Dependency relation

In computer science, in particular in concurrency theory, a dependency relation is a binary relation that is finite,[1]^:4 symmetric, and reflexive;[1]^:6 i.e. a finite tolerance relation. That is, it is a finite set of ordered pairs ${\displaystyle D}$, such that

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

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

If ${\displaystyle \Sigma }$ (also called alphabet) denotes the set on which ${\displaystyle D}$ is defined, then the independency induced by ${\displaystyle D}$ is the binary relation ${\displaystyle I}$

${\displaystyle I=(\Sigma \times \Sigma )\setminus D}$

That is, the independency is the set of all ordered pairs that are not in ${\displaystyle D}$. The independency relation is symmetric and irreflexive. Conversely, given any symmetric and irreflexive relation ${\displaystyle I}$ on a finite alphabet, the relation

${\displaystyle D=(\Sigma \times \Sigma )\setminus I}$

is a dependency relation.

The pairs ${\displaystyle (\Sigma ,D)}$ and ${\displaystyle (\Sigma ,I)}$, or the triple ${\displaystyle (\Sigma ,D,I)}$ (with ${\displaystyle I}$ induced by ${\displaystyle D}$) are sometimes called the concurrent alphabet or the reliance alphabet. In this case, elements ${\displaystyle x,y\in \Sigma }$ are called dependent if ${\displaystyle xDy}$ holds, and independent, else (i.e. if ${\displaystyle xIy}$ holds).[1]^:6

Given a reliance alphabet ${\displaystyle (\Sigma ,D,I)}$, a symmetric and irreflexive relation ${\displaystyle \doteq }$ can be defined on the free monoid ${\displaystyle \Sigma ^{*}}$ of all possible strings of finite length by: ${\displaystyle xaby\doteq xbay}$ for all strings ${\displaystyle x,y\in \Sigma ^{*}}$ and all independent symbols ${\displaystyle a,b\in I}$. The equivalence closure of ${\displaystyle \doteq }$ is denoted ${\displaystyle \equiv }$, or ${\displaystyle \equiv _{(\Sigma ,D,I)}}$, and called ${\displaystyle (\Sigma ,D,I)}$-equivalence. Informally, ${\displaystyle p\equiv q}$ holds if the string ${\displaystyle p}$ can be transformed into ${\displaystyle q}$ by a finite sequence of swaps of adjacent independent symbols. The equivalence classes of ${\displaystyle \equiv }$ are called traces,[1]^:7–8 and are studied in trace theory.