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\begin{document}

\title{Petri Nets, Configuration Structures\\ and Higher Dimensional
       Automata\thanks{To appear in Proceedings CONCUR '99.}}
\author{R.J. van Glabbeek\thanks{This work was supported by ONR under
			grant number N00014-92-J-1974.}}
\institute{Computer Science Department, Stanford University\\
			Stanford, CA 94305-9045, USA.\\
			{\tt http://theory.stanford.edu/\~{}rvg/}\\
			{\tt rvg@cs.stanford.edu}}
\maketitle

\noindent
In this talk, translations between several models of concurrent systems
are reviewed c.q.\ proposed. The models considered capture causality,
branching time, and their interplay, and these features are preserved by
the translations. To the extent that the models are intertranslatable,
this yields support for the point of view that they are all different
representations of the same phenomena. The translations can then be
applied to reformulate any issue that arises in the context of one
model into one expressed in another model, which might be more
suitable for analysing that issue. To the extent that the models are
not intertranslatable, my investigations are aimed at classifying them
w.r.t.\ their expressiveness in modelling phenomena in concurrency.
The results are summarised in the figure at the end of this paper.

Starting point is the work of {\sc Nielsen, Plotkin \& Winskel}
\cite{NPW81}, in which safe Petri nets are translated, through the
intermediate stages of occurrence nets, prime event structures with a
binary conflict relation, and their families of configurations, into a
class of Scott domains.\vspace{1ex}

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\centerline{\raise 1em\box\graph}

\section{From nets to configurations}

In {\sc Van Glabbeek \& Plotkin} \cite{GP95} extensions of the
translations above to unsafe Petri nets have been studied. For this
purpose two different schools of thought in interpreting the causal
behaviour of nets needed to be distinguished, which we called the {\em
individual} and {\em collective token} philosophy. Their difference is
illustrated by the following net.
\vspace{-4ex}
\begin{figure}[htb]
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\vspace{-5ex}
\end{figure}
According to the individual token philosophy, A has an execution in
which the action $b$ causally depends on $a$, whereas according to the
collective token philosophy, $a$ and $b$ are always causally independent.

In {\sc Meseguer, Montanari \& Sassone} \cite{MMS92}, the unfolding
from \cite{NPW81}, translating safe nets into the subclass of
occurrence nets, is extended to arbitrary nets, while preserving
the individual token interpretation.  It follows that under this
interpretation prime event structures are expressive enough to
represent all processes expressible by Petri nets.

Under the collective token interpretation there turn out to be nets
whose causal behaviour cannot be faithfully represented by a prime
event structure. Representative examples are the two nets below,
\begin{figure}[htb]
\vspace{-4ex}
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\vspace{-5ex}
\end{figure}
modelling what I often call {\em disjunctive causality} and
{\em resolvable conflict}, respectively.

In {\sc Winskel} \cite{Wi87a} a more general notion of event structure is
proposed, extending the prime event structures with a binary conflict
relation from \cite{NPW81}, along with matching generalisations of the
families of configurations and Scott domains. These event structures
capture disjunctive causality, but not resolvable conflict.

The families of configurations of Winskel's event structures were
introduced merely to facilitate the construction of the Scott domains
associated to these event structures. In {\sc Van Glabbeek \& Goltz}
\cite{GG90} we found it convenient to use such families as a model of
concurrency in its own right. In this context the families were
called {\em configuration structures}. A configuration structure can
be given by a set of {\em events}, modelling occurrences of actions
the represented system may perform, possibly a {\em labelling
function}, associating actions to events, and a collection of sets of
events, the {\em configurations}, modelling the states of the
represented system, and satisfying a number of closure conditions. A
configuration represents the state in which the events it contains
have occurred. The closure conditions ensure that each configuration
structure can be regarded as the family of configurations of an event
structure.

