224,683 research outputs found
An ILP-based Proof System for the Crossing Number Problem
Formally, approaches based on mathematical programming are able to find provably optimal solutions.
However, the demands on a verifiable formal proof are typically much higher than the guarantees
we can sensibly attribute to implementations of mathematical programs. We consider this in the context of the crossing number problem, one of the most prominent problems in topological graph theory. The problem asks for the minimum number of edge crossings in any drawing of a given graph. Graph-theoretic proofs for this problem are known to be notoriously hard to obtain. At the same time, proofs even for very specific graphs are often of interest in crossing number research, as they can, e.g., form the basis for inductive proofs.
We propose a system to automatically generate a formal proof based on an ILP computation. Such a proof is (relatively) easily verifiable, and does not require the understanding of any complex ILP codes. As such, we hope our proof system may serve as a showcase for the necessary steps and central design goals of how to establish formal proof systems based on mathematical programming formulations
Succinct Malleable NIZKs and an Application to Compact Shuffles
Depending on the application, malleability in cryptography can be viewed as either a flaw or — especially if sufficiently understood and restricted — a feature. In this vein, Chase, Kohlweiss, Lysyanskaya, and Meiklejohn recently defined malleable zero-knowledge proofs, and showed how to control the set of allowable transformations on proofs. As an application, they construct the first compact verifiable shuffle, in which one such controlled-malleable proof suffices to prove the correctness of an entire multi-step shuffle. Despite these initial steps, a number of natural open problems remain: (1) their construction of controlled-malleable proofs relies on the inherent malleability of Groth-Sahai proofs and is thus not based on generic primitives; (2) the classes of allowable transformations they can support are somewhat restrictive; and (3) their construction of a compactly verifiable shuffle has proof size O(N 2 + L) (where N is the number of votes and L is the number of mix authorities), whereas in theory such a proof could be of size O(N + L). In this paper, we address these open problems by providing a generic construction of controlledmalleable proofs using succinct non-interactive arguments of knowledge, or SNARGs for short. Our construction has the advantage that we can support a very general class of transformations (as we no longer rely on the transformations that Groth-Sahai proofs can support), and that we can use it to obtain a proof of size O(N + L) for the compactly verifiable shuffle
Efficient Bayesian Learning in Social Networks with Gaussian Estimators
We consider a group of Bayesian agents who try to estimate a state of the
world through interaction on a social network. Each agent
initially receives a private measurement of : a number picked
from a Gaussian distribution with mean and standard deviation one.
Then, in each discrete time iteration, each reveals its estimate of to
its neighbors, and, observing its neighbors' actions, updates its belief using
Bayes' Law.
This process aggregates information efficiently, in the sense that all the
agents converge to the belief that they would have, had they access to all the
private measurements. We show that this process is computationally efficient,
so that each agent's calculation can be easily carried out. We also show that
on any graph the process converges after at most steps, where
is the number of agents and is the diameter of the network. Finally, we
show that on trees and on distance transitive-graphs the process converges
after steps, and that it preserves privacy, so that agents learn very
little about the private signal of most other agents, despite the efficient
aggregation of information. Our results extend those in an unpublished
manuscript of the first and last authors.Comment: Added coauthor. Added proofs for fast convergence on trees and
distance transitive graphs. Also, now analyzing a notion of privac
Waterproof: educational software for learning how to write mathematical proofs
In order to help students learn how to write mathematical proofs, we
developed the educational software called Waterproof
(https://github.com/impermeable/waterproof). Waterproof is based on the Coq
proof assistant. As students type out their proofs in the program, it checks
the logical soundness of each proof step and provides additional guiding
feedback. Contrary to Coq proofs, proofs written in Waterproof are similar in
style to handwritten ones: proof steps are denoted using controlled natural
language, the structure of proofs is made explicit by enforced signposting, and
chains of inequalities can be used to prove larger estimates. To achieve this,
we developed the Coq library coq-waterproof. The library extends Coq's default
tactics using the Ltac2 tactic language. We include many code snippets in this
article to increase the number of available Ltac2 examples. Waterproof has been
used to supplement teaching the course Analysis 1 at the TU/e for a couple of
years. Students started using Waterproof's controlled formulations of proof
steps in their handwritten proofs as well; the explicit phrasing of these
sentences helps to clarify the logical structure of their arguments.Comment: The Waterproof software can be found at
https://github.com/impermeable/waterproof . This article pertains to
Waterproof version 0.6.1. The Coq library coq-waterproof can be found at
https://github.com/impermeable/coq-waterproof . This article pertains to
coq-waterproof version 1.2.
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