Algorithms and lower bounds for de Morgan formulas of low-communication leaf gates

The class $FORMULA[s] \circ \mathcal{G}$ consists of Boolean functions computable by size-$s$ de Morgan formulas whose leaves are any Boolean functions from a class $\mathcal{G}$. We give lower bounds and (SAT, Learning, and PRG) algorithms for $FORMULA[n^{1.99}]\circ \mathcal{G}$, for classes $\mathcal{G}$ of functions with low communication complexity. Let $R^{(k)}(\mathcal{G})$ be the maximum $k$-party NOF randomized communication complexity of $\mathcal{G}$. We show: (1) The Generalized Inner Product function $GIP^k_n$ cannot be computed in $FORMULA[s]\circ \mathcal{G}$ on more than $1/2+\varepsilon$ fraction of inputs for $$s = o \! \left ( \frac{n^2}{ \left(k \cdot 4^k \cdot {R}^{(k)}(\mathcal{G}) \cdot \log (n/\varepsilon) \cdot \log(1/\varepsilon) \right)^{2}} \right).$$ As a corollary, we get an average-case lower bound for $GIP^k_n$ against $FORMULA[n^{1.99}]\circ PTF^{k-1}$. (2) There is a PRG of seed length $n/2 + O\left(\sqrt{s} \cdot R^{(2)}(\mathcal{G}) \cdot\log(s/\varepsilon) \cdot \log (1/\varepsilon) \right)$ that $\varepsilon$-fools $FORMULA[s] \circ \mathcal{G}$. For $FORMULA[s] \circ LTF$, we get the better seed length $O\left(n^{1/2}\cdot s^{1/4}\cdot \log(n)\cdot \log(n/\varepsilon)\right)$. This gives the first non-trivial PRG (with seed length $o(n)$) for intersections of $n$ half-spaces in the regime where $\varepsilon \leq 1/n$. (3) There is a randomized $2^{n-t}$-time $\#$SAT algorithm for $FORMULA[s] \circ \mathcal{G}$, where $$t=\Omega\left(\frac{n}{\sqrt{s}\cdot\log^2(s)\cdot R^{(2)}(\mathcal{G})}\right)^{1/2}.$$ In particular, this implies a nontrivial #SAT algorithm for $FORMULA[n^{1.99}]\circ LTF$. (4) The Minimum Circuit Size Problem is not in $FORMULA[n^{1.99}]\circ XOR$. On the algorithmic side, we show that $FORMULA[n^{1.99}] \circ XOR$ can be PAC-learned in time $2^{O(n/\log n)}$

Circuit Lower Bounds for MCSP from Local Pseudorandom Generators

The Minimum Circuit Size Problem (MCSP) asks if a given truth table of a Boolean function f can be computed by a Boolean circuit of size at most theta, for a given parameter theta. We improve several circuit lower bounds for MCSP, using pseudorandom generators (PRGs) that are local; a PRG is called local if its output bit strings, when viewed as the truth table of a Boolean function, can be computed by a Boolean circuit of small size. We get new and improved lower bounds for MCSP that almost match the best-known lower bounds against several circuit models. Specifically, we show that computing MCSP, on functions with a truth table of length N, requires - N^{3-o(1)}-size de Morgan formulas, improving the recent N^{2-o(1)} lower bound by Hirahara and Santhanam (CCC, 2017), - N^{2-o(1)}-size formulas over an arbitrary basis or general branching programs (no non-trivial lower bound was known for MCSP against these models), and - 2^{Omega (N^{1/(d+2.01)})}-size depth-d AC^0 circuits, improving the superpolynomial lower bound by Allender et al. (SICOMP, 2006). The AC^0 lower bound stated above matches the best-known AC^0 lower bound (for PARITY) up to a small additive constant in the depth. Also, for the special case of depth-2 circuits (i.e., CNFs or DNFs), we get an almost optimal lower bound of 2^{N^{1-o(1)}} for MCSP

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)

One-Way Functions and a Conditional Variant of MKTP

One-way functions (OWFs) are central objects of study in cryptography and computational complexity theory. In a seminal work, Liu and Pass (FOCS 2020) proved that the average-case hardness of computing time-bounded Kolmogorov complexity is equivalent to the existence of OWFs. It remained an open problem to establish such an equivalence for the average-case hardness of some natural NP-complete problem. In this paper, we make progress on this question by studying a conditional variant of the Minimum KT-complexity Problem (MKTP), which we call McKTP, as follows. 1) First, we prove that if McKTP is average-case hard on a polynomial fraction of its instances, then there exist OWFs. 2) Then, we observe that McKTP is NP-complete under polynomial-time randomized reductions. 3) Finally, we prove that the existence of OWFs implies the nontrivial average-case hardness of McKTP. Thus the existence of OWFs is inextricably linked to the average-case hardness of this NP-complete problem. In fact, building on recently-announced results of Ren and Santhanam [Rahul Ilango et al., 2021], we show that McKTP is hard-on-average if and only if there are logspace-computable OWFs

