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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク History monoid ： ウィキペディア英語版 History monoid 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 rendezvous 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.〔M.W. Shields "Concurrent Machines", ''Computer Journal'', (1985) 28 pp. 449–465.〕 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. ==Product monoids and projection== Let :$A=(\backslash Sigma\_1,\backslash Sigma\_2,\backslash ldots,\backslash Sigma\_n)$ denote an ''n''-tuple of alphabets $\backslash Sigma\_k$. Let $P(A)$ denote all possible combinations of finite-length strings from the alphabets: :$P(A)=\backslash Sigma\_1^$ * \times \Sigma_2^ * \times \cdots \times \Sigma_n^ * (In more formal language, $P(A)$ is the Cartesian product of the free monoids of the $\backslash Sigma\_k$. The superscript star is the Kleene star.) Composition in the product monoid is component-wise, so that, for :$\backslash bold=(u\_1,u\_2,\backslash ldots,u\_n)\; \backslash ,$ and :$\backslash bold=(v\_1,v\_2,\backslash ldots,v\_n)\; \backslash ,$ then :$\backslash bold=(u\_1v\_1,u\_2v\_2,\backslash ldots,u\_nv\_n)\; \backslash ,$ for all $\backslash bold,\; \backslash bold$ in $P(A)$. Define the union alphabet to be :$\backslash Sigma=\backslash Sigma\_1\; \backslash cup\; \backslash Sigma\_2\; \backslash cup\; \backslash cdots\; \backslash cup\; \backslash Sigma\_n.\; \backslash ,$ (The union here is the set union, not the disjoint union.) Given any string $w\backslash in\; \backslash Sigma^$ *, we can pick out just the letters in some $\backslash Sigma\_k^$ * using the corresponding string projection $\backslash pi\_k:\backslash Sigma^$ *\to\Sigma_k^ *. A distribution $\backslash pi:\backslash Sigma^$ *\to P(A) is the mapping that operates on $w\backslash in\; \backslash Sigma^$ * with all of the $\backslash pi\_k$, separating it into components in each free monoid: :$\backslash pi(w)\backslash mapsto\; (\backslash pi\_1(w),\; \backslash pi\_2(w),\; \backslash ldots\; ,\; \backslash pi\_n(w)).\; \backslash ,$ 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「History monoid」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.