In \cite{GP95} we proposed to drop the closure conditions, thereby
obtaining a more general model of concurrency, capturing both
disjunctive causality and resolvable conflict. The resulting
configuration structures are, up to isomorphism, the {\em extensional
Chu spaces} of {\sc Gupta \& Pratt} \cite{GP93a}, but equipped with a
slightly different computational interpretation. Through suitable
translations we showed that these configuration structures are equally
expressive as general Petri nets without self-loops. Such nets are
called {\em pure}. To this end we defined a {\em 1-occurrence net} to
be a Petri net in which each transition can fire at most once, and we
showed how any (pure) Petri net can be converted into a (pure)
1-occurrence net, using a construction we called {\em 1-unfolding}.
We argued that this conversion preserves essential features of the
represented system like causality and branching time. It may convert a
finite net into an infinite one, however. The translations between pure
1-occurrence nets and configuration structures take the transitions of
the net to be the events of the configuration structure and vice versa;
this way a configuration structure can be fully recovered from its
Petri net representation. Our translations also extend the
correspondence between flow nets and flow event structures proposed in
{\sc Boudol} \cite{Bo90}.

{\em ST-configuration structures} are a further generalisation of
configuration\linebreak structures in which the configurations may
contain certain events `partially' (in case they are currently being
executed). They are (a mild generalisation of) what are called {\em
local event structures} in {\sc Hoogers, Kleijn \& Thiagarajan}
\cite{HKT93}. In forthcoming work, Gordon Plotkin and I extend the
translations between pure nets and configuration structures to
translations between arbitrary Petri nets and ST-configuration
structures, thus showing that also these models are equally
expressive.  The same was done, using a different construction, for
general Petri nets without autoconcurrency in \cite{HKT93}.
We also propose a matching generalisation of the model of
event structures. \nocite{Gun92b}

\section{Scott domains versus process graphs}

In \cite{NPW81} a ``curious mismatch'' is observed between the domains
that result from translating nets or event structures, and the ones
originally studied by {\sc Scott} \cite{Scott70}.  Although
mathematically of the same nature, a domain that arises through the
translations of \cite{NPW81} represents a single concurrent system,
namely the same one represented by the Petri net or event structure it
originated from.  In domain theory, on the other hand, processes show
up at best as the {\em elements} of a domain.  Thus the use of domains
to represent concurrent systems is novel in \cite{NPW81}.

In most models of concurrency, attention is restricted to {\em
discrete} processes, i.e.\ processes that can perform only finitely
many actions in a finite time. Petri nets are commonly interpreted to
represent discrete processes---this comes with the common definitions
of the {\em firing rule}. On prime event structures the axiom of {\em
finite causes} restricts attention to the structures representing
discrete systems, and in Winskel's general event structures
discreteness is obtained by the way the notion of a configuration of
an event structure is defined.  A Scott domain is a partially ordered
set, satisfying certain conditions.  The {\em finite} elements in such
a domain are the ones that dominate only finitely many other
elements. A discrete Scott domain (resulting from translating a discrete
event structure) has the property that its infinitary part is
redundant, in the sense that it can be recovered in full from its
finitary part (the partial suborder of its finite elements).  The
finitary part of a domain can, without loss of information, be
trivially represented, and is often displayed, as an unlabelled rooted
graph. Therefore I argue that the correspondence between event
structures and domains proposed in \cite{NPW81}, and generalised to
all event structures in \cite{Wi87a}, can equivalently, or maybe
better, be regarded as a correspondence between event structures and a
class of unlabelled transition systems or {\em process graphs}. As
remarked in \cite{vG88a}, this correspondence can trivially be extended
to {\em labelled} event structures and transitions systems; the latter
are easier to label than domains. It follows immediately that process
graphs, or labelled transition systems, are at least as capable of
expressing causality as labelled event structures.

The computational interpretation of domains, inherited from
that of event structures, naturally applies to the process graphs
corresponding with those domains.  These graphs capture causality
through confluence of squares of transitions. This computational
interpretation can be extended to process graphs that do not
correspond to event structures or Scott domains. It can be seen as an
enrichment of the classical interpretation of process graphs.
Just like any process graph can be unfolded into a tree, while
preserving its interleaving interpretation, I propose a causality
respecting unfolding of arbitrary process graphs into so-called {\em
history preserving} ones, which preserves transition squares. History
preserving process graphs generalise the Scott domains originating
from the general event structures of \cite{Wi87a}. They can also model
phenomena like resolvable conflict that are not expressible by
these event structures.

