6,147 research outputs found
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 that can perform
unstructured computations on the input string . Let be
the locality of the circuit, a parameter that bounds the combined strength of
the oracle functions , and
be the set of -cliques and the set of complete -partite graphs,
respectively (similarly to [Razborov, 1985]). Our results can be informally
stated as follows.
1. For an appropriate extension of depth- monotone circuits with local
oracles, we show that the size of the smallest circuits separating
(triangles) and (complete bipartite graphs) undergoes two phase
transitions according to .
2. For , arbitrary depth, and , we
prove that the monotone circuit size complexity of separating the sets
and is , 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 -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 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 . 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 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 it is
consistent with theory that computational class , where is one of
the pairs: and , and , and
. In other words, these theories cannot establish
infinitely often circuit upper bounds for the corresponding problems. This is
of interest because the weaker theory 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
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 -uniform hypergraph . For , let and . We show that with probability
tending to 1 as , the largest intersecting subhypergraph of
has size , for any . This lower bound on is
asymptotically best possible for . For this range of and ,
we are able to show stability as well.
A different behavior occurs when . In this case, the lower bound on
is almost optimal. Further, for the small interval , the largest intersecting subhypergraph of
has size , provided that .
Together with previous work of Balogh, Bohman and Mubayi, these results
settle the asymptotic size of the largest intersecting family in
, for essentially all values of and
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 aniÔes 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 diseaseâs 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 transportersâ 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 transportersâ 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 aniĂ”es atravĂ©s de membranas celulares, com recurso a canais proteicos, Ă© fundamental em vĂĄrios processos biolĂłgicos. Assim, o funcionamento deficiente desta complexa maquinaria celular estĂĄ relacionado com o aparecimento de vĂĄrias canalopatias como a Fibrose CĂstica (FC), associada ao transporte deficiente de cloreto atravĂ©s do canal CFTR. Os tratamentos atuais para esta doença apenas minoram os seus sintomas, sendo necessĂĄrio desenvolver tratamentos alternativos, como, por exemplo, as terapias de substituição de canal. Este facto, ao longo das Ășltimas dĂ©cadas, tem motivado o desenvolvimento de molĂ©culas capazes de procederem ao reconhecimento e, posteriormente, ao transporte passivo de aniĂ”es em vesĂculas lipĂdicas e em modelos celulares de FC. No entanto, uma molĂ©cula com atividade de transporte depende de um delicado equilĂbrio entre sua a lipofilia e a afinidade para o aniĂŁo. Neste contexto, esta tese situa-se na interface entre QuĂmica supramolecular, medicinal e computacional. Os estudos teĂłricos que aqui se reportam consistiram em cĂĄlculos de mecĂąnica quĂąntica conjugados com simulaçÔes de dinĂąmica molecular baseados em campos de forças clĂĄssicos. Foram investigadas duas sĂ©ries de molĂ©culas trĂpodes (capĂtulos II e V), uma de transportadores derivados de decalina (capĂtulo III), e quatro sĂ©ries de esquaramidas (capĂtulo IV). Os resultados
estruturais e energĂ©ticos obtidos contribuĂram para compreender, ao nĂvel atĂłmico, a interação destas molĂ©culas com modelos de membranas, bem como dos respetivos mecanismos de transporte de aniĂ”es. As simulaçÔes de dinĂąmica molecular de difusĂŁo passiva foram realizadas com os complexos de cloreto colocados na fase aquosa ou dentro da bicamada
fosfolipĂdica, permitindo o estudo da capacidade de um transportador permear a interface ĂĄgua/lĂpido e de se difundir no meio altamente empacotado da bicamada. Por outro lado, a avaliação da distribuição do potencial electroestĂĄtico na superfĂcie eletrĂłnica das molĂ©culas correlaciona-se com as suas constantes de associação com aniĂ”es. Nos capĂtulos IV e V descrevem-se tambĂ©m simulaçÔes de dinĂąmica molecular constrangidas, realizadas com esquaramidas lineares e derivados trĂpodes fluorinados, respetivamente. Estas simulaçÔes
permitiram a estimativa das barreiras de energia livre associadas Ă difusĂŁo destas duas sĂ©ries de molĂ©culas atravĂ©s do modelo de membrana por reconstrução do potencial de força mĂ©dia ao longo da normal Ă bicamada. As barreiras energĂ©ticas para estas duas sĂ©ries de molĂ©culas sĂŁo consistentes com os dados experimentais de transporte e lipofilia. Adicionalmente, uma simulação reportada no capĂtulo V mostra, pela primeira vez, um transportador neutro a facilitar o transporte de cloreto atravĂ©s de uma bicamada fosfolipĂdica.
No Ăąmbito da QuĂmica supramolecular, no capĂtulo VI reporta-se o desenvolvimento de parĂąmetros de campo de forças para ligaçÔes de calcogĂ©nio. Estas ligaçÔes, tal como as ligaçÔes de halogĂ©nio, tambĂ©m permitem o reconhecimento e transporte transmembranar de aniĂ”es, surgindo como potenciais alternativas Ă s ligaçÔes de hidrogĂ©nio convencionais estudadas
nos capĂtulos 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) atravĂ©s do COMPETE 2020 â Programa Operacional Competitividade e Internacionalização.Programa Doutoral em Biomedicin
Algorithms and lower bounds for de Morgan formulas of low-communication leaf gates
The class consists of Boolean functions
computable by size- de Morgan formulas whose leaves are any Boolean
functions from a class . We give lower bounds and (SAT, Learning,
and PRG) algorithms for , for classes
of functions with low communication complexity. Let
be the maximum -party NOF randomized communication
complexity of . We show:
(1) The Generalized Inner Product function cannot be computed in
on more than fraction of inputs
for As a corollary, we get an average-case lower bound for
against .
(2) There is a PRG of seed length that -fools . For
, we get the better seed length . This gives the first
non-trivial PRG (with seed length ) for intersections of half-spaces
in the regime where .
(3) There is a randomized -time SAT algorithm for , where In particular, this implies a nontrivial
#SAT algorithm for .
(4) The Minimum Circuit Size Problem is not in .
On the algorithmic side, we show that can be
PAC-learned in time
Near-optimal small-depth lower bounds for small distance connectivity
We show that any depth- circuit for determining whether an -node graph
has an -to- path of length at most must have size
. The previous best circuit size lower bounds for this
problem were (due to Beame, Impagliazzo, and Pitassi
[BIP98]) and (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- circuits of size
for this problem (and strengthening our bound even to
would require proving that undirected connectivity is not in )
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]
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