Time-Space Lower Bounds for Simulating Proof Systems with Quantum and Randomized Verifiers

A line of work initiated by Fortnow in 1997 has proven model-independent time-space lower bounds for the $\mathsf{SAT}$ problem and related problems within the polynomial-time hierarchy. For example, for the $\mathsf{SAT}$ problem, the state-of-the-art is that the problem cannot be solved by random-access machines in $n^c$ time and $n^{o(1)}$ space simultaneously for $c < 2\cos(\frac{\pi}{7}) \approx 1.801$. We extend this lower bound approach to the quantum and randomized domains. Combining Grover's algorithm with components from $\mathsf{SAT}$ time-space lower bounds, we show that there are problems verifiable in $O(n)$ time with quantum Merlin-Arthur protocols that cannot be solved in $n^c$ time and $n^{o(1)}$ space simultaneously for $c < \frac{3+\sqrt{3}}{2} \approx 2.366$, a super-quadratic time lower bound. This result and the prior work on $\mathsf{SAT}$ can both be viewed as consequences of a more general formula for time lower bounds against small-space algorithms, whose asymptotics we study in full. We also show lower bounds against randomized algorithms: there are problems verifiable in $O(n)$ time with (classical) Merlin-Arthur protocols that cannot be solved in $n^c$ randomized time and $n^{o(1)}$ space simultaneously for $c < 1.465$, improving a result of Diehl. For quantum Merlin-Arthur protocols, the lower bound in this setting can be improved to $c < 1.5$.Comment: 38 pages, 5 figures. To appear in ITCS 202

Easiness Amplification and Uniform Circuit Lower Bounds

We present new consequences of the assumption that time-bounded algorithms can be "compressed" with non-uniform circuits. Our main contribution is an "easiness amplification" lemma for circuits. One instantiation of the lemma says: if n^{1+e}-time, tilde{O}(n)-space computations have n^{1+o(1)} size (non-uniform) circuits for some e > 0, then every problem solvable in polynomial time and tilde{O}(n) space has n^{1+o(1)} size (non-uniform) circuits as well. This amplification has several consequences: * An easy problem without small LOGSPACE-uniform circuits. For all e > 0, we give a natural decision problem, General Circuit n^e-Composition, that is solvable in about n^{1+e} time, but we prove that polynomial-time and logarithmic-space preprocessing cannot produce n^{1+o(1)}-size circuits for the problem. This shows that there are problems solvable in n^{1+e} time which are not in LOGSPACE-uniform n^{1+o(1)} size, the first result of its kind. We show that our lower bound is non-relativizing, by exhibiting an oracle relative to which the result is false. * Problems without low-depth LOGSPACE-uniform circuits. For all e > 0, 1 < d < 2, and e < d we give another natural circuit composition problem computable in tilde{O}(n^{1+e}) time, or in O((log n)^d) space (though not necessarily simultaneously) that we prove does not have SPACE[(log n)^e]-uniform circuits of tilde{O}(n) size and O((log n)^e) depth. We also show SAT does not have circuits of tilde{O}(n) size and log^{2-o(1)}(n) depth that can be constructed in log^{2-o(1)}(n) space. * A strong circuit complexity amplification. For every e > 0, we give a natural circuit composition problem and show that if it has tilde{O}(n)-size circuits (uniform or not), then every problem solvable in 2^{O(n)} time and 2^{O(sqrt{n log n})} space (simultaneously) has 2^{O(sqrt{n log n})}-size circuits (uniform or not). We also show the same consequence holds assuming SAT has tilde{O}(n)-size circuits. As a corollary, if n^{1.1} time computations (or O(n) nondeterministic time computations) have tilde{O}(n)-size circuits, then all problems in exponential time and subexponential space (such as quantified Boolean formulas) have significantly subexponential-size circuits. This is a new connection between the relative circuit complexities of easy and hard problems

Why a simple herding model may generate the stylized facts of daily returns: Explanation and estimation

