Mail address
LSV, CNRS & ENS de Cachan
61, avenue du Président Wilson
94235 CACHAN Cedex
France

Office
RH-B-002 Email
zetzsche(at)lsv(dot)fr

Since November 2015, I am a Postdoc at the LSV Cachan,
funded by a fellowship of the DAAD (German Academic Exchange Service). Before
that, I obtained a PhD with Prof. Dr. Roland Meyer
in the Concurrency Theory Group
in Kaiserslautern. I defended my dissertation on June 19th, 2015. Until December
2010, I studied Computer Science (with a minor in Mathematics) at
Universität Hamburg.

My research is mostly concerned with questions of expressiveness and
decidability for infinite-state systems. Topics I have worked on include (in
no particular order):

Abstract
This work studies which storage mechanisms in automata permit
decidability of the reachability problem. The question is formalized
using valence automata, an abstract model that generalizes automata with
storage. For each of a variety of storage mechanisms, one can choose a
(typically infinite) monoid $M$ such that valence automata over $M$ are
equivalent to (one-way) automata with this type of storage.
In fact, many interesting storage mechanisms can be realized by monoids defined
by finite graphs, called graph monoids.
Hence, we study for which graph
monoids the emptiness problem for valence automata is decidable. A particular
model realized by graph monoids is that of Petri nets with a pushdown stack.
For these, decidability is a long-standing open question and we do not answer
it here.
However, if one excludes subgraphs corresponding to this model, a
characterization can be achieved. This characterization yields a new extension
of Petri nets with a decidable reachability problem. Moreover, we provide a
description of those storage mechanisms for which decidability remains open.
This leads to a natural model that generalizes both pushdown Petri nets and
priority multicounter machines.
The cases that are proven decidable together make up a natural and apparently
new extension of Petri nets with decidable reachability. It is finally shown
that this model can be combined with another such extension by Atig and Ganty:
We present a further decidability result that subsumes both of these Petri net
extensions.

@MISC{Zetzsche2016b,
AUTHOR = {Zetzsche, Georg},
TITLE = {The Emptiness Problem for Valence Automata over Graph Monoids},
NOTE = {Submitted to journal},
YEAR = {2016}
}

Abstract
We investigate the monoid of transformations that are induced by
sequences of writing to and reading from a queue storage. We describe
this monoid by means of a confluent and terminating semi-Thue system
and study some of its basic algebraic properties, e.g., conjugacy.
Moreover, we show that while several properties concerning its
rational subsets are undecidable, their uniform membership problem is
NL-complete. Furthermore, we present an algebraic characterization of
this monoid's recognizable subsets. Finally, we prove that it is not
Thurston-automatic.

@MISC{HuschenbettKuskeZetzsche2016a,
AUTHOR = {Huschenbett, Martin and Kuske, Dietrich and Zetzsche, Georg},
TITLE = {The Monoid of Queue Actions},
NOTE = {Submitted to journal},
YEAR = {2016}
}

Abstract
The separability problem for word languages of a class $\C$ by languages of
a class $\S$ asks, for two given languages $I$ and $E$ from $\C$, whether
there exists a language $S$ from $\S$ that includes $I$ and excludes $E$,
that is, $I \subseteq S$ and $S\cap E = \emptyset$. In this work, we assume
some mild closure properties for $\C$ and study for which such classes $\C$,
separability by piecewise testable languages (PTL) is decidable. We
characterize these classes in terms of decidability of (two variants of) an
unboundedness problem. From this we deduce that separability by PTL is
decidable for a number of language classes, such as the context-free
languages and languages of labeled vector addition systems. Furthermore, it
follows that separability by PTL is decidable if and only if one can compute
for any language of the class its downward closure wrt.\ the \subword
ordering (i.e., if the set of \subwords of any language of the class
is effectively regular).
The obtained decidability results contrast some undecidability
results. In fact, for all the (non-regular) language classes we
present as examples with decidable separability, it is undecidable
whether a given language is a PTL itself.
Our characterization involves a result of independent interest,
which states that for \emph{any} kind of languages $I$ and $E$,
non-separability is equivalent to the existence of common patterns
in $I$ and $E$.

@MISC{CzerwinskiMartensRooijenZeitounZetzsche2015a,
AUTHOR = {Czerwi{\'{n}}ski, Wojciech and Martens, Wim and van Rooijen, Lorijn and Zeitoun, Marc and Zetzsche, Georg},
TITLE = {A Characterization for Decidable Separability by Piecewise Testable Languages},
NOTE = {Submitted to journal},
YEAR = {2015}
}

Abstract
For a language $L$, we consider its cyclic closure, and more generally the
language $C^k(L)$, which consists of all words obtained by partitioning words
from $L$ into $k$ factors and permuting them. We prove that the classes of
ET0L and EDT0L languages are closed under the operators $C^k$. This both
sharpens and generalises Brandstädt's result that if $L$ is context-free then
$C^k(L)$ is context-sensitive and not context-free in general for $k\geq 3$.
We also show that the cyclic closure of an indexed language is indexed.

@MISC{BroughCiobanuElderZetzsche2016a,
AUTHOR = {Brough, Tara and Ciobanu, Laura and Elder, Murray and Zetzsche, Georg},
TITLE = {Permutations of context-free, ET0L and indexed languages},
NOTE = {Accepted for Discrete Mathematics {\&} Theoretical Computer Science},
YEAR = {2016}
}

Abstract
We study what languages can be constructed from a non-regular
language $L$ using Boolean operations and (synchronized) rational
transductions. If all rational transductions are allowed, one can
construct the whole arithemtical hierarchy relative to $L$. If only
synchronized rational transductions are allowed, we present
non-regular languages that allow to construct at least languages
arbitrarily high in the arithmetical hierarchy and we present
non-regular languages that allow to construct only decidable
languages.
A consequence of the results is that aside from the regular
languages, no full trio generated by a single language is closed under
complementation.
Our construction also shows that there is a fixed rational Kripke
frame such that assigning an arbitrary non-regular language to some
variable allows the definition of any language from the arithmetical
hierarchy in the corresponding Kripke structure using multimodal
logic.

