Universal Algebra and Computational Complexity Lecture 3
Universal Algebra and Computational Complexity Lecture 3
Universal Algebra and Computational Complexity Lecture 3
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Clone Membership Problem (CLO)<br />
INPUT: An algebra A <strong>and</strong> an operation g : A k → A.<br />
QUESTION: Is g ∈ Clo A?<br />
Obvious algorithm: Determine whether g ∈ Sg A(Ak )(pr k 1 , . . . , pr k k ).<br />
The running time is bounded by a polynomial in ||A (Ak) ||.<br />
Can show<br />
log ||A (Ak) || ≤ n k ||A|| ≤ (||g|| + ||A||) 2 .<br />
Hence the running time is bounded by the exponential of a polynomial in<br />
the size of the input (A, g). I.e., CLO ∈ EXPTIME.<br />
Ross Willard (Waterloo) <strong>Algebra</strong> <strong>and</strong> <strong>Complexity</strong> Třešť, September 2008 12 / 31
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