Symmetric Formulas for Products of Permutations
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We study the formula complexity of the word problem ๐ถ๐๐๐ฝ_S_n,k : {0,1}^kn^2โ{0,1}: given n-by-n permutation matrices M_1,โฆ,M_k, compute the (1,1)-entry of the matrix product M_1โฏ M_k. An important feature of this function is that it is invariant under action of S_n^k-1 given by (ฯ_1,โฆ,ฯ_k-1)(M_1,โฆ,M_k) = (M_1ฯ_1^-1,ฯ_1M_2ฯ_2^-1,โฆ,ฯ_k-2M_k-1ฯ_k-1^-1,ฯ_k-1M_k). This symmetry is also exhibited in the smallest known unbounded fan-in {๐ ๐ญ๐ฃ,๐ฎ๐ฑ,๐ญ๐ฎ๐ณ}-formulas for ๐ถ๐๐๐ฝ_S_n,k, which have size n^O(log k). In this paper we prove a matching n^ฮฉ(log k) lower bound for S_n^k-1-invariant formulas computing ๐ถ๐๐๐ฝ_S_n,k. This result is motivated by the fact that a similar lower bound for unrestricted (non-invariant) formulas would separate complexity classes ๐ญ๐ข^1 and ๐ซ๐๐๐๐๐บ๐ผ๐พ. Our more general main theorem gives a nearly tight n^d(k^1/d-1) lower bound on the G^k-1-invariant depth-d {๐ฌ๐ ๐ฉ,๐ ๐ญ๐ฃ,๐ฎ๐ฑ,๐ญ๐ฎ๐ณ}-formula size of ๐ถ๐๐๐ฝ_G,k for any finite simple group G whose minimum permutation representation has degreeย n. We also give nearly tight lower bounds on the G^k-1-invariant depth-d {๐ ๐ญ๐ฃ,๐ฎ๐ฑ,๐ญ๐ฎ๐ณ}-formula size in the case where G is an abelian group.
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