@MISC{ZetzscheKuskeLohrey2015a,
AUTHOR = {Zetzsche, Georg and Kuske, Dietrich and Lohrey, Markus},
TITLE = {On {Boolean} closed full trios and rational {Kripke} frames},
NOTE = {Accepted for Theory of Computing Systems},
YEAR = {2016}
}

@MISC{KoenigLohreyZetzsche2015a,
AUTHOR = {K{\"o}nig, Daniel and Lohrey, Markus and Zetzsche, Georg},
TITLE = {Knapsack and subset sum problems in nilpotent, polycyclic, and co-context-free groups},
NOTE = {Accepted for Contemporary Mathematics},
YEAR = {2016}
}

Abstract
The downward closure of a language is the set of all (not necessarily
contiguous) subwords of its members. It is well-known that the downward
closure of every language is regular. Moreover, recent results show
that downward closures are computable for quite powerful system models.
One advantage of abstracting a language by its downward closure is that then,
equivalence and inclusion become decidable. In this work, we study the
complexity of these two problems. More precisely, we consider the following
decision problems: Given languages $K$ and $L$ from classes $\C$ and $\D$,
respectively, does the downward closure of $K$ include (equal) that of $L$?
These problems are investigated for finite automata, one-counter automata,
context-free grammars, and reversal-bounded counter automata. For each
combination, we prove a completeness result either for fixed or for arbitrary
alphabets. Moreover, for Petri net languages, we show that both problems are
Ackermann-hard and for higher-order pushdown automata of order $k$, we prove
hardness for complements of nondeterministic $k$-fold exponential time.

@MISC{Zetzsche2016a,
AUTHOR = {Zetzsche, Georg},
TITLE = {The complexity of downward closure comparisons},
YEAR = {2016},
NOTE = {To appear in Proc. of the 43rd International Colloquium on Automata, Languages and Programming (ICALP 2016)}
}

Abstract
We study the computational and descriptional complexity
of the following transformation: Given a one-counter automaton (OCA) A,
construct a nondeterministic finite automaton (NFA) B
that recognizes an abstraction of the language of A:
its (1) downward closure, (2) upward closure, or (3) Parikh image.
For the Parikh image over a fixed alphabet and for the upward and
downward closures, we find polynomial-time algorithms that compute
such an NFA. For the Parikh image with the alphabet as part of
the input, we find a quasi-polynomial time algorithm and prove
a completeness result: we construct a sequence of OCA that admits
a polynomial-time algorithm iff there is one for all OCA.
For all three abstractions, it was previously unknown
if appropriate NFA of sub-exponential size exist.

@MISC{AtigChistikovHofmanKumarSaivasanZetzsche2015a,
AUTHOR = {Atig, Mohamed Faouzi and Chistikov, Dmitry and Hofman, Piotr and Kumar, K Narayan and Saivasan, Prakash and Zetzsche, Georg},
TITLE = {Complexity of regular abstractions of one-counter languages},
YEAR = {2016},
NOTE = {To appear in Proc. of the Thirty-First Annual ACM/IEEE Symposium on Logic in Computer Science (LICS 2016)}
}

Abstract
First-order logic with the reachability predicate (FOR)
is an important means of specification in system analysis.
Its decidability status is known for some individual
types of infinite-state systems such as pushdown (decidable)
and vector addition systems (undecidable).
This work aims at a general understanding of which types of
systems admit decidability. As a unifying model, we employ
valence systems over graph monoids, which feature a
finite-state control and are parameterized by a monoid to
represent their storage mechanism. As special cases, this
includes pushdown systems, various types of counter systems
(such as vector addition systems) and combinations thereof.
Our main result is a complete characterization of those
graph monoids where FOR is decidable for the resulting
transition systems.

@MISC{DOsualdoMeyerZetzsche2016a,
AUTHOR = {D'Osualdo, Emanuele and Meyer, Roland and Zetzsche, Georg},
TITLE = {First-order logic with reachability for infinite-state systems},
NOTE = {To appear in Proc. of the Thirty-First Annual ACM/IEEE Symposium on Logic in Computer Science (LICS 2016)},
YEAR = {2016}
}

Abstract
It is shown that the knapsack problem, which was introduced by Myasnikov et al.
for arbitrary finitely generated groups, can be solved in NP for graph
groups. This result even holds if the group elements are represented in a
compressed form by SLPs, which generalizes the classical NP-completeness
result of the integer knapsack problem. We also prove general transfer results:
NP-membership of the knapsack problem is passed on to finite
extensions, HNN-extensions over finite associated subgroups, and amalgamated
products with finite identified subgroups.

@INPROCEEDINGS{LohreyZetzsche2016a,
AUTHOR = {Lohrey, Markus and Zetzsche, Georg},
TITLE = {Knapsack in Graph Groups, HNN-Extensions and Amalgamated Products},
PAGES = {50:1--50:14},
SERIES = {Leibniz International Proceedings in Informatics (LIPIcs)},
EDITOR = {Nicolas Ollinger and Heribert Vollmer},
ADDRESS = {Dagstuhl, Germany},
PUBLISHER = {Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
VOLUME = {47},
BOOKTITLE = {Proc. of the 33rd International Symposium on Theoretical Aspects of Computer Science (STACS 2016)},
YEAR = {2016},
DOI = {10.4230/LIPIcs.STACS.2016.50}
}

Abstract
The downward closure of a word language is the set
of all (not necessarily contiguous) subwords of its
members. It is well-known that the downward closure of
any language is regular. While the downward closure
appears to be a powerful abstraction, algorithms for
computing a finite automaton for the downward closure
of a given language have been established only for few
language classes.
This work presents a simple general method for computing
downward closures. For language classes that are closed
under rational transductions, it is shown that the
computation of downward closures can be reduced to
checking a certain unboundedness property.
This result is used to prove that downward closures
are computable for (i) every language class with
effectively semilinear Parikh images that are closed
under rational transductions, (ii) matrix languages,
and (iii) indexed languages (equivalently, languages
accepted by higher-order pushdown automata of order~2).

