66 research outputs found
Interactive Coding with Constant Round and Communication Blowup
No full textThe problem of constructing error-resilient interactive protocols was introduced in the seminal works of Schulman (FOCS 1992, STOC 1993). These works show how to convert any two-party interactive protocol into one that is resilient to constant-fraction of error, while blowing up the communication by only a constant factor. Since these seminal works, there have been many followup works which improve the error rate, the communication rate, and the computational efficiency.
All these works only consider only an increase in communication complexity and did not consider an increase in round complexity. This work is the first one that considers the blowup of round complexity in noisy setting. While techniques from other papers can be easily adapted encode protocols with arbitrarily round complexity this coding schemes will lead to large(and usually unbounded) increase in round complexity of the protocol.
In this work, we show how to convert any protocol Π, with no a priori known communication bound, into an error-resilient protocol Π', with comparable computational efficiency, that is resilient to constant fraction of adversarial error, while blowing up both the communication complexity and the round complexity by at most a constant factor. We consider the model where in each round each party may send a message of arbitrary length, where the length of the messages and the length of the protocol may be adaptive, and may depend on the private inputs of the parties and on previous communication. We consider the adversarial error model, where ε-fraction of the communication may be corrupted, where we allow each corruption to be an insertion or deletion (in addition to toggle).
In addition, we try to minimize the blowup parameters: In particular, we construct such Π' with (1+Õ(ε^(1/4))) blowup in communication and O(1) blowup in rounds. We also show how to reduce the blowup in rounds at the expense of increasing the blowup in communication, and construct Π' where both the blowup in rounds and communication, approaches one (i.e., no blowup) as ε approaches zero. We give "evidence" that our parameters are "close to" optimal
Noisy Radio Network Lower Bounds via Noiseless Beeping Lower Bounds
No full textMuch of today’s communication is carried out over large wireless systems with different input-output behaviors. In this work, we compare the power of central abstractions of wireless communication through the general notion of boolean symmetric f-channels: In every round of the f-channel, each of its n parties decides to either broadcast or not, and the channel outputs f(m), where m is the number of broadcasting parties.
Our first result is that the well studied beeping channel, where f is the threshold-1 function, is not stronger than any other f-channel. To this end, we design a protocol over the f-channel and prove that any protocol that simulates it over the beeping channel blows up the round complexity by a factor of Ω(log n). Our lower bound technique may be of independent interest, as it essentially generalizes the popular fooling set technique by exploiting a "local" relaxation of combinatorial rectangles.
Curiously, while this result shows the limitations of a noiseless channel, namely, the beeping channel, we are able to use it to show the limitations of the noisy version of many other channels. This includes the extensively studied single-hop radio network model with collisions-as-silence (CAS), which is equivalent to the f-channel with f(m) = 1 iff m = 1.
In particular, our second and main result, obtained from the first, shows that converting CAS protocols to noise resilient ones may incur a large performance overhead, i.e., no constant rate interactive code exists. To this end, we design a CAS protocol and prove that any protocol that simulates it over the noisy CAS model with correlated stochastic noise, blows up the round complexity by a factor of Ω(log n). We mention that the Ω(log n) overhead in both our results is tight
Protecting Single-Hop Radio Networks from Message Drops
No full textSingle-hop radio networks (SHRN) are a well studied abstraction of communication over a wireless channel. In this model, in every round, each of the n participating parties may decide to broadcast a message to all the others, potentially causing collisions. We consider the SHRN model in the presence of stochastic message drops (i.e., erasures), where in every round, the message received by each party is erased (replaced by ⊥) with some small constant probability, independently.
Our main result is a constant rate coding scheme, allowing one to run protocols designed to work over the (noiseless) SHRN model over the SHRN model with erasures. Our scheme converts any protocol Π of length at most exponential in n over the SHRN model to a protocol Π' that is resilient to constant fraction of erasures and has length linear in the length of Π.
We mention that for the special case where the protocol Π is non-adaptive, i.e., the order of communication is fixed in advance, such a scheme was known. Nevertheless, adaptivity is widely used and is known to hugely boost the power of wireless channels, which makes handling the general case of adaptive protocols Π both important and more challenging. Indeed, to the best of our knowledge, our result is the first constant rate scheme that converts adaptive protocols to noise resilient ones in any multi-party model
Barriers for Rank Methods in Arithmetic Complexity
No full textArithmetic complexity, the study of the cost of computing polynomials via additions and multiplications, is considered (for many good reasons) simpler to understand than Boolean complexity, namely computing Boolean functions via logical gates. And indeed, we seem to have significantly more lower bound techniques and results in arithmetic complexity than in Boolean complexity. Despite many successes and rapid progress, however, foundational challenges, like proving super-polynomial lower bounds on circuit or formula size for explicit polynomials, or super-linear lower bounds on explicit 3-dimensional tensors, remain elusive.
