On monotone circuits with local oracles and clique lower bounds

We investigate monotone circuits with local oracles [K., 2016], i.e., circuits containing additional inputs $y_i = y_i(\vec{x})$ that can perform unstructured computations on the input string $\vec{x}$. Let $\mu \in [0,1]$ be the locality of the circuit, a parameter that bounds the combined strength of the oracle functions $y_i(\vec{x})$, and $U_{n,k}, V_{n,k} \subseteq \{0,1\}^m$ be the set of $k$-cliques and the set of complete $(k-1)$-partite graphs, respectively (similarly to [Razborov, 1985]). Our results can be informally stated as follows. 1. For an appropriate extension of depth-$2$ monotone circuits with local oracles, we show that the size of the smallest circuits separating $U_{n,3}$ (triangles) and $V_{n,3}$ (complete bipartite graphs) undergoes two phase transitions according to $\mu$. 2. For $5 \leq k(n) \leq n^{1/4}$, arbitrary depth, and $\mu \leq 1/50$, we prove that the monotone circuit size complexity of separating the sets $U_{n,k}$ and $V_{n,k}$ is $n^{\Theta(\sqrt{k})}$, under a certain restrictive assumption on the local oracle gates. The second result, which concerns monotone circuits with restricted oracles, extends and provides a matching upper bound for the exponential lower bounds on the monotone circuit size complexity of $k$-clique obtained by Alon and Boppana (1987).Comment: Updated acknowledgements and funding informatio

Consistency of circuit lower bounds with bounded theories

Proving that there are problems in $\mathsf{P}^\mathsf{NP}$ that require boolean circuits of super-linear size is a major frontier in complexity theory. While such lower bounds are known for larger complexity classes, existing results only show that the corresponding problems are hard on infinitely many input lengths. For instance, proving almost-everywhere circuit lower bounds is open even for problems in $\mathsf{MAEXP}$. Giving the notorious difficulty of proving lower bounds that hold for all large input lengths, we ask the following question: Can we show that a large set of techniques cannot prove that $\mathsf{NP}$ is easy infinitely often? Motivated by this and related questions about the interaction between mathematical proofs and computations, we investigate circuit complexity from the perspective of logic. Among other results, we prove that for any parameter $k \geq 1$ it is consistent with theory $T$ that computational class ${\mathcal C} \not \subseteq \textit{i.o.}\mathrm{SIZE}(n^k)$, where $(T, \mathcal{C})$ is one of the pairs: $T = \mathsf{T}^1_2$ and ${\mathcal C} = \mathsf{P}^\mathsf{NP}$, $T = \mathsf{S}^1_2$ and ${\mathcal C} = \mathsf{NP}$, $T = \mathsf{PV}$ and ${\mathcal C} = \mathsf{P}$. In other words, these theories cannot establish infinitely often circuit upper bounds for the corresponding problems. This is of interest because the weaker theory $\mathsf{PV}$ already formalizes sophisticated arguments, such as a proof of the PCP Theorem. These consistency statements are unconditional and improve on earlier theorems of [KO17] and [BM18] on the consistency of lower bounds with $\mathsf{PV}$

NP-hardness of circuit minimization for multi-output functions

Can we design efficient algorithms for finding fast algorithms? This question is captured by various circuit minimization problems, and algorithms for the corresponding tasks have significant practical applications. Following the work of Cook and Levin in the early 1970s, a central question is whether minimizing the circuit size of an explicitly given function is NP-complete. While this is known to hold in restricted models such as DNFs, making progress with respect to more expressive classes of circuits has been elusive. In this work, we establish the first NP-hardness result for circuit minimization of total functions in the setting of general (unrestricted) Boolean circuits. More precisely, we show that computing the minimum circuit size of a given multi-output Boolean function f : {0,1}^n ? {0,1}^m is NP-hard under many-one polynomial-time randomized reductions. Our argument builds on a simpler NP-hardness proof for the circuit minimization problem for (single-output) Boolean functions under an extended set of generators. Complementing these results, we investigate the computational hardness of minimizing communication. We establish that several variants of this problem are NP-hard under deterministic reductions. In particular, unless ? = ??, no polynomial-time computable function can approximate the deterministic two-party communication complexity of a partial Boolean function up to a polynomial. This has consequences for the class of structural results that one might hope to show about the communication complexity of partial functions

