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ALPS’07 -- Groups and Complexity
, 2007
"... The connection between groups and recursive (un)decidability has a long history, going back to the early 1900s. Also, various polynomial-time algorithms have been known in group theory for a long time. However the impact of more general computational complexity (e.g., NP-completeness or PSpace-compl ..."
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The connection between groups and recursive (un)decidability has a long history, going back to the early 1900s. Also, various polynomial-time algorithms have been known in group theory for a long time. However the impact of more general computational complexity (e.g., NP-completeness or PSpace-completeness) has been relatively small and recent. These lectures review a sampling of older facts about algorithmic problems in group theory, and then present more recent results about the connection with complexity: isoperimetric functions and NP; Thompson groups, boolean circuits, and coNP; Thompson monoids and circuit complexity; Thompson groups, reversible computing, and #P; distortion of Thompson groups within Thompson monoids, and one-way permutations. We are especially interested in deep connections between computational complexity and group theory. By “connection ” we do not just mean analyzing the computational complexity of algorithms about groups. We are more interested in algebraic characterizations of complexity classes in terms of group theory, i.e., in finding a “mirror image” of all of complexity theory within group theory. Conversely, we are interested in the computational nature of concepts that appear at first purely algebraic.
The R- and L-orders of the Thompson-Higman monoid Mk,1 and their complexity
, 812
"... We study the monoid generalization Mk,1 of the Thompson-Higman groups, and we characterize the R- and the L-preorder of Mk,1. Although Mk,1 has only one non-zero J-class and k−1 non-zero D-classes, the R- and the L-preorder are complicated; in particular,
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We study the monoid generalization Mk,1 of the Thompson-Higman groups, and we characterize the R- and the L-preorder of Mk,1. Although Mk,1 has only one non-zero J-class and k−1 non-zero D-classes, the R- and the L-preorder are complicated; in particular, <R is dense (even within an L-class), and <L is dense (even within an R-class). We study the computational complexity of the R- and the L-preorder. When inputs are given by words over a finite generating set of Mk,1, the R- and the L-preorder decision problems are in P. The main result of the paper is that over a “circuit-like ” generating set, the R-preorder decision problem of Mk,1 is Π P 2-complete, whereas the L-preorder decision problem is coNP-complete. We also prove related results about circuits: For combinational circuits, the surjectiveness problem is Π P 2-complete, whereas the injectiveness problem is coNP-complete. 1
