Intro duction to Trace Theory
tutorial
Antoni Mazurkiewicz
Institute of Computer Science Polish Academy of Sciences
ul Ordona Warszawa and
Institute of Informatics Jagiellonian University
ul Nawo jki Krak ow Poland
amaz wars ipipan waw pl
Novemb er
Abstract
The concept of traces has b een intro duced for describing non sequential b e
haviour of concurrent systems via its sequential observations Traces represent con
current pro cesses in the same way as strings represent sequential ones The theory
of traces is used as a to ol for reasoning ab out non sequential pro cesses as a matter
of fact using this theory one can get a calculus of concurrent pro cesses analogous to
the ones available for sequential systems In this intro ductory tutorial we discuss a
numb er of issues that are at the foundations of trace theory Among the discussed
topics are algebraic prop erties of traces dep endence graphs histories xed p oint
calculus for nding the b ehaviour of nets comp ositionality and mo dularity
Key words concurrency traces pro cesses partial ordering non sequential
systems
Intro duction
The theory of traces has b een motivated by the theory of Petri Nets and the theory of
formal languages and automata
Already in s Carl Adam Petri has develop ed the foundations for the theory of
concurrent systems In his seminal work he has presented a mo del which is based on
the communication of interconnected sequential systems He has also presented an entirely
Mazurkiewicz Intro duction to Trace Theory
new set of notions and problems that arise when dealing with concurrent systems His
mo del or actually a family of net based mo dels that has b een since generically referred
to as Petri Nets has provided b oth an intuitive informal framework for representing basic
situations of concurrent systems and a formal framework for the mathematical analysis of
concurrent systems The intuitive informal framework has b een based on a very convenient
graphical representation of net based systems while the formal framework has b een based
on the token game resulting from this graphical representation
The notion of a nite automaton seen as a restricted typ e of Turing Machine has
by then b ecome a classical mo del of a sequential system The strength of automata
theory is based on the simple elegance of the underlying mo del which admits p owerful
mathematical to ols in the investigation of b oth the structural prop erties expressed by the
underlying graph like mo del and the b ehavioural prop erties based on the notion of the
language of a system The notion of language has provided a valuable link with the theory
of free monoids
The original attempt of the theory of traces was to use the well develop ed to ols
of formal language theory for the analysis of concurrent systems where the notion of
concurrent system is understo o d in very much the same way as it is done in the theory of
Petri Nets The idea was that in this way one will get a framework for reasoning ab out
concurrent systems which for a numb er of imp ortant problems would b e mathematically
more convenient that some approaches based on formalizations of the token game
In s when the basic theory of traces has b een formulated the most p opular
approach to deal with concurrency was interleaving In this approach concurrency is
replaced by non determinism where concurrent execution of actions is treated as non
deterministic choice of the order of executions of those actions Although the interleaving
approach is quite adequate for many problems it has a numb er of serious pitfalls We
will brie y discuss here some of them b ecause those were imp ortant drawbacks that we
wanted to avoid in trace theory
By reducing concurrency to non determinism one assigns two di erent meanings to
the term non deterministic choice the choice b etween two or more p ossible actions
that exclude each other and the lack of information ab out the order of two or more
actions that are executed indep endently of each other Since in the theory of concurrent
systems we are interested not only in the question what is computed but also in the
question how it is computed this identi cation may b e very misleading
This disadvantage of the interleaving approach is well visible in the treatment of
re nement see e g It is widely recognized that re nement is one of the basic trans
formations of any calculus of concurrent systems from theoretical metho dological and
practical p oint of view This transformation must preserve the basic relationship b etween
a system and its b ehaviour the b ehaviour of the re ned system is the re ned b ehaviour
of the original system This requirement is not resp ected if the b ehaviour of the system
is represented by interleaving
Mazurkiewicz Intro duction to Trace Theory
Also in considerations concerning inevitability see the interleaving approach
leads to serious problems In non deterministic systems where a choice b etween two
partners may b e rep eated in nite numb er of times a run discriminating against one of
the partners is p ossible an action of an apriori chosen partner is not inevitable On
the other hand if the two partners rep eat their actions independently of each other an
action of any of them is inevitable However in the interleaving approach one do es not
distinguish b etween these two situations b oth are describ ed in the same way Then as
a remedy against this confusion a sp ecial notion of fairness has to b e intro duced where
in fact this notion is outside of the usual algebraic means for the description of system
b ehaviour
Finally the identi cation of non deterministic choice and concurrency b ecomes a
real drawback when considering serializability of transactions To keep consistency
of a database to which a numb er of users have concurrent access the database manager
has to allow concurrent execution of those transactions that do not interfere with each
other To this end it is necessary to distinguish transactions that are in con ict from those
which are indep endent of each other the identi cation of the non deterministic choice in
the case of con icting transactions with the choice of the execution order in case of
indep endent transactions leads to serious di culties in the design of database managing
systems
Ab ove we have sketched some of the original motivations that led to the formulation of
the theory of traces in Since then this theory has b een develop ed b oth in breadth and
in depth and this volume presents the state of the art of the theory of traces Some of the
developments have followed the initial motivation coming from concurrent systems while
other fall within the areas such as formal language theory theory of partially commutative
monoids graph grammars combinatorics of words etc
In our pap er we discuss a numb er of notions and results that have played a crucial
role in the initial development of the theory of traces and which in our opinion are still
quite relevant
Preliminary notions
Let X b e a set and R b e a binary relation in X R is an ordering relation in X or X
is ordered by R if R is re exive xR x for all x X transitive xR y R z implies xR z
and antisymmetric xR y and y R x implies x y if moreover R is connected for any
x y X either xR y or y R x then R is said to b e a linear or total ordering A set
together with an ordering relation is an ordered set Let X b e a set ordered by relation
R Any subset Y of X is then ordered by the restriction of R to Y i e by R Y Y
The set f n g will b e denoted by By an alphabet we shall understand
a nite set of symbols letters alphab ets will b e denoted by greek capital letters e g
etc Let b e an alphab et the set of all nite sequences of elements of willl
Mazurkiewicz Intro duction to Trace Theory
b e denoted by such sequences will b e referred to as strings over Small initial
letters a b with p ossible sub or sup erscripts will denote strings small nal latin
letters w u v etc will denote strings If u a a a is a string n is called
1 2 n
the length of u and is denoted by juj For any strings a a a b b b the
1 2 n 1 2 m
concatenation a a a b b b is the string a a a b b b
1 2 n 1 2 m 1 2 n 1 2 m
Usually sequences a a a are written as a a a Consequently a will denote
1 2 n 1 2 n
symb ol a as well as the string consisting of single symb ol a the prop er meaning will b e
always understo o d by a context String of length containing no symb ols is denoted
by The set of strings over an alphab et together with the concatenation op eration and
the empty string as the neutral element will b e referred to as the monoid of strings over
or the free monoid generated by Symb ol is used to denote the monoid of strings
over as well the set itself
We say that symb ol a occurs in string w if w w aw for some strings w w For
1 2 1 2
each string w de ne Alph w the alphabet of w as the set of all symb ols o ccurring in w
By w a we shall understand the numb er of o ccurrences of symb ol a in string w
Let b e an alphab et and w b e a string over an arbitrary alphab et then w
denotes the string projection of w onto de ned as follows
if w
u if w ua a
w
u a if w ua a
Roughly sp eaking pro jection onto deletes from strings all symb ols not in The
subscript in is omitted if is understo o d by a context
The right cancel lation of symb ol a in string w is the string w a de ned as follows
a
w if a b
w b a
w a b otherwise
for all strings w and symb ols a b It is easy to prove that pro jection and cancellation
commute
w a w a
for all strings w and symb ols a
