In
computer science
Computer science is the study of computation, automation, and information. Computer science spans theoretical disciplines (such as algorithms, theory of computation, information theory, and automation) to Applied science, practical discipli ...
, a trace is a set of
strings, wherein certain letters in the string are allowed to
commute
Commute, commutation or commutative may refer to:
* Commuting, the process of travelling between a place of residence and a place of work
Mathematics
* Commutative property, a property of a mathematical operation whose result is insensitive to th ...
, but others are not. It generalizes the concept of a string, by not forcing the letters to always be in a fixed order, but allowing certain reshufflings to take place. Traces were introduced by
Pierre Cartier and
Dominique Foata
Dominique Foata (born October 12, 1934) is a mathematician who works in enumerative combinatorics. With Pierre Cartier and Marcel-Paul Schützenberger he pioneered the modern approach to classical combinatorics, that lead, in part, to the curre ...
in 1969 to give a combinatorial proof of
MacMahon's master theorem. Traces are used in theories of
concurrent computation, where commuting letters stand for portions of a job that can execute independently of one another, while non-commuting letters stand for locks,
synchronization points or
thread joins.
[Sándor & Crstici (2004) p.161]
The trace monoid or free partially commutative monoid is a
monoid
In abstract algebra, a branch of mathematics, a monoid is a set equipped with an associative binary operation and an identity element. For example, the nonnegative integers with addition form a monoid, the identity element being 0.
Monoid ...
of traces. In a nutshell, it is constructed as follows: sets of commuting letters are given by an
independency relation. These induce an equivalence relation of equivalent strings; the elements of the equivalence classes are the traces. The equivalence relation then partitions up the
free monoid (the set of all strings of finite length) into a set of equivalence classes; the result is still a monoid; it is a
quotient monoid and is called the ''trace monoid''. The trace monoid is
universal
Universal is the adjective for universe.
Universal may also refer to:
Companies
* NBCUniversal, a media and entertainment company
** Universal Animation Studios, an American Animation studio, and a subsidiary of NBCUniversal
** Universal TV, a t ...
, in that all dependency-homomorphic (see below) monoids are in fact
isomorphic
In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word i ...
.
Trace monoids are commonly used to model
concurrent computation, forming the foundation for
process calculi. They are the object of study in
trace theory
In mathematics and computer science, trace theory aims to provide a concrete mathematical underpinning for the study of concurrent computation and process calculi. The underpinning is provided by an algebraic definition of the free partially c ...
. The utility of trace monoids comes from the fact that they are isomorphic to the monoid of
dependency graphs; thus allowing algebraic techniques to be applied to
graphs, and vice versa. They are also isomorphic to
history monoids, which model the history of computation of individual processes in the context of all scheduled processes on one or more computers.
Trace
Let
denote the free monoid, that is, the set of all strings written in the alphabet
. Here, the asterisk denotes, as usual, the
Kleene star. An
independency relation on
then induces a (symmetric) binary relation
on
, where
if and only if there exist
, and a pair
such that
and
. Here,
and
are understood to be strings (elements of
), while
and
are letters (elements of
).
The trace is defined as the reflexive transitive closure of
. The trace is thus an equivalence relation on
, and is denoted by
, where
is the dependency relation corresponding to
that is
and conversely
Clearly, different dependencies will give different equivalence relations.
The
transitive closure implies that
if and only if there exists a sequence of strings
such that
and
and
for all
. The trace is stable under the monoid operation on
(
concatenation
In formal language theory and computer programming, string concatenation is the operation of joining character strings end-to-end. For example, the concatenation of "snow" and "ball" is "snowball". In certain formalisations of concatenat ...
) and is therefore a
congruence relation
In abstract algebra, a congruence relation (or simply congruence) is an equivalence relation on an algebraic structure (such as a group, ring, or vector space) that is compatible with the structure in the sense that algebraic operations done ...
on
.
The trace monoid, commonly denoted as
, is defined as the quotient monoid
:
The homomorphism
:
is commonly referred to as the
natural homomorphism or canonical homomorphism. That the terms ''natural'' or ''canonical'' are deserved follows from the fact that this morphism embodies a universal property, as discussed in a later section.
One will also find the trace monoid denoted as
where
is the independency relation. Confusingly, one can also find the commutation relation used instead of the independency relation (it differs by including all the diagonal elements).
