Computational Complexity of Functions

TL;DR

This paper discusses the computational complexity of functions, extending classical theorems with new proofs that incorporate recent advances in machine models, and provides a tighter analysis of complexity bounds.

Contribution

It offers an improved proof of classical complexity theorems by integrating recent machine model extensions, allowing for tighter complexity bounds without restrictions.

Findings

Extended classical theorems to machines with separate input and working tapes

Achieved tighter complexity bounds with no overhead beyond an additive constant

Incorporated recent advances in machine models to generalize previous results

Abstract

Below is a translation from my Russian paper. I added references, unavailable to me in Moscow. Similar results have been also given in [Schnorr Stumpf 75] (see also [Lynch 75]). Earlier relevant work (classical theorems like Compression, Speed-up, etc.) was done in [Tseitin 56, Rabin 59, Hartmanis Stearns 65, Blum 67, Trakhtenbrot 67, Meyer Fischer 72]. I translated only the part with the statement of the results. Instead of the proof part I appended a later (1979, unpublished) proof sketch of a slightly tighter version. The improvement is based on the results of [Meyer Winklmann 78, Sipser 78]. Meyer and Winklmann extended earlier versions to machines with a separate input and working tape, thus allowing complexities smaller than the input length (down to its log). Sipser showed the space-bounded Halting Problem to require only additive constant overhead. The proof in the appendix below employs both advances to extend the original proofs to machines with a fixed alphabet and a separate input and working space. The extension has no (even logarithmic) restrictions on complexity and no overhead (beyond an additive constant). The sketch is very brief and a more detailed exposition is expected later: [Seiferas Meyer].