History monoid

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In mathematics and computer science, a history monoid is a way of representing the histories of concurrently running computer processes as a collection of strings, each string representing the individual history of a process. The history monoid provides a set of synchronization primitives (such as locks, mutexes or thread joins) for providing rendez-vous points between a set of independently executing processes or threads.

History monoids occur in the theory of concurrent computation, and provide a low-level mathematical foundation for process calculi, such as CSP the language of communicating sequential processes, or CCS, the calculus of communicating systems. History monoids were first presented by M.W. Shields[1]

History monoids are isomorphic to trace monoids (free partially commutative monoids) and to the monoid of dependency graphs. As such, they are free objects and are universal. The history monoid is a type of semi-abelian categorical product in the category of monoids.

Contents

[edit] Product monoids and projection

Let

A=(\Sigma_1,\Sigma_2,\cdots,\Sigma_n)

denote an n-tuple of alphabets Σk, and let P(A) denote the Cartesian product of free monoids

P(A)=\Sigma_1^* \times \Sigma_2^* \times \cdots \times \Sigma_n^*

where \Sigma_k^* is the free monoid for the alphabet Σk. That is, it is the collection of all finite-length strings formed from the letters of Σk; the superscript symbol * is the Kleene star. Composition in the product monoid is component-wise, so that, for

\bold{u}=(u_1,u_2,\cdots,u_n)

and

\bold{v}=(v_1,v_2,\cdots,v_n)

then

\bold{uv}=(u_1v_1,u_2v_2,\cdots,u_nv_n)

for all \bold{u}, \bold{v} in P(A). Define the union alphabet to be

\Sigma=\Sigma_1 \cup \Sigma_2 \cup \cdots \cup \Sigma_n

Here, the union should be understood to be the set union and should not be confused with the disjoint union. A distribution is the projection mapping

\pi:\Sigma^*\to P(A)

taking a string w\in \Sigma^* to

\pi(w)\mapsto (\pi_1(w), \pi_2(w), \cdots , \pi_n(w))

Here, the individual maps \pi_k:\Sigma^*\to\Sigma_k^* are to be understood to be string projections. That is, πk(w) is the string where all letters not in Σk have been removed.

[edit] Histories

For every a\in\Sigma, the tuple π(a) is called the elementary history of a. It serves as an indicator function for the inclusion of a letter a in an alphabet Σk. That is,

\pi(a)=(a_1,a_2,\cdots,a_n)

where

a_i=\begin{cases} a \mbox{ if } a\in \Sigma_k \\ \varepsilon \mbox { otherwise } \end{cases}

Here, \varepsilon denotes the empty string. The history monoid H(A) is the free monoid generated by elementary histories. It is clearly a submonoid of the product monoid P(A). The elements of H(A) are called global histories, and the projections of a global history are called individual histories.

[edit] Connection to computer science

The use of the word history in this context, and the connection to concurrent computing, can be understood as follows. An individual history is a record of the sequence of states of a process (or thread or machine); the alphabet Σk is the set of states of the process.

A letter that occurs in two or more alphabets serves as a synchronization primitive between the various individual histories. That is, if such a letter occurs in one individual history, it must also occur in another history, and serves to "tie" or "rendez-vous" them together.

Consider, for example, Σ1 = {a,b,c} and Σ2 = {a,d,e}. The union alphabet is of course Σ = {a,b,c,d,e}. The elementary histories are (a,a), (b,\varepsilon), (c,\varepsilon), (\varepsilon,d) and (\varepsilon,e). In this example, an individual history of the first process might be "bcbcc" while the individual history of the second machine might be "ddded". Both of these individual histories are represented by the global history "bcbdddcced", since the projection of this string onto the individual alphabets yields the individual histories. In the global history, the letter b can be considered to commute with the letters d and e, in that these can be rearranged without changing the individual histories. Such commutation is simply a statement that the first and second processes are running concurrently, and are unordered with respect to each other; they have not (yet) exchanged any messages or performed any synchronization.

The letter a serves as a synchronization primitive, as its occurrence marks a spot in both the global and individual histories, that cannot be commuted across. Thus, while the letters b and c can be re-ordered past d and e, they cannot be reordered past a. Thus, the global history "bcdabe" and the global history "bdcaeb" both have as individual histories "bcab" and "dae", indicating that the execution of d may happen before or after c. However, the letter a is synchronizing, so that e is guaranteed to happen after c, even though e is in a different process than c.

[edit] Properties

The history monoid is isomorphic to the trace monoid, and as such, is a type of semi-abelian categorical product in the category of monoids. In particular, the history monoid H(\Sigma_1,\Sigma_2,\cdots,\Sigma_n) is isomorphic to the trace monoid \mathbb{M}(D) with the dependency relation given by

D=\left(\Sigma_1\times\Sigma_1\right)\cup \left(\Sigma_2\times\Sigma_2\right)\cup \cdots \cup \left(\Sigma_n\times\Sigma_n\right)

In simple terms, this is just the formal statement of the informal discussion given above: the letters in an alphabet Σk can be commutatively re-ordered past the letters in an alphabet Σj, unless they are letters that occur in both alphabets. Thus, traces are exactly global histories, and vice-versa.

[edit] References

  1. ^ M.W. Shields "Concurrent Machines", Computer Journal, (1985) 28 pp.449-465.
  • Antoni Mazurkiewicz, "Introduction to Trace Theory", pp 3-41, in The Book of Traces, V. Diekert, G. Rozenberg, eds. (1995) World Scientific, Singapore ISBN 9810220588
  • Volker Diekert, Yves Métivier, "Partial Commutation and Traces", In G. Rozenberg and A. Salomaa, editors, Handbook of Formal Languages, Vol. 3, Beyond Words, pages 457--534. Springer-Verlag, Berlin, 1997.