Problem
Research questions and friction points this paper is trying to address.
one-tape Turing machine
accepting-path counting
time threshold
#P_1 completeness
generating function
Accepting-Path Counting at the One-Tape $n\log n$ Threshold
🤖 AI Summary
This project investigates the complexity boundaries of counting accepting paths for single-tape Turing machines. By integrating single-tape time gap theory, Tadaki's linear counting theorem, and Beame's universal counting machine adaptation technique, it establishes $n \log n$ as the precise complexity threshold for accepting path counting. The study demonstrates that below this threshold, the generating function for accepting paths is rational; upon reaching it, the problem abruptly becomes #P-complete, enabling the construction of instances exhibiting uncomputable exponential growth rates. This work characterizes the phase transition of accepting path counting from simple computability to #P-completeness, thereby delineating the computational complexity boundaries inherent to single-tape Turing machines.
📝 Abstract
We observe that the classical $n\log n$ time threshold for one-tape Turing machines is also a threshold for their accepting-path counts. Below it, every nondeterministic one-tape machine running in strong $o(n\log n)$ time has a rational ordinary generating function of accepting-path counts. At strong $O(n\log n)$ time, the situation changes completely: there is a fixed one-tape machine whose accepting-path function is complete for $\#\mathsf P_1$, the tally analogue of $\#\mathsf P$, under parsimonious polynomial-time tally reductions. A second construction within the same time bound gives positive accepting-path counts with a noncomputable exponential growth rate. The rationality result combines the one-tape time gap with the linear-time counting theorem of Tadaki, Yamakami and Lin. The completeness proof adapts the linear-time universal counting machine of Beame et al. to the one-tape setting.
Innovation
Methods, ideas, or system contributions that make the work stand out.
one-tape Turing machine
accepting-path counting
n log n threshold
#P_1-completeness
generating function
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