938 research outputs found
Stretching demi-bits and nondeterministic-secure pseudorandomness
We develop the theory of cryptographic nondeterministic-secure pseudorandomness beyond the point reached by Rudich's original work [25], and apply it to draw new consequences in average-case complexity and proof complexity. Specifically, we show the following: Demi-bit stretch: Super-bits and demi-bits are variants of cryptographic pseudorandom generators which are secure against nondeterministic statistical tests [25]. They were introduced to rule out certain approaches to proving strong complexity lower bounds beyond the limitations set out by the Natural Proofs barrier of Razborov and Rudich [23]. Whether demi-bits are stretchable at all had been an open problem since their introduction. We answer this question affirmatively by showing that: every demi-bit b : {0, 1}n â {0, 1}n+1 can be stretched into sublinear many demi-bits bâČ: {0, 1}n â {0, 1}n+nc , for every constant 0 < c < 1. Average-case hardness: Using work by Santhanam [26], we apply our results to obtain new averagecase Kolmogorov complexity results: we show that Kpoly[n-O(1)] is zero-error average-case hard against NP/poly machines iff Kpoly[n-o(n)] is, where for a function s(n) : N â N, Kpoly[s(n)] denotes the languages of all strings x â {0, 1}n for which there are (fixed) polytime Turing machines of description-length at most s(n) that output x. Characterising super-bits by nondeterministic unpredictability: In the deterministic setting, Yao [31] proved that super-polynomial hardness of pseudorandom generators is equivalent to ("nextbit") unpredictability. Unpredictability roughly means that given any strict prefix of a random string, it is infeasible to predict the next bit. We initiate the study of unpredictability beyond the deterministic setting (in the cryptographic regime), and characterise the nondeterministic hardness of generators from an unpredictability perspective. Specifically, we propose four stronger notions of unpredictability: NP/poly-unpredictability, coNP/poly-unpredictability, â©-unpredictability and âȘunpredictability, and show that super-polynomial nondeterministic hardness of generators lies between â©-unpredictability and âȘunpredictability. Characterising super-bits by nondeterministic hard-core predicates: We introduce a nondeterministic variant of hard-core predicates, called super-core predicates. We show that the existence of a super-bit is equivalent to the existence of a super-core of some non-shrinking function. This serves as an analogue of the equivalence between the existence of a strong pseudorandom generator and the existence of a hard-core of some one-way function [8, 12], and provides a first alternative characterisation of super-bits. We also prove that a certain class of functions, which may have hard-cores, cannot possess any super-core
LIPIcs, Volume 251, ITCS 2023, Complete Volume
LIPIcs, Volume 251, ITCS 2023, Complete Volum
Exponential Separations Using Guarded Extension Variables
We study the complexity of proof systems augmenting resolution with inference rules that allow, given a formula ? in conjunctive normal form, deriving clauses that are not necessarily logically implied by ? but whose addition to ? preserves satisfiability. When the derived clauses are allowed to introduce variables not occurring in ?, the systems we consider become equivalent to extended resolution. We are concerned with the versions of these systems without new variables. They are called BC?, RAT?, SBC?, and GER?, denoting respectively blocked clauses, resolution asymmetric tautologies, set-blocked clauses, and generalized extended resolution. Each of these systems formalizes some restricted version of the ability to make assumptions that hold "without loss of generality," which is commonly used informally to simplify or shorten proofs.
Except for SBC?, these systems are known to be exponentially weaker than extended resolution. They are, however, all equivalent to it under a relaxed notion of simulation that allows the translation of the formula along with the proof when moving between proof systems. By taking advantage of this fact, we construct formulas that separate RAT? from GER? and vice versa. With the same strategy, we also separate SBC? from RAT?. Additionally, we give polynomial-size SBC? proofs of the pigeonhole principle, which separates SBC? from GER? by a previously known lower bound. These results also separate the three systems from BC? since they all simulate it. We thus give an almost complete picture of their relative strengths
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
Mining Butterflies in Streaming Graphs
This thesis introduces two main-memory systems sGrapp and sGradd for performing the fundamental analytic tasks of biclique counting and concept drift detection over a streaming graph. A data-driven heuristic is used to architect the systems. To this end, initially, the growth patterns of bipartite streaming graphs are mined and the emergence principles of streaming motifs are discovered. Next, the discovered principles are (a) explained by a graph generator called sGrow; and (b) utilized to establish the requirements for efficient, effective, explainable, and interpretable management and processing of streams. sGrow is used to benchmark stream analytics, particularly in the case of concept drift detection.
sGrow displays robust realization of streaming growth patterns independent of initial conditions, scale and temporal characteristics, and model configurations. Extensive evaluations confirm the simultaneous effectiveness and efficiency of sGrapp and sGradd. sGrapp achieves mean absolute percentage error up to 0.05/0.14 for the cumulative butterfly count in streaming graphs with uniform/non-uniform temporal distribution and a processing throughput of 1.5 million data records per second. The throughput and estimation error of sGrapp are 160x higher and 0.02x lower than baselines. sGradd demonstrates an improving performance over time, achieves zero false detection rates when there is not any drift and when drift is already detected, and detects sequential drifts in zero to a few seconds after their occurrence regardless of drift intervals
Semitopology: a new topological model of heterogeneous consensus
A distributed system is permissionless when participants can join and leave
the network without permission from a central authority. Many modern
distributed systems are naturally permissionless, in the sense that a central
permissioning authority would defeat their design purpose: this includes
blockchains, filesharing protocols, some voting systems, and more. By their
permissionless nature, such systems are heterogeneous: participants may only
have a partial view of the system, and they may also have different goals and
beliefs. Thus, the traditional notion of consensus -- i.e. system-wide
agreement -- may not be adequate, and we may need to generalise it.