In our approach a language is de ned by an alphab et and a set of strings over
i e a subset of Frequently a language will b e identi ed with its set of strings
however in contrast to many other approaches two languages with the same set of strings
but with di erent alphab ets will b e considered as di erent in particular two singleton
languages containing only empty strings but over di erent alphab ets are considered as
di erent Languages will b e denoted by initial latin capital letters e g A B with
p ossible subscripts and sup erscripts
Mazurkiewicz Intro duction to Trace Theory
If A B are languages over a common alphab et then their concatenation AB is the
language fuv j u A v B g over the same alphab et If w is a string A is a language
then omitting braces around singletons
w A fw u j u Ag Aw fuw j u Ag
The p ower of a language A is de ned recursively
0 n+1 n
A f g A A A
for each n and the iteration A of A is the union
1
n
A
n=0
Extend the pro jection function from strings to languages de ning for each language A
and any alphab et the pro jection of A onto as the language A de ned as
A f u j u Ag
For each string w elements of the set
Pref w fu j v uv w g
are called pre xes of w Obviously Pref w contains and w For any language A over
de ne
Pref A Pref w
w 2A
Elements of Pref A are called pre xes of A Obviously A Pref A Language A
is pre x closed if A Pref A Clearly for any language A the language Pref A is
pre x closed The pre x relation is a binary relation v in such that u v w if and
only u Pref w It is clear that the pre x relation is an ordering relation in any set of
strings
Dep endency and traces
By a dependency we shall mean any nite re exive and symmetric relation i e a nite
set of ordered pairs D such that if a b is in D then b a and a a are in D Let D
b e a dep endency the the domain of D will b e denoted by and called the alphabet of
D
D Given dep endency D call the relation I D independency induced
D D D
by D Clearly indep endency is a symmetric and irre exive relation In particular the
empty relation identity relation in and the full relation in the relation are
dep endencies the rst has empty alphab et the second is the least dep endency in the
third is the greatest dep endency in Clearly the union and intersection of a nite
numb er of dep endencies is a dep endency It is also clear that each dep endency is the
union of a nite numb er of full dep endencies since any symmetric and re exive relation
is the union of its cliques
Mazurkiewicz Intro duction to Trace Theory
2 2
Example The relation D fa bg fa cg is a dep endency fa b cg I
D D
f b c c b g 2
In this section we cho ose dep endency as a primary notion of the trace theory however
for other purp oses it may b e more convenient to take as a basic notion concurrent alphabet
i e any pair D where is an alphab et and D is a dep endency or any pair I
where is an alphab et and I is an indep endency or reliance alphabet i e any triple
D I where is an alphab et D is a dep endency and I is indep endency induced by
D
Let D b e a dep endency de ne the trace equivalence for D as the least congruence
in the monoid such that for all a b
D
D
a b I ab ba
D D
Equivalence classes of are called traces over D the trace represented by string w
D
is denoted by w By we shall denote the set f w j w g and by the
D D D D
D
set f a j a g
D D
2 2
Example For dep endency D fa bg fa cg the trace over D represented by string
abbca is abbca fabbca abcba acbbag 2
D
By de nition a single trace arises by identifying all strings which di er only in the
ordering of adjacent indep endent symb ols The quotient monoid M D is called
D
D
the trace monoid over D and its elements the traces over D Clearly M D is generated
by In the monoid M D some symb ols from commute in contrast to the monoid
D
of strings for that reason M D is also called a free partial ly commutative monoid over
D As in the case of the monoid of strings we use the symb ol M D to denote the monoid
itself as well as the set of all traces over D It is clear that in case of full dep endency i e if
D is a single clique traces reduce to strings and M D is isomorphic with the free monoid
of strings over We are going to develop the algebra of traces along the same lines as
D
it has b een done in case of of strings Let us recall that the mapping
D D
such that
w w
D D
is a homomorphism of onto M D called the natural homomorphism generated by the
equivalence
D
Now we shall give some simple facts ab out the trace equivalence and traces Let D
b e xed from now on subscripts D will b e omitted if it causes no ambiguity I will b e
the indep endency induced by D will b e the domain of D all symb ols will b e symb ols
in all strings will b e strings over and all traces will b e traces over D unless
D D
explicitly stated otherwise
It is clear that u v implies Alph u Alph v thus for all strings w we can
de ne Alph w as Alph w Denote by a binary relation in such that u v if
Mazurkiewicz Intro duction to Trace Theory
and only if there are x y and a b I such that u xaby v xbay it is not
di cult to prove that is the symmetric re exive and transitive closure of In other
words u v if and only if there exists a sequence w w w n such that
0 1 n
w u w v and for each i i n w w
0 n i 1 i
Permutation is the least congruence in such that for all symb ols a b
ab ba
If u v we say that u is a p ermutation of v Comparing with we see at once that
u v implies u v
R
De ne the mirror image w of string w as follows
R R R
ua a u
for any string u and symb ol a It is clear that the following implication the mirror rule
holds
R R
u v u v
R R
thus we can de ne w as equal to w
Since obviously u v implies u a v a and is the transitive and re exive
closure of we have the following prop erty the cancellation prop erty of traces
u v u a v a
thus we can de ne w a as w a for all strings w and symb ols a
We have also the following projection rule
u u v
v
for any alphab et b eing a consequence of the obvious implication u u v
v
Let u v b e strings a b b e symb ols a b If ua v b then applying twice rule
we get u b v a denoting u b by w by the cancellation rule again we get u
ua a v b a v a b w b Similarly we get v w a Since ua v b w ba w ab
and by the de nition of traces ab ba Thus we have the following implication for all
strings u v and symb ols a b
ua v b a b a b I w u w b v w a
Obviously u v xuy xv y If xuy xv y by the cancellation rule xu xv
R R R R R R
by the mirror rule u x v x again by the cancellation rule u v and again by
the mirror rule u v Hence from the mirror prop erty and the cancellation prop erty it
follows the following implication
xuy xv y u v
Extend indep endency relation from symb ols to strings de ning strings u v to b e
indep endent u v I if Alph u Alph v I Notice that the empty string is
indep endent of any other w I trivially holds for any string w
Mazurkiewicz Intro duction to Trace Theory
Prop osition For al l strings w u v and symbol a Alph v uav w a a v I
Proof By induction If v the implication is clear Otherwise there is symb ol
b a and string x such that v xb By Prop osition a b I and uaxb w a By
cancellation rule since b a we get uax w b a By induction hyp othesis a x I
Since a b I a xb I which proves the prop osition 2
The next theorem is a trace generalization of the Levi Lemma for strings which
reads for all strings u v x y if uv xy then there exists a string w such that either
uw x w y v or xw u w v y In case of traces this useful lemma admits more
symmetrical form
Theorem Levi Lemma for traces For any strings u v x y such that uv xy there
exist strings z z z z such that z z I and
1 2 3 4 2 3
u z z v z z x z z y z z
1 2 3 4 1 3 2 4
Proof Let u v x y b e strings such that uv xy If y the prop osition is proved by
setting z u z z z v Let w b e a string and e b e a symb ol such that y w e
1 2 4 3
0 00 0 00
Then there can b e two cases either there are strings v v such that v v ev and
00 0 00 0 00 00
e Alph v or there are strings u u such that u u eu and e Alph u v
0 00
In case by the cancellation rule we have uv v xw by induction hyp othesis
0 0 0 0
there are z z z z such that
1 2 3 4
0 0 0 0 0 00 0 0 0 0
z w z z x z z v z z uv z
4 2 3 1 4 3 2 1
0 0 0 0 0 0 00
and z z I Set z z z z e z z z z By Prop osition e v I
1 2 3 4
2 3 1 2 3 4
0 0 0 0 0
hence e z I and e z I Since e z I ez z e and it is easy to check
3 4 4 4 4
0 0 0
that equivalences are satis ed We have also z z I and e z I it yields
2 3 3
0 0
I which proves z z I e z z
2 3
3 2
0 00
In case by cancellation rule we have u u v xw and by induction hyp othesis there
0 0 0 0
are z z z z such that
1 2 3 4
0 0 0 00 0 0 0 0 0 0
u z z u v z z x z z w z z
1 2 3 4 1 3 2 4
0 0 0 0 0 0
and z z I Set as ab ove z z z z e z z z z By Prop osition
1 2 3 4
2 3 1 2 3 4
00 0 0 0 0 0 0
v I hence e z inI and e z I Since ue z z e z ez z z equiva e u
2 4
3 4 2 4 2 4
0 0 0
lences are satis ed By induction hyp othesis z z I together with e z inI it
2 3 3
implies z z I 2
2 3
Let u v b e traces over the same dep endency Trace u is a pre x of trace v and
v is a dominant of u if there exists trace x such that ux v Similarly to the case
of strings the relation to b e a pre x in the set of all traces over a xed dep endency is
Mazurkiewicz Intro duction to Trace Theory
y
x
z
3
z
4
z
1
z
2
u v
Figure Graphical interpretation of Levi Lemma for traces
a
R
ab ac
R
abb abc
R
abbc
abbca
2 2
Figure Structure of pre xes of abbca for D fa bg fa cg D
Mazurkiewicz Intro duction to Trace Theory
an ordering relation however in contrast to the pre x relation in the set of strings the