Examples
Consider the alphabet
. A possible dependency relation is
:
The corresponding independency is
:
Therefore, the letters
commute. Thus, for example, a trace equivalence class for the string
would be
:
The equivalence class
is an element of the trace monoid.
Properties
The cancellation property states that equivalence is maintained under
right cancellation. That is, if
, then
. Here, the notation
denotes right cancellation, the removal of the first occurrence of the letter ''a'' from the string ''w'', starting from the right-hand side. Equivalence is also maintained by left-cancellation. Several corollaries follow:
* Embedding:
if and only if
for strings ''x'' and ''y''. Thus, the trace monoid is a syntactic monoid.
* Independence: if
and
, then ''a'' is independent of ''b''. That is,
. Furthermore, there exists a string ''w'' such that
and
.
* Projection rule: equivalence is maintained under
string projection, so that if
, then
.
A strong form of
Levi's lemma holds for traces. Specifically, if
for strings ''u'', ''v'', ''x'', ''y'', then there exist strings
and
such that
for all letters
and
such that
occurs in
and
occurs in
, and
:
:
Universal property
A dependency morphism (with respect to a dependency ''D'') is a morphism
:
to some monoid ''M'', such that the "usual" trace properties hold, namely:
:1.
implies that
:2.
implies that
:3.
implies that
:4.
and
imply that
Dependency morphisms are universal, in the sense that for a given, fixed dependency ''D'', if
is a dependency morphism to a monoid ''M'', then ''M'' is
isomorphic
In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word i ...
to the trace monoid
. In particular, the natural homomorphism is a dependency morphism.
Normal forms
There are two well-known normal forms for words in trace monoids. One is the ''
lexicographic'' normal form, due to Anatolij V. Anisimov and
Donald Knuth
Donald Ervin Knuth ( ; born January 10, 1938) is an American computer scientist, mathematician, and professor emeritus at Stanford University. He is the 1974 recipient of the ACM Turing Award, informally considered the Nobel Prize of computer sc ...
, and the other is the ''Foata'' normal form due to
Pierre Cartier and
Dominique Foata
Dominique Foata (born October 12, 1934) is a mathematician who works in enumerative combinatorics. With Pierre Cartier and Marcel-Paul Schützenberger he pioneered the modern approach to classical combinatorics, that lead, in part, to the curre ...
who studied the trace monoid for its
combinatoric
Combinatorics is an area of mathematics primarily concerned with counting, both as a means and an end in obtaining results, and certain properties of finite structures. It is closely related to many other areas of mathematics and has many app ...
s in the 1960s.
[Section 2.3, Diekert and Métivier 1997.]
Unicode's
Normalization Form Canonical Decomposition (NFD) is an example of a lexicographic normal form - the ordering is to sort consecutive characters with non-zero canonical combining class by that class.
Trace languages
Just as a formal language can be regarded as a subset of
, the set of all possible strings, so a trace language is defined as a subset of
all possible traces.
Alternatively, but equivalently, a language
is a trace language, or is said to be consistent with dependency ''D'' if
:
where
:
is the trace closure of a set of strings.
See also
*
Trace cache
Notes
References
General references
*
*
* Antoni Mazurkiewicz, "Introduction to Trace Theory", pp 3–41, in ''The Book of Traces'', V. Diekert, G. Rozenberg, eds. (1995) World Scientific, Singapore
* Volker Diekert, ''Combinatorics on traces'',
LNCS 454, Springer, 1990, , pp. 9–29
* {{citation , last1=Sándor , first1=Jozsef , last2=Crstici , first2=Borislav , title=Handbook of number theory II , location=Dordrecht , publisher=Kluwer Academic , year=2004 , isbn=1-4020-2546-7 , pages=32–36 , zbl=1079.11001
Seminal publications
* Pierre Cartier and Dominique Foata, ''Problèmes combinatoires de commutation et réarrangements'', Lecture Notes in Mathematics 85, Springer-Verlag, Berlin, 1969
Free 2006 reprintwith new appendixes
* Antoni Mazurkiewicz, ''Concurrent program schemes and their interpretations'', DAIMI Report PB 78, Aarhus University, 1977
Semigroup theory
Formal languages
Free algebraic structures
Combinatorics
Trace theory