@INPROCEEDINGS{Zetzsche2015b,
AUTHOR = {Zetzsche, Georg},
TITLE = {An Approach to Computing Downward Closures},
EDITOR = {Magn{\'{u}}s M. Halld{\'{o}}rsson and Kazuo Iwama and Naoki Kobayashi and Bettina Speckmann},
SERIES = {LNCS},
VOLUME = {9135},
PAGES = {440--451},
PUBLISHER = {Springer},
ADDRESS = {Berlin Heidelberg},
YEAR = {2015},
BOOKTITLE = {Proc. of the 42nd International Colloquium on Automata, Languages and Programming (ICALP 2015)},
DOI = {10.1007/978-3-662-47666-6_35}
}

Abstract
The downward closure of a language $L$ of words is the set of all (not
necessarily contiguous) subwords of members of $L$. It is well known that the
downward closure of any language is regular. Although the downward
closure seems to be a promising abstraction, there are only few language
classes for which an automaton for the downward closure is known to be
computable.
It is shown here that for stacked counter automata, the downward closure is
computable. Stacked counter automata are finite automata with a storage
mechanism obtained by \emph{adding blind counters} and \emph{building stacks}.
Hence, they generalize pushdown and blind counter automata.
The class of languages accepted by these automata are precisely those in the
hierarchy obtained from the context-free languages by alternating two closure
operators: imposing semilinear constraints and taking the algebraic extension.
The main tool for computing downward closures is the new concept of Parikh
annotations. As a second application of Parikh annotations, it is shown that
the hierarchy above is strict at every level.

@INPROCEEDINGS{Zetzsche2015a,
AUTHOR = {Zetzsche, Georg},
TITLE = {Computing downward closures for stacked counter automata},
PAGES = {743--756},
SERIES = {Leibniz International Proceedings in Informatics (LIPIcs)},
EDITOR = {Ernst W. Mayr and Nicolas Ollinger},
ADDRESS = {Dagstuhl, Germany},
PUBLISHER = {Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
VOLUME = {30},
BOOKTITLE = {Proc. of the 32nd International Symposium on Theoretical Aspects of Computer Science (STACS 2015)},
YEAR = {2015},
DOI = {10.4230/LIPIcs.STACS.2015.743}
}

Abstract
We investigate the monoid of transformations that are induced by
sequences of writing to and reading from a queue storage. We describe
this monoid by means of a confluent and terminating semi-Thue system
and study some of its basic algebraic properties, e.g., conjugacy.
Moreover, we show that while several properties concerning its
rational subsets are undecidable, their uniform membership problem is
NL-complete. Furthermore, we present an algebraic characterization of
this monoid's recognizable subsets. Finally, we prove that it is not
Thurston-automatic.

@INPROCEEDINGS{HuschenbettKuskeZetzsche2014a,
AUTHOR = {Huschenbett, Martin and Kuske, Dietrich and Zetzsche, Georg},
TITLE = {The Monoid of Queue Actions},
BOOKTITLE = {Proc. of the 39th International Symposium on Mathematical Foundations of Computer Science (MFCS 2014)},
VOLUME = {8634},
EDITOR = {Csuhaj-Varj{\'{u}}, Erzs{\'{e}}bet and Dietzfelbinger, Martin and {\'{E}}sik, Zolt{\'{a}}n},
SERIES = {LNCS},
PUBLISHER = {Springer},
ADDRESS = {Berlin Heidelberg},
PAGES = {340--351},
YEAR = {2014},
DOI = {10.1007/978-3-662-44522-8_29}
}

Abstract
A Boolean closed full trio is a class of languages that is closed under the
Boolean operations (union, intersection, and complementation) and rational
transductions. It is well-known that the regular languages constitute such a
Boolean closed full trio. It is shown here that every such language class that
contains any non-regular language already includes the whole arithmetical
hierarchy (and even the one relative to this language).
A consequence of this result is that aside from the regular languages, no full
trio generated by one language is closed under complementation.
Our construction also shows that there is a fixed rational Kripke frame such
that assigning an arbitrary non-regular language to some variable allows the
definition of any language from the arithmetical hierarchy in the
corresponding Kripke structure using multimodal logic.

@INPROCEEDINGS{LohreyZetzsche2014a,
AUTHOR = {Lohrey, Markus and Zetzsche, Georg},
TITLE = {On {Boolean} closed full trios and rational {Kripke} frames},
BOOKTITLE = {Proc. of the 31st International Symposium on Theoretical Aspects of Computer Science (STACS 2014)},
PAGES = {530--541},
SERIES = {Leibniz International Proceedings in Informatics (LIPIcs)},
VOLUME = {25},
EDITOR = {Mayr, Ernst W. and Portier, Natacha},
PUBLISHER = {Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
ADDRESS = {Dagstuhl, Germany},
YEAR = {2014},
DOI = {10.4230/LIPIcs.STACS.2014.530}
}

Abstract
Valence automata are a generalization of various models of automata with
storage. Here, each edge carries, in addition to an input word, an element of
a monoid. A computation is considered valid if multiplying the monoid elements
on the visited edges yields the identity element. By choosing suitable
monoids, a variety of automata models can be obtained as special valence
automata.
This work is concerned with the accepting power of valence automata.
Specifically, we ask for which monoids valence automata can accept only
context-free languages or only languages with semilinear Parikh image,
respectively.
First, we present a characterization of those graph products (of monoids) for
which valence automata accept only context-free languages. Second, we provide a
necessary and sufficient condition for a graph product of copies of the
bicyclic monoid and the integers to yield only languages with semilinear Parikh
image when used as a storage mechanism in valence automata. Third, we show that
all languages accepted by valence automata over torsion groups have a
semilinear Parikh image.

@INPROCEEDINGS{BuckheisterZetzsche2013a,
AUTHOR = {Buckheister, P. and Zetzsche, Georg},
TITLE = {Semilinearity and Context-Freeness of Languages Accepted by Valence Automata},
YEAR = {2013},
BOOKTITLE = {Proc. of the 38th International Symposium on Mathematical Foundations of Computer Science (MFCS 2013)},
EDITOR = {Chatterjee, Krishnendu and Sgall, Jir{\'{i}}},
PUBLISHER = {Springer},
ADDRESS = {Berlin Heidelberg},
PAGES = {231--242},
VOLUME = {8087},
SERIES = {LNCS},
DOI = {10.1007/978-3-642-40313-2_22}
}

Abstract
It is shown that membership in rational subsets of wreath
products $H \wr V$ with $H$ a finite group and $V$ a virtually
free group is decidable. On the other hand, it is shown that
there exists a fixed finitely generated submonoid in the wreath
product $\mathbb{Z}\wr\mathbb{Z}$ with an undecidable
membership problem.

