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| In [[mathematics]] and [[computer science]], a '''history monoid''' is a way of representing the histories of concurrently running computer [[process (computer science)|processes]] as a collection of [[string (computer science)|strings]], each string representing the individual history of a process. The history monoid provides a set of [[synchronization primitive]]s (such as [[lock (computer science)|locks]], [[mutex]]es or [[thread join]]s) for providing rendezvous points between a set of independently executing processes or threads.
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| 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.<ref>M.W. Shields "Concurrent Machines", ''Computer Journal'', (1985) '''28''' pp. 449–465.</ref>
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| History monoids are isomorphic to [[trace monoid]]s (free partially commutative monoids) and to the [[monoid]] of [[dependency graph]]s. As such, they are [[free object]]s and are [[universal property|universal]]. The history monoid is a type of semi-abelian [[categorical product]] in the [[category (mathematics)|category]] of monoids.
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| ==Product monoids and projection==
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| Let
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| :<math>A=(\Sigma_1,\Sigma_2,\ldots,\Sigma_n)</math>
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| denote an ''n''-tuple of [[alphabet (computer science)|alphabets]] <math>\Sigma_k</math>. Let <math>P(A)</math> denote all possible combinations of finite-length strings from the alphabets:
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| :<math>P(A)=\Sigma_1^* \times \Sigma_2^* \times \cdots \times \Sigma_n^*</math>
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| (In more formal language, <math>P(A)</math> is the [[Cartesian product]] of the [[free monoid]]s of the <math>\Sigma_k</math>. The superscript star is the [[Kleene star]].) Composition in the product monoid is component-wise, so that, for
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| :<math>\bold{u}=(u_1,u_2,\ldots,u_n) \, </math>
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| and
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| :<math>\bold{v}=(v_1,v_2,\ldots,v_n) \, </math>
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| then
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| :<math>\bold{uv}=(u_1v_1,u_2v_2,\ldots,u_nv_n) \, </math>
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| for all <math>\bold{u}, \bold{v}</math> in <math>P(A)</math>. Define the union alphabet to be
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| :<math>\Sigma=\Sigma_1 \cup \Sigma_2 \cup \cdots \cup \Sigma_n. \,</math>
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| (The union here is the [[set union]], not the [[disjoint union]].) Given any string <math>w\in \Sigma^*</math>, we can pick out just the letters in some <math>\Sigma_k^*</math> using the corresponding [[string projection]] <math>\pi_k:\Sigma^*\to\Sigma_k^*</math>. A '''distribution''' <math>\pi:\Sigma^*\to P(A)</math> is the mapping that operates on <math>w\in \Sigma^*</math> with all of the <math>\pi_k</math>, separating it into components in each free monoid:
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| :<math>\pi(w)\mapsto (\pi_1(w), \pi_2(w), \ldots , \pi_n(w)). \,</math>
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| ==Histories==
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| For every <math>a\in\Sigma</math>, the tuple <math>\pi(a)</math> is called the '''elementary history''' of ''a''. It serves as an [[indicator function]] for the inclusion of a letter ''a'' in an alphabet <math>\Sigma_k</math>. That is,
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| :<math>\pi(a)=(a_1,a_2,\ldots,a_n)</math>
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| where
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| :<math>a_i=\begin{cases}
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| a \mbox{ if } a\in \Sigma_k \\
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| \varepsilon \mbox { otherwise }.
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| \end{cases}</math>
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| Here, <math>\varepsilon</math> denotes the [[empty string]]. The '''history monoid''' <math>H(A)</math> is the [[free monoid]] generated by elementary histories. It is clearly a submonoid of the product monoid <math>P(A)</math>. The elements of <math>H(A)</math> are called '''global histories''', and the projections of a global history are called '''individual histories'''.
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| ==Connection to computer science==
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| 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 [[state (computer science)|states]] of a process (or [[thread (computer science)|thread]] or [[Turing machine|machine]]); the alphabet <math>\Sigma_k</math> is the set of states of the process.
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| 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 "rendezvous" them together.
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| Consider, for example, <math>\Sigma_1=\{a,b,c\}</math> and <math>\Sigma_2=\{a,d,e\}</math>. The union alphabet is of course <math>\Sigma=\{a,b,c,d,e\}</math>. The elementary histories are <math>(a,a)</math>, <math>(b,\varepsilon)</math>, <math>(c,\varepsilon)</math>, <math>(\varepsilon,d)</math> and <math>(\varepsilon,e)</math>. In this example, an individual history of the first process might be <math>bcbcc</math> while the individual history of the second machine might be <math>ddded</math>. Both of these individual histories are represented by the global history <math>bcbdddcced</math>, since the projection of this string onto the individual alphabets yields the individual histories. In the global history, the letter <math>b</math> can be considered to commute with the letters <math>d</math> and <math>e</math>, 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.
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| The letter <math>a</math> 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 <math>b</math> and <math>c</math> can be re-ordered past <math>d</math> and <math>e</math>, they cannot be reordered past <math>a</math>. Thus, the global history <math>bcdabe</math> and the global history <math>bdcaeb</math> both have as individual histories <math>bcab</math> and <math>dae</math>, indicating that the execution of <math>d</math> may happen before or after <math>c</math>. However, the letter <math>a</math> is synchronizing, so that <math>e</math> is guaranteed to happen after <math>c</math>, even though <math>e</math> is in a different [[process (computer science)|process]] than <math>c</math>.
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| ==Properties==
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| The history monoid is isomorphic to the [[trace monoid]], and as such, is a type of semi-abelian [[categorical product]] in the [[category (mathematics)|category]] of monoids. In particular, the history monoid <math>H(\Sigma_1,\Sigma_2,\ldots,\Sigma_n)</math> is isomorphic to the trace monoid <math>\mathbb{M}(D)</math> with the [[dependency relation]] given by
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| :<math>D=\left(\Sigma_1\times\Sigma_1\right)\cup
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| \left(\Sigma_2\times\Sigma_2\right)\cup \cdots \cup
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| \left(\Sigma_n\times\Sigma_n\right).</math>
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| In simple terms, this is just the formal statement of the informal discussion given above: the letters in an alphabet <math>\Sigma_k</math> can be commutatively re-ordered past the letters in an alphabet <math>\Sigma_j</math>, unless they are letters that occur in both alphabets. Thus, traces are exactly global histories, and vice-versa.
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| ==Notes==
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| <references/>
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| ==References==
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| * Antoni Mazurkiewicz, "Introduction to Trace Theory", pp 3–41, in ''The Book of Traces'', V. Diekert, G. Rozenberg, eds. (1995) World Scientific, Singapore ISBN 981-02-2058-8
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| * Volker Diekert, Yves Métivier, "[http://citeseer.ist.psu.edu/diekert97partial.html 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.
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| [[Category:Concurrency (computer science)]]
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| [[Category:Semigroup theory]]
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| [[Category:Formal languages]]
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| [[Category:Free algebraic structures]]
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