Several brands of transition systems enriched with some auxiliary
structure to capture causality have been proposed as models of
concurrency, cf.\ the {\em asynchronous transition systems} of {\sc
Shields} \cite{Sh85} and {\sc Bednarczyk} \cite{Bz87}, the {\em
behaviour structures} of {\sc Rabinovich \& Trakhtenbrot} \cite{RT88},
the {\em concurrent transition systems} of {\sc Stark} \cite{Sta89}
and {\sc Droste}, \cite{Dr92} and the {\em transition systems with
independence} of {\sc Nielsen \& Winskel} \cite{WN95}. In each of
these cases the added structure does not fundamentally increase their
expressiveness: after a suitable behaviour-preserving unfolding, the
causalities expressed by this added structure are completely
determined by the underlying transition system, which always forms a
history preserving process graph.

Event automata, studied by {\sc Pinna \& Poign\'e} \cite{PP95}, fit
between configuration structures and ST-configuration structures.
Through appropriate translations these can be shown to be equally
expressive as the so-called {\em configuration-deterministic} process
graphs. Graphs which are not configuration deterministic do not
correspond to nets or event-oriented models. Interestingly,
translating back and forth between event automata and process graphs
may repeatedly increase the number of events of the event automaton
representation of the system in question, namely by spitting events
into subevents that occur in disconnected parts of the system
representation.  Hence these translations cannot be expressed as
reflexions or coreflexions in a suitable categorical framework.

\section{Higher dimensional automata}

The concurrent interpretation of process graphs allows one to think of
squares and cubes as being ``filled in''.  {\sc Pratt} \cite{Pr91a}
proposes a {\em geometric model of concurrency}, refining this
approach by not necessarily filling in {\em all} squares and cubes,
but explicitly filling in only those that one wants to represent
concurrency. Alternative formalisations of this idea appear in {\sc
Van Glabbeek} \cite{vG91}, {\sc Goubault \& Jensen} \cite{GJ92} and
{\sc Cattani \& Sassone} \cite{CS96}.  Although the resulting model of
{\em higher dimensional automata} is more complicated than that of
plain automata or process graphs, it is more expressive as well. The
Petri net below,
\begin{figure}[htb]
\vspace{-4ex}
%\input{3mutex}
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\vspace{-5ex}
\end{figure}
for instance, is expressible by a higher dimensional automaton in the
form of a cube, as displayed above, of which all 6 sides are filled in
(representing the 6 possible concurrent firings of two transitions,
either before or after the third one fires), but the interior is not.
The causal behaviour of this system cannot be represented by a process
graph, and hence neither by an event automaton, nor by a pure Petri net.
Process graphs and the mentioned transition systems with extra
structure to capture causality can be regarded as one- and
two-dimensional automata, respectively.

A representation of higher dimensional automata in which the names of
both events and actions are incorporated, is given by {\em labelled step
transition systems} (LSTSs), see also {\sc Badouel} \cite{Ba96}. These
naturally unfold into (alternative representations of)
ST-configuration structures. Petri nets translate to LSTSs by taking
their marking graphs; LSTSs translate to higher dimensional automata
as in \cite{vG91} by forgetting event names (but remembering their
labels).  

{\sc Ehrenfeucht \& Rozenberg} \cite{ER90} characterised which
process graphs can be obtained as the marking graphs of a safe nets.
% These graphs are called {\em elementary transition systems} in \cite{NRT92}.
Likewise, {\sc Mukund} \cite{Mu92} characterised which LSTSs
can be obtained as the step marking graphs of general Petri nets.
Both papers also yield translations back from (step) transition systems
to nets, but only for systems in the characterised class. More general
translations from LSTSs to nets can be obtained through unfolding.
\vspace{3ex}\vfill

%\input{structures}
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  }%
}%
\centerline{\raise 1em\box\graph}

%\bibliography{/kilby/u1/rvg/lib/abbreviations,/kilby/u1/rvg/lib/new,/kilby/u1/rvg/lib/dbase}
\begin{thebibliography}{10}

\bibitem{Ba96}
{\sc E.~Badouel} (1996):
\newblock {\em Splitting of actions, higher-dimensional automata, and net synthesis.}
\newblock Technical Report RR-3490, Inria, France.