@INPROCEEDINGS{LohreySteinbergZetzsche2013a,
AUTHOR = {Lohrey, Markus and Steinberg, Benjamin and Zetzsche, Georg},
TITLE = {Rational Subsets and Submonoids of Wreath Products},
BOOKTITLE = {Proc. of the 40th International Colloquium on Automata, Languages and Programming (ICALP 2013)},
EDITOR = {Fedor V. Fomin and R{\={u}}si{\c{n}}{\v{s}} Freivalds and Marta Kwiatkowska and David Peleg},
PUBLISHER = {Springer},
ADDRESS = {Berlin Heidelberg},
VOLUME = {7966},
SERIES = {LNCS},
YEAR = {2013},
PAGES = {361--372},
DOI = {10.1007/978-3-642-39212-2_33}
}

Abstract
We consider the computational power of silent transitions in one-way automata
with storage. Specifically, we ask which storage mechanisms admit a
transformation of a given automaton into one that accepts the same language
and reads at least one input symbol in each step.
We study this question using the model of valence automata. Here, a finite
automaton is equipped with a storage mechanism that is given by a monoid.
This work presents generalizations of known results on silent transitions.
For two classes of monoids, it provides characterizations of those monoids that
allow the removal of silent transitions. Both classes are defined by
graph products of copies of the bicyclic monoid and the group of integers. The
first class contains pushdown storages as well as the blind counters while the
second class contains the blind and the partially blind counters.

@INPROCEEDINGS{Zetzsche2013a,
AUTHOR = {Zetzsche, Georg},
TITLE = {Silent Transitions in Automata with Storage},
BOOKTITLE = {Proc. of the 40th International Colloquium on Automata, Languages and Programming (ICALP 2013)},
EDITOR = {Fedor V. Fomin and R{\={u}}si{\c{n}}{\v{s}} Freivalds and Marta Kwiatkowska and David Peleg},
PUBLISHER = {Springer},
ADDRESS = {Berlin Heidelberg},
VOLUME = {7966},
SERIES = {LNCS},
YEAR = {2013},
PAGES = {434--445},
DOI = {10.1007/978-3-642-39212-2_39}
}

Abstract
In each grammar model, it is an important question whether erasing productions
are necessary to generate all languages. Using the concept of grammars with
control languages by Salomaa, which offers a uniform treatment of a variety of
grammar models, we present a condition on the class of control languages that
guarantees that erasing productions are avoidable in the resulting grammar
model. On the one hand, this generalizes the previous result that in Petri net
controlled grammars, erasing productions can be eliminated. On the other hand,
it allows us to infer that the same is true for vector grammars.

@INPROCEEDINGS{Zetzsche2011c,
AUTHOR = {Zetzsche, Georg},
TITLE = {A Sufficient Condition for Erasing Productions to Be Avoidable},
BOOKTITLE = {Proc. of the 15th International Conference on Developments in Language Theory (DLT 2011)},
EDITOR = {Giancarlo Mauri and Alberto Leporati},
PUBLISHER = {Springer},
ADDRESS = {Berlin Heidelberg},
VOLUME = {6795},
SERIES = {LNCS},
PAGES = {452--463},
YEAR = {2011},
DOI = {10.1007/978-3-642-22321-1_39}
}

Abstract
During recent decades, classical models in language theory have been extended
by control mechanisms defined by monoids. We study which monoids cause the
extensions of context-free grammars, finite automata, or finite state
transducers to exceed the capacity of the original model. Furthermore, we
investigate when, in the extended automata model, the nondeterministic variant
differs from the deterministic one in capacity. We show that all these
conditions are in fact equivalent and present an algebraic characterization. In
particular, the open question of whether every language generated by a valence
grammar over a finite monoid is context-free is provided with a positive
answer.

@INPROCEEDINGS{Zetzsche2011b,
AUTHOR = {Zetzsche, Georg},
TITLE = {On the Capabilities of Grammars, Automata, and Transducers Controlled by Monoids},
BOOKTITLE = {Proc. of the 38th International Colloquium on Automata, Languages and Programming (ICALP 2011)},
EDITOR = {Luca Aceto et al.},
PUBLISHER = {Springer},
ADDRESS = {Berlin Heidelberg},
VOLUME = {6756},
SERIES = {LNCS},
PAGES = {222--233},
YEAR = {2011},
DOI = {10.1007/978-3-642-22012-8_17}
}

Abstract
Three open questions in the theory of regulated
rewriting are addressed. The first is whether every
permitting random context grammar has a non-erasing
equivalent. The second asks whether the same is true
for matrix grammars without appearance checking. The
third concerns whether permitting random context
grammars have the same generative capacity as matrix
grammars without appearance checking.
The main result is a positive answer to the first
question. For the other two, conjectures are
presented. It is then deduced from the main result
that at least one of the two holds.

@INPROCEEDINGS{Zetzsche2010,
AUTHOR = {Zetzsche, Georg},
TITLE = {On Erasing Productions in Random Context Grammars},
BOOKTITLE = {Proc. of the 37th International Colloquium on Automata, Languages and Programming (ICALP 2010)},
EDITOR = {S. Abramsky et al.},
VOLUME = {6199},
SERIES = {LNCS},
PUBLISHER = {Springer},
ADDRESS = {Berlin Heidelberg},
PAGES = {175--186},
YEAR = {2010},
DOI = {10.1007/978-3-642-14162-1_15}
}

Abstract
It is shown that applying linear erasing to a Petri net
language yields a language generated by a non-erasing
matrix grammar. The proof uses Petri net controlled
grammars. These are context-free grammars, where the
application of productions has to comply with a firing
sequence in a Petri net. Petri net controlled grammars are
equivalent to arbitrary matrix grammars (without
appearance checking), but a certain restriction on them
(linear Petri net controlled grammars) leads to the class
of languages generated by non-erasing matrix grammars.
It is also shown that in Petri net controlled grammars
(with final markings and arbitrary labeling), erasing
rules can be eliminated, which yields a reformulation of
the problem of whether erasing rules in matrix grammars
can be eliminated.

@INPROCEEDINGS{Zetzsche09,
AUTHOR = {Zetzsche, Georg},
TITLE = {Erasing in {Petri} Net Languages and Matrix Grammars},
BOOKTITLE = {Proc. of the 13th International Conference on Developments in Language Theory (DLT 2009)},
EDITOR = {Diekert, Volker and Nowotka, Dirk},
PAGES = {490--501},
VOLUME = {5583},
YEAR = {2009},
SERIES = {LNCS},
PUBLISHER = {Springer},
ADDRESS = {Berlin Heidelberg},
DOI = {10.1007/978-3-642-02737-6_40}
}

Abstract
We compare various modes of firing transitions in Petri nets
and define classes of languages defined this way. We define
languages through steps, i. e. sets of transitions, maximal
steps, multi-steps, and maximal multi-steps of transitions
in Petri nets, but in a different manner than those defined
in [Burk 81a,Burk 83], by considering labeled transitions.
We will show that we obtain a hierarchy of families of
languages defined by multiple use of transition in firing
transitions in a single multistep. Except for the maximal
multi-steps all classes can be simulated by sequential
firing of transitions.