pre x relation for traces orders sets of traces in general only partially
2 2
The structure of pre xes of the trace abbca for dep endency fa bg fa cg is given
in the Figure
Prop osition Let w be a trace and u v be pre xes of w Then there exist the
greatest common pre x and the least common dominant of u and v
Proof Since u v are pre xes of w there are traces x y such that ux v y Then
by Levi Lemma for traces there are strings z z z z such that u z z x z z v
1 2 3 4 1 2 3 4
z z y z z and z z are indep endent Then z is the greatest common pre x and
1 3 2 4 2 3 1
z z z is the least common dominant of u v Indeed z is a common pre x of b oth
1 2 3 1
u v it is the greates common pre x since any symb ol in z do es not o ccur in z and
2 3
any symb ol in z do es not o ccur in z hence any extension of z is not a pre x either of
3 2 1
z z u or of z z v Thus any trace b eing a pre x of u and of v must b e a
1 2 1 3
pre x of z Similarly z z z is a common dominant of u v but any prop er pre x
1 1 2 3
of z z z is either not dominating u if it do es not contain a symb ol from z or not
1 2 3 2
dominating v if it do es not contain a symb ol from z 2
3
Let us close this section with a characterization of trace monoids by homomorphisms
so called dep endency morphisms
A dependency morphism w r t D is any homomorphism from the monoid of strings
over onto another monoid such that
D
A w w
A a b I ab ba
A ua v u v a
A ua v b a b a b I
Lemma Let be a dependency morphism u v a b If ua v b and
a b then there exists w such that u w b and v w a
Proof Let u v a b b e such as assumed in the ab ove Lemma and let
ua v b
Applying twice A to we get
u b v a
Mazurkiewicz Intro duction to Trace Theory
Set w u b Then
w a u b a
ua b since a b
v from by A
and
w b u b b
v a b from
v b a since a b
u from by A
2
Lemma Let be dependency morphisms w r t the same dependency Then x
y x y for al l x y
Proof Let D b e a dep endency and let b e two concurrency mappings w r t dep en
dency D x y and x y If x then by A y and clearly x y
If x then y Thus x ua y v b for some u v a b and we have
ua v b
There can b e two cases In the rst case we have a b then by A u v and by
induction hyp othesis u v Thus ua v b which proves x y In
the second case a b By A a b I By Lemma we get u w b v w a
for some w By induction hyp othesis
u w b v w a
hence
ua w ba v b w ab
By A since a b I
ba ab
hence w ba w ab which proves x y 2
Theorem If are dependency morphisms w r t the same dependency onto monoids
M N then M is isomorphic with N
Mazurkiewicz Intro duction to Trace Theory
Proof By Lemma the mapping de ned by the equality
w w
for all w is a homomorphism from M onto N Since has its inverse namely
1
w w
which is a homomorphism also is a homomorphism from N onto M is an isomorphism
2
Theorem The natural homomorphism of the monoid of strings over onto the
D
monoid of traces over D is a dependency morphism
Proof We have to check conditions A A for w w Condition A is obviously
D
satis ed condition A follows directly from the de nition of trace monoids Condition
A follows from the cancellation prop erty condition A follows from 2
Corollary If is a dep endency morphism w r t D onto M then M is isomorphic
with the monoid of traces over D and the isomorphism is induced by the mapping such
that a a for all a Proof It follows from Theorem and Theorem
D D D
2
Dep endence graphs
Dep endence graphs are thought as graphical representations of traces which make explicit
the ordering of symb ol o ccurrences within traces It turns out that for a given dep endency
the algebra of traces is isomorphic with the algebra of dep endency graphs as de ned
b elow Therefore it is only a matter of taste which ob jects are chosen for representing
concurrent pro cesses equivalence classes of strings or lab elled graphs
Let D b e a dep endency relation Dep endency graphs over D or d graphs for short
are nite oriented acyclic graphs with no des lab elled with symb ols from in such a
D
way that two no des of a d graph are connected with an arc if and only if they are di erent
and lab elled with dep endent symb ols Formally a triple
V R
is a dep endence graph d graph over D if
V is a nite set of no des of
R V V the set of arcs of
V the lab elling of D
Mazurkiewicz Intro duction to Trace Theory
a
A
A
A
A
b b
A
AU
A
c
A
A
A
b
A
A
A
A
AU
R
a
2 2
Figure A dep endence graph over D fa bg fa cg
such that
+
R id acyclicity
V
1
v v R R id v v D D connectivity
1 2 V 1 2
0 00 0 00
Two d graphs are isomorphic if there exists a bijection b etween their no des
preserving lab elling and arc connections As usual two isomorphic graphs are identi ed
all subsequent prop erties of d graphs are formulated up to isomorphism The empty d
graph will b e denoted by and the set of all isomorphism classes of d graphs over
D by
D
2 2
Example Let D fa bg fa cg Then the no de lab elled graph V R with
V f g
R f g
a b c b a
is a d graph It is isomorphic to the graph in Fig
The following fact follows immediately from the de nition
0 00 0 0 00
Prop osition Let D D be dependencies V R be a d graph over D V R be
00 0 00 0 00
a d graph over D If D D then R R 2
Mazurkiewicz Intro duction to Trace Theory
a a
H
H
Hj
A
c c
A
A
A
P
b b
P
A
P
P
AU
P
P
P
P
Pq
a a
R R
R R
b b
Figure D graphs comp osition
A vertex of a d graph with no arcs leaving it leading to it is said to b e a maxi
mum minimum resp vertex Clearly a d graph has at least one maximum and one
minimum vertex Since d graphs are acyclic the transitive and re exive closure of d
graph arc relation is an ordering Thus each d graph uniquely determines an ordering
of symb ol o ccurrences This ordering will b e discussed later on for the time b eing we
only mention that considering d graphs as descriptions of nonsequential pro cesses and
dep endency relation as a mo del of causal dep endency this ordering may b e viewed as
causal ordering of pro cess events
Observe that in case of full dep endency D i e if D S S the arc relation of
D D
any d graph g over D is the linear order of vertices of g in case of minimum dep endency D
i e when D is an identity relation any d graph over D consists of a numb er of connected
comp onents each of them b eing a linearly ordered set of vertices lab elled with a common
symb ol
De ne the comp osition of d graphs as follows for all graphs in the comp o
1 2 D
sition with with is a graph arising from the disjoint union of and by
1 2 1 2 1 2
adding to it new arcs leading from each no de of to each no de of provided they are
1 2
lab elled with dep endent symb ols Formally V R i there are instances
1 2
V R V R of resp ectively such that
1 1 1 2 2 2 1 2
V V V V V
1 2 1 2
R R R R
1 2 12
1 2
where R denotes a binary relation in V such that
12
v v R v V v V v v D
1 2 12 1 1 2 2 1 2
Example In Fig the comp osition of two d graphs is presented thin arrows represent
arcs added in the comp osition
Mazurkiewicz Intro duction to Trace Theory
Prop osition The composition of d graphs is a d graph
Proof In view of and it is clear that the comp osition of two d graphs
1 2
is a nite graph with no des lab elled with symb ols from It is acyclic since and
D 1
are acyclic and by in there is no arcs leading from no des of to no des of
2 2 1
Let v v b e no des of with v v D If b oth of them are no des of or of
1 2 1 2 1
then by D connectivity of comp onents and by they are also joined in If v is a
2 1
no de of and v is a no de of then by and they are joined in which proves
1 2 2
D connectivity of 2
Comp osition of d graphs can b e viewed as sequential as well as parallel com
p osing two indep endent d graphs i e d graphs such that any no de of one of them is
indep endent of any no de of the other we get the result intuitively understo o d as paral
lel comp osition while comp osing two dep endent d graphs i e such that any no de of one
of them is dep endent of any lab el of the other the result can b e viewed as the sequential
or serial comp osition In general d graph comp osition is a mixture of parallel and
sequential comp ositions where some but not all no des of one d graph are dep endent
on some no des of the other and some of them are not The nature of comp osition of
d graphs dep ends on the nature of the underlying dep endency relation
Denote the empty graph with no no des by e The set of all d graphs over a dep en
dency D with comp osition de ned as ab ove and with the empty graph as a distinguished
element forms an algebra denoted by G D
Theorem The G D is a monoid
Proof Since the empty graph is obviously the neutral left and right element w r t to
the comp osition it su ces to show that the comp osition of d graphs is asso ciative Let
for i V R b e a representant of d graph such that V V for i j
i i i i i j
By simple calculation we prove that is isomorphic to the d graph V R
1 2 3
with
V V V V
1 2 3
R R R R R R R
1 2 3 12 13 23
1 2 3
where R denotes a binary relation in V such that
ij
v v R v V v V v v D
1 2 ij 1 i 2 j 1 2
and the same result we obtain for 2
1 2 3
Let D b e a dep endency For each string w over S denote by hw i the d graph
D D
de ned recursively
h i e hw ai hw i a f a a g
D D D
Mazurkiewicz Intro duction to Trace Theory
for all strings w and symb ols a In other words hw ai arises from the graph hw i by adding
to it a new no de lab elled with symb ol a and new arcs leading to it from all vertices of
hw i lab elled with symb ols dep endent of a
2 2
D graph habbcai with dep endency D fa bg fa cg is presented in Fig
D
Let dep endency D b e xed from now on and let I hw i denote I hw i re
D D D
sp ectively
