The Nonsolvable Group Technique
Bounded-width branching programs offer interesting connections between complexity classes and algebraic structures. Let k ≥ 2 and n ≥ 1 be integers. A width-k branching program over n-bit inputs prescribes manipulation of a pebble placed on a track of k squares. First the pebble is place d on square 1. Then a sequence of instructions is executed. Each instruction is simple: it tells you to examine an input bit and then move the pebble to another (possibly the same) square, where to which square the pebble will be moved depends on the examined bit, the current location of the pebble, and the step of the computation. The program accepts the input if and only if the pebble is not on square 1 at the end.
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