@INPROCEEDINGS{JantzenZetzsche2008a,
AUTHOR = {Jantzen, Matthias and Zetzsche, Georg},
TITLE = {Labeled Step Sequences in Petri Nets},
BOOKTITLE = {Proc. of the International Conference on Applications and Theory of Petri nets (PETRI NETS 2008)},
ADDRESS = {Berlin Heidelberg},
YEAR = {2008},
PAGES = {270--287},
EDITOR = {van Hee, Kees M. and Valk, R{\"u}diger},
PUBLISHER = {Springer},
VOLUME = {5062},
SERIES = {LNCS},
DOI = {10.1007/978-3-540-68746-7_19}
}

Abstract
It is shown that membership in rational subsets of wreath
products $H \wr V$ with $H$ a finite group and $V$ a virtually
free group is decidable. On the other hand, it is shown that
there exists a fixed finitely generated submonoid in the wreath
product $\mathbb{Z}\wr\mathbb{Z}$ with an undecidable
membership problem.

@ARTICLE{LohreySteinbergZetzsche2015a,
AUTHOR = {Lohrey, Markus and Steinberg, Benjamin and Zetzsche, Georg},
TITLE = {Rational subsets and submonoids of wreath products},
JOURNAL = {Information and Computation},
VOLUME = {243},
PAGES = {191--204},
YEAR = {2015},
DOI = {10.1016/j.ic.2014.12.014}
}

Toward Understanding the Generative Capacity of Erasing Rules in Matrix Grammars
International Journal of Foundations of Computer Science 22(2), 2011
Special Issue on DLT 2009 [ Show BibTeXHide BibTeX
| Show abstractHide abstract
| DOI ]

Abstract
This article presents approaches to the open problem of whether erasing
rules can be eliminated in matrix grammars. The class of languages
generated by non-erasing matrix grammars is characterized by the newly
introduced linear Petri net grammars. Petri net grammars are known to be
equivalent to arbitrary matrix grammars (without appearance checking). In
linear Petri net grammars, the marking has to be linear in size with
respect to the length of the sentential form. The characterization by
linear Petri net grammars is then used to show that applying linear erasing
to a Petri net language yields a language generated by a non-erasing matrix
grammar. It is also shown that in Petri net grammars (with final markings
and arbitrary labeling), erasing rules can be eliminated, which yields two
reformulations of the problem of whether erasing rules in matrix grammars
can be eliminated.

@ARTICLE{Zetzsche2011a,
AUTHOR = {Zetzsche, Georg},
TITLE = {Toward Understanding the Generative Capacity of Erasing Rules in Matrix Grammars},
JOURNAL = {International Journal of Foundations of Computer Science},
VOLUME = {22},
NUMBER = {2},
PAGES = {411--426},
YEAR = {2011},
DOI = {10.1142/S0129054111008118}
}

Abstract
The previously introduced multiset language classes
defined by multiset pushdown automata are being
explored with respect to their closure properties and
alternative characterizations.

@ARTICLE{KudlekTotzkeZetzsche2009a,
AUTHOR = {Kudlek, Manfred and Totzke, Patrick and Zetzsche, Georg},
TITLE = {Properties of Multiset Language Classes Defined by Multiset Pushdown Automata},
JOURNAL = {Fundamenta Informaticae},
VOLUME = {93},
PAGES = {235--244},
NUMBER = {1-3},
YEAR = {2009},
DOI = {10.3233/FI-2009-0099}
}

Abstract
Multiset finite Automata, a model equivalent to
regular commutative grammars, are extended with a
multiset store and the accepting power of this
extended model of computation is investigated. This
type of multiset automata come in two flavours,
varying only in the ability of testing the storage
for emptiness. This paper establishes normal forms
and relates the derived language classes to each
other as well as to known multiset language classes.

@ARTICLE{KudlekTotzkeZetzsche2009,
AUTHOR = {Kudlek, Manfred and Totzke, Patrick and Zetzsche, Georg},
TITLE = {Multiset Pushdown Automata},
JOURNAL = {Fundamenta Informaticae},
VOLUME = {93},
PAGES = {221--233},
NUMBER = {1-3},
YEAR = {2009},
DOI = {10.3233/FI-2009-0098}
}

Abstract
We present a generalization of finite automata using
Petri nets as control, called Concurrent Finite Automata
for short. Several modes of acceptance, defined by final
markings of the Petri net, are introduced, and their
equivalence is shown. The class of languages obtained by
l-free concurrent finite automata contains both the class
of regular sets and the class of Petri net languages
defined by final marking, and is contained in the class
of context-sensitive languages.

@ARTICLE{FaJaKuRoZe2008,
AUTHOR = {Farwer, Berndt and Jantzen, Matthias and Kudlek, Manfred and R{\"o}lke, Heiko and Zetzsche, Georg},
TITLE = {Petri Net Controlled Finite Automata},
JOURNAL = {Fundamenta Informaticae},
VOLUME = {85},
PAGES = {111--121},
NUMBER = {1-4},
YEAR = {2008}
}

Abstract
This paper presents results regarding the various
relations among the language classes defined by
Concurrent Finite Automata, relations to other language
classes, as well as decidability and closure properties.

@ARTICLE{JantzenKudlekZetzsche2008,
AUTHOR = {Jantzen, Matthias and Kudlek, Manfred and Zetzsche, Georg},
TITLE = {Language Classes Defined by Concurrent Finite Automata},
JOURNAL = {Fundamenta Informaticae},
VOLUME = {85},
PAGES = {267--280},
NUMBER = {1-4},
YEAR = {2008}
}

Abstract
This work studies which storage mechanisms in automata permit
decidability of the reachability problem. The question is formalized
using valence automata, an abstract model that generalizes automata with
storage. For each of a variety of storage mechanisms, one can choose a
(typically infinite) monoid $M$ such that valence automata over $M$ are
equivalent to (one-way) automata with this type of storage.
In fact, many interesting storage mechanisms can be realized by monoids defined
by finite graphs, called graph monoids. Hence, we study for which graph
monoids the emptiness problem for valence automata is decidable. A particular
model realized by graph monoids is that of Petri nets with a pushdown stack.
For these, decidability is a long-standing open question and we do not answer
it here.
However, if one excludes subgraphs corresponding to this model, a
characterization can be achieved. This characterization yields a new extension
of Petri nets with a decidable reachability problem. Moreover, we provide a
description of those storage mechanisms for which decidability remains open.
This leads to a natural model that generalizes both pushdown Petri nets and
priority multicounter machines.