At the same time (and possibly for similar reasons), we have plenty more excuses, in the form of "barrier results" for failing to prove basic lower bounds in Boolean complexity than in arithmetic complexity. Efforts to find barriers to arithmetic lower bound techniques seem harder, and despite some attempts we have no excuses of similar quality for these failures in arithmetic complexity. This paper aims to add to this study.
In this paper we address rank methods, which were long recognized as encompassing and abstracting almost all known arithmetic lower bounds to-date, including the most recent impressive successes. Rank methods (under the name of flattenings) are also in wide use in algebraic geometry for proving tensor rank and symmetric tensor rank lower bounds. Our main results are barriers to these methods. In particular,
1. Rank methods cannot prove better than (2^d)*n^(d/2) lower bound on the tensor rank of any d-dimensional tensor of side n. (In particular, they cannot prove super-linear, indeed even >8n tensor rank lower bounds for any 3-dimensional tensors.)
2. Rank methods cannot prove (d+1)n^(d/2) on the Waring rank of any n-variate polynomial of degree d. (In particular, they cannot prove such lower bounds on stronger models, including depth-3 circuits.)
The proofs of these bounds use simple linear-algebraic arguments, leveraging connections between the symbolic rank of matrix polynomials and the usual rank of their evaluations. These techniques can perhaps be extended to barriers for other arithmetic models on which progress has halted.
To see how these barrier results directly inform the state-of-art in arithmetic complexity we note the following.
First, the bounds above nearly match the best explicit bounds we know for these models, hence offer an explanations why the rank methods got stuck there. Second, the bounds above are a far cry (quadratically away) from the true complexity (e.g. of random polynomials) in these models, which if achieved (by any methods), are known to imply super-polynomial formula lower bounds.
We also explain the relation of our barrier results to other attempts, and in particular how they significantly differ from the recent attempts to find analogues of "natural proofs" for arithmetic complexity. Finally, we discuss the few arithmetic lower bound approaches which fall outside rank methods, and some natural directions our barriers suggest
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Computation over the Noisy Broadcast Channel with Malicious Parties
No full textWe study the n-party noisy broadcast channel with a constant fraction of malicious parties. Specifically, we assume that each non-malicious party holds an input bit, and communicates with the others in order to learn the input bits of all non-malicious parties. In each communication round, one of the parties broadcasts a bit to all other parties, and the bit received by each party is flipped with a fixed constant probability (independently for each recipient). How many rounds are needed?
Assuming there are no malicious parties, Gallager gave an (n log log n)-round protocol for the above problem, which was later shown to be optimal. This protocol, however, inherently breaks down in the presence of malicious parties.
We present a novel n ⋅ ̃(√{log n})-round protocol, that solves this problem even when almost half of the parties are malicious. Our protocol uses a new type of error correcting code, which we call a locality sensitive code and which may be of independent interest. Roughly speaking, these codes map "close" messages to "close" codewords, while messages that are not close are mapped to codewords that are very far apart.
We view our result as a first step towards a theory of property preserving interactive coding, i.e., interactive codes that preserve useful properties of the protocol being encoded. In our case, the naive protocol over the noiseless broadcast channel, where all the parties broadcast their input bit and output all the bits received, works even in the presence of malicious parties. Our simulation of this protocol, unlike Gallager’s, preserves this property of the original protocol
Round-Vs-Resilience Tradeoffs for Binary Feedback Channels
No full textIn a celebrated result from the 60’s, Berlekamp showed that feedback can be used to increase the maximum fraction of adversarial noise that can be tolerated by binary error correcting codes from 1/4 to 1/3. However, his result relies on the assumption that feedback is "continuous", i.e., after every utilization of the channel, the sender gets the symbol received by the receiver. While this assumption is natural in some settings, in other settings it may be unreasonable or too costly to maintain.
In this work, we initiate the study of round-restricted feedback channels, where the number r of feedback rounds is possibly much smaller than the number of utilizations of the channel. Error correcting codes for such channels are protocols where the sender can ask for feedback at most r times, and, upon a feedback request, it obtains all the symbols received since its last feedback request. We design such error correcting protocols for both the adversarial binary erasure channel and for the adversarial binary corruption (bit flip) channel. For the erasure channel, we give an exact characterization of the round-vs-resilience tradeoff by designing a (constant rate) protocol with r feedback rounds, for every r, and proving that its noise resilience is optimal.