Prop osition For each d graph over D there is a string w such that hw i
Proof It is clear for the empty d graph By induction let b e a non empty d graph
remove from one of its maximum vertices It is clear that the resulting graph is a d
graph By induction hyp othesis there is a string w such that hw i is isomorphic with this
restricted d graph then hw ai where a is the lab el of the removed vertex is isomorphic
with the original graph 2
Therefore the mapping hi is a surjection
Prop osition Mapping G D de ned by w hw i is a dependency mor
phism w r t D
Proof By Prop osition mapping is a surjection By an easy induction we prove that
for each strings u v
uv huv i hui hv i u v
and clearly h i e Thus is a homomorphism onto G D Condition A is
obviously satis ed If a b I d graph habi has no arcs hence it is isomorphic with hbai
It proves A By de nition d graph huai for string u and symb ol a has a maximum
vertex lab elled with a if huai hv i then hv i has also a maximum vertex lab elled with
a removing these vertices from b oth graphs results in isomorphic graphs it is easy to
see that removing such a vertex from hv i results in hv ai Hence hui hv ai which
proves A If huai hv bi and a b then has at least two maximum vertices one
of them lab elled with a and the second lab elled with b Since b oth of them are maximum
vertices there is no arc joining them It proves a b I and A is satis ed 2
Theorem The trace monoid M D is isomorphic with the monoid G D of d graphs
over dependency D the isomorphism is induced by the bijection such that a hai
D D
for al l a D
Mazurkiewicz Intro duction to Trace Theory
2 2 2 2 2 2 2
fa cg fc dg fa b cg fb c dg fa bg fa cg fc dg
a a
d
b b
H
1 2
Hj
c c c
d d d
Figure Trace pro jections
Proof It follows from Corollary to Theorem 2
The ab ove theorem claims that traces and d graphs over the same dep endency can
b e viewed as two faces of the same coin the same concepts can b e expressed in two
di erent ways sp eaking ab out traces the algebraic character of the concept is stressed
up on while sp eaking ab out d graphs its causality or ordering features are emphasized
We can consider some graph theoretical features of traces e g connectivity of traces
as well as some algebraic prop erties of dep endence graphs e g comp osition of d graphs
Using this isomorphism one can prove facts ab out traces using graphical metho ds and
the other way around prove some graph prop erties using algebraic metho ds In fact
dual nature of traces algebraic and graph theoretical was the principal motivation to
intro duce them for representing concurrent pro cesses
An illustration of graph theoretical prop erties of traces consider the notion useful
in the other part of this b o ok of connected trace a trace is connected if the d graph
corresp onding to it is connected The algebraic de nition of this notion could b e the
following trace t is connected if there is no non empty traces t t with t t t and such
1 2 1 2
that Alph t Alph t I A connected component of trace t is the trace corresp onding
1 2
to a connected comp onent of the d graph corresp onding to t the algebraic de nition of
this notion is much more complex
As the second illustration of graph representation of traces let us represent in
2 2
graphical form pro jection of trace abcd with dep endency fa b cg fb c dg onto
1
2 2
dep endency fa cg fc dg and pro jection of the same trace onto dep endency
2
2 2 2
fa bg fa cg fc dg Fig Pro jection is onto dep endency with smaller al
1
phab et than the original symb ol b is deleted under pro jection pro jection preserves
2
all symb ols but delete some dep endencies namely dep endency b c and b d
Mazurkiewicz Intro duction to Trace Theory
Histories
The concepts presented in this section originate in pap ers of M W Shields Sh Sh Sh
The main idea is to represent non sequential pro cesses by a collection of individual his
tories of concurrently running comp onents an individual history is a string of events
concerning only one comp onent and the global history is a collection of individual ones
This approach app ealing directly to the intuitive meaning of parallel pro cessing is partic
ularly well suited to CSP like systems where individual comp onents run indep endently
of each other with one exception an event concerning a numb er of in CSP at most
two comp onents can o ccur only coincidently in all these comp onents handshaking
or rendez vous synchronization principle The presentation and the terminology used
here have b een adjusted to the present purp oses and di er from those of the authors
Let b e a n tuple of nite alphab ets Denote by P the
1 2 n
pro duct monoid
1 2 n
The comp osition in P is comp onentwise if u u u u v v v v
1 2 n 1 2 n
then
uv u v u v u v
1 1 2 2 n n
for all u v
1 2 n
Let and let b e the pro jecton from onto for i n
1 2 n i i
By the distribution in we understand here the mapping
1 2 n
de ned by the equality
w w w w
1 2 n
For each a the tuple a will b e called the elementary history of a Thus the
elementary history of a is the n tuple a a a such that
1 2 n
a if a
i
a
i
otherwise
Thus elementary history a is n tuple consising of one symb ol strings a that o ccur
in p ositions corresp onding to comp onents containing symb ol a and of empty strings that
o ccur in the remaining p ositions
Let H b e the submonoid of P generated by the set of all elementary histories
in P Elements of H will b e called global histories or simply histories and
comp onents of global histories individual histories in
Example Let fa bg fa cg Then fa b cg and the
1 2 1 2
elementary histories of are a a b c The pair
w abba aca
Mazurkiewicz Intro duction to Trace Theory
is a global history in since
w a a b b c a a
abbca
The pair abba cca is not a history since it cannot b e obtained by comp osition of ele
mentary histories 2
From the p oint of view of concurrent pro cesses the subalgebra of histories can b e
interpreted as follows There is n sequential comp onents of a concurrent system each of
them is capable to execute sequentially some elementary actions creating in this way
a sequential history of the comp onent run All of comp onents can act indep endently of
each other but an action common to a numb er of comp onents can b e executed by all of
them coincidently There is no other synchronization mechanisms provided in the system
Then the join action of all comp onents is a n tuple of individual histories of comp onents
such individual histories are consistent with each other i e roughly sp eaking they
can b e combined into one global history The following theorem o ers a formal criterion
for such a consistency
Prop osition The distribution in is a homomorphism from the monoid of strings
over onto the monoid of histories H
Proof It is clear that the distribution of any string over is a history in H It
is also clear that any history in H can b e obtained as the distribution from a string
over since any history is a comp osition of elementary histories a a a
1 2 m
a a a By prop erties of pro jection it follows the congruency of the distribution
1 2 m
uv u v 2
2 2 2
The distribution in is Theorem Let D
1 2 n
n 2 1
a dependency morphism w r t D onto the monoid of histories H
Proof Denote the assumed dep endency by D and the induced indep endency by I By
Prop osition the distribution is a homomorphism onto the monoid of histories By its
de nition we have at once w w which proves A By the de nition of
dep endency two symb ols a b are dep endent i there is index i such that a as well as b
b elongs to i e a b are indep endent i there is no index i such that contains b oth
i i
of them It proves that if a b I then ab ba since for all i n
a if a
i
b if b
ab ba
i
i i
in the remaining cases
Thus A holds To prove A assume ua v then we have ua v for all
i i
i n hence ua a v a since pro jection and cancellation commute
i i
ua a v a i e u v a for all i which implies u v a
i i i i
It proves A Supp ose now ua v b if it were i with a b it would b e
i i
u a v b which is imp ossible Therefore there is no such i i e a b I 2
i i
Mazurkiewicz Intro duction to Trace Theory
Theorem Monoid of histories H is isomorphic with the monoid of
1 2 n
2 2 2
traces over dependency
1 2 n
Proof It follows from the Theorem and Corollary to Theorem 2
Therefore similarly to the case of d graphs histories can b e viewed as yet another
third face of traces to represent a trace by a history the underlying dep endency should
b e converted to a suitable tuple of alphab ets One p ossibility but not unique is to take
cliques of dep endency as individual alphab ets and then to make use of Theorem for
2
constructing histories As an example consider trace abbca over dep endency fa bg
2
fa cg as in the previous sections then the system comp onents are fa bg fa cg and
the history corresp onding to trace abbca is the history abba aca
Ordering of symb ol o ccurrences
Traces dep endence graphs and histories are the same ob jects from the algebraic p oint of
view since monoids of them are isomorphic they b ehave in the same way under com
p osition However actually they are di erent ob jects their di erence is visible when
considering how symb ols are ordered in corresp onding ob jects traces graphs and histo
ries
There are two kinds of ordering de ned by strings the ordering of symb ols o ccurring
in a string called the o ccurrence ordering and the ordering of pre xes of a strings
the pre x ordering Lo oking at strings as at mo dels of sequential pro cesses the rst
determines the order of event o ccurrences within a pro cess The second is the order of
