$$ \newcommand \Propose {\mathit{propose}} \newcommand \Soft {\mathit{soft}} \newcommand \Cert {\mathit{cert}} \newcommand \Late {\mathit{late}} \newcommand \Redo {\mathit{redo}} \newcommand \Down {\mathit{down}} \newcommand \Next {\mathit{next}} \newcommand \CommitteeSize {\mathrm{CommitteeSize}} \newcommand \CommitteeThreshold {\mathrm{CommitteeThreshold}} \newcommand \Proposal {\mathrm{Proposal}} \newcommand \Digest {\mathrm{Digest}} \newcommand \Encoding {\mathrm{Encoding}} \newcommand \Hash {\mathrm{Hash}} $$

Elementary Data Types

A period \( p \) is a 64-bit integer.

A step \( s \) is an 8-bit integer.

Steps are named for clarity and are defined as follows:

The following functions are defined on \( s \):

• \( \CommitteeSize(s) \) is a 64-bit integer defined as follows:

$$ \CommitteeSize(s) = \left\{ \begin{array}{rl} 20 & : s = \Propose \\ 2990 & : s = \Soft \\ 1500 & : s = \Cert \\ 500 & : s = \Late \\ 2400 & : s = \Redo \\ 6000 & : s = \Down \\ 5000 & : \text{otherwise} \end{array} \right. $$

• \( \CommitteeThreshold(s) \) is a 64-bit integer defined as follows:

$$ \CommitteeThreshold(s) = \left\{ \begin{array}{rl} 0 & : s = \Propose \\ 2267 & : s = \Soft \\ 1112 & : s = \Cert \\ 320 & : s = \Late \\ 1768 & : s = \Redo \\ 4560 & : s = \Down \\ 3838 & : \text{otherwise} \end{array} \right. $$

A proposal-value is a tuple \( v = (I, p, \Digest(e), \Hash(\Encoding(e))) \) where:

• \( I \) is an address (the “original proposer”),

• \( p \) is a period (the “original period”),

• \( \Hash \) is some cryptographic hash function.

The special proposal-value where all fields are the zero-string is called the bottom proposal \( \bot \).