On Approximating Total Variation Distance

Total variation distance (TV distance) is a fundamental notion of distance between probability distributions. In this work, we introduce and study the computational problem of determining the TV distance between two product distributions over the domain {0, 1}n. We establish the following results. 1. Exact computation of TV distance between two product distributions is #P-complete. This is in stark contrast with other distance measures such as KL, Chi-square, and Hellinger which tensorize over the marginals. 2. Given two product distributions P and Q with marginals of P being at least 1/2 and marginals of Q being at most the respective marginals of P, there exists a fully polynomial-time randomized approximation scheme (FPRAS) for computing the TV distance between P and Q. In particular, this leads to an efficient approximation scheme for the interesting case when P is an arbitrary product distribution and Q is the uniform distribution. We pose the question of characterizing the complexity of approximating the TV distance between two arbitrary product distributions as a basic open problem in computational statistics

Total Variation Distance Estimation Is as Easy as Probabilistic Inference

In this paper, we establish a novel connection between total variation (TV) distance estimation and probabilistic inference. In particular, we present an efficient, structure-preserving reduction from relative approximation of TV distance to probabilistic inference over directed graphical models. This reduction leads to a fully polynomial randomized approximation scheme (FPRAS) for estimating TV distances between distributions over any class of Bayes nets for which there is an efficient probabilistic inference algorithm. In particular, it leads to an FPRAS for estimating TV distances between distributions that are defined by Bayes nets of bounded treewidth. Prior to this work, such approximation schemes only existed for estimating TV distances between product distributions. Our approach employs a new notion of $partial$ couplings of high-dimensional distributions, which might be of independent interest.Comment: 24 page

The complexity and applications of circuit minimization

The main topic of this thesis is the study of the Minimum Circuit Size Problem (MCSP). MCSP is a decision problem: Given the truth table of a finite Boolean function and a size parameter 0 <= theta <= 2^n - 1 (in binary), is there a Boolean circuit C of size at most theta such that C computes f? MCSP is an important problem due to its unexpected connections to many other areas of complexity theory, such as learning, pseudorandomness, and average-case complexity. In a similar manner, Kolmogorov complexity poses the following question: Given a string x, what is the size of the smallest program that outputs x when run on the empty string? While Kolmogorov complexity cannot be computed, in general, there are resource-bounded versions of Kolmogorov complexity that are computable and have been very influential in complexity theory, just like MCSP is. In this work, we study MCSP and some of its Kolmogorov complexity counterparts, and answer the following two questions. 1. What is the complexity of MCSP? That is, how easy or difficult is it to compute MCSP? We prove MCSP lower bounds against several restricted models of computation. These, among others, include formulas, branching programs, constant-depth circuits, and one-tape Turing machines. Almost all of the above lower bounds are the first of their kind for MCSP, and almost match the state-of-the-art lower bounds against these models. 2. What are some interesting applications of MCSP? Here, we show that the average-case hardness of a conditional Kolmogorov complexity counterpart of MCSP is almost equivalent to the existence of one-way functions (OWFs). Along with the concurrent and independent work by Liu and Pass, this is one of the first natural NP-complete problems whose average-case hardness is shown to be (almost) equivalent to the existence of OWFs.Open Acces