pro cess states since each pre x of a string can b e interpreted as a partial execution of the
pro cess interpreting by this string and such a partial execution determines uniquely an
intermediate state of the pro cess Both orderings although di erent are closely related
given one of them the second can b e reconstructed
Traces are intended to b e generalizations of strings hence the ordering given by traces
should b e a generalization of that given by strings Actually it is the case as we could
exp ect the ordering resulting from a trace is partial In this chapter we shall discuss in
details the ordering resulting from trace approach
We start from string ordering since it gives us some to ols and notions used next
for de ning the trace ordering We shall consider b oth kinds of ordering rst the pre x
ordering and next the o ccurrence ordering Next we consider ordering de ned within
dep endence graphs nally ordering of symb ols supplied by histories will b e discussed
In the whole section D will denote a xed dep endency relation b e its alphab et
I the indep endency relation induced by D All strings are assumed to b e strings over
The set of all traces over D will b e denoted by the same symb ol as the monoid of traces
over D i e by M D
Mazurkiewicz Intro duction to Trace Theory
Thus in contrast to linearly ordered pre xes of a string the set P r ef t is ordered
by only partially Interpreting a trace as a single run of a system its pre x structure can
b e viewed as the partially ordered set of all intermediate global system states reached
during this run In this interpretation represents the initial state incomparable pre xes
represent states arising in e ect of events o ccurring concurrently and the ordering of pre x
represents the temp oral ordering of system states
Occurrence ordering in strings At the b eginning of this section we give a couple
of recursive de nition of auxiliary notions All of them de ne some functions on in
D
0
these de nitions w denotes always a string over and e e e symb ols in
D D
Let a b e a symb ol w b e a string the numb er of o ccurrences of a in w is here denoted
by w a Thus a w a a w a w b a w a for all strings w and symb ols
a b The occurrence set of w is a subset of
Occ w f a n j a n w a g
It is clear that a string and its p ermutation have the same o ccurrence set since they
have the same o ccurrence numb er of symb ols hence all representants of a trace over an
arbitrary dep endency have the same o ccurrence sets
Example The o ccurrence set of string abbca is the set
f a a b b c g
2
Let R and A b e an alphab et de ne the pro jection R of R onto A as
A
follows
R f a n R j a Ag
A
It is easy to show that
Occ w Occ w
A A
Thus Occ w Occ w
A
The occurrence ordering in a string w is an ordering Ord w in the o ccurrence set of
w de ned recursively as follows
Ord Ord w a Ord w Occ w a f a w a g
Thus Ord w Occ w Occ w It is obvious that Ord w is a linear ordering for
any string w
Occurrence ordering in traces In this section we generalize the notion of o ccur
rence ordering from strings to traces Let D b e a dep endency As it has b een already
noticed u v implies Occ u Occ v for all strings u v Thus the o ccurrence set
for a trace can b e set as the o ccurrence set of its arbitrary representant De ne now
Mazurkiewicz Intro duction to Trace Theory
the o ccurrence ordering of a trace w as the intersection of o ccurrence orderings of all
representants of w
Ord w Ord u
uw
This de nition is correct since all instances of a trace have the common o ccurrence set
and the intersection of di erent linear ordering in a common set is a partial ordering of
this set Thus according to this de nition ordering of all representants of a single trace
form all p ossible linearizations extensions to the linear ordering of the trace o ccurrence
ordering From this de nition it follows that the o ccurrence ordering of a trace is the
greatest ordering common for all its representants In the rest of this section we give
some prop erties of this ordering and its alternative de nitions
We can interprete the ordering Ord w from the p oint of view of concurrency as
follows Supp ose there is a numb er of observers lo oking at the same concurrent pro cess
represented by a trace Observation made by each of them is sequential each of them
sees only one representant of this trace If two observers discover an opp osite ordering
of the same events in the observed pro cess it means that these events are actually not
ordered in the pro cess and that the di erence noticed by observers results only b ecause
their sp eci c p oints of view Thus such an ordering should not b e taken into account
and consequently two events can b e considered as really ordered in the pro cess if and
only if they are ordered in the same way in all p ossible observations of the pro cess all
observers agree on the same ordering
D graph ordering Consider now d graphs over D Let V R b e a dep en
dence graph Let as in case of strings a denotes the numb er of vertices of lab elled
with a De ne the o ccurrence set of as the set
Occ f a n j n a g
We say thet a vertex v precedes vertex u in if there is an arc leading from v to u in
Let v a n denotes a vertex of lab elled with a which is preceded in by precisely n
vertices lab elled with a Observe that for each element a n in the o ccurrence set of
there exists precisely one vertex v a n
Occurrence relation for is the binary relation Q in Occ such that
a n b m Q v a n v b m R
2
Example The diagram of the o ccurrence relation for the d graph habbcai over fa bg
2
fa cg is given in Fig 2
Since Q is acyclic its transitive and re exive closure of Q is an ordering relation
in the set Occ this ordering will b e called the occurrence ordering of and will b e
denoted by Ord
Prop osition For each string w the ordering Ord w is an extension of Ord hw i to
a linear ordering
Mazurkiewicz Intro duction to Trace Theory
a
A
A
A
A
b
A
AU
A
c
A
A
A
b
A
A
A
A
AU
R
a
2 2
Figure The o ccurrence relation for d graph habbcai over D fa bg fa cg
D
Proof It follows easily from the de nition of arc connections in d graphs by comparing
it with the de nition of Ord w given by 2
Prop osition Let be a d graph and let S be an extension of Ord to a linear
ordering Then there is a string u such that S Ord u and g hui
Proof By induction If is the empty graph then set u Let b e not empty and
let S b e the extension of Ord g to a linear ordering Let a n b e the maximum element
of S Then has a maximum vertex lab elled with a Delete from this vertex and
denote the resulting graph by delete also the maximum element from S and denote the
resulting ordering by R Clearly R is the linear extension of by induction hyp othesis
there is a string say w such that R Ord w and hw i But then S Ord w a and
hw ai Thus u w a meets the requirement of the prop osition 2
Theorem For al l strings w Ord w Ord hw i
Proof Obvious in view of the de nition of the trace ordering and prop ositions and
2
This theorem states that the intersection of linear orderings of representants of a
trace is the same ordering as the transitive and re exive closure of the ordering of the
d graph corresp onding to that trace
History ordering Now let b e n tuple of nite alphab ets
1 2 n
2 2 2
D b e the pro jection on and let b e the
1 2 n i i
1 2 n
distribution function
Mazurkiewicz Intro duction to Trace Theory
Fact For any string w over and each i n
Ord w Ord w
i i
Proof Equality holds for w Let w ue for string u and symb ol e If
e equality holds by induction hyp othesis since w u Let then
i i i
a By de nition Ord ua Ord u Occ ua f a u a g By induction
i
hyp othesis Ord u Ord u by Occ ua Occ ua a implies
i i i i i
u a u a hence Ord ua Occ ua f a u a g Ord ua It
i i i i
ends the pro of 2
Fact For each string w over
n
Occ w Occ w
i
i=1
Proof Since by de nition Occ w Occ w and Occ w Occ w for
i i i
S
n
each i n hence Occ w Occ w Let e k Occ w hence a k
i
i=1
Occ w for i such that a thus a k Occ w for this i but it means that
i i i
S
n
Occ w which completes the pro of 2 a k
i
i=1
Let w w w b e a global history de ne
1 2 n
n
Ord w Ord w w w
i 1 2 n
i=1
The following theorem close the comparison of di erent typ es of de ning trace or
dering
S
2
Theorem Let D let w be distribution function in and let
1 2 n
i
hw i be the d graph hw i Then for each string w Ord w Ord hw i
D
Proof Let D b e such as assumed ab ove let hw i denotes hw i and let w
D
w w w Thus for each i w w Since Occ w Occ hw i and
1 2 n i i
Occ w Occ w we have Occ w Occ hw i Any two symb ols in are dep endent
i i i
S
n
hence Ord w Ord hw i Therefore Ord w Ord hw i and since Ord hw i is
i i
i=1
S
n
an ordering Ord w Ord hw i To prove the inverse inclusion observe that
i
i=1
0 0 00 00 0 00
if there is an arc in an o ccurrence graph from e k to e k then e e D it
0 00 0 0 00 00
means that there is i such that e e and that e k e k Ord w hence
i i
0 0 00 00
e k e k Ord w It ends the pro of 2
The history ordering de ned by is illustrated in Fig
Mazurkiewicz Intro duction to Trace Theory
a b b c c e
e
e
a
e
e e
b b
e
e
e
e e
c c
e
e
Figure History ordering e follows a
Mazurkiewicz Intro duction to Trace Theory
It explains the principle of history ordering Let us interprete ordering Ord w in
similar way as we have done it in case of traces Each individual history of a comp onent
can b e viewed as the result of a lo cal observation of a pro cess limited to events b elonging
to the rep ertoire of the comp onent From such a viewp oint an observer can notice only