@INPROCEEDINGS{Zetzsche2015c,
AUTHOR = {Zetzsche, Georg},
TITLE = {The Emptiness Problem for Valence Automata or: Another Decidable Extension of Petri Nets},
EDITOR = {Miko{\l}aj Boja{\'{n}}czyk and S{\l}awomir Lasota and Igor Potapov},
SERIES = {LNCS},
VOLUME = {9328},
PAGES = {166--178},
PUBLISHER = {Springer},
ADDRESS = {Berlin Heidelberg},
BOOKTITLE = {Proc. of the 9th International Workshop on Reachability Problems (RP 2015)},
YEAR = {2015},
DOI = {10.1007/978-3-319-24537-9_15}
}

Abstract
Two kinds of multiset automata with a storage attached, varying
only in their ability of testing the storage for emptiness, are
introduced, as well as normal forms. Their accepting power and
relation to other multiset languages classes is investigated.

@INPROCEEDINGS{KudlekTotzkeZetzsche2008,
AUTHOR = {Kudlek, Manfred and Totzke, Patrick and Zetzsche, Georg},
TITLE = {Multiset Storage Automata},
BOOKTITLE = {Proc. of the Workshop on Concurrency, Specification and Programming (CS{\&}P 2008)},
EDITOR = {Burkhard, H.-D. and Czaja, Ludwik and Lindemann, G. and Skowron, A.},
VOLUME = {2},
PAGES = {265--277},
YEAR = {2008}
}

Abstract
The previously introduced multiset language classes defined by
multiset storage automata are being explored with respect to their
closure properties and alternative characterizations.

@INPROCEEDINGS{KudlekTotzkeZetzsche2008a,
AUTHOR = {Kudlek, Manfred and Totzke, Patrick and Zetzsche, Georg},
TITLE = {Properties of Multiset Language Classes Defined by Multiset Storage Automata},
BOOKTITLE = {Proc. of the Workshop on Concurrency, Specification and Programming (CS{\&}P 2008)},
EDITOR = {Burkhard, H.-D. and Czaja, Ludwik and Lindemann, G. and Skowron, A.},
VOLUME = {2},
PAGES = {278--288},
YEAR = {2008}
}

@INPROCEEDINGS{KudlekZetzsche2010,
AUTHOR = {Kudlek, Manfred and Zetzsche, Georg},
TITLE = {Concurrent finite automata and related language classes (an overview)},
BOOKTITLE = {Proc. of the Workshop Automata, Formal Languages and Algebraic Systems (AFLAS 2008)},
YEAR = {2010},
EDITOR = {Ito, Masami and Kobayashi, Yuji and Kunitaka, Shoji},
PUBLISHER = {World Scientific},
ADDRESS = {New Jersey},
PAGES = {103--113},
DOI = {10.1142/9789814317610_0008}
}

@INPROCEEDINGS{FaJaKuRoZe2007,
AUTHOR = {Farwer, Berndt and Jantzen, Matthias and Kudlek, Manfred and R{\"o}lke, Heiko and Zetzsche, Georg},
TITLE = {On Concurrent Finite Automata},
BOOKTITLE = {Proc. of the Workshop on Concurrency, Specification and Programming (CS{\&}P 2007)},
EDITOR = {Czaja, Ludwik},
VOLUME = {1},
PAGES = {180--190},
YEAR = {2007}
}

@INPROCEEDINGS{JantzenKudlekZetzsche2007a,
AUTHOR = {Jantzen, Matthias and Kudlek, Manfred and Zetzsche, Georg},
TITLE = {On Languages Accepted by Concurrent Finite Automata},
BOOKTITLE = {Proc. of the Workshop on Concurrency, Specification and Programming (CS{\&}P 2007)},
EDITOR = {Czaja, Ludwik},
VOLUME = {2},
PAGES = {321--332},
YEAR = {2007}
}

Abstract
A valence automaton over a monoid M is a finite automaton in
which each edge carries an input word and an element of M. A
word is then accepted if there is a run that spells the word
such that the product of the monoid elements is the identity.
By choosing appropriate monoids M, one can obtain various
kinds of automata with storage as special valence automata.
Examples include pushdown automata, blind multicounter
automata, and partially blind multicounter automata.
Therefore, valence automata offer a framework to generalize
results on such automata with storage. This talk will present
recent advances in this direction. The addressed questions
include: For which monoids can we accept non-regular
languages? For which monoids can we determinize automata? For
which monoids do we have a Parikh's Theorem (as for pushdown
automata)?

Abstract
For each grammar model with regulated rewriting, it is an important question
whether erasing productions add to its expressivity. In some cases, however,
this has been a longstanding open problem.
In recent years,
several results have been obtained that clarified the generative capacity of erasing
productions in some grammar models with classical types of regulated rewriting.
The aim of this talk is to give an overview of these results.

Other talks

Monoids as Storage Mechanisms
Given at Seminar of the INFINI group at LSV in December 2015, Cachan, France [ Show abstractHide abstract
| Slides ]

Abstract
The investigation of models extending finite automata by some storage mechanism
is a central theme in theoretical computer science. Choosing an appropriate
storage mechanism can yield a model that is expressive enough to capture a
given behavioral aspect while admitting desired means of analysis.
It is therefore a central concern to understand which storage mechanisms have
which properties regarding expressiveness and (algorithmic) analysis. This
talk presents a line of research that aims for general insights in this
direction. In other words: How does the structure of the storage mechanism
influences expressiveness and analysis of the resulting model?
In order to study this question, one needs a model in which the storage
mechanism appears as a parameter. Such a model is available in valence
automata, where the storage mechanism is given by a (typically infinite)
monoid. Choosing a suitable monoid then yields models such as Turing machines,
pushdown automata, vector addition systems, or combinations thereof.
This talk surveys a selection of results that characterize storage mechanisms
with certain desirable properties, such as deciability of reachability,
semilinearity of Parikh images, and avoidability of epsilon-transitions.

