Table Of ContentLecture Notes in Computer Science 6742
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
Franz Winkler (Ed.)
Algebraic
Informatics
4th International Conference, CAI 2011
Linz, Austria, June 21-24, 2011
Proceedings
1 3
VolumeEditor
FranzWinkler
JohannesKeplerUniversity
ResearchInstituteforSymbolicComputation(RISC)
AltenbergerStraße69,4040,Linz,Austria
E-mail:[email protected]
ISSN0302-9743 e-ISSN1611-3349
ISBN978-3-642-21492-9 e-ISBN978-3-642-21493-6
DOI10.1007/978-3-642-21493-6
SpringerHeidelbergDordrechtLondonNewYork
LibraryofCongressControlNumber:2011928784
CRSubjectClassification(1998):F.4,I.1.3,F.1.1,F.4.1,F.4.3,F.4.2
LNCSSublibrary:SL1–TheoreticalComputerScienceandGeneralIssues
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Preface
CAI 2011 was the 4th International Conference on Algebraic Informatics. The
three previous conferences, CAI 2005, CAI 2007, and CAI 2009, were all held
at the Department of Mathematics of the Aristotle University of Thessaloniki
in Greece. At CAI 2009, the Steering Committee asked me to organize CAI
2011 at RISC in Linz/Hagenberg, Austria; a proposal which I gladly accepted.
In fact the focus of CAI is in very close correlation to the research topics at
RISC,seeingmathematicsandcomputerscienceasoneacademicfieldwithclose
internal interactions and mutual dependencies. Thus, CAI 2011 continued the
tradition established by previous CAIs, namely, to bring together researchers
from theoretical computer science and constructive algebra. The goal of this
endeavorwas to enhance the understanding of syntactic and semantic problems
byalgebraicmodels;andalsotopropagatetheapplicationofmoderntechniques
frominformaticsinalgebraiccomputation.CAI2011triedtoachievethesegoals
via invited lectures, tutorials, and contributed research talks.
As stated in the call for papers and on the homepage of CAI 2011, the
topics of interest included algebraic semantics, formal power series, syntactic
objects,algebraicpicture processing,finite and infinite computations,acceptors
and transducers for discrete structures, decision problems, algebraic characteri-
zationoflogicaltheories,processalgebra,algebraicalgorithms,algebraiccoding
theory, algebraic aspects of cryptography, term rewriting, algebraic aspects of
number theory.
The programof CAI 2011 consisted of 4 invited lectures and 13 contributed
talks. With 1 exception, all these presentations are reflected in the proceed-
ings. Additionally 2 tutorials were given: by A. Middeldorp and F. Neurauter
on “Termination and Complexity of Rewrite Systems,” and by P. Padawitz on
“Co/Algebraic Modelling and Verification at Work.” Unfortunatey, during the
preparation phase of CAI 2011 one of our leading colleagues and designated
memberoftheProgramCommittee,StephenL.Bloom,passedaway.Inthefirst
lectureattheopeningofCAI2011,Z.E´sik,memberoftheSteeringCommittee,
recalled the life and academic achievements of Stephen L. Bloom.
IamgratefultoagreatnumberofcolleaguesformakingCAI2011asuccessful
event: the members of the Steering Committee, the colleagues in the Program
Committee, my co-workers in the Local Committee, and also Alfred Hofmann
and his team at Springer LNCS.
June 2011 Franz Winkler
Organization
CAI 2011 was organized by the Research Institute for Symbolic Computation,
Johannes Kepler University Linz.