Erd\H{o}s-Ko-Rado for random hypergraphs: asymptotics and stability

We investigate the asymptotic version of the Erd\H{o}s-Ko-Rado theorem for the random $k$-uniform hypergraph $\mathcal{H}^k(n,p)$. For $2 \leq k(n) \leq n/2$, let $N=\binom{n}k$ and $D=\binom{n-k}k$. We show that with probability tending to 1 as $n\to\infty$, the largest intersecting subhypergraph of $\mathcal{H}^k(n,p)$ has size $(1+o(1))p\frac kn N$, for any $p\gg \frac nk\ln^2\!\left(\frac nk\right)D^{-1}$. This lower bound on $p$ is asymptotically best possible for $k=\Theta(n)$. For this range of $k$ and $p$, we are able to show stability as well. A different behavior occurs when $k = o(n)$. In this case, the lower bound on $p$ is almost optimal. Further, for the small interval $D^{-1}\ll p \leq (n/k)^{1-\varepsilon}D^{-1}$, the largest intersecting subhypergraph of $\mathcal{H}^k(n,p)$ has size $\Theta(\ln (pD)N D^{-1})$, provided that $k \gg \sqrt{n \ln n}$. Together with previous work of Balogh, Bohman and Mubayi, these results settle the asymptotic size of the largest intersecting family in $\mathcal{H}^k(n,p)$, for essentially all values of $p$ and $k$

Randomness and intractability in Kolmogorov complexity

We introduce randomized time-bounded Kolmogorov complexity (rKt), a natural extension of Levin's notion [Leonid A. Levin, 1984] of Kolmogorov complexity. A string w of low rKt complexity can be decompressed from a short representation via a time-bounded algorithm that outputs w with high probability. This complexity measure gives rise to a decision problem over strings: MrKtP (The Minimum rKt Problem). We explore ideas from pseudorandomness to prove that MrKtP and its variants cannot be solved in randomized quasi-polynomial time. This exhibits a natural string compression problem that is provably intractable, even for randomized computations. Our techniques also imply that there is no n^{1 - epsilon}-approximate algorithm for MrKtP running in randomized quasi-polynomial time. Complementing this lower bound, we observe connections between rKt, the power of randomness in computing, and circuit complexity. In particular, we present the first hardness magnification theorem for a natural problem that is unconditionally hard against a strong model of computation

Transporte transmembranar de aniões por moléculas sintéticas: uma investigação por métodos de modelação molecular