The Emptiness Problem for Valence Automata or: Another Decidable Extension of Petri Nets
Given at RP 2015, Warsaw, Poland [ Show abstractHide abstract
| Slides ]

Abstract
This work studies which storage mechanisms in automata permit
decidability of the reachability problem. The question is formalized
using valence automata, an abstract model that generalizes automata with
storage. For each of a variety of storage mechanisms, one can choose a
(typically infinite) monoid $M$ such that valence automata over $M$ are
equivalent to (one-way) automata with this type of storage.
In fact, many interesting storage mechanisms can be realized by monoids defined
by finite graphs, called graph monoids. Hence, we study for which graph
monoids the emptiness problem for valence automata is decidable. A particular
model realized by graph monoids is that of Petri nets with a pushdown stack.
For these, decidability is a long-standing open question and we do not answer
it here.
However, if one excludes subgraphs corresponding to this model, a
characterization can be achieved. This characterization yields a new extension
of Petri nets with a decidable reachability problem. Moreover, we provide a
description of those storage mechanisms for which decidability remains open.
This leads to a natural model that generalizes both pushdown Petri nets and
priority multicounter machines.

Abstract
The downward closure of a word language is the set of all (not necessarily
contiguous) subwords of its members. It is known that the downward closure of
every language is regular. However, algorithms for computing a finite automaton
for the downward closure of a given language are known only for few language
classes. This work presents a simple general approach to this problem. It is
used to prove that downward closures are computable for (i)~every language
class with effectively semilinear Parikh images that is closed under rational
transductions, (ii)~matrix languages, and (iii)~indexed languages
(equivalently, languages accepted by higher-order pushdown automata of
order~2).

Abstract
The downward closure of a word language is the set
of all (not necessarily contiguous) subwords of its
members. It is well-known that the downward closure of
any language is regular. While the downward closure
appears to be a powerful abstraction, algorithms for
computing a finite automaton for the downward closure
of a given language have been established only for few
language classes.
This work presents a simple general method for computing
downward closures. For language classes that are closed
under rational transductions, it is shown that the
computation of downward closures can be reduced to
checking a certain unboundedness property.
This result is used to prove that downward closures
are computable for (i) every language class with
effectively semilinear Parikh images that are closed
under rational transductions, (ii) matrix languages,
and (iii) indexed languages (equivalently, languages
accepted by higher-order pushdown automata of order~2).

Downward Closures of Indexed Languages
Given at HOPA 2015, Kyoto, Japan [ Slides ]

Abstract
The downward closure of a language $L$ of words is the set of all (not
necessarily contiguous) subwords of members of $L$. It is well known that the
downward closure of any language is regular. Although the downward
closure seems to be a promising abstraction, there are only few language
classes for which an automaton for the downward closure is known to be
computable.
It is shown here that for stacked counter automata, the downward closure is
computable. Stacked counter automata are finite automata with a storage
mechanism obtained by \emph{adding blind counters} and \emph{building stacks}.
Hence, they generalize pushdown and blind counter automata.
The class of languages accepted by these automata are precisely those in the
hierarchy obtained from the context-free languages by alternating two closure
operators: imposing semilinear constraints and taking the algebraic extension.
The main tool for computing downward closures is the new concept of Parikh
annotations. As a second application of Parikh annotations, it is shown that
the hierarchy above is strict at every level.

Effectively regular downward closures
Given at LSV Seminar at ENS Cachan in October 2014, Cachan, France [ Show abstractHide abstract
| Slides ]

Abstract
The downward closure of a language is the set of all (not
necessarily contiguous) subwords of its members. It is a
well-known consequence of Higman's Lemma that the downward
closure of every language is regular.
Aside from encoding interesting counting properties, the
downward closure constitutes a promising abstraction: If L is
the set of action sequences of a system, then the downward
closure of L is precisely what is observed through a lossy
channel, i.e. when actions can go unnoticed arbitrarily. Hence,
if the downward closure is available as a regular language,
the equivalence and even inclusion of system behaviors can be
decided with respect to such observations.
However, there are only few classes of languages for which it is
known how to compute the downward closure of a given language as
a finite automaton. This talk presents new approaches to this
problem.

Expressiveness and analysis of valence automata over graph monoids
Given at FORMAT Workshop 07/2014, Kaiserslautern, Germany [ Slides ]

Abstract
Recent work on automata with abstract storage revealed a class of
storage mechanisms that proves quite expressive and amenable to various
kinds of algorithmic analysis. The storage mechanisms in this class are
obtained by \emph{building stacks} and \emph{adding blind counters}.
The former is to construct a new mechanism that stores a stack whose
entries are configurations of an old mechanism. One can then manipulate
the topmost entry, pop it if empty, or start a new one on top. Adding a
blind counter to an old mechanism yields a new mechanism in which the
old one and a blind counter can be used simultaneously. We call the
resulting model \emph{stacked counter automaton}.
This talk presents results on expressivity, Parikh images, membership
problems, and the computability of downward closures.

Abstract
A Boolean closed full trio is a class of languages that is closed under the
Boolean operations (union, intersection, and complementation) and rational
transductions. It is well-known that the regular languages constitute such a
Boolean closed full trio. It is shown here that every such language class that
contains any non-regular language already includes the whole arithmetical
hierarchy (and even the one relative to this language).
A consequence of this result is that aside from the regular languages, no full
trio generated by one language is closed under complementation.
Our construction also shows that there is a fixed rational Kripke frame such
that assigning an arbitrary non-regular language to some variable allows the
definition of any language from the arithmetical hierarchy in the
corresponding Kripke structure using multimodal logic.

Abstract
A Boolean closed full trio is a class of languages that is closed under Boolean
operations (union, intersection, and complement) and rational transductions. It
is well-known that the regular languages constitute such a Boolean closed full
trio. We present a result stating that every such language class that
contains any non-regular language already contains the whole arithmetical
hierarchy.
Our construction also shows that there is a fixed rational Kripke frame such
that assigning an arbitrary non-regular language to some variable allows the
interpretation of any language from the arithmetical hierarchy in the
corresponding Kripke structure.

Recent advances on valence automata as a generalization of automata with storage
(joint work with Phoebe Buckheister)
Given at Theorietag 2013, Ilmenau, Germany [ Show abstractHide abstract
| Slides ]

Abstract
A valence automaton over a monoid $M$ is a finite automaton
in which each edge carries an input word and an element of
$M$. A word is then accepted if there is a run that spells
the word such that the product of the monoid elements is the
identity.
By choosing suitable monoids $M$, one can obtain various
kinds of automata with storage as special valence automata.
Examples include pushdown automata, blind multicounter
automata, and partially blind multicounter automata.
Therefore, valence automata offer a framework to generalize
results on such automata with storage.
This talk will present recent advances in this direction. The addressed
questions include: For which monoids do we have a Parikh's Theorem (as for
pushdown automata)? For which monoids can we avoid silent transitions?