Steering Committee
Jean Berstel, Marne-la-Vall´ee
Symeon Bozapalidis, Thessaloniki
Zolt´an E´sik, Szeged
Werner Kuich, Vienna
Arto Salomaa, Turku
Program Committee
Erhard Aichinger, Linz
Armin Biere, Linz
Symeon Bozapalidis, Thessaloniki
Bruno Courcelle, Bordeaux
Wan Fokkink, Amsterdam
Zolt´an Fu¨lo¨p, Szeged
Ulrike Golas, Berlin
Kevin Hammond, St. Andrews
Sˇtˇepˇan Holub, Prague
Jan Willem Klop, Amsterdam
Barbara Ko¨nig, Duisburg
Laura Kova´cs, Vienna
Salvador Lucas, Valencia
Damian Niwinski, Warsaw
Attila Petho˝, Debrecen
George Rahonis, Thessaloniki
Simona Ronchi Della Rocca, Turin
Davide Sangiorgi, Bologna
Wolfgang Schreiner, Linz
Mikhail Volkov, Ekaterinburg
Franz Winkler, Linz (Chair)
VIII Organization
Referees
E. Aichinger B. Ko¨nig
A. Biere E. Kopczynski
Y. Bilu L. Kova´cs
E. Bonelli A. Kreuzer
S. Bozapalidis S. Lucas
L. Braud A. Mahboubi
A. Cano D. Masulovic
B. Courcelle O. Matz
U. De Liguoro T. Mossakowski
M. Deneufchaˆtel D. Niwinski
A. Di Bucchianico K. Ogata
C. Do¨nch A. Petho˝
W. Fokkink M. Pradella
Z. Fu¨lo¨p P. Prihoda
J. Fuss G. Rahonis
S. Galbraith E. Rodr´ıguez Carbonell
D. Giammarresi S. Ronchi Della Rocca
U. Golas D. Sangiorgi
A. Griggio W. Schreiner
Y. Guiraud S. Sheinvald
K. Hammond I. Shparlinski
Sˇ. Holub Y. Stamatiou
A. Kalampakas E. Teske-Wilson
T. Kazana E. Verbeek
W.-F. Ke M. Volkov
M. Klazar F. Winkler
J.W. Klop
Organizing Committee
Christian Do¨nch
Franz Lichtenberger
Johannes Middeke
Chaˆu Ngoˆ
Wolfgang Schreiner
Franz Winkler (Chair)
Sponsors
Bundesministerium fu¨r Wissenschaft und Forschung (BMWF)
Johannes Kepler Universita¨t Linz (JKU)
Linzer Hochschulfonds (LHF)
Land Ober¨osterreich
Table of Contents
Invited Papers
Joint Spectral Radius Theory for Automated Complexity Analysis of
Rewrite Systems ................................................. 1
Aart Middeldorp, Georg Moser, Friedrich Neurauter,
Johannes Waldmann, and Harald Zankl
From Grammars and Automata to Algebras and Coalgebras........... 21
Peter Padawitz
Theme and Variations on the Concatenation Product................. 44
Jean-E´ric Pin
Some Combinatorial Applications of Gro¨bner Bases .................. 65
Lajos R´onyai and Tama´s M´esza´ros
Contributed Papers
Comparing Necessary Conditions for Recognizability of
Two-dimensional Languages....................................... 84
Marcella Anselmo and Maria Madonia
Typed Monoids – An Eilenberg-Like Theorem for Non Regular
Languages ...................................................... 97
Christoph Behle, Andreas Krebs, and Stephanie Reifferscheid
Codes and CombinatorialStructures from Circular Planar Nearrings ... 115
Anna Benini, Achille Frigeri, and Fiorenza Morini
Independence of Hyperlogarithms over Function Fields via Algebraic
Combinatorics................................................... 127
Matthieu Deneufchaˆtel, G´erard H.E. Duchamp,
Vincel Hoang Ngoc Minh, and Allan I. Solomon
Quantifier Elimination over Finite Fields Using Gro¨bner Bases......... 140
Sicun Gao, Andr´e Platzer, and Edmund M. Clarke
F-Rank-Width of (Edge-Colored)Graphs ........................... 158
Mamadou Moustapha Kant´e and Michael Rao
An Algorithm for Computing a Basis of a Finite Abelian Group ....... 174
Gregory Karagiorgos and Dimitrios Poulakis
X Table of Contents
Rewriting in Varieties of Idempotent Semigroups..................... 185
Ondˇrej Kl´ıma, Miroslav Korbela´ˇr, and Libor Pola´k
Simplifying Algebraic Functional Systems ........................... 201
Cynthia Kop
Hadamard Matrices, Designs and Their Secret-Sharing Schemes........ 216
Christos Koukouvinos, Dimitris E. Simos, and Zlatko Varbanov
I-RiSC: An SMT-Compliant Solver for the Existential Fragment of
Real Algebra .................................................... 230
Ulrich Loup and Erika A´brah´am
Variable Tree Automata over Infinite Ranked Alphabets .............. 247
Irini-Eleftheria Mens and George Rahonis
Author Index.................................................. 261
Joint Spectral Radius Theory for Automated
(cid:2)
Complexity Analysis of Rewrite Systems
Aart Middeldorp1, Georg Moser1, Friedrich Neurauter1,
Johannes Waldmann2, and Harald Zankl1
1 Instituteof Computer Science, Universityof Innsbruck,Austria
2 Fakult¨at Informatik, Mathematik undNaturwissenschaften, Hochschule fu¨r
Technik,Wirtschaft undKulturLeipzig, Germany
Abstract. Matrixinterpretationscanbeusedtoboundthederivational
complexity of term rewrite systems. In particular, triangular matrix in-
terpretations overthenatural numbersare known toinducepolynomial
upper bounds on the derivational complexity of (compatible) rewrite
systems. Recently two different improvements were proposed, based on
the theory of weighted automata and linear algebra. In this paper we
strengthen and unify these improvements by using joint spectral radius
theory.
