Transcendent Logic: A Self-Modifying Expression Tree

Target Function

Self-Modification

Mutation strength 0.30

Annealing speed 0.995

Generations / sec 4

State

Generation

Fitness (MSE)

–

Nodes

Temperature

–

variable x constant + − × sin / cos abs / neg / x²

How it works

The tree on the right is a program: an expression tree over variable x built from operators (+, −, ×, sin, cos, abs, neg, x²). Evaluating it at some x produces a number — the program's output. Each generation the system proposes a random edit to its own structure:

Replace a random subtree with a freshly grown one
Perturb a constant's value
Swap an operator for a same-arity sibling (sin↔cos, +↔−↔×)
Wrap a node in a new unary operator
Simplify a node away, replacing it with one of its own children

The edit is scored by mean-squared error against a target function sampled at 25 points. It is accepted or rejected by the simulated-annealing rule:

Δ = fitness(new) − fitness(old)
accept if Δ ≤ 0
accept with probability P = exp(−Δ / T)  if Δ > 0
T ← T · cooling_rate   (cools every generation)

Early on, high temperature T lets the tree accept worse edits — exploring wildly different structures. As T decays, only genuine improvements survive, and the structure settles into a local optimum. This is the same accept/reject logic used by simulated annealing and genetic programming to search program-space instead of a fixed parameter vector: the "logic" being optimized is the tree's own shape, not just numbers inside it.

The green curve (target) and amber curve (the tree's current output) on the right show the fit converging as generations pass.

Generation 0 Fitness –

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