events lo cal to the comp onent he is situated at remaining actions are invisible for him
Thus b eing lo calized at i th comp onent he notices the ordering Ord w only To
i
discover the global ordering of events in the whole history all such individual histories
have to b e put together then events observed coincidently from di erent comp onents
form a sort of links b etween individual observations and make p ossible to discover an
ordering b etween events lo cal to separate and remote comp onents The rule is an event
00 00 0 0
o ccurrence a n follows another event o ccurrence a n if there is a sequence of
0 0 00 00
event o ccurrences b eginning with a n and ending with a n in which every element
follows its predecessor according to the individual history containing b oth of them Such
a principle is used by historians to establish a chronology of events concerning remote and
separate historical ob jects
Let us comment and compare all three ways of de ning ordering within graphs traces
and histories They have b een de ned in the following way
Ord hw i as the transitive and re exive closure of arc relation of hw i
Ord w as intersection of linear orderings of all representants of w
Ord w as the transitive closure of the union of linear orderings of all
w comp onents
i
All of them describ e pro cesses as partially ordered sets of event o ccurrences They
have b een de ned in di erent way in natural way for graphs as the intersection of indi
vidually observed sequential orderings in case of traces and as the union of individually
observed sequential orderings in case of histories In case of traces observations were
complete containing every action but sometimes discovering di erent orderings of some
events in case of histories observations were incomplete limited only to lo cal events but
discovering always the prop er ordering of noticed events In the case of traces the way
of obtaining the global ordering is by comparing all individual observations and rejecting
a sub jective and irrelevant information by intersection of individual orderings in case
of histories all individual observations are collected together to gain complementary and
relevant information by unifying individual observations While the second metho d has
b een compared to the metho d of historians the rst one can b e compared to that of
physicists In all three cases the results are the same it gives an additional argument for
the ab ove mentioned de nitions and indicates their soundness
Trace languages
Let D b e a dep endency by a trace language over D we shall mean any set of traces over
D The set of all trace laguages over D will b e denoted by T For a given set L of D
Mazurkiewicz Intro duction to Trace Theory
strings over denote by L the set f w j w Lg and for a given set T of traces
D D D
S
over D denote by T the set fw j w T g Clearly
D
L L and T T
D D
for all string languages L and trace languages T over D String language L such
S
that L L is called to b e existential ly consistent with dep endency D The
D
comp osition T T of trace languages T T over the same dep endency D is the set ft t j
1 2 1 2 1 2
t T t T g The iteration of trace language T over D is de ned in the same way as
1 2 2 2
the iteration of string language
1
n
T T
n=0
where
0 n+1 n
T and T T T
D
The following prop osition results directly from the de nitions
Prop osition For any dependency D any string languages L L L over and for
1 2 D
any family fL g of string languages over
i i2I D
D
L L L L
1 2 D 1 2 D
D
L L L L
1 2 1 D 2 D
L L L L
1 2 D 1 2 D
D
L L
i D i D
i2I i2I
L L
D
D
2
Before de ning the synchronization of languages which is the most imp ortant op
eration on languages for the present purp ose let us intro duce the notion of pro jection
adjusted for traces and dep endencies
Let C b e a dep endency C D and let t b e a trace over D then the trace projection
t of t onto C is a trace over C de ned as follows
C
if t
C D
t if t t e e
t
C 1 1 D C
C
t e if t t e e
C 1 C 1 D C
Intuitively trace pro jection onto C deletes from traces all symb ols not in and weakens
C
the dep endency within traces Thus the pro jection of a trace over dep endency D onto a
dep endency C which is smaller but has the same alphab et as D is a weaker trace
containing more representants the the original one
The following prop osition is easy to prove
Mazurkiewicz Intro duction to Trace Theory
Prop osition Let C D be dependencies D C Then u v u v
C D D D
2
De ne the synchronization of the string language L over with the string language
1 1
L over as the string language L k L over such that
2 2 1 2 1 2
w L w L w L k L
2 1 1 2
2 1
Prop osition Let L L L L be string languages be the alphabet of L be the
1 2 3 1 1 2
alphabet of L let fL g be a family of string languages over a common alphabet and
2 i i2I
let w be a string over Then
1 2
L k L L
L k L L k L
1 2 2 1
L k L k L L k L k L
1 2 3 1 2 3
f g k f g f g
[
1 2 1 2
L k L L k L
i i
i2I i2I
L k L fw g L f w g k L f w g
1 2 1 2
1 2
fw g L k L f w gL k f w gL
1 2 1 2
1 2
S
0
Proof Equalities are obvious Let w w L k L it means that
i
i2I
S S
0
L k w L and w L i e w L and w L then i I
i i i
i2I i2I
L which proves To prove let u L k L fw g it means that there exists v
1 2
such that v L k L and u v w by de nition of synchronization v L and
1 2 1
1
v L b ecause w is a string over this is equivalent to v w L f w g
2 1 2 1
2 1 1
and v w L f w g it means that u L f w g k L f w g Pro of of
2 1 2
2 2 1 2
is similar 2
De ne the synchronization of the trace language T over D with the trace language
1 1
T over D as the trace language T k T over D D such that
2 2 1 2 1 2
t T k T t T t T
1 2 D 1 D 2
1 2
Prop osition For any dependencies D D and any string languages L over L
1 2 1 D 2
1
over
D
2
L k L L k L
1 D 2 D 1 2 D [D
1 2 1 2
Proof
L k L ft j t L t L g
1 D 2 D D 1 D D 2 D
1 2 1 1 2 2
f w j w L w L g
D [D S 1 S 2
1 2
D D
1 2
f w j w L k L g
D [D 1 2
1 2
fw j w L k L g
1 2 D [D
1 2
L k L
1 2 D [D
1 2 2
Mazurkiewicz Intro duction to Trace Theory
a
a c a
Q
Q
Qs
c
b
k
R
c
e
b
Q
Q
Qs
e
d b
d
Figure Synchronization of two singleton trace languages
Prop osition Let T T T T be trace languages let D be the dependency of T
1 2 3 1 1
D be the dependency of T let fT g be a family of trace languages over a common
2 2 i i2I
dependency and let t be a trace over D D Then
1 2
T k T T
T k T T k T
1 2 2 1
T k T k T T k T k T
1 2 3 1 2 3
f g k f g f g
D D D [D
1 2 1 2
T k T T k T
i i
i2I i2I
T k T ftg T f t g k T f t g
1 2 1 D 2 D
1 2
ftg T k T f t gT k f t gT
1 2 D 1 D 2
1 2
Proof It is similar to that of Prop osition
2 2 2 2
Example Let D fa b dg fb c dg D fa b cg fa c eg T
1 2 1
f cabd g T f acbe g The synchronization T k T is the language f acbde g
D 2 D 1 2 D
1 2
2 2
over dep endency D D D fa b c dg fa c eg Synchronization of these two
1 2
trace languages is illustrated in Fig where traces are represented by corresp onding
d graphs without arcs resulting by transitivity from others
Observe that in a sense the synchronization op eration is inverse w r to pro jection
namely we have for any trace t over D D the equality
1 2
ftg f t g k f t g
D D
1 2
Observe also that for any traces t over D t over D the synchronization ft g k ft g
1 1 2 2 1 2
is either empty or a singleton set Thus the synchronization can b e viewed as a partial
op eration on traces In general it is not so in case of strings synchronization of two
singleton string languages may not b e a singleton E g the synchronization of the string
Mazurkiewicz Intro duction to Trace Theory
language fabg over fa bg with the string language fcdg over fc dg is the string language
fabcd acbd cabd acdb cadb cdabg while the synchronization of the trace language f ab g
2 2
over fa bg with the trace language f cd g over fc dg is the singleton trace language
2 2
f abcd g over fa bg fc dg It is yet another justi cation of using traces instead of
strings for representing runs of concurrent systems
Synchronization op eration can b e easily extended to an arbitrary numb er of argu
ments for a family of trace languages fT g with T b eing a trace language over D we
i i2I i i
S
de ne k T as the trace language over D such that
i2I i i
i2I
t k T t T
i2I i i2I D i
i
Fact If T T are trace languages over a common dependency then T k T T T
1 2 1 2 1 2
Proof Let T T b e trace languages over dep endency D then t T k T if and only
1 2 1 2
if t T and t T but since t is a trace over D t t hence t T and
D 1 D 2 D 1
t T 2
2
A trace language T is pre x closed if s v t T s T
Prop osition If T T are pre x closed trace languages then so is their synchroniza
1 2
tion T k T
1 2
Proof Let T T b e pre x closed trace languages over dep endencies D D resp ectively
1 2 1 2
0 00 0 00
Let t T k T and let t b e a pre x of t then there is t such that t t t But
1 2
0 00 0 00
then by prop erties of pro jection t t t t t T and t
D D D D 1 D
1 1 1 1 2
0 00 0 00 0 0
t t t t T since T T are pre x closed t T and t
D D D 2 1 2 D 1 D
2 2 2 1 2
0
T hence by de nition t T k T 2
2 1 2
Fixed p oint equations Let X X X Y b e families of sets n and let
1 2 n
f X X X Y f is monotone if
1 2 n
0 00 0 0 0 00 00 00
i X X f X X X f X X X
i i 1 2 n 1 2 n
0 00
for each X X X i n
i
i i
Fact Concatenation union iteration and synchronization operations on string trace
langages are monotone
Proof Obvious in view of corresp onding de nitions 2
Fact Superposition composition of any number of monotone operations is monotone
Mazurkiewicz Intro duction to Trace Theory
Proof Clear 2
Let D D D D b e dep endencies n and let
1 2 n
D D
D
D
n
1 2
f
f is congruent if
0 00 0 0 0 00 00 00
