Implicit and Explicit dependencies

February 18, 2012

Shadows of the Truth by Alexandre Borovik offers an interesting perspective upon human learning experience. The book is abundant with examples of concepts being challenging at the early age, but clarified later (sometimes, much later) in life. One of my stumbling blocks was the definition of implicit and explicit dependencies. Now, with modest help of relational thinking (that is, basic dependency theory) I can report some progress.

Intuition behind implicit and explicit dependencies is clear. For example, given the two functions

y(x,t) = x^2 + t
x(t) = t^2

then, in the formula for y we notice two variables x and t , which suggests that y explicitly depends on t . Compare it with

y(x,t) = x^2
x(t) = t^2

where formula for y involves variable x only. Since, at the second line we have x expressed in terms of t , y still depends on t , but the dependence is implicit.

The concept of implicit and explicit dependencies surfaces in many places, for example Partial Derivative Equations and Noether conservation theorems, which both are part of undergraduate math and physics curriculum. Nevertheless, most textbooks take this concept for granted, perhaps implying that mathematically mature reader should have no problems understanding it. Wikipedia offers couple dedicated articles: Dependent and Independent Variables giving overview and intuition, and more ambitious Time-Invariant System with an attempt to formal definition.

The concept of time-invariant system belongs to physics, the idea being that if we shift the time variable t , then it doesn’t affect time-invariant system behavior. This is illustrated by “formal example” in the middle of the page, where by comparing values of y(x,t) with two arguments t vs. t+\delta they suggest that y(x,t) = t x(t) is time-dependent, while y(x,t) = 10 x(t) is not. Compared to math, physics standards of rigor are lax, so it takes little effort to find a flaw. Take x(t) = t , then y(x,t) = t x(t) = {x(t)}^2 so y(x,t) is time-invariant with proper choice of x(t) !

Can relational dependency theory suggest any insight? By glancing over standard definitions of functional dependency:

\forall y_{1} \forall y_{2} \forall x : S(x, y_{1}) \wedge S(x, y_{2}) \implies y_{1} = y_{2}

and independence:

\exists y_{1} \exists y_{2} \exists x : S(x, y_{1}) \wedge S(x, y_{2}) \wedge y_{1} \not= y_{2}

it becomes obvious that dependency concepts hinge upon equality/inequality (amended with some quantifiers, perhaps), and not upon domain algebraic structure (time-shifts). Let’s examine closely two examples:

y(x,t) = t-x
x(t) = t^2-2 t


y(x,t) = x^2-3 x
x(t) = t^2-2 t

Tabulating values at points t=0,1,2,3 we’ll get relations

R=[t  x  y]
   0  0  0
   1  -1  2
   2  0  2
   3  3  0


S=[t  x  y]
   0  0  0
   1  -1  4
   2  0  0
   3  3  0

correspondingly. The second relation S honors FDs t->x and x->y (and by Armstrong transitivity t->y), while the first one R does only t->x and t->y. Therefore, the formal definition of variable y being not [explicitly] dependent of t is equivalent to the absence of functional dependency x->y — if not counter intuitive, then terminologically confusing to say the least!


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