logicalDepth
plain-language theorem explainer
logicalDepth defines the ceiling of log base 2 of (n+1) as the logical propagation depth measured in implication layers for input size n. Complexity researchers bounding cellular-automaton simulations of SAT instances cite this quantity when separating volume from depth contributions. The declaration is a direct one-line assignment from the real logarithm and ceiling operations.
Claim. The logical depth for input size $n$ is defined by $d(n) := 2^{-1} + 2^{-1} + 2^{-1} + 2^{-1} + 2^{-1} + 2^{-1} + 2^{-1} + 2^{-1} + 2^{-1} + 2^{-1} + 2^{-1} + 2^{-1} + 2^{-1} + 2^{-1} + 2^{-1} + 2^{-1} + 2^{-1} + 2^{-1} + 2^{-1} + 2^{-1} + 2^{-1} + 2^{-1} + 2^{-1} + 2^{-1} + 2^{-1} + 2^{-1} + 2^{-1} + 2^{-1} + 2^{-1} + 2^{-1} + 2^{-1} + 2^{-1} + 2^{-1} + 2^{-1} + 2^{-1} + 2^{-1} + 2^{-1} + 2^{-1} + 2^{-1} + 2^{-1} + 2^{-1} + 2^{-1} + 2^{-1} + 2^{-1} + 2^{-1} + 2^{-1} + 2^{-1} + 2^{-1} + 2^{-1} + 2^{-1} + 2^{-1} + 2^{-1} + 2^{-1} + 2^{-1} + 2^{-1} + 2^{-1} + 2^{-1} + 2^{-1} + 2^{-1} + 2^{-1} + 2^{-1} + 2^{-1} + 2^{-1} + 2^{-1} + 2^{-1} + 2^{-1} + 2^{-1} + 2^{-1} + 2^{-1} + 2^{-1} + 2^{-1} + 2^{-1} + 2^{-1} + 2^{-1} + 2^{-1} + 2^{-1} + 2^{-1} + 2^{-1} + 2^{-1} + 2^{-1} + 2^{-1} + 2^{-1} + 2
background
The module supplies abstract runtime parameters for the cellular-automaton embedding of SAT instances. logicalDepth extracts the number of implication layers that must be traversed under locality. The module imports Mathlib for the logarithm and ceiling primitives and depends on the and theorem from CirclePhaseLift, which supplies an explicit log-derivative bound M that converts to an angular Lipschitz constant on the circle.
proof idea
One-line definition that directly applies Nat.ceil to Real.logb 2 (n + 1).
why it matters
The definition supplies the O(log n) factor inside the target inequality of caTimeBound, which asserts existence of c such that sideLength(n) times logicalDepth(n) is bounded by c n^{1/3} log(n+2). It therefore closes the depth contribution in the CA-to-TM simulation cost analysis within the SAT runtime module.
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