The paper proposes an elementary agent-based asset pricing model that, invoking the two trader types of fundamentalists and chartists, comprises four features: (i) price determination by excess demand; (ii) a herding mechanism that gives rise to a macroscopic adjustment equation for the market fractions of the two groups; (iii) a rush towards fundamentalism when the price misalignment becomes too large; and (iv) a stronger noise component in the demand per chartist trader than in the demand per fundamentalist trader, which implies a structural stochastic volatility in the returns. Combining analytical and numerical methods, the interaction between these elements is studied in the phase plane of the price and a majority index. In addition, the model is estimated by the method of simulated moments, where the choice of the moments reflects the basic stylized facts of the daily returns of a stock market index. A (parametric) bootstrap procedure serves to set up an econometric test to evaluate the model's goodness-of-fit, which proves to be highly satisfactory. The bootstrap also makes sure that the estimated structural parameters are well identified. --structural stochastic volatility,method of simulated moments,autocorrelation pattern,fat tails,bootstrapped p-values

Dagstuhl News January - December 2006

"Dagstuhl News" is a publication edited especially for the members of the Foundation "Informatikzentrum Schloss Dagstuhl" to thank them for their support. The News give a summary of the scientific work being done in Dagstuhl. Each Dagstuhl Seminar is presented by a small abstract describing the contents and scientific highlights of the seminar as well as the perspectives or challenges of the research topic

Classifying Problems into Complexity Classes

A fundamental problem in computer science is, stated informally: Given a problem, how hard is it?. We measure hardness by looking at the following question: Given a set A whats is the fastest algorithm to determine if “x ∈ A? ” We measure the speed of an algorithm by how long it takes to run on inputs of length n, as a function of n. For example, sorting a list of length n can be done in roughly n log n steps. Obtaining a fast algorithm is only half of the problem. Can you prove that there is no better algorithm? This is notoriously difficult; however, we can classify problems into complexity classes where those in the same class are roughly equally hard. In this chapter we define many complexity classes and describing natural problems that are in them. Our classes go all the way from regular languages to various shades of undecidable. We then summarize all that is known about these classes.

Bounded Relativization

Relativization is one of the most fundamental concepts in complexity theory, which explains the difficulty of resolving major open problems. In this paper, we propose a weaker notion of relativization called bounded relativization. For a complexity class ?, we say that a statement is ?-relativizing if the statement holds relative to every oracle ? ? ?. It is easy to see that every result that relativizes also ?-relativizes for every complexity class ?. On the other hand, we observe that many non-relativizing results, such as IP = PSPACE, are in fact PSPACE-relativizing. First, we use the idea of bounded relativization to obtain new lower bound results, including the following nearly maximum circuit lower bound: for every constant ? > 0, BPE^{MCSP}/2^{?n} ? SIZE[2?/n]. We prove this by PSPACE-relativizing the recent pseudodeterministic pseudorandom generator by Lu, Oliveira, and Santhanam (STOC 2021). Next, we study the limitations of PSPACE-relativizing proof techniques, and show that a seemingly minor improvement over the known results using PSPACE-relativizing techniques would imply a breakthrough separation NP ? L. For example: - Impagliazzo and Wigderson (JCSS 2001) proved that if EXP ? BPP, then BPP admits infinitely-often subexponential-time heuristic derandomization. We show that their result is PSPACE-relativizing, and that improving it to worst-case derandomization using PSPACE-relativizing techniques implies NP ? L. - Oliveira and Santhanam (STOC 2017) recently proved that every dense subset in P admits an infinitely-often subexponential-time pseudodeterministic construction, which we observe is PSPACE-relativizing. Improving this to almost-everywhere (pseudodeterministic) or (infinitely-often) deterministic constructions by PSPACE-relativizing techniques implies NP ? L. - Santhanam (SICOMP 2009) proved that pr-MA does not have fixed polynomial-size circuits. This lower bound can be shown PSPACE-relativizing, and we show that improving it to an almost-everywhere lower bound using PSPACE-relativizing techniques implies NP ? L. In fact, we show that if we can use PSPACE-relativizing techniques to obtain the above-mentioned improvements, then PSPACE ? EXPH. We obtain our barrier results by constructing suitable oracles computable in EXPH relative to which these improvements are impossible