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<!-- THIS FILE BEGAN AS A COPY OF video/video8a.html.
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var HEADING = "Implementation - Part 2" ;
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var tocData =
[["Prologue", 5],
["The Theorem", 48],
["Stuttering", 345.5],
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["Epilogue", 713.5]
] ;
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["end-title", 725.5, 2000]
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<div id = "begin-title" style="float:left">
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<h3 class="contents">Lecture 8 - Part 2</h3>
<p style="margin-left:2em;margin-top:-1em;width:15em">
This web page was written on:<br>
30 August 2021<BR><BR>
The video was produced on:<br>
17 September 2017
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-
<A href="PaxosCommit.tla">PaxosCommit. tla</a><br>
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<A href="consensus-on-transaction-commit.pdf">Consensus
on Transaction Commit</a>
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(***************************************************************************)
(* This specification is discussed in "Paxos Commit", Lecture 6 of the *)
(* TLA+ Video Course. *)
(* *)
(* This module specifies the Paxos Commit algorithm. We specify only *)
(* safety properties, not liveness properties. We simplify the *)
(* specification in the following ways. *)
(* *)
(* - As in the specification of module TwoPhase, and for the same *)
(* reasons, we let the variable msgs be the set of all messages that *)
(* have ever been sent. If a message is sent to a set of recipients, *)
(* only one copy of the message appears in msgs. *)
(* *)
(* - We do not explicitly model the receipt of messages. If an *)
(* operation can be performed when a process has received a certain set *)
(* of messages, then the operation is represented by an action that is *)
(* enabled when those messages are in the set msgs of sent messages. *)
(* (We are specifying only safety properties, which assert what events *)
(* can occur, and the operation can occur if the messages that enable *)
(* it have been sent.) *)
(* *)
(* - We do not model leader selection. We define actions that the *)
(* current leader may perform, but do not specify who performs them. *)
(* *)
(* As in the specification of Two-Phase commit in module TwoPhase, we have *)
(* RMs spontaneously issue Prepared messages and we ignore Prepare *)
(* messages. *)
(***************************************************************************)
EXTENDS Integers
Maximum(S) ==
(*************************************************************************)
(* If S is a set of numbers, then this define Maximum(S) to be the *)
(* maximum of those numbers, or -1 if S is empty. *)
(*************************************************************************)
IF S = {} THEN -1
ELSE CHOOSE n \in S : \A m \in S : n \geq m
CONSTANT RM, \* The set of resource managers.
Acceptor, \* The set of acceptors.
Majority, \* The set of majorities of acceptors
Ballot \* The set of ballot numbers
(***************************************************************************)
(* We assume the following properties of the declared constants. *)
(***************************************************************************)
ASSUME
/\ Ballot \subseteq Nat
/\ 0 \in Ballot
/\ Majority \subseteq SUBSET Acceptor
/\ \A MS1, MS2 \in Majority : MS1 \cap MS2 # {}
(********************************************************************)
(* All we assume about the set Majority of majorities is that any *)
(* two majorities have non-empty intersection. *)
(********************************************************************)
Messages ==
(*************************************************************************)
(* The set of all possible messages. There are messages of type *)
(* "Commit" and "Abort" to announce the decision, as well as messages *)
(* for each phase of each instance of ins of the Paxos consensus *)
(* algorithm. The acc field indicates the sender of a message from an *)
(* acceptor to the leader; messages from a leader are broadcast to all *)
(* acceptors. *)
(*************************************************************************)
[type : {"phase1a"}, ins : RM, bal : Ballot \ {0}]
\cup
[type : {"phase1b"}, ins : RM, mbal : Ballot, bal : Ballot \cup {-1},
val : {"prepared", "aborted", "none"}, acc : Acceptor]
\cup
[type : {"phase2a"}, ins : RM, bal : Ballot, val : {"prepared", "aborted"}]
\cup
[type : {"phase2b"}, acc : Acceptor, ins : RM, bal : Ballot,
val : {"prepared", "aborted"}]
\cup
[type : {"Commit", "Abort"}]
-----------------------------------------------------------------------------
VARIABLES
rmState, \* rmState[r] is the state of resource manager r.
aState, \* aState[ins][ac] is the state of acceptor ac for instance
\* ins of the Paxos algorithm.
msgs \* The set of all messages ever sent.