Keywords: derivational complexity, matrix interpretations, weighted
automata, joint spectral radius.
1 Introduction
This paper is concerned with automated complexity analysis of term rewrite
systems. Given a terminating rewrite system, the aim is to obtain information
about the maximal length of rewrite sequences in terms of the size of the initial
term. This is knownas derivationalcomplexity. Developing methods for bound-
ing the derivational complexity of rewrite systems has become an active and
competitive1 researcharea in the past few years (e.g. [6, 11–15, 19, 21]).
Matrix interpretations [4] are a popular method for automatically proving
termination of rewrite systems. They can readily be used to establish upper
bounds on the derivational complexity of compatible rewrite systems. However,
in general, matrix interpretations induce exponential (rather than polynomial)
upperbounds.Inordertoobtainpolynomialupperbounds,thematricesusedin
a matrix interpretation must satisfy certain (additional) restrictions, the study
of which is the central concern of [14, 15, 19].
So what are the conditions for polynomial boundedness of a matrix inter-
pretation? In the literature,two different approacheshave emerged. On the one
hand,thereistheautomata-basedapproachof[19],wherematricesareviewedas
(cid:2) This research is supported by FWF (Austrian Science Fund) project P20133.
Friedrich Neurauter is supported by a grant of theUniversity of Innsbruck.
1 http://www.termination-portal.org/wiki/Complexity
F.Winkler(Ed.):CAI2011,LNCS6742,pp.1–20,2011.
(cid:2)c Springer-VerlagBerlinHeidelberg2011
2 A. Middeldorp et al.
weighted(word)automatacomputinga weightfunction,whichisrequiredtobe
polynomially bounded. The resultis a complete characterization(i.e., necessary
and sufficient conditions) of polynomially bounded matrix interpretations over
N. On the other hand, there is the algebraic approachpursued in [15] (originat-
ing from[14]) thatcanhandle matrix interpretationsoverN,Q,andR but only
provides sufficient conditions for polynomial boundedness. In what follows, we
shall see, however,that these two seemingly different approaches can be unified
and strengthened with the help of joint spectral radius theory [9, 10], a branch
of mathematics dedicated to studying the growth rate of products of matrices
taken from a set.
Theremainderofthis paperisorganizedasfollows.Inthenextsectionwere-
callpreliminariesfromlinearalgebraandtermrewriting.Wegiveabriefaccount
of the matrix method for proving termination of rewrite systems. In Section 3
weintroducethealgebraicapproachforcharacterizingthe polynomialgrowthof
matrix interpretations. We improve upon the results of [15] by considering the
minimal polynomial associated with the component-wise maximum matrix of
the interpretation. We further show that the joint spectral radius of the matri-
cesintheinterpretationprovidesabettercharacterizationofpolynomialgrowth
and provide conditions for the decidability of the latter. Section 4 is devoted to
automata-based methods for characterizing the polynomial growth of matrix
interpretations. We revisit the characterization results of [19] and provide pre-
cise complexity statements. In Section 5 we unify the two approaches and show
that, in theory at least, the joint spectral radius theory approach subsumes the
automata-based approach. Automation of the results presented in earlier sec-
tions is the topic of Section 6. To this end we extend the results from [15, 19].
We also provide experimental results. We conclude with suggestions for future
researchin Section 7.
2 Preliminaries
As usual, we denote by N, Z, Q and R the sets of natural, integer, rational and
real numbers. Given D ∈ {N,Z,Q,R} and m ∈ D, >D denotes the standard
order of the respective domain and Dm abbreviates {x∈D |x(cid:2)m}.
Linear Algebra: Let R be a ring (e.g., Z, Q, R). The ring of all n-dimensional
squarematrices overRisdenotedbyRn×nandthepolynomial ring innindeter-
minates x1,...,xn by R[x1,...,xn](cid:2). In the special case n = 1, any polynomial
p ∈ R[x] can be written as p(x) = dk=0akxk for some d ∈ N. For the largest
k such that ak (cid:3)=0, we call akxk the leading term of p, ak its leading coefficient
and k its degree. The polynomial p is said to be monic if its leading coefficient
is one. It is said to be linear, quadratic, cubic if its degree is one, two, three.
IncaseRisequippedwithapartialorder(cid:2),thecomponent-wiseextensionof
this orderto Rn×n is alsodenotedas(cid:2).We saythatamatrix Ais non-negative
ifA(cid:2)0 anddenote the setofallnon-negativen-dimensionalsquarematricesof
Zn×n by Nn×n. The n×n identity matrix is denoted by In and the n×n zero
matrix is denoted by 0n. We simply write I and 0 if n is clear fromthe context.