This is a challenge: how should we understand what heterogeneous consensus
is; what mathematical framework might this require; and how can we use this to
build understanding and mathematical models of robust, effective, and secure
permissionless systems in practice?
We analyse heterogeneous consensus using semitopology as a framework. This is
like topology, but without the restriction that intersections of opens be open.
Semitopologies have a rich theory which is related to topology, but with its
own distinct character and mathematics. We introduce novel well-behavedness
conditions, including an anti-Hausdorff property and a new notion of `topen
set', and we show how these structures relate to consensus. We give a
restriction of semitopologies to witness semitopologies, which are an
algorithmically tractable subclass corresponding to Horn clause theories,
having particularly good mathematical properties. We introduce and study
several other basic notions that are specific and novel to semitopologies, and
study how known quantities in topology, such as dense subsets and closures,
display interesting and useful new behaviour in this new semitopological
context
Pointwise convergence for the Schr\"odinger equation [After Xiumin Du and Ruixiang Zhang]
This expository essay accompanied the author's presentation at the
S\'eminaire Bourbaki on 01 April 2023. It describes the breakthrough work of
Du--Zhang on the Carleson problem for the Schr\"odinger equation, together with
background material in multilinear harmonic analysis.Comment: 72 pages, 6 figures, comments welcome
Sum-Of-Squares Lower Bounds for the Minimum Circuit Size Problem
We prove lower bounds for the Minimum Circuit Size Problem (MCSP) in the Sum-of-Squares (SoS) proof system. Our main result is that for every Boolean function f: {0,1}? ? {0,1}, SoS requires degree ?(s^{1-?}) to prove that f does not have circuits of size s (for any s > poly(n)). As a corollary we obtain that there are no low degree SoS proofs of the statement NP ? P/poly.
We also show that for any 0 < ? < 1 there are Boolean functions with circuit complexity larger than 2^{n^?} but SoS requires size 2^{2^?(n^?)} to prove this. In addition we prove analogous results on the minimum monotone circuit size for monotone Boolean slice functions.
Our approach is quite general. Namely, we show that if a proof system Q has strong enough constraint satisfaction problem lower bounds that only depend on good expansion of the constraint-variable incidence graph and, furthermore, Q is expressive enough that variables can be substituted by local Boolean functions, then the MCSP problem is hard for Q
Wylder and Wynona: A Graphic Novel Retelling of the Grimm\u27s Fairy Tale The Little Brother and Sister
This thesis provides insight on the psychoanalysis of fairy tales, namely the Grimm fairy tale, âThe Little Brother and Sister.â The critical analysis defines theories about the human mind popularized by Sigmund Freud and especially Carl Jung. Modern psychologist Paul Moxnes applies Jungâs theories about character archetypes in fairy tales to his modern study about âdeep rolesâ which solidifies the important relationship between human psychology, fairy tales, and fairy tale retellings. With a deeper understanding of the psychological implications of character archetypes, the creative retelling portion of this thesis rewrites the old version of the brother and sister characters from âThe Little Brother and Sisterâ into characters that demonstrate agency and self-empowerment in a graphic novel retelling titled Wylder and Wynona
From Houses of Worship to Worship in Houses: The Social Construction of Sacred Places in Early 21st Century China
While the concept of worship in houses can be traced back to the Christian house church places in Dura Europos between 233 and 256 AD during the Roman Empire, after the foundation of the People's Republic of China in 1949, this kind of church spaces began to appear all across the country. Characterized by the absence of a formal iconic church building or interior, existing types of secular architectural spaces (apartments, offices, basements, etc.) were rented by the Christian community and converted into sacred spaces.
Space is susceptible to manipulations caused by human actions. Now what happens if space is manipulated to house not merely a different function but transcendence? As French Marxist philosopher and sociologist Henri Lefebvre's argument in The Production of Space (1991), space is not only a social product but also a complex social construction, based on values and the social production of meanings, which affects spatial practices and perceptions. An existing space, he says, may outlive its original purpose and the raison d'Ă©tre which initially determined its forms, functions, and structures; it may thus, in a sense, become vacant and susceptible to being diverted, re-appropriated, and utilized for a different purpose than its original intent.
With my analysis of the worship places of urban house churches in early 21st-century China from the perspective of urban context and architectural space (foregrounded by the development of informal church space in the historical context of Chinese society and politics), this research shows how religious metaphors function as the productive mediators in the process of knowledge transfer between architectural and other professional discourses by bringing back social imagination to the politically neutral spaces of every day; de facto reconstructing the social through transduction of the metaphor of informal spaces
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