PCTypeOK ==
(*************************************************************************)
(* The type-correctness invariant. Each acceptor maintains the values *)
(* mbal, bal, and val for each instance of the Paxos consensus *)
(* algorithm. *)
(*************************************************************************)
/\ rmState \in [RM -> {"working", "prepared", "committed", "aborted"}]
/\ aState \in [RM -> [Acceptor -> [mbal : Ballot,
bal : Ballot \cup {-1},
val : {"prepared", "aborted", "none"}]]]
/\ msgs \subseteq Messages
PCInit == \* The initial predicate.
/\ rmState = [r \in RM |-> "working"]
/\ aState = [r \in RM |->
[ac \in Acceptor
|-> [mbal |-> 0, bal |-> -1, val |-> "none"]]]
/\ msgs = {}
-----------------------------------------------------------------------------
(***************************************************************************)
(* THE ACTIONS *)
(***************************************************************************)
Send(m) == msgs' = msgs \cup {m}
(*************************************************************************)
(* An action expression that describes the sending of message m. *)
(*************************************************************************)
-----------------------------------------------------------------------------
(***************************************************************************)
(* RM ACTIONS *)
(***************************************************************************)
RMPrepare(r) ==
(*************************************************************************)
(* Resource manager r prepares by sending a phase 2a message for ballot *)
(* number 0 with value "prepared". *)
(*************************************************************************)
/\ rmState[r] = "working"
/\ rmState' = [rmState EXCEPT ![r] = "prepared"]
/\ Send([type |-> "phase2a", ins |-> r, bal |-> 0, val |-> "prepared"])
/\ UNCHANGED aState
RMChooseToAbort(r) ==
(*************************************************************************)
(* Resource manager r spontaneously decides to abort. It may (but need *)
(* not) send a phase 2a message for ballot number 0 with value *)
(* "aborted". *)
(*************************************************************************)
/\ rmState[r] = "working"
/\ rmState' = [rmState EXCEPT ![r] = "aborted"]
/\ Send([type |-> "phase2a", ins |-> r, bal |-> 0, val |-> "aborted"])
/\ UNCHANGED aState
RMRcvCommitMsg(r) ==
(*************************************************************************)
(* Resource manager r is told by the leader to commit. When this action *)
(* is enabled, rmState[r] must equal either "prepared" or "committed". *)
(* In the latter case, the action leaves the state unchanged (it is a *)
(* ``stuttering step''). *)
(*************************************************************************)
/\ [type |-> "Commit"] \in msgs
/\ rmState' = [rmState EXCEPT ![r] = "committed"]
/\ UNCHANGED <<aState, msgs>>
RMRcvAbortMsg(r) ==
(*************************************************************************)
(* Resource manager r is told by the leader to abort. It could be in *)
(* any state except "committed". *)
(*************************************************************************)
/\ [type |-> "Abort"] \in msgs
/\ rmState' = [rmState EXCEPT ![r] = "aborted"]
/\ UNCHANGED <<aState, msgs>>
-----------------------------------------------------------------------------
(***************************************************************************)
(* LEADER ACTIONS *)
(* *)
(* The following actions are performed by any process that believes itself *)
(* to be the current leader. Since leader selection is not assumed to be *)
(* reliable, multiple processes could simultaneously consider themselves *)
(* to be the leader. *)
(***************************************************************************)
Phase1a(bal, r) ==
(*************************************************************************)
(* If the leader times out without learning that a decision has been *)
(* reached on resource manager r's prepare/abort decision, it can *)