Ion transport across cell membranes, via protein membrane channels, is crucial in several biological processes. Thus, the malfunctioning of this complex cellular machinery is linked with several channelopathies, such as cystic fibrosis (CF), associated with the deficient chloride transport through the CFTR channel. As present treatments only aim to manage the disease’s symptoms, alternative treatments are needed, such as channel replacement therapies. Over the last decades, this fact has motivated the development of synthetic anion transporters able to recognise and later promote the passive anion transport in lipid vesicles or even in CF cell models. However, the design of small drug-like transporters is still not straightforward and depends on an intricate equilibrium between the transporters’ binding affinity and lipophilicity. In this context, this thesis interfaces between the supramolecular, medicinal and computational fields of Chemistry. The theoretical investigations reported in this thesis consisted on quantum calculations together with molecular dynamics simulations based on classical force fields. The following series of molecules were investigated: two series of tripodal molecules (chapters II and V), a series of decalin-based transporters (chapter III) and four series of squaramide-based compounds (chapter IV). The structural and energetic insights allowed to understand, at the atomistic level, the interaction of synthetic molecules with membrane models as well as the anion transport mechanisms. The molecular dynamics simulations of passive diffusion were carried out with chloride complexes placed either in the water phase or inside the phospholipid bilayer, allowing the investigation of the transporters’ ability to permeate the water/lipid interface and to diffuse within the highly packed bilayer medium. Moreover, the assessment of the electrostatic surface potential of the transporters yielded insights that generally correlate well with anion binding constants. In chapters IV and V, constrained molecular dynamics simulations with linear squaramides and fluorinated tripodal derivatives are reported, respectively. These simulations allowed to estimate the free energy barriers associated with the translocation of these two series of molecules across the membrane model, with the reconstruction of the potential of mean force along the bilayer normal. The energetic barriers assessed for both series of molecules agree well with their lipophilicities and experimental anion transport data. Furthermore, a simulation reported in chapter V shows, for the first time, a neutral transporter facilitating the translocation of chloride across a phospholipid bilayer. Within the scope of supramolecular Chemistry, chapter VI reports the development of force field parameters for chalcogen bonding interactions. These bonding interaction, as well as halogen bonds, can also be used for the recognition and transmembrane transport of anions, becoming potential alternatives to the ubiquitous hydrogen bonds studied in the previous chapters.O transporte de aniões através de membranas celulares, com recurso a canais proteicos, é fundamental em vários processos biológicos. Assim, o funcionamento deficiente desta complexa maquinaria celular está relacionado com o aparecimento de várias canalopatias como a Fibrose Cística (FC), associada ao transporte deficiente de cloreto através do canal CFTR. Os tratamentos atuais para esta doença apenas minoram os seus sintomas, sendo necessário desenvolver tratamentos alternativos, como, por exemplo, as terapias de substituição de canal. Este facto, ao longo das últimas décadas, tem motivado o desenvolvimento de moléculas capazes de procederem ao reconhecimento e, posteriormente, ao transporte passivo de aniões em vesículas lipídicas e em modelos celulares de FC. No entanto, uma molécula com atividade de transporte depende de um delicado equilíbrio entre sua a lipofilia e a afinidade para o anião. Neste contexto, esta tese situa-se na interface entre Química supramolecular, medicinal e computacional. Os estudos teóricos que aqui se reportam consistiram em cálculos de mecânica quântica conjugados com simulações de dinâmica molecular baseados em campos de forças clássicos. Foram investigadas duas séries de moléculas trípodes (capítulos II e V), uma de transportadores derivados de decalina (capítulo III), e quatro séries de esquaramidas (capítulo IV). Os resultados estruturais e energéticos obtidos contribuíram para compreender, ao nível atómico, a interação destas moléculas com modelos de membranas, bem como dos respetivos mecanismos de transporte de aniões. As simulações de dinâmica molecular de difusão passiva foram realizadas com os complexos de cloreto colocados na fase aquosa ou dentro da bicamada fosfolipídica, permitindo o estudo da capacidade de um transportador permear a interface água/lípido e de se difundir no meio altamente empacotado da bicamada. Por outro lado, a avaliação da distribuição do potencial electroestático na superfície eletrónica das moléculas correlaciona-se com as suas constantes de associação com aniões. Nos capítulos IV e V descrevem-se também simulações de dinâmica molecular constrangidas, realizadas com esquaramidas lineares e derivados trípodes fluorinados, respetivamente. Estas simulações permitiram a estimativa das barreiras de energia livre associadas à difusão destas duas séries de moléculas através do modelo de membrana por reconstrução do potencial de força média ao longo da normal à bicamada. As barreiras energéticas para estas duas séries de moléculas são consistentes com os dados experimentais de transporte e lipofilia. Adicionalmente, uma simulação reportada no capítulo V mostra, pela primeira vez, um transportador neutro a facilitar o transporte de cloreto através de uma bicamada fosfolipídica. No âmbito da Química supramolecular, no capítulo VI reporta-se o desenvolvimento de parâmetros de campo de forças para ligações de calcogénio. Estas ligações, tal como as ligações de halogénio, também permitem o reconhecimento e transporte transmembranar de aniões, surgindo como potenciais alternativas às ligações de hidrogénio convencionais estudadas nos capítulos anteriores.Mais ainda, os estudos apresentados nesta tese foram realizados com recursos computacionais adquiridos sob o projeto P2020-PTDC/QEQ-SUP/4283/2014, financiado pela FCT e pelo Fundo Europeu de Desenvolvimento Regional (FEDER) através do COMPETE 2020 − Programa Operacional Competitividade e Internacionalização.Programa Doutoral em Biomedicin

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 $2^{O(n/\log n)}$

Near-optimal small-depth lower bounds for small distance connectivity

We show that any depth-$d$ circuit for determining whether an $n$-node graph has an $s$-to-$t$ path of length at most $k$ must have size $n^{\Omega(k^{1/d}/d)}$. The previous best circuit size lower bounds for this problem were $n^{k^{\exp(-O(d))}}$ (due to Beame, Impagliazzo, and Pitassi [BIP98]) and $n^{\Omega((\log k)/d)}$ (following from a recent formula size lower bound of Rossman [Ros14]). Our lower bound is quite close to optimal, since a simple construction gives depth-$d$ circuits of size $n^{O(k^{2/d})}$ for this problem (and strengthening our bound even to $n^{k^{\Omega(1/d)}}$ would require proving that undirected connectivity is not in $\mathsf{NC^1}.$) Our proof is by reduction to a new lower bound on the size of small-depth circuits computing a skewed variant of the "Sipser functions" that have played an important role in classical circuit lower bounds [Sip83, Yao85, H{\aa}s86]. A key ingredient in our proof of the required lower bound for these Sipser-like functions is the use of \emph{random projections}, an extension of random restrictions which were recently employed in [RST15]. Random projections allow us to obtain sharper quantitative bounds while employing simpler arguments, both conceptually and technically, than in the previous works [Ajt89, BPU92, BIP98, Ros14]