Designing such error correcting protocols for the corruption channel is substantially more involved. We show that obtaining the optimal resilience, even with one feedback round (r = 1), requires settling (proving or disproving) a new, seemingly unrelated, "clean" combinatorial conjecture, about the maximum cut in weighted graphs versus the "imbalance" of an average cut. Specifically, we prove an upper bound on the optimal resilience (impossibility result), and show that the existence of a matching lower bound (a protocol) is equivalent to the correctness of our conjecture
Information Dissemination via Broadcasts in the Presence of Adversarial Noise
No full textWe initiate the study of error correcting codes over the multi-party adversarial broadcast channel. Specifically, we consider the classic information dissemination problem where n parties, each holding an input bit, wish to know each other’s input. For this, they communicate in rounds, where, in each round, one designated party sends a bit to all other parties over a channel governed by an adversary that may corrupt a constant fraction of the received communication. We mention that the dissemination problem was studied in the stochastic noise model since the 80’s.
While stochastic noise in multi-party channels has received quite a bit of attention, the case of adversarial noise has largely been avoided, as such channels cannot handle more than a 1/n-fraction of errors. Indeed, this many errors allow an adversary to completely corrupt the incoming or outgoing communication for one of the parties and fail the protocol. Curiously, we show that by eliminating these "trivial" attacks, one can get a simple protocol resilient to a constant fraction of errors. Thus, a model that rules out such attacks is both necessary and sufficient to get a resilient protocol.
The main shortcoming of our dissemination protocol is its length: it requires Θ(n²) communication rounds whereas n rounds suffice in the absence of noise. Our main result is a matching lower bound of Ω(n²) on the length of any dissemination protocol in our model. Our proof first "gets rid" of the channel noise by converting it to a form of "input noise", showing that a noisy dissemination protocol implies a (noiseless) protocol for a version of the direct sum gap-majority problem. We conclude the proof with a tight lower bound for the latter problem, which may be of independent interest
Optimal Short-Circuit Resilient Formulas
No full textWe consider fault-tolerant boolean formulas in which the output of a faulty
gate is short-circuited to one of the gate's inputs. A recent result by Kalai
et al. (FOCS 2012) converts any boolean formula into a resilient formula of
polynomial size that works correctly if less than a fraction of the gates
(on every input-to-output path) are faulty. We improve the result of Kalai et
al., and show how to efficiently fortify any boolean formula against a fraction
of short-circuit gates per path, with only a polynomial blowup in size.
We additionally show that it is impossible to obtain formulas with higher
resilience and sub-exponential growth in size.
Towards our results, we consider interactive coding schemes when noiseless
feedback is present; these produce resilient boolean formulas via a
Karchmer-Wigderson relation. We develop a coding scheme that resists up to a
fraction of corrupted transmissions in each direction of the interactive
channel. We further show that such a level of noise is maximal for coding
schemes with sub-exponential blowup in communication. Our coding scheme takes a
surprising inspiration from Blockchain technology.Comment: The second version corrects an error in the theorems of Section 3.
Accepted to JAC
Satirizing Ethics
No full textI explored three late-night satirical monologues from late-night television following the Orlando Nightclub Shooting on 12 June 2016. I examined the monologues using a social constructionist approach to understand what lessons could be drawn from the event and how a satirist references the nomos when tackling an issue in the world. Satire uses comedic tools to point out follies within society and each host used his or her platform on late-night television to address issues they believed needed to be addressed following the shooting. What I found was that each host aimed to set an agenda in their monologues by emphasizing a) certain facts and information about the event and by b) presenting the audience with a way of viewing what happened and why the issue of gun violence and hate crimes in American needs to be addressed to ensure incidents like these do not happened again
Exponential Separation Between Powers of Regular and General Resolution over Parities
No full textProving super-polynomial lower bounds on the size of proofs of unsatisfiability of Boolean formulas using resolution over parities is an outstanding problem that has received a lot of attention after its introduction by Itsykson and Sokolov [Dmitry Itsykson and Dmitry Sokolov, 2014]. Very recently, Efremenko, Garlík and Itsykson [Klim Efremenko et al., 2023] proved the first exponential lower bounds on the size of ResLin proofs that were additionally restricted to be bottom-regular. We show that there are formulas for which such regular ResLin proofs of unsatisfiability continue to have exponential size even though there exist short proofs of their unsatisfiability in ordinary, non-regular resolution. This is the first super-polynomial separation between the power of general ResLin and that of regular ResLin for any natural notion of regularity.
Our argument, while building upon the work of Efremenko et al. [Klim Efremenko et al., 2023], uses additional ideas from the literature on lifting theorems
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