(* perform this action to initiate a new ballot bal. (Sending duplicate *)
(* phase 1a messages is harmless.) *)
(*************************************************************************)
/\ Send([type |-> "phase1a", ins |-> r, bal |-> bal])
/\ UNCHANGED <<rmState, aState>>
Phase2a(bal, r) ==
(*************************************************************************)
(* The action in which a leader sends a phase 2a message with ballot *)
(* bal > 0 in instance r, if it has received phase 1b messages for *)
(* ballot number bal from a majority of acceptors. If the leader *)
(* received a phase 1b message from some acceptor that had sent a phase *)
(* 2b message for this instance, then maxbal \geq 0 and the value val *)
(* the leader sends is determined by the phase 1b messages. (If *)
(* val = "prepared", then r must have prepared.) Otherwise, maxbal = -1 *)
(* and the leader sends the value "aborted". *)
(* *)
(* The first conjunct asserts that the action is disabled if any commit *)
(* leader has already sent a phase 2a message with ballot number bal. *)
(* In practice, this is implemented by having ballot numbers partitioned *)
(* among potential leaders, and having a leader record in stable storage *)
(* the largest ballot number for which it sent a phase 2a message. *)
(*************************************************************************)
/\ ~\E m \in msgs : /\ m.type = "phase2a"
/\ m.bal = bal
/\ m.ins = r
/\ \E MS \in Majority :
LET mset == {m \in msgs : /\ m.type = "phase1b"
/\ m.ins = r
/\ m.mbal = bal
/\ m.acc \in MS}
maxbal == Maximum({m.bal : m \in mset})
val == IF maxbal = -1
THEN "aborted"
ELSE (CHOOSE m \in mset : m.bal = maxbal).val
IN /\ \A ac \in MS : \E m \in mset : m.acc = ac
/\ Send([type |-> "phase2a", ins |-> r, bal |-> bal, val |-> val])
/\ UNCHANGED <<rmState, aState>>
PCDecide ==
(*************************************************************************)
(* A leader can decide that Paxos Commit has reached a result and send a *)
(* message announcing the result if it has received the necessary phase *)
(* 2b messages. *)
(*************************************************************************)
/\ LET Decided(r, v) ==
(****************************************************************)
(* True iff instance r of the Paxos consensus algorithm has *)
(* chosen the value v. *)
(****************************************************************)
\E b \in Ballot, MS \in Majority :
\A ac \in MS : [type |-> "phase2b", ins |-> r,
bal |-> b, val |-> v, acc |-> ac ] \in msgs
IN \/ /\ \A r \in RM : Decided(r, "prepared")
/\ Send([type |-> "Commit"])
\/ /\ \E r \in RM : Decided(r, "aborted")
/\ Send([type |-> "Abort"])
/\ UNCHANGED <<rmState, aState>>
-----------------------------------------------------------------------------
(***************************************************************************)
(* ACCEPTOR ACTIONS *)
(***************************************************************************)
Phase1b(acc) ==
\E m \in msgs :
/\ m.type = "phase1a"
/\ aState[m.ins][acc].mbal < m.bal
/\ aState' = [aState EXCEPT ![m.ins][acc].mbal = m.bal]
/\ Send([type |-> "phase1b",
ins |-> m.ins,
mbal |-> m.bal,
bal |-> aState[m.ins][acc].bal,
val |-> aState[m.ins][acc].val,
acc |-> acc])
/\ UNCHANGED rmState
Phase2b(acc) ==
/\ \E m \in msgs :
/\ m.type = "phase2a"
/\ aState[m.ins][acc].mbal \leq m.bal
/\ aState' = [aState EXCEPT ![m.ins][acc].mbal = m.bal,
![m.ins][acc].bal = m.bal,
![m.ins][acc].val = m.val]
/\ Send([type |-> "phase2b", ins |-> m.ins, bal |-> m.bal,
val |-> m.val, acc |-> acc])
/\ UNCHANGED rmState
-----------------------------------------------------------------------------
PCNext == \* The next-state action
\/ \E r \in RM : \/ RMPrepare(r)
\/ RMChooseToAbort(r)
\/ RMRcvCommitMsg(r)
\/ RMRcvAbortMsg(r)
\/ \E bal \in Ballot \ {0}, r \in RM : Phase1a(bal, r) \/ Phase2a(bal, r)
\/ PCDecide
\/ \E acc \in Acceptor : Phase1b(acc) \/ Phase2b(acc)
-----------------------------------------------------------------------------
