On the depth complexity of formulas

The problem of minimizing the depth of formulas by equivalence preserving transformations is formalized in a general algebraic setting. For a particular algebraic system ∑₀ specific methods of a dynamic programming nature are developed for proving lower bounds on depth. Such lower bounds for ∑₀ automatically imply the same results for the systems of (i) arithmetic computations with addition and multiplication only, and (ii) computations over finite languages using union and concatenation. The specific lower bounds obtained are (i) depth 2 n-o(n) for the permanent, (ii) depth (0.25+o(1))log²n for the symmetric polynomials and (iii) depth 1.16log n for a problem of formula size n.