Semilinearity and Context-Freeness of Languages Accepted by Valence Automata
(joint work with Phoebe Buckheister)
Given at MFCS 2013, Klosterneuburg, Austria [ Show abstractHide abstract
| Slides ]

Abstract
Valence automata are a generalization of various models of automata with
storage. Here, each edge carries, in addition to an input word, an element of
a monoid. A computation is considered valid if multiplying the monoid elements
on the visited edges yields the identity element. By choosing suitable
monoids, a variety of automata models can be obtained as special valence
automata.
This work is concerned with the accepting power of valence automata.
Specifically, we ask for which monoids valence automata can accept only
context-free languages or only languages with semilinear Parikh image,
respectively.
First, we present a characterization of those graph products (of monoids) for
which valence automata accept only context-free languages. Second, we provide a
necessary and sufficient condition for a graph product of copies of the
bicyclic monoid and the integers to yield only languages with semilinear Parikh
image when used as a storage mechanism in valence automata. Third, we show that
all languages accepted by valence automata over torsion groups have a
semilinear Parikh image.

Abstract
It is shown that membership in rational subsets of wreath
products $H \wr V$ with $H$ a finite group and $V$ a virtually
free group is decidable. On the other hand, it is shown that
there exists a fixed finitely generated submonoid in the wreath
product $\mathbb{Z}\wr\mathbb{Z}$ with an undecidable
membership problem.

Abstract
We consider the computational power of silent transitions in one-way automata
with storage. Specifically, we ask which storage mechanisms admit a
transformation of a given automaton into one that accepts the same language
and reads at least one input symbol in each step.
We study this question using the model of valence automata. Here, a finite
automaton is equipped with a storage mechanism that is given by a monoid.
This work presents generalizations of known results on silent transitions.
For two classes of monoids, it provides characterizations of those monoids that
allow the removal of silent transitions. Both classes are defined by
graph products of copies of the bicyclic monoid and the group of integers. The
first class contains pushdown storages as well as the blind counters while the
second class contains the blind and the partially blind counters.

Abstract
A valence automaton over a monoid $M$ is a finite automaton in
which each edge carries an input word and an element of $M$. A
word is then accepted if there is a run that spells the word
such that the product of the monoid elements is the identity.
By choosing appropriate monoids $M$, one can obtain various
kinds of automata with storage as special valence automata.
Examples include pushdown automata, blind multicounter
automata, and partially blind multicounter automata. Therefore,
valence automata offer a framework to generalize results on
such automata with storage. This talk will present recent
results on valence automata. The addressed questions include:
For which monoids can we accept non-regular languages? For
which monoids can we determinize automata? For which monoids
can we avoid silent edges (i.e., those which read no input
symbol)?

Abstract
During recent decades, classical models in language theory have been extended
by control mechanisms defined by monoids. We study which monoids cause the
extensions of context-free grammars, finite automata, or finite state
transducers to exceed the capacity of the original model. Furthermore, we
investigate when, in the extended automata model, the nondeterministic variant
differs from the deterministic one in capacity. We show that all these
conditions are in fact equivalent and present an algebraic characterization. In
particular, the open question of whether every language generated by a valence
grammar over a finite monoid is context-free is provided with a positive
answer.

Abstract
In each grammar model, it is an important question whether erasing productions
are necessary to generate all languages. Using the concept of grammars with
control languages by Salomaa, which offers a uniform treatment of a variety of
grammar models, we present a condition on the class of control languages that
guarantees that erasing productions are avoidable in the resulting grammar
model. On the one hand, this generalizes the previous result that in Petri net
controlled grammars, erasing productions can be eliminated. On the other hand,
it allows us to infer that the same is true for vector grammars.

On the Capabilities of Grammars, Automata, and Transducers Controlled by Monoids
Given at ICALP 2011, Zürich, Switzerland [ Show abstractHide abstract
]

Abstract
During recent decades, classical models in language theory have been extended
by control mechanisms defined by monoids. We study which monoids cause the
extensions of context-free grammars, finite automata, or finite state
transducers to exceed the capacity of the original model. Furthermore, we
investigate when, in the extended automata model, the nondeterministic variant
differs from the deterministic one in capacity. We show that all these
conditions are in fact equivalent and present an algebraic characterization. In
particular, the open question of whether every language generated by a valence
grammar over a finite monoid is context-free is provided with a positive
answer.

Abstract
Three open questions in the theory of regulated
rewriting are addressed. The first is whether every
permitting random context grammar has a non-erasing
equivalent. The second asks whether the same is true
for matrix grammars without appearance checking. The
third concerns whether permitting random context
grammars have the same generative capacity as matrix
grammars without appearance checking.
The main result is a positive answer to the first
question. For the other two, conjectures are
presented. It is then deduced from the main result
that at least one of the two holds.

Abstract
It is shown that applying linear erasing to a Petri net
language yields a language generated by a non-erasing
matrix grammar. The proof uses Petri net controlled
grammars. These are context-free grammars, where the
application of productions has to comply with a firing
sequence in a Petri net. Petri net controlled grammars are
equivalent to arbitrary matrix grammars (without
appearance checking), but a certain restriction on them
(linear Petri net controlled grammars) leads to the class
of languages generated by non-erasing matrix grammars.
It is also shown that in Petri net controlled grammars
(with final markings and arbitrary labeling), erasing
rules can be eliminated, which yields a reformulation of
the problem of whether erasing rules in matrix grammars
can be eliminated.

Abstract
We compare various modes of firing transitions in Petri nets
and define classes of languages defined this way. We define
languages through steps, i. e. sets of transitions, maximal
steps, multi-steps, and maximal multi-steps of transitions
in Petri nets, but in a different manner than those defined
in [Burk 81a,Burk 83], by considering labeled transitions.
We will show that we obtain a hierarchy of families of
languages defined by multiple use of transition in firing
transitions in a single multistep. Except for the maximal
multi-steps all classes can be simulated by sequential
firing of transitions.

Abstract
We present a generalization of finite automata using Petri nets
as control. Acceptance is defined by final markings of the
Petri net. The class of languages obtained by $\lambda$-free
concurrent finite automata contains both the class of regular
sets and the class of Petri net languages defined by final
marking.