Relations between diagonalization, proof systems, and complexity gaps

AbstractIn this paper we study diagonal processes over time bounded computations of one-tape Turing machines by diagonalizing only over those machines for which there exist formal proofs that they operate in the given time bound. This replaces the traditional “clock” in resource bounded diagonalization by formal proofs about running times and establishes close relations between properties of proof systems and existence of sharp time bounds for one-tape Turing machine complexity classes. These diagonalization methods also show that the Gap Theorem for resource bounded computations can hold only for those complexity classes which differ from the corresponding provable complexity classes. Furthermore, we show that there exist recursive time bounds T(n) such that the class of languages for which we can formally prove the existence of Turing machines which accept them in time T(n) differs from the class of languages accepted by Turing machines for which we can prove formally that they run in time T(n). We also investigate the corresponding problems for tape bound computations and discuss the difference time and tapebounded computations

One-Tape Turing Machine and Branching Program Lower Bounds for MCSP

For a size parameter s: ? ? ?, the Minimum Circuit Size Problem (denoted by MCSP[s(n)]) is the problem of deciding whether the minimum circuit size of a given function f : {0,1}? ? {0,1} (represented by a string of length N : = 2?) is at most a threshold s(n). A recent line of work exhibited "hardness magnification" phenomena for MCSP: A very weak lower bound for MCSP implies a breakthrough result in complexity theory. For example, McKay, Murray, and Williams (STOC 2019) implicitly showed that, for some constant ?? > 0, if MCSP[2^{??? n}] cannot be computed by a one-tape Turing machine (with an additional one-way read-only input tape) running in time N^{1.01}, then P?NP. In this paper, we present the following new lower bounds against one-tape Turing machines and branching programs: 1) A randomized two-sided error one-tape Turing machine (with an additional one-way read-only input tape) cannot compute MCSP[2^{???n}] in time N^{1.99}, for some constant ?? > ??. 2) A non-deterministic (or parity) branching program of size o(N^{1.5}/log N) cannot compute MKTP, which is a time-bounded Kolmogorov complexity analogue of MCSP. This is shown by directly applying the Ne?iporuk method to MKTP, which previously appeared to be difficult. 3) The size of any non-deterministic, co-non-deterministic, or parity branching program computing MCSP is at least N^{1.5-o(1)}. These results are the first non-trivial lower bounds for MCSP and MKTP against one-tape Turing machines and non-deterministic branching programs, and essentially match the best-known lower bounds for any explicit functions against these computational models. The first result is based on recent constructions of pseudorandom generators for read-once oblivious branching programs (ROBPs) and combinatorial rectangles (Forbes and Kelley, FOCS 2018; Viola 2019). En route, we obtain several related results: 1) There exists a (local) hitting set generator with seed length O?(?N) secure against read-once polynomial-size non-deterministic branching programs on N-bit inputs. 2) Any read-once co-non-deterministic branching program computing MCSP must have size at least 2^??(N)

Unbounded-error quantum computation with small space bounds

We prove the following facts about the language recognition power of quantum Turing machines (QTMs) in the unbounded error setting: QTMs are strictly more powerful than probabilistic Turing machines for any common space bound $s$ satisfying $s(n)=o(\log \log n)$. For "one-way" Turing machines, where the input tape head is not allowed to move left, the above result holds for $s(n)=o(\log n)$. We also give a characterization for the class of languages recognized with unbounded error by real-time quantum finite automata (QFAs) with restricted measurements. It turns out that these automata are equal in power to their probabilistic counterparts, and this fact does not change when the QFA model is augmented to allow general measurements and mixed states. Unlike the case with classical finite automata, when the QFA tape head is allowed to remain stationary in some steps, more languages become recognizable. We define and use a QTM model that generalizes the other variants introduced earlier in the study of quantum space complexity.Comment: A preliminary version of this paper appeared in the Proceedings of the Fourth International Computer Science Symposium in Russia, pages 356--367, 200

An in-between "implicit" and "explicit" complexity: Automata

Implicit Computational Complexity makes two aspects implicit, by manipulating programming languages rather than models of com-putation, and by internalizing the bounds rather than using external measure. We survey how automata theory contributed to complexity with a machine-dependant with implicit bounds model