\bibitem{Bz87}
{\sc M.~Bednarczyk} (1987):
\newblock {\em Categories of asynchronous systems}.
\newblock PhD thesis, Computer Science, University of Sussex, Brighton.

\bibitem{Bo90}
{\sc G.~Boudol} (1990):
\newblock {\em Flow event structures and flow nets.}
\newblock In I.~Guessarian, editor: {\sl Semantics of Systems of Concurrent Processes, Proceedings LITP Spring School on Theoretical Computer Science, {\rm La Roche Posay, France}}, {\sl \rm LNCS} 469, Springer, pp. 62--95.

\bibitem{CS96}
{\sc G.L. Cattani \& V.~Sassone} (1996):
\newblock {\em Higher dimensional transition systems.}
\newblock In {\sl {\rm Proceedings $11^{th}$ Annual IEEE Symposium on} Logic in Computer Science {\rm (LICS 96), New Brunswick, USA}}, IEEE Computer Society Press, pp. 55--62.

\bibitem{Dr92}
{\sc M.~Droste} (1992):
\newblock {\em Concurrent automata and domains.}
\newblock {\sl International Journal of Foundations of Computer Science} 3(4), pp. 389--418.

\bibitem{ER90}
{\sc A.~Ehrenfeucht \& G.~Rozenberg} (1990):
\newblock {\em Partial 2-structures.}
\newblock {\sl Acta Informatica} 27(4), pp. 315--368.

\bibitem{vG88a}
{\sc R.J.~van Glabbeek} (1988):
\newblock {\em An operational non-interleaved process graph semantics of {CCSP} \rm (abstract).}
\newblock In E.-R. Olderog, U.~Goltz \& R.J. van Glabbeek, editors: {\sl Combining compositionality and concurrency, summary of a GMD-workshop, K\"onigswinter, March 1988}, Arbeitspapiere der GMD 320, pp. 18--19.

\bibitem{vG91}
{\sc R.J.~van Glabbeek} (1991):
\newblock {\em Bisimulations for higher dimensional automata.}
\newblock Email message, July 7, '91.
\newblock Available at {\tt http://theory.stanford.edu/\-\~{}rvg/hda}.

\bibitem{GG90}
{\sc R.J.~van Glabbeek \& U.~Goltz} (1990):
\newblock {\em Refinement of actions in causality based models.}
\newblock In J.W. de~Bakker, W.P.~de Roever \& G.~Rozenberg, editors: {\sl Proceedings REX Workshop on Stepwise Refinement of Distributed Systems: Models, Formalism, Correctness, {\rm Mook, The Netherlands, May/June 1989}}, {\sl \rm LNCS} 430, Springer, pp. 267--300.

\bibitem{GP95}
{\sc R.J.~van Glabbeek \& G.D. Plotkin} (1995):
\newblock {\em Configuration structures (extended abstract).}
\newblock In D.~Kozen, editor: {\sl {\rm Proceedings $10^{th}$ Annual IEEE Symposium on} Logic in Computer Science {\rm (LICS 95), San Diego, USA}}, IEEE Computer Society Press, pp. 199--209.

\bibitem{GJ92}
{\sc E.~Goubault \& T.~Jensen} (1992):
\newblock {\em Homology of higher dimensional automata.}
\newblock In W.R. Cleaveland, editor: {\sl Proceedings CONCUR 92, {\rm Stony Brook, NY, USA}}, {\sl \rm LNCS} 630, Springer, pp. 254--268.

\bibitem{Gun92b}
{\sc J.~Gunawardena} (1992):
\newblock {\em Causal automata.}
\newblock {\sl Theoretical Computer Science} 101, pp. 265--288.

