The Complexity of Solving Equations Over Finite Groups

  • Authors:

  • Mikael Goldmann;Alexander Russell

  • Affiliations:

  • -;-

  • Venue:
  • COCO '99 Proceedings of the Fourteenth Annual IEEE Conference on Computational Complexity
  • Year:
  • 1999

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Abstract

We study the computational complexity of solving systems of equations over a finite group. An equation over a group G is an expression of the form w(1) w(2) ... w(k) = e where each w(i) is either a variable, an inverted variable, or group constant and e is the identity element of G. A solution to such an equation is an assignment of the variables (to values in G) which realizes the equality. A system of equations is a collection of such equations; a solution is then an assignment which simultaneously realizes each equation.We demonstrate that the problem of determining if a (single) equation has a solution is NP-complete for all non-solvable groups G. For nilpotent groups, this same problem is shown to be in P. The analogous problem for systems of such equations is shown to be NP-complete if G is non-Abelian, and in P otherwise. Finally, we observe some connections between these languages and the theory of non-uniform automata.