Compression theorem

Compression theorem In computational complexity theory, the compression theorem is an important theorem about the complexity of computable functions.

The theorem states that there exists no largest complexity class, with computable boundary, which contains all computable functions.

Compression theorem Given a Gödel numbering {displaystyle varphi } of the computable functions and a Blum complexity measure {displaystyle Phi } where a complexity class for a boundary function {displaystyle f} is defined as {displaystyle mathrm {C} (f):={varphi _{i}in mathbf {R} ^{(1)}|(forall ^{infty }x),Phi _{i}(x)leq f(x)}.} Then there exists a total computable function {displaystyle f} so that for all {displaystyle i} {displaystyle mathrm {Dom} (varphi _{i})=mathrm {Dom} (varphi _{f(i)})} and {displaystyle mathrm {C} (varphi _{i})subsetneq mathrm {C} (varphi _{f(i)}).} References Salomaa, Arto (1985), "Theorem 6.9", Computation and Automata, Encyclopedia of Mathematics and Its Applications, vol. 25, Cambridge University Press, pp. 149–150, ISBN 9780521302456. Zimand, Marius (2004), "Theorem 2.4.3 (Compression theorem)", Computational Complexity: A Quantitative Perspective, North-Holland Mathematics Studies, vol. 196, Elsevier, p. 42, ISBN 9780444828415. P ≟ NP  This theoretical computer science–related article is a stub. You can help Wikipedia by expanding it.

Categories: Computational complexity theoryStructural complexity theoryTheorems in the foundations of mathematicsTheoretical computer science stubs