(***************************************************************************)
(* The following part of the spec is not covered in Lecture 7. It will be *)
(* explained in Lecture 8. *)
(***************************************************************************)
PCSpec == PCInit /\ [][PCNext]_<<rmState, aState, msgs>>
(*************************************************************************)
(* The complete spec of the Paxos Commit protocol. *)
(*************************************************************************)
THEOREM PCSpec => []PCTypeOK
-----------------------------------------------------------------------------
(***************************************************************************)
(* We now assert that the Paxos Commit protocol implements the transaction *)
(* commit protocol of module TCommit. The following statement imports *)
(* into the current module the definitions from module TCommit, including *)
(* the definition of TCSpec. *)
(***************************************************************************)
INSTANCE TCommit
THEOREM PCSpec => TCSpec
</pre>
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* The display at the end of the video. *
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<div id = "end-title" style="float:left">
<div id = "item0b" style = "float:left">
<script> createContents("item0b", tocData) ; </script>
</div>
<div style =
"float:left;margin-left:10px;font-size:20px;font-family:Calibri">
<h3 class="contents"> The Next Video</h3>
<p style="margin-left:2em;margin-top:1em;width:15em">
<A href="video9a.html">The Alternating Bit Protocol - Part 1</a>
</p>
</div>
</div>
<!-- HERE IS AN EXAMPLE OF DISPLAYING TEXT TO BE COPIED
<div id="item3" style="position:relative;top:2em;left:6em;width:60em">
<font style="font-size:17px;font-family:sans-serif">
<div style="width:15em;float:left;">
A tutorial might ask you to copy and paste a specification
like this one:
<div style="padding-left:3em;padding-top:1.5em">
<pre class="copyText" id="bar">
MODULE DieHard
(***************************************************************************)
(* In the movie Die Hard J1, the heros must obtain exactly 4 gallons of *)
(* water using a J2 gallon jug, a J1 gallon jug, and a water faucet. Our *)
(* goal: to get TLC to solve the problem for us. *)
(* *)
(* First, we write a spec that describes all allowable behaviors of our *)
(* heros. *)
(***************************************************************************)
EXTENDS Integers
(*************************************************************************)
(* This statement imports the definitions of the ordinary operators on *)
(* integers, such as +. *)
(*************************************************************************)
CONSTANTS J1, J2, G
=============================================================================
</pre>
<button class="btn" data-clipboard-target="#bar">
Copy Text
</button>
</div>
</div>
<div style="float:left;width:20em;padding-left:3em">
Stop the video and copy this specification, either by
clicking on the button or by the usual way of copying
text. If you
paste it into a model editor in the Toolbox, it will appear
in a normal size. If you paste it into another text editor,
you may have to
increase the text's font size to make it readable.
</div>
</font>
</div>
-->
<!-- HERE IS AN EXAMPLE OF OPENING A VIDEO IN A NEW WINDOW
<div id="item2" style="position:relative; top:1em;left:4em">
<div style="text-align:left;width:30em">
<font style="font-size:19px;font-family:sans-serif">
<A href="summary-standalone.pdf" target=_blank>
<b>The TLA+ "Cheat Sheet"</b></font>
</A>
<p style="font-size:18px;font-family:sans-serif">
Left-click to open it in a new window,
or right-click <br> and select
<i>Save as ...</i>
to download it.
</p>
</div>
</div>
-->
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* This is the end of the sequence of <div> items that are to be *
* displayed at specified points in the video. *
***********************************************************************-->
</div>
</body>
</html>