Dependency relation - Citizendia

In mathematics and computer science, a dependency relation is a binary relation that is finite, symmetric, and reflexive. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and Computer science (or computing science) is the study and the Science of the theoretical foundations of Information and Computation and their In Mathematics, a binary relation (or a dyadic or 2-place relation) is an arbitrary association of elements within a set or with elements of In Mathematics, a Binary relation R over a set X is symmetric if it holds for all a and b in X that In Set theory, a Binary relation can have among other properties reflexivity or irreflexivity. That is, it is a finite set of ordered pairs D, such that

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

In general, dependency relations are not transitive; thus, they generalize the notion of an equivalence relation by discarding transitivity. In Mathematics, a Binary relation R over a set X is transitive if whenever an element a is related to an element b In Mathematics, an equivalence relation is a Binary relation between two elements of a set which groups them together as being "equivalent"

Let $Σ$ denote the alphabet of all the letters of D. In Computer science, an alphabet is a usually finite set of characters or digits Then the independency induced by D is the binary relation I

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

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 $(Σ, D)$ and $(Σ, I)$ , or the triple $(Σ, D, I)$ (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. In Abstract algebra, the free monoid on a set A is the Monoid whose elements are all the finite sequences (or strings) of zero or The elements of the equivalence classes induced by the independency are called traces, and are studied in trace theory. In Mathematics, given a set X and an Equivalence relation ~ on X, the equivalence class of an element a in X In Mathematics and Computer science, a trace is a set of strings, wherein certain letters in the string are allowed to commute but others are not In Mathematics and Computer science, trace theory aims to provide a concrete mathematical underpinning for the study of Concurrent computation and

Examples

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

$\begin{matrix} D &=& \{a,b\}\times\{a,b\} \quad \cup \quad \{a,c\}\times\{a,c\} \\ &=& \{a,b\}^2 \cup \{a,c\}^2 \\ &=& \{ (a,b),(b,a),(a,c),(c,a),(a,a),(b,b),(c,c)\} \end{matrix}$

The corresponding independency is

$I_D=\{(b,c)\,,\,(c,b)\}$

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

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