$$ \newcommand \Soft {\mathit{soft}} \newcommand \Cert {\mathit{cert}} \newcommand \Bundle {\mathrm{Bundle}} \newcommand \Vote {\mathrm{Vote}} \newcommand \FilterTimeout {\mathrm{FilterTimeout}} \newcommand \DeadlineTimeout {\mathrm{DeadlineTimeout}} \newcommand \Next {\mathit{next}} $$

State Transitions

After receiving message events or a time events, the player may update some components of its state.

New Round

When a player observes that a new round \( (r, 0) \) has begun, the player sets

• \( \bar{s} := s \),

• \( \bar{v} := \bot \),

• \( p := 0 \),

• \( s := 0 \).

Specifically, if a new round has begun, then

$$ N((r-i, p, s, \bar{s}, V, P, \bar{v}), L, \ldots) = ((r, 0, 0, s, V’, P’, \bot), L’, \ldots) $$

for some \( i > 0 \).

Important

IMPLEMENTATION:

New round reference implementation.

New Period

When a player observes that a new period \( (r, p) \) has begun, the player sets

• \( \bar{s} := s \),

• \( s := 0 \).

Also, the player sets \( \bar{v} := v \) if the player has observed \( \Bundle(r, p-1, s, v) \) given some values \( s > \Cert \) (or \( s = \Soft \)), \( v \neq \bot \); if none exist, the player sets \( \bar{v} := \sigma(S, r, p-i) \) if it exists, where \( p-i \) was the player’s period immediately before observing the new period; and if none exist, the player does not update \( \bar{v} \).

In other words, if \( \Bundle(r, p-1, s, v) \in V’ \) for some \( v \neq \bot, s > \Cert \) or \( s = \Soft \), then

$$ N((r, p-i, s, \bar{s}, V, P, \bar{v}), L, \ldots) = ((r, p, 0, s, V’, P, v), L’, \ldots); $$

and otherwise, if \( \Bundle(r, p-1, s, \bot) \in V’ \) for some \( s > \Cert \) with \( \sigma(S, r, p-i) \) defined, then

$$ N((r, p-i, s, \bar{s}, V, P, \bar{v}), L, \ldots) = ((r, p, 0, s, V’, P, \sigma(S, r, p-i)), L’, \ldots); $$

and otherwise

$$ N((r, p-i, s, \bar{s}, V, P, \bar{v}), L, \ldots) = ((r, p, 0, s, V’, P, \bar{v}), L’, \ldots); $$

for some \( i > 0 \) (where \( S = (r, p-i, s, \bar{s}, V, P, \bar{v}) \)).

Important

IMPLEMENTATION:

New period reference implementation.

Garbage Collection

When a player observes that either a new round or a new period \( (r, p) \) has begun, then the player garbage-collects old state.

In other words,

$$ N((r-i, p-i, s, \bar{s}, V, P, \bar{v}), L, \ldots) = ((r, p, \bar{s}, 0, V’ \setminus V^\ast_{r, p}, P’ \setminus P^\ast_{r, p}, \bar{v}), L, \ldots) $$

where

$$ \begin{aligned} V^\ast_{r, p} &= \{\Vote(I, r’, p’, \bar{s}, v) | \Vote \in V, r’ < r\} \\\ &\cup \{\Vote(I, r’, p’, \bar{s}, v) | \Vote \in V, r’ = r, p’ + 1 < p\} \end{aligned} $$

and \( P^\ast_{r, p} \) is defined similarly.

New Step

A player may also update its step after receiving a timeout event.

On observing a timeout event of \( \FilterTimeout(p) \) for a period \( p \), the player sets \( s := \Cert \).

On observing a timeout event of \( \DeadlineTimeout(p) \) for a period \( p \), the player sets \( s := \Next_0 \).

On observing a timeout event of \( \DeadlineTimeout(p) + 2^{s_t}\lambda + u \) where \( u \in [0, 2^{s_t}\lambda) \) sampled uniformly at random, the player sets \( s := s_t \).

Important

IMPLEMENTATION:

New step reference implementation.

In other words,

$$ \begin{aligned} &N((r, p, s, \bar{s}, V, P, \bar{v}), L, t(\FilterTimeout(p), p)) \\ &\qquad = ((r, p, \Cert, \bar{s}, V, P, \bar{v}), L’, \ldots) \\[0.35em] &N((r, p, s, \bar{s}, V, P, \bar{v}), L, t(\DeadlineTimeout(p), p)) \\ &\qquad = ((r, p, \Next_0, \bar{s}, V, P, \bar{v}), L’, \ldots) \\[0.35em] &N((r, p, s, \bar{s}, V, P, \bar{v}), L, t(\DeadlineTimeout(p) + 2^{s_t}\lambda + u, p)) \\ &\qquad = ((r, p, \Next_{s_t}, \bar{s}, V, P, \bar{v}), L’, \ldots). \end{aligned} $$