i X X f X X X f X X X
D D D D
i i i i 1 2 n 1 2 n
0 00
D
i
i n for all X X
i i
Fact Concatenation union iteration and synchronization operations on string lan
gages are congruent
Proof It follows from Prop osition and Prop osition 2
Fact Superposition composition of any number of congruent operations is congruent
Proof Clear 2
Let D D D D b e dep endencies n and let
1 2 n
D D
D
D
n 2 1
f
b e congruent Denote by f the function
D
T T T T
D D D D
n
1 2
f
D
de ned by the equality
f L L L f L L L
1 2 n D D 1 D 2 D n D
n
1 2
This de nition is correct by congruency of f
X X
We say that the set x is the least xed p oint of function f if f x x
0 0 0
and for any set x with f x x the inclusion x x holds
0
D D
Theorem Let D be a dependency and f be monotone and congruent
If L is the least xed point of f then L is the least xed point of f
0 0 D D
Mazurkiewicz Intro duction to Trace Theory
Proof Let D f and L b e such as de ned in the formulation of Theorem First observe
0
0
that for any L with f L L we have L L Indeed let L b e the least set of strings
0
0 0 0 0 0 0
such that f L L then L L Since f is monotone f f L f L hence f L
0 0 0 0
meets also the inclusion and consequently L f L hence L f L and by the
0
de nition of L we have L L L Now
0 0
L f L f L
0 D 0 D D 0 D
by congruency of f Thus L is a xed p oint of f We show that L is the
0 D D 0 D
least xed p oint of f Let T b e a trace language over D such that f T T
D D
S S S
and set L T Thus L T and T L L By we have f L
D D
S S S
f L by congruency of f we have f L f L by de nition of L we
D D D D
S S S S
have f L f T and by de nition of T we get f T T and again
D D D D
S
by de nition of L T L Collecting all ab ove relations together we get f L L
Thus as we have already proved L L hence L L T It proves L to b e
0 0 D D 0
the least xed p oint of f 2
D
This theorem can b e easily generalized for tuples of monotonic and congruent func
tions As we have already mention functions built up from variables and constants by
means of union concatenation and iteration op erations can serve as examples of mono
tonic and congruent functions Theorem allows to lift the well known and broadly
applied metho d of de ning string languages by xed p oint equations to the case of trace
languages It o ers also a useful and handy to ol for analysis concurrent systems repre
sented by elementary net systems as shown in the next section
It is worthwhile to note that under the same assumptions as in Theorem M can
0
b e the greatest xed p oint of a function f while M is not the greatest xed p oint of
0
f
Elementary Net Systems
In this section we show how the trace theory is related to net mo dels of concurrent
systems First elementary net systems intro duced in on the base of Petri nets and
will b e presented The b ehaviour of such systems will b e represented by string languages
describing various sequences of system actions that can b e executed while the system
is running Next the trace b ehaviour will b e de ned and compared with the string
b ehaviour Finally comp ositionality of b ehaviours will b e shown the b ehaviour of a
net comp osed from other nets as comp onents turns out to b e the synchronization of the
comp onents b ehaviour
Elementary net systems By an elementary net system called simply net from
now on we understand any ordered quadruple
0
N P E F m
N N N N
Mazurkiewicz Intro duction to Trace Theory
x
e c
b
x
d
a
Figure Example of an elementary net system
where P E are nite disjoint sets of places and transitions resp F P E
N N N N N
0
E P the ow relation and m P It is assumed that D om F C od F
N N N N N
N
1
P E there is no isolated places and transitions and F F Any subset of
N N
0
P is called a marking of N m is called the initial marking A place in a marking is
N
N
said to b e marked or carrying a token
As all other typ es of nets elementary net systems are represented graphically using
b oxes to represent transitions circles to represent places and arrows leading from circles
to b oxes or from b oxes to circles to represent the ow relation in such a representation
circles corresp onding to places in the initial marking are marked with dots In Fig the
net P E F m with
P f g
E fa b c d eg
F f a a b b c c
d d b b e e g
m f g
is represented graphically
P
De ne Pre Post Prox as functions from E to such that for all a E as follows
Pre a fp j p a F g
N N
Post a fp j e p F g
N N
Prox a Pre a Post a
N N N
places in Pre a are called the entry places of a those in Post a are called the
N N
exit places of a and those in Prox a are neighbours of a the set Prox a is the
N N
1
neighb ourho o d of a in N The assumption F F means that no place can b e
an entry and an exit of the same transition The subscript N is omitted if the net is understo o d
Mazurkiewicz Intro duction to Trace Theory
P P
N
Transition function of net N is a partial function E such that
N N
N
Pre a m Post a m
1 1
m a m
N 1 2
m m Pre a Post a
2 1
As usual subscript N is omitted if the net is understo o d this convention will hold for
all subsequent notions and symb ols related to them
Clearly this de nition is correct since for each marking m and transition a there
exists at most one marking equal to m a If m a is de ned for marking m and
transition a we say that a is enabled at marking m We say that the transition function
describ es single steps of the system
P P
Reachability function of net N is the function E de ned recursively
N
by equalities
m m
N
m w a m w a
N
N N
P
for all m w E a E
P
Behavioural function of net N is the function E de ned by w
N N
0
m w for all w E
As usual the subscript N is omitted everywhere the net N is understo o d
The set of strings S D om is the sequential b ehaviour of net N the set
N
of markings R C od is the set of reachable markings of N Elements of S will b e
N
called execution sequences of N execution sequence w such that w m P is said
to lead to m
The sequential behaviour of N is clearly a pre x closed language and as any pre x
closed language it is ordered into a tree by the pre x relation Maximal linearly ordered
subsets of S can b e viewed as sequential observations of the b ehaviour of N i e obser
vations made by observers capable to see only a single event o ccurrence at a time The
ordering of symb ols in strings of S re ects not only the ob jective causal ordering of
event o ccurrences but also a sub jective observational ordering resulting from a sp eci c
view over concurrent actions Therefore the structure of S alone do es not allow to decide
whether the di erence in ordering is caused by a con ict resolution a decision made in
the system or by di erent observations of concurrency In order to extract from S the
causal ordering of event o ccurrences we must supply S with an additional information as
such information we take here the dep endency of events
Dependency relation for net N is de ned as the relation D E E such that
N
a b D Prox a Prox b
N N N
Intuitively sp eaking two transitions are dep endent if either they share a common entry
place then they comp ete for taking a token away from this place or they share a
Mazurkiewicz Intro duction to Trace Theory
common exit place and then they comp ete for putting a token into the place or an entry
place of one transition is the exit place of the other and then one of them waits for the
other Transitions are indep endent if their neighb ourho o ds are disjoint If b oth such
transitions are enabled at a marking they can b e executed concurrently indep endently
of each other
De ne the non sequential b ehaviour of net N as the trace language S and denote it
D
by B That is the non sequential b ehaviour of a net arises from its sequential b ehaviour
N
by identifying some execution sequences as it follows from the dep endency de nition
two such sequences are identi ed if they di er in the order of indep endent transitions
execution
Sequential and non sequential b ehaviour consistency is guaranteed by the following
prop osition where the equality of values of two partial functions means that either b oth
of them are de ned and then their values are equal or b oth of them are unde ned
Prop osition For any net N the behavioural function is congruent w r to D i e
for any strings u v E u v implies u v
D
Proof Let u v it su ces to consider the case u xaby and v xbay for in
D
dep endent transitions a b and arbitrary x y E Because of the b ehavioural function
de nition we have to prove only
m a b m b a
for an arbitrary marking m By simple insp ection we conclude that b ecause of indep en
dency of a and b i e b ecause of disjointness of neighb ourho o ds of a and b b oth sides of
the ab ove condition are equal to a con guration
m Pre a Pre b Post a Post b
or b oth of them are unde ned 2
This is another motivation for intro ducing traces for describing concurrent systems
b ehaviour from the p oint of view of reachability p ossibilities all equivalent execution
sequences are the same As we have shown earlier ring traces can b e viewed as partially
ordered sets of symb ol o ccurrences two o ccurrences are not ordered if they represent
transitions that can b e executed indep endently of each other Thus the ordering within
ring traces is determined by mutual dep endencies of events and then it re ects the causal
relationship b etween event o ccurrences rather than the non ob jective ordering following
from the string representation of net activity
Two traces are said to b e consistent if b oth of them are pre xes of a common trace
A trace language T is complete if any two consistent traces in T are pre xes of a trace
