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Harnessing the unwieldy MEX function
"... A pair of integer sequences that split Z>0 is often–especially in the context of combinatorial game theory–defined recursively by An = mex{Ai,Bi: 0 ≤ i < n}, Bn = An+Cn (n ≥ 0), where mex (Minimum EXcludant) of a subset S of nonnegative integers is the smallest nonnegative integer not in S, and C: Z ..."
Abstract

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A pair of integer sequences that split Z>0 is often–especially in the context of combinatorial game theory–defined recursively by An = mex{Ai,Bi: 0 ≤ i < n}, Bn = An+Cn (n ≥ 0), where mex (Minimum EXcludant) of a subset S of nonnegative integers is the smallest nonnegative integer not in S, and C: Z≥0 → Z≥0. Given x, y ∈ Z>0, a typical problem is to decide whether x = An, y = Bn. For general functions Cn, the best algorithm for this decision problem is exponential in the input size Ω(log x + log y). We prove constructively that the problem is actually polynomial for the wide class of approximately linear functions Cn. This solves constructively and efficiently the complexity question of a number of previously analyzed takeaway games of various authors.