\bibitem{GP93a}
{\sc V.~Gupta \& V.R. Pratt} (1993):
\newblock {\em Gates accept concurrent behavior.}
\newblock In {\sl Proc. 34th Ann. IEEE Symp. on Foundations of Comp. Sci.}, pp. 62--71.
\newblock More material on Chu spaces can be found at {\tt http://boole.stanford.edu/\-chuguide.html}.

\bibitem{HKT93}
{\sc P.W. Hoogers, H.C.M. Kleijn \& P.S. Thiagarajan} (1993):
\newblock {\em Local event structures and {Petri} nets.}
\newblock In E.~Best, editor: {\sl Proceedings CONCUR 93, {\rm Hildesheim, Germany}}, {\sl \rm LNCS} 715, Springer, pp. 462--476.

\bibitem{MMS92}
{\sc J.~Meseguer, U.~Montanari \& V.~Sassone} (1992):
\newblock {\em On the semantics of {Petri} nets.}
\newblock In W.R. Cleaveland, editor: {\sl Proceedings CONCUR 92, {\rm Stony Brook, NY, USA}}, {\sl \rm LNCS} 630, Springer, pp. 286--301.

\bibitem{Mu92}
{\sc M.~Mukund} (1992):
\newblock {\em Petri nets and step transition systems.}
\newblock {\sl International Journal of Foundations of Computer Science} 3(4), pp. 443--478.

\bibitem{NPW81}
{\sc M.~Nielsen, G.D. Plotkin \& G.~Winskel} (1981):
\newblock {\em Petri nets, event structures and domains, part {I}.}
\newblock {\sl Theoretical Computer Science} 13(1), pp. 85--108.

\bibitem{NRT92}
{\sc M.~Nielsen, G.~Rozenberg \& P.S. Thiagarajan} (1992):
\newblock {\em Elementary transition systems.}
\newblock {\sl Theoretical Computer Science} 96, pp. 3--33.

\bibitem{PP95}
{\sc G.M. Pinna \& A.~Poign\'e} (1995):
\newblock {\em On the nature of events: another perspective in concurrency.}
\newblock {\sl Theoretical Computer Science} 138(2), pp. 425--454.

\bibitem{Pr91a}
{\sc V.R. Pratt} (1991):
\newblock {\em Modeling concurrency with geometry.}
\newblock In {\sl Proc. 18th Ann. ACM Symposium on Principles of Programming Languages}, pp. 311--322.

\bibitem{RT88}
{\sc A.~Rabinovich \& B.A. Trakhtenbrot} (1988):
\newblock {\em Behavior structures and nets.}
\newblock {\sl Fundamenta Informaticae} 11(4), pp. 357--404.

\bibitem{Scott70}
{\sc D.~Scott} (1970):
\newblock {\em Outline of a mathematical theory of computation.}
\newblock In {\sl Proceedings of the 4$^{th}$ Annual Princeton Conference on Information Sciences and Systems}, pp. 169--176.

\bibitem{Sh85}
{\sc M.W. Shields} (1985):
\newblock {\em Concurrent machines.}
\newblock {\sl The Computer Journal} 28(5), pp. 449--465.

\bibitem{Sta89}
{\sc E.W. Stark} (1989):
\newblock {\em Concurrent transition systems.}
\newblock {\sl Theoretical Computer Science} 64, pp. 221--269.

\bibitem{Wi87a}
{\sc G.~Winskel} (1987):
\newblock {\em Event structures.}
\newblock In W.~Brauer, W.~Reisig \& G.~Rozenberg, editors: {\sl Petri Nets: Applications and Relationships to Other Models of Concurrency, Advances in Petri Nets 1986, Part II; Proceedings of an Advanced Course, {\rm Bad Honnef, September 1986}}, {\sl \rm LNCS} 255, Springer, pp. 325--392.

\bibitem{WN95}
{\sc G.~Winskel \& M.~Nielsen} (1995):
\newblock {\em Models for concurrency.}
\newblock In S.~Abramsky, D.M. Gabbay \& T.S.E. Maibaum, editors: {\sl Handbook of Logic in Computer Science}, volume 4: Semantic Modelling, chapter~1. Oxford University Press.

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