in T
Mazurkiewicz Intro duction to Trace Theory
x
e c
b b
x
d
a
Figure Decomp osition of a net into two sequential comp onents
Prop osition The trace behaviour of any net is a complete trace language
Proof Let B b e the trace b ehaviour of net N I b e the indep endence induced by D and
let w u B and let w x uy for some strings x y Then by Levi Lemma for traces
there are strings v v v v such that v v w v v x v v u v v y and v v
1 2 3 4 1 2 3 4 1 3 2 4 2 3
I Let m v Since w D om v v D om since u v v D om
1 1 2 1 3
hence m v is de ned and m v is de ned since v v I v v v v v v
2 3 2 3 1 2 3 1 3 2
moreover m v v is de ned m v v is de ned and m v v m v v It
2 3 3 2 2 3 3 2
means that v v v v v v D om i e w v uv D om It ends the pro of 2
1 2 3 1 3 2 3 2
Comp osition of nets and their b ehaviours Here we are going to present a
mo dular way of nding the b ehaviour of nets It will b e shown how to construct the
b ehaviour of a net from b ehaviours of its parts The rst step is to decomp ose a given
net into mo dules next to nd their b ehaviours and nally to put them together nding
in this way the b ehaviour of the original net
Let I f ng we say that a net N P E F m is comp osed of nets
N P E F m i I and write
i i i i i
N N N N
1 2 n
if i j implies P P N are pairwise place disjoint and
i j i
n n n n
m F m E F P E P
i i i i
i=1 i=1 i=1 i=1
The net in Fig is comp osed from two nets presented in Fig
Mazurkiewicz Intro duction to Trace Theory
Comp osition of nets could b e de ned without assuming the disjointness of sets of
places instead we could use the disjoint union of sets of places rather than the set
theoretical union in the comp osition de nition However it would require a new de nition
of nets with two nets considered identical if there is an isomorphism identifying nets with
di erent sets of places and preserving the ow relation and the initial marking For sake
of simplicity we assume sets of places of the comp osition comp onents to b e disjoint and
we use the usual set theoretical union in our de nition
Prop osition Composition of nets is associative and commutative
Proof Obvious 2
0
Let I f ng and let N N N N N N P E F m for
1 2 n i i i i
i
all i I Let Pre Post Prox b e denoted by Pre Post Prox moreover let b e
N N N i i i i
i i i
the pro jection functions from E onto E Finally let m m P for any m P
i i i
S
P and Prop osition Let Q R P and set Q Q P R R P If P
i i I i I
i2I
the members of the family fP g are pairwise disjoint then
i i2I
Q m i I Q m
i i
R Q m i I R Q m
i i i
m Q R i I m Q R
i i i
R m Q i I R m Q
i i i
S S
and Q Q R R
i i
i2I i2I
Proof Clear 2
S
n
Prop osition D D
N N
i=1 i
Proof Let a b D N by de nition Prox a Prox b by Prox a
Prox b Q Prox a Prox b Q hence by Prop osition a b D
i2I i i i
S S
n n
Prox a Prox b Q Q Prox a Prox b
i i i
i=1 i=1
S
n
a b D 2
N
i=1
i
Prop osition Let denotes the transition function of N Then
i i
0 00 0 00 0 00
m a m i I a E m a m a E m m
i i i
i i i i
0 00
for al l m m P and a E
Mazurkiewicz Intro duction to Trace Theory
Proof It is a direct consequence of Prop osition 2
From the ab ove prop osition it follows by an easy induction
0 00 0 00
m w m i I m w m
i
i i i
The main theorem of this section is the following
Theorem
k k B k B B B
N N N N +N ++N
n n
2 1 1 2
0
Proof Set m fw E j m w mg for each m P and m
i
0 0
fw E j m w mg for each m P by Prop osition m w m
i i
i i i
0
m w m thus w m w m It means that
i2I i i i2I i i i
i i
m m k m k k m
1 1 2 2 n n
Thus by de nition of the b ehaviour the pro of is completed 2
This theorem allows us to nd the b ehaviour of a net knowing b ehaviours of its
comp onents One can exp ect the b ehaviour of comp onents to b e easier to nd than that
of the whole net As an example let us nd the b ehaviour of the net in Fig using
its decomp osition into two comp onents as in Fig These comp onents are sequential
from the theory of sequential systems it is known that their b ehaviour can b e describ ed
by the least solutions of equations
X X be b
1 1
X X abc ab a d
2 2
for languages X X over alphab ets fb eg fa b c dg resp ectively By de nition of trace
1 2
b ehaviour of nets by Theorem and Theorem and we have
X X be b
1 D 1 D D D D
1 1 1 1 1
X X abc ab a d
2 D 2 D D D D D D
2 2 2 2 2 2 2
2 2
with D fb eg D fa b c dg The equation for the synchronization X k X is
1 2 1 2
by prop erties of synchronization as follows
X k X X be b k X abc ab a d
1 D 2 D 1 D D D D 2 D D D D D D
1 2 1 1 1 1 2 2 2 2 2 2
and after some easy calculations using Prop osition we get
X k X X k X abc abe ab a d
1 2 D 1 2 D D
Mazurkiewicz Intro duction to Trace Theory
a
2
a
1
R
a
b
n
1
R
b b
2 m
Figure An example of atomic net
2 2
with D D D fb eg fa b c dg Observe that comp osition of two comp onents
1 2
that are sequential results in a net with some indep endent transitions in the consid
ered case the indep endent pairs of transitions are a e c e d e and their symmetric
images
There exists a standard set of very simple nets namely nets with a singleton set of
places the b ehaviours of which can b e considered as known Such nets are called atomic
or atoms an arbitrary net can b e comp osed from such nets hence the b ehaviour of any
net can b e comp osed from the b ehaviours of atoms Thus giving the b ehaviours of atoms
we supply another de nition of the elementary net system b ehaviour More formally let
0
N P E F m b e a net for each p P the net
0
N fpg Prox p F m
p p p
where
0 0
F f e p j e Pre p g f p e j e Post p g m m fpg
p p
is called an atom of N determined by p Clearly the following prop osition holds
Prop osition Each net is the composition of al l its atoms
The b ehaviour of atoms can b e easily found Namely let the atom N b e de ned as
0
N fpg A Z F m where A fe j e p F g Z fe j p e F g Say that N is
0 0
marked if m fpg and unmarked otherwise i e if m Then by the de nition of
b ehavioural function trace b ehaviour B is the trace language ZA Z if it is
N D
0
2
marked and AZ A if it is unmarked where D A Z
D
Conclusions
Some basic notions of trace theory has b een brie y presented In the next chapters of this
b o ok this theory will b e made broader and deep er the intention of the present chapter
Mazurkiewicz Intro duction to Trace Theory
x
e c
b b
e c
b b
x
d
a a
Figure Atoms of net in Fig
was to show some initial ideas that motivated the whole enterprise In the sequel the
reader will b e able to get acquainted with the development trace theory as a basis of non
standard logic as well as with the basic and involved results from the theory of monoids
with prop erties of graph representations of traces and with generalization of the notion
of nite automaton that is consistent with the trace approach All this work shows that
the concurrency issue is still challenging and stimulating fruitful research
Acknowledgements The author is greatly indebted to Professor Grzegorz
Rozenb erg for his help and encouragement without them this pap er will not app ear
Also the supp ort of Esprit CALIBAN Working Group is gratefully acknowledged
References
Aalb ersb erg IJ J Studies in Trace Theory Ph D Dissertation Leiden University
Bakker de J W Ro ever de W P Rozenb erg G eds Stepwise Re nement of
Distributed Systems Pro c of REX Workshop Lecture Notes in Computer Science
Ehrenfeucht A Rozenb erg G On the Structure of Dependence Graphs Rapp ort
Dept of Computer Science University of Leiden
Fle M P and Roucairol G On Serializability of Iterated Transactions Pro c
ACM SIGACT SIGOPS Symp on Princ of Distrib Comp Ottawa
Mazurkiewicz Intro duction to Trace Theory
Grab owski J On partial languages Fundamenta Informaticae
Gyory G Knuth E Ronyai L Grammatical Projections Elementary
Constructions Working Pap er I I MTA SZTAKI Budap est
Hoare C A R Communicating Sequential Processes Communications of the ACM
vol
Janicki R Mazurkiewicz traces semantics for communicating sequential processes
Pro c th Europ ean Workshop on Appl and Theory of Petri Nets Aarhus
Knuth E Petri nets and trace languages Pro c st Europ Conf on Parallel and
Distr Pro cessing Toulouse
Levi F W On semigroups Bull of the Calcutta Math So c
Mazurkiewicz A Concurrent Program Schemes and Their Interpretations DAIMI
Techn Rep of Aarhus University PB
Mazurkiewicz A Traces histories graphs instances of a process monoid
Lecture Notes in Computer Science
Mazurkiewicz A Semantics of Concurrent Systems A Modular Fixed Point Trace
Approach LNCS
Mazurkiewicz A Ochmanski E Penczek W Concurrent Systems and
Inevitability Theoretical Computer Science
Petri C A Fundamentals of a Theory of Asynchronous Information Flow Pro c of
IFIP Congress North Holland Amsterdam
Petri C A Concepts of Net Theory Pro ceedings of the Symp osium and Summer
Scho ol on MFCS High Tatras
Petri C A Non Sequential Processes GMD ISF Rep ort Gesellschaft fuer
Mathematik und Datenverarb eitung mbH Bonn
Rozenb erg G Behaviour of elementary net systems Advances in Petri Nets
Part I Pro ceedings of an Advanced Course Lecture Notes in Computer Science
Rozenb erg G Thiagara jan P S Petri Nets Basic Notions Structure and
Behaviour Lecture Notes in Computer Science
Shields M W Non Sequential Behaviour Part Int Rep ort CSR Dept of
Computer Science University of Edinburgh
Thiagara jan P S Elementary net systems Advances in Petri Nets Part I
Pro ceedings of an Advanced Course Lecture Notes in Computer Science
Mazurkiewicz Intro duction to Trace Theory
Zielonka W Proving assertions about paral lel programs by means of traces ICS
PAS Rep ort Inst of Comp Science Polish Academy of Sciences Warszawa