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We consider the computational complexity of some problems dealing with matrix rank.<br /> Let E, S be subsets of a commutative ring R.<br />Let x1, x2, ..., xt be variables. Given a matrix M = M(x1, x2, ..., xt)<br />with entries chosen from E union {x1, x2, ..., xt}, we want to determine<br />maxrankS(M) = max rank M(a1, a2, ... , at)<br />and<br />minrankS(M) = min rank M(a1, a2, ..., at). <br />There are also variants of these problems that specify more about the<br />structure of M, or instead of asking for the minimum or maximum rank, <br />ask if there is some substitution of the variables that makes the matrix<br /> invertible or noninvertible.<br />Depending on E, S, and on which variant is studied, the complexity<br />of these problems can range from polynomial-time solvable to random<br />polynomial-time solvable to NP-complete to PSPACE-solvable to<br />unsolvable.

Let M be a fixed finite monoid. We consider the problem of implementing a data type containing a vector x = (x 1 , x 2 , . . . , x n ) ∈ M n , initially (1, 1, . . . , 1) with two kinds of operations, for each i ∈ {1, . . . , n}, a ∈ M, an operation change i,a which changes x i , to a and a single operation product returning n i=1 x i . This is the dynamic word problem. If we in addition for each j ∈ 1, . . . , n have an operation prefix j ; returning n i=1 x i , we talk about the dynamic prefix problem. We analyze the complexity of these problems in the cell probe or decision assignment tree model for two natural cell sizes, 1 bit and log n bits. We obtain a classification of the complexity based on algebraic properties of M .

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