$$ \newcommand \Bundle {\mathrm{Bundle}} \newcommand \Soft {\mathit{soft}} \newcommand \Cert {\mathit{cert}} \newcommand \Proposal {\mathrm{Proposal}} $$

Resynchronization Attempt

Where specified, a player attempts to resynchronize.

A resynchronization attempt involves the following stages.

First, the player broadcasts its freshest bundle, if one exists.

A player’s freshest bundle is a complete bundle defined as follows:

• \( \Bundle(r, p, \Soft, v) \subset V \) for some \( v \), if it exists, or else

• \( \Bundle(r, p-1, s, \bot) \subset V \) for some \( s > \Cert \), if it exists, or else

• \( \Bundle(r, p-1, s, v) \subset V \) for some \( s > \Cert, v \neq \bot \), if it exists.

⚙️ IMPLEMENTATION

Freshness relation reference implementation.

Second, if the player broadcasted a bundle \( \Bundle(r, p, s, v) \), and \( v \neq \bot \), then the player broadcasts \( \Proposal(v) \) if the player has it.

Specifically, a resynchronization attempt:

• Corresponds to no additional outputs if no freshest bundle exists

$$ N(S, L, \ldots) = (S’, L’, \ldots), $$

• Corresponds to a broadcast of the freshest bundle after a relay output and before any subsequent broadcast outputs, if said bundle exists, no matching proposal exists

$$ N(S, L, \ldots) = (S’, L’, (\ldots, \Bundle^\ast(r, p, s, v), \ldots)), $$

• Otherwise corresponds to a broadcast of both a bundle and its associated proposal after a relay output and before any subsequent broadcast outputs

$$ N(S, L, \ldots) = (S’, L’, (\ldots, \Bundle^\ast(r, p, s, v), \Proposal(v), \ldots)). $$