Parametric Study of the Gaits of a Quadruped Robot Using Hildebrand Diagrams

156 σ.Στην παρούσα Διπλωματική Εργασία, μελετάται – εκτενώς – το πρόβλημα της συστηματικής περιγραφής της «μόνιμης κατάστασης» της εκάστοτε προσομοιωμένης κίνησης ενός επίπεδου, υποεπενεργούμενου, τετράποδου ρομπότ. Ιδιαίτερη σημασία, αποδίδεται στη διερεύνηση της επίδρασης κάθε μίας εκ των διάφορων παραμέτρων που συνθέτουν το τετράποδο ρομπότ, το περιβάλλον του, τις αρχικές συνθήκες της κίνησής του ή τη λειτουργία του συστήματος ελέγχου του, στο είδος και τα χαρακτηριστικά του επιλεγόμενου, από αυτό, διασκελισμού. Αυτός ο διασκελισμός, επαναλαμβανόμενος, συνθέτει την προαναφερόμενη μόνιμη κατάσταση της κίνησης του ρομπότ. Τέλος, αναπτύσσεται μία διαδικασία που υπολογίζει τα μοτίβα που δύνανται να περιγράψουν αφαιρετικά την εκάστοτε μόνιμη κίνηση του ρομπότ, αφού πρώτα εισαχθεί μία συγκεκριμένη κατηγοριοποίηση των εκτελούμενων διασκελισμών. Όσον αφορά στο θεωρητικό σκέλος της εργασίας, αξιοποιώντας τη μεθοδολογία των διαγραμμάτων Hildebrand, που χρησιμοποιούνται ευρέως στο πεδίο της Βιολογίας για την περιγραφή των βηματισμών των ζώων με πόδια, αναπτύσσεται ένας χαμηλού – επιπέδου αλγόριθμος που υπολογίζει τα ποσοτικά χαρακτηριστικά του αντιπροσωπευτικότερου διασκελισμού κάθε δυνατής μόνιμης κίνησης του ρομπότ. Ο αλγόριθμος που συντάσσεται για τον υπολογισμό των διαγραμμάτων Hildebrand, αποτελείται από τρία μέρη. Το πρώτο μέρος, αφορά στον υπολογισμό της χρονικής στιγμής μετάβασης από τη «μεταβατική» στη μόνιμη κατάσταση. Το δεύτερο, αφορά στη συλλογή των χρονικών στιγμών απογείωσης και προσγείωσης των ποδιών. Το τρίτο, και τελευταίο, σχετίζεται με τον αριθμητικό υπολογισμό των συστατικών στοιχείων των διαγραμμάτων. Για την περίπτωση των διδιάστατων τετράποδων ρομπότ, αυτά, είναι τρία, και, αφορούν στα (δύο, το πλήθος) ποσοστά φόρτισης των ποδιών και την ποσοστιαία χρονική διαφορά των προσγειώσεών τους.In this Diploma Thesis, the problem of the systematic description of the gait, that, when repeated creates the steady state part of a given simulated quadruped robot motion, is addressed. Of great importance, is considered to be the inspection of the effect of each parameter, that emerges either in the design process of the robot, the modeling of its environment, the portraiture of its initial motion conditions, or in its control system configuration, in the computed characteristics of the selected, by the quadruped robot, gait. Subsequently, a procedure is developed, such that, using this, one is able to derive the pattern that abstractly describes a given steady state motion part. The theoretical part of this thesis, is deployed around the adoption of the Hildebrand diagrams methodology, which emerges in the field of Biology in cases where the compact description of the gait, which a legged animal uses in its motion, is necessary. What is done, is the derivation of a low – level algorithmic process that carries out the Hildebrand diagram computation (i.e., its quantitative characteristics) of the most typical gait that is respective of a given simulated robot motion. The algorithm that is employed for the computation of the Hildebrand diagrams, consists of three parts. The first, refers to the computation of the instance that indicates the end of the “transient” state and the beginning of the “steady” state of its movement. The second, concerns the collection of the instance values that are respective to the touch – down and lift – off events of the two pairs of legs (hind and fore). Conclusively, the third part, contains the method for the calculation of the numeric values that compose a Hildebrand diagram. In the case of planar four – legged robots, these are three, namely, the hind and fore duty factors and the phase relationship between the ground contact instances of the leg pairs. Each of those three quantities, is considered as a percentage of the stride duration.Μυρισιώτης Η. Δημήτριο

On Approximating Total Variation Distance

Total variation distance (TV distance) is a fundamental notion of distance between probability distributions. In this work, we introduce and study the computational problem of determining the TV distance between two product distributions over the domain $\{0,1\}^n$. We establish the following results. 1. Exact computation of TV distance between two product distributions is $\#\mathsf{P}$-complete. This is in stark contrast with other distance measures such as KL, Chi-square, and Hellinger which tensorize over the marginals. 2. Given two product distributions $P$ and $Q$ with marginals of $P$ being at least $1/2$ and marginals of $Q$ being at most the respective marginals of $P$, there exists a fully polynomial-time randomized approximation scheme (FPRAS) for computing the TV distance between $P$ and $Q$. In particular, this leads to an efficient approximation scheme for the interesting case when $P$ is an arbitrary product distribution and $Q$ is the uniform distribution. We pose the question of characterizing the complexity of approximating the TV distance between two arbitrary product distributions as a basic open problem in computational statistics.Comment: 22 pages, 1 figur