# Theory Uv_Defs

section ‹The UniformVoting Algorithm›
theory Uv_Defs
imports Heard_Of.HOModel "../Consensus_Types" "../Quorums"
begin

text ‹The contents of this file have been taken almost verbatim from the
Heard Of Model AFP entry. The only difference is that the types have been
changed.›

subsection ‹Model of the algorithm›

abbreviation "nSteps ≡ 2"

definition phase where "phase (r::nat) ≡ r div nSteps"

definition step where "step (r::nat) ≡ r mod nSteps"

text ‹
The following record models the local state of a process.
›

record 'val pstate =
last_obs :: 'val                ― ‹current value held by process›
agreed_vote :: "'val option"    ― ‹value the process voted for, if any›
decide :: "'val option"  ― ‹value the process has decided on, if any›

text ‹
Possible messages sent during the execution of the algorithm, and characteristic
predicates to distinguish types of messages.
›

datatype 'val msg =
Val 'val
| ValVote 'val "'val option"
| Null  ― ‹dummy message in case nothing needs to be sent›

definition isValVote where "isValVote m ≡ ∃z v. m = ValVote z v"

definition isVal where "isVal m ≡ ∃v. m = Val v"

text ‹
Selector functions to retrieve components of messages. These functions
have a meaningful result only when the message is of appropriate kind.
›

fun getvote where
"getvote (ValVote z v) = v"

fun getval where
"getval (ValVote z v) = z"
| "getval (Val z) = z"

definition UV_initState where
"UV_initState p st ≡ (agreed_vote st = None) ∧ (decide st = None)"

text ‹
We separately define the transition predicates and the send functions
for each step and later combine them to define the overall next-state relation.
›

definition msgRcvd where  ― ‹processes from which some message was received›
"msgRcvd (msgs:: process ⇀ 'val msg) = {q . msgs q ≠ None}"

definition smallestValRcvd where
"smallestValRcvd (msgs::process ⇀ ('val::linorder) msg) ≡
Min {v. ∃q. msgs q = Some (Val v)}"

text ‹
In step 0, each process sends its current ‹last_obs› value.

It updates its ‹last_obs› field to the smallest value it has received.
If the process has received the same value ‹v› from all processes
from which it has heard, it updates its ‹agreed_vote› field to ‹v›.
›

definition send0 where
"send0 r p q st ≡ Val (last_obs st)"

definition next0 where
"next0 r p st (msgs::process ⇀ ('val::linorder) msg) st' ≡
(∃v. (∀q ∈ msgRcvd msgs. msgs q = Some (Val v))
∧ st' = st ⦇ agreed_vote := Some v, last_obs := smallestValRcvd msgs ⦈)
∨ ¬(∃v. ∀q ∈ msgRcvd msgs. msgs q = Some (Val v))
∧ st' = st ⦇ last_obs := smallestValRcvd msgs ⦈"

text ‹
In step 1, each process sends its current ‹last_obs› and ‹agreed_vote› values.
›

definition send1 where
"send1 r p q st ≡ ValVote (last_obs st) (agreed_vote st)"

definition valVoteRcvd where
"valVoteRcvd (msgs :: process ⇀ 'val msg) ≡
{q . ∃z v. msgs q = Some (ValVote z v)}"

definition smallestValNoVoteRcvd where
"smallestValNoVoteRcvd (msgs::process ⇀ ('val::linorder) msg) ≡
Min {v. ∃q. msgs q = Some (ValVote v None)}"

definition someVoteRcvd where
― ‹set of processes from which some vote was received›
"someVoteRcvd (msgs :: process ⇀ 'val msg) ≡
{ q . q ∈ msgRcvd msgs ∧ isValVote (the (msgs q)) ∧ getvote (the (msgs q)) ≠ None }"

definition identicalVoteRcvd where
"identicalVoteRcvd (msgs :: process ⇀ 'val msg) v ≡
∀q ∈ msgRcvd msgs. isValVote (the (msgs q)) ∧ getvote (the (msgs q)) = Some v"

definition x_update where
"x_update st msgs st' ≡
(∃q ∈ someVoteRcvd msgs . last_obs st' = the (getvote (the (msgs q))))
∨ someVoteRcvd msgs = {} ∧ last_obs st' = smallestValNoVoteRcvd msgs"

definition dec_update where
"dec_update st msgs st' ≡
(∃v. identicalVoteRcvd msgs v ∧ decide st' = Some v)
∨ ¬(∃v. identicalVoteRcvd msgs v) ∧ decide st' = decide st"

definition next1 where
"next1 r p st msgs st' ≡
x_update st msgs st'
∧ dec_update st msgs st'
∧ agreed_vote st' = None"

text ‹
The overall send function and next-state relation are simply obtained as
the composition of the individual relations defined above.
›

definition UV_sendMsg where
"UV_sendMsg (r::nat) ≡ if step r = 0 then send0 r else send1 r"

definition UV_nextState where
"UV_nextState r ≡ if step r = 0 then next0 r else next1 r"

definition (in quorum_process) UV_commPerRd where
"UV_commPerRd HOrs ≡ ∀p. HOrs p ∈ Quorum"

definition UV_commGlobal where
"UV_commGlobal HOs ≡ ∃r. ∀p q. HOs r p = HOs r q"

subsection ‹The \emph{UniformVoting} Heard-Of machine›

text ‹
We now define the HO machine for \emph{Uniform Voting} by assembling the
algorithm definition and its communication predicate. Notice that the
coordinator arguments for the initialization and transition functions are
unused since \emph{UniformVoting} is not a coordinated algorithm.
›

definition (in quorum_process) UV_HOMachine where
"UV_HOMachine = ⦇
CinitState =  (λp st crd. UV_initState p st),
sendMsg =  UV_sendMsg,
CnextState = (λr p st msgs crd st'. UV_nextState r p st msgs st'),
HOcommPerRd = UV_commPerRd,
HOcommGlobal = UV_commGlobal
⦈"

abbreviation (in quorum_process)
"UV_M ≡ (UV_HOMachine::(process, 'val::linorder pstate, 'val msg) HOMachine)"

end