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This Markovian simulation of Linear Leios computes the probability of EB certifications as RBs are produced.
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This Markovian model of Linear Leios computes the probability of EB certifications as RBs are produced.
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## Contents
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-[Approach](#approach)
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-[Model](#model)
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-[Parameters](#parameters)
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-[Example](#example)
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-[Usage](#usage)
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-[Building](#building)
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## Approach
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## Model
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The protocol state is represented by three quantities.
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- The number of RBs that have been produced.
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- The number of EBs that have been produced.
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- Whether an honest RB was produced.
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- Whether a certificate is ready for inclusion in the next RB.
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Time is tracked in terms of block-forging opportunties instead of in terms of slots.
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Transitions occur in several substeps:
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1.*Forge RB:* create a new RB.
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2.*Certify:* include a certificate in the RB.
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3.*Forge EB:* create a new EB.
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4.*Vote:* cast votes to reach a quorum.
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### Substep 1: Forge RB
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The adversarial model assumes that adversaries obstain from producing RBs, EBs, and votes. Let $f_\text{adv}$ be the fraction of stake held by the adversary. Two transitions are possible:
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-*Forge a new RB:* probability $1 - f_\text{adv}$.
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-*Abstain from forging a new RB:* probability $f_\text{adv}$.
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### Substep 2: Certify
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Provided that a quorum of votes have endorsed the EB, the following conditions are required for certifying it in the RB that was just forged.
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-*RB spacing:* The previous RB must have been forged at least $3 L_\text{hdr} + L_\text{vote} + L_\text{diff}$ slots previously.
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- The distribution of gaps between RBs follows the negative-binomial distribution.
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-*Quorum:* The voting yielded a quorum.
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-*Not an adversary:* There is no RB if an adversary is the designated block producer.
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### Substep 3: Forge EB
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Provided that an honest RB exists, an EB can be forged if the node has received the previous EB and computed the ledger state.
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- Because of their membership in the previous vote, a fraction $n_\text{comm} / n_\text{pools}$ of the pools have already updated their ledger state.
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- Of the pools not having voted on the block we define $p_\text{late}$ as the probability that the EB has arrived too late to compute the ledger state needed to produce the next EB.
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### Substep 4: Vote
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Obtaining a successful quorum of votes is distributed according to bernoulli trials where the expected number of successes is the mean committee size and the success probabilities of individual pools accord with the stake distribution. Those success probabilities are modified in several ways:
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(to be written)
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-*Adversaries do not vote:* probability $f_\text{adv}$.
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-*RB header arrives within $L_\text{hdr}$ slots:* probability $p_\text{rb}$.
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-*EB is fully validated within $3 L_\text{hdr} + L_\text{vote}$ slots*: probability $p_\text{eb}$.
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