# The network complexity and the Turing machine complexity of finite functions

Article

Received:

- 96 Downloads
- 63 Citations

## Summary

Let *L(f)* be the network complexity of a Boolean function *L(f)*. For any *n*-ary Boolean function *L(f)* let \(TC(f) = min\{ T_p^{\bar A} (n){\text{ (}}\parallel p\parallel + 1gS_p^{\bar A} {\text{(}}n{\text{):}}res_p^{\bar A} {\text{(}}n{\text{) = }}f\} \). Hereby *p* ranges over all relative Turing programs and *Ā* ranges over all oracles such that given the oracle *Ā*, the restriction of *p* to inputs of length *n* is a program for *L(f)*. ∥p∥ is the number of instructions of *p*. *T*_{ p }^{ Ā }(*n*) is the time bound and *S*_{ p }^{ Ā }of the program *p* relative to the oracle *Ā* on inputs of length *n*. Our main results are (1) *L(f)* ≦ O(TC(L*(f)*)), (2) TC*(f)* ≦ O(L*(f)*^{2}^{2+ɛ}) for every ɛ ⋙ O.

## Keywords

Information System Operating System Data Structure Communication Network Information Theory
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

## Preview

Unable to display preview. Download preview PDF.

## References

- 1.Fischer, M. J.: Lectures on network complexity. Preprint Universität Frankfurt, June 1974Google Scholar
- 2.Hennie, F. C., Stearns, R. E.: Two tape simulation of multitape turing machines. J. ACM
**13**, 533–546 (1966)Google Scholar - 3.Lupanov, O. B.: Complexity of formula realisation of functions of logical algebra. Prob. Cybernetics
**3**(1962)Google Scholar - 4.Paterson, M. S., Fischer, M. J., Meyer, A. R.: An improved overlap argument for on-line multiplication. In: Complexity of Computation. SIAM AMS Proceedings
**7**, 97–111 (1974)Google Scholar - 5.Savage, J. E.: Computational work and time on finite machines. J. ACM
**19**, 660–674 (1972)Google Scholar - 6.Schnorr, C. P.: Lower bounds for the product of time and space requirements of Turing machine computations. Proc. Symposium on the Mathematical Foundations of Computer Science 1973, High Tatras. CSSR, Math. Inst. Slovak Academy of Sciences 1973, P. 153–161Google Scholar

## Copyright information

© Springer-Verlag 1976