ofthe follow-
ing values. Consider the set CA(n) of radius-1 cellular automata with n states.
2
Let
– inv(n) denote the smallest number m such that the inverse map of every
reversible CA in CA(n) can be defined using range-m neighborhood,
– bal(n) denote the smallest number l such that every non-surjective CA in
CA(n) has an unbalanced word of length l,
– orph(n) denote the smallest number l such that every non-surjective CA in
CA(n) has an orphan of length l, and
