Rent's Rule and Information Equilibrium

Another application of information equilibrium

Rent’s rule is a heuristic relationship between the number of “pins” (“terminals” below) and the number of “gates” in logic blocks in a computing architecture — one that is surprisingly robust. There is even a theoretical basis that is surprisingly similar to the derivation of the information equilibrium condition:

The paper goes on to add a parameter p that is related to the topology of the circuit. Taking the information equilibrium approach we would posit that if T and G are in information equilibrium T ⇄ G — i.e. the information in the distribution of states of the gates “G” is equal to the information in the distribution of the states of the terminals “T” then:

\(\displaystyle \frac{dT}{dG} = p \; \frac{T}{G}\)

such that

\(\left( \frac{T}{T_{0}}\right) = \left( \frac{G}{G_{0}}\right)^{p}\)

where T₀ and G₀ are constants of integration. Alternatively:

\(T = t \; G^{p}\)

The topology parameter p is the information transfer index.

What is also interesting is that in “Region II” for larger values of G (figure from here where B for blocks is used instead of G for gates, and P for pins is used instead of T for terminals) the value starts to fall below the heuristic estimate:

In the information transfer framework, this is indicative of non-ideal information transfer and we would derive (using Gronwall’s inequality):

\(T \leq t \; G^{p}\)

This gives us another analogy to work with when understanding economic relationships (e.g. pins = demand, gates = supply). As a side note, I worked out some time ago that you can understand a transistor (including Early voltage and compression) using the information transfer framework as well.