text: Formalizing control planes. An LLM control plane Rω is a potentially randomized algorithm. It is parameterized by 
  a string ω, called the parameters. It utilizes some number n of LLMs denoted by M. We will mostly focus on the 
  case of n = 2, and, for reasons that will be clear in a moment, use Ms (“strong”) and Mw (“weak”) to denote the two 
  underlying LLMs. Then inference on an input x ∈ X for some set X of allowed queries is performed by computing 
  a response via y ←$ RM 
  ω (x). Here we use ←$ to denote running R with fresh random coins; we use ← when R is 
  deterministic. We focus on inference for a single query, but it is straightforward to extend our abstraction for control 
  planes to include sessions: the controller would maintain state across invocations, potentially adapting its behavior as a 
  function of a sequence of queries and responses. 
  
  LLM control planes should, in general, be relatively computationally lightweight, at least compared to the underlying 
  LLMs. This is particularly so in the cost-motivated usage of control planes, as a computationally or financially expensive 
  control plane would eat into cost savings incurred by utilizing cheaper underlying LLMs for some queries. For example, 
  predictive binary routers use relatively simple classifiers to determine which of Ms or Mw should be used to respond to a 
  query. 