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In mathematics and computer science, a dependency relation is a binary relation that is finite, symmetric, and reflexive. That is, it is a finite set of ordered pairs D, such that

• If $(a,b)\backslash in\; D$ then $(b,a)\; \backslash in\; D$ (symmetric)
• If $a$ is an element of the set on which the relation is defined, then $(a,a)\; \backslash in\; D$ (reflexive)

In general, dependency relations are not transitive; thus, they generalize the notion of an equivalence relation by discarding transitivity.

Let $\backslash Sigma$ denote the alphabet of all the letters of D. Then the independency induced by D is the binary relation I

$I\; =\; \backslash Sigma\; \backslash times\; \backslash Sigma\; -\; D$

That is, the independency is the set of all ordered pairs that are not in D. Clearly, the independency is symmetric and irreflexive.

The pairs $(\backslash Sigma,\; D)$ and $(\backslash Sigma,\; I)$, or the triple $(\backslash Sigma,\; D,\; I)$ (with I induced by D) are sometimes called the concurrent alphabet or the reliance alphabet.

The pairs of letters in an independency relation induce an equivalence relation on the free monoid of all possible strings of finite length. The elements of the equivalence classes induced by the independency are called traces, and are studied in trace theory.

Examples

Consider the alphabet $\backslash Sigma=\backslash \{a,b,c\backslash \}$. A possible dependency relation is

$\backslash begin\{matrix\}\; D$

&=& \{a,b\}\times\{a,b\} \quad \cup \quad \{a,c\}\times\{a,c\} \\
&=& \{a,b\}^2 \cup \{a,c\}^2 \\
&=& \{ (a,b),(b,a),(a,c),(c,a),(a,a),(b,b),(c,c)\} 

\end{matrix}

The corresponding independency is

$I\_D=\backslash \{(b,c)\backslash ,,\backslash ,(c,b)\backslash \}$

Therefore, the letters $b,c$ commute, or are independent of one-another.

This article is licensed under the GNU Free Documentation License. It uses material from Wikipedia

This article is licensed under the GNU Free Documentation License. It uses material from Wikipedia