Understanding Network Connections

Butler Lampson

There are lots of protocols for establishing connections (or equivalently, doing at-most-once message delivery) across a network that can delay, reorder, duplicate and lose packets. Most of the popular ones are based on three-way handshake, but some use clocks or extra stable storage operations to reduce the number of messages required. It’s hard to understand why the protocols work, and there are almost no correctness proofs; even careful specifications are rare.

I will give a specification for at-most-once message delivery, an informal account of the main problems an implementation must solve and the common features that most implementations share, and outlines of proofs for three implementations. The specifications and proofs based on Lamport’s methods for using abstraction functions to understand concurrent systems, and I will say something about how his methods can be applied to many other problems of practical interest.

Understanding Network Connections

Butler Lampson

October 30, 1995

This is joint work with Nancy Lynch.

The errors in this talk are mine, however.

Overview

Specify at-most-once message delivery.

Describe other features we want from an implementation

Give a framework for thinking about implementations.

Show how to prove correctness of an implementation.

The Problem

Network Connections
or
Reliable At-Most-Once Messages

Messages are delivered in fifo order.

A message is not delivered more than once.

A message is acked only if delivered.

A message or ack is lost only if it is being sent between crash and recovery.

Pragmatics

“Everything should be made as simple as possible, but no simpler.”
A. Einstein

Make progress: regardless of crashes,
if both ends stay up a waiting message is sent, and
otherwise both parties become idle.

Idle at no cost: an idle agent
has no state that changes for each message, and
doesn’t send any packets.

Minimize stable storage operations — <<1 per message.

Use channels that are easy to implement:
They may lose, duplicate, or reorder messages.

Pragmatics

Some pragmatic issues we won’t discuss:
Retransmission policy.
Detecting failure of an attempt to send or ack, by timing it out.

Describing a System

A system is defined by a safety and a liveness property:

Safety: nothing bad ever happens. Defined by a state machine:

A set of states. A state is a pair (external state, internal state)
A set of initial states.
A set of transitions from one state to another.

Liveness: something good eventually happens.

An action is a named set of transitions; actions partition the transitions.

For instance: put(m); get(m); crash_s

A history is a possible sequence of actions, starting from an initial state.

The behavior of the system is the set of possible histories.

An external action is one in which the external state changes. Correspondingly there are external histories and behaviors.

Defining Actions

An action is:

A name, possibly with parameters: put(“red”).

A guard, a predicate on the state which must be true for this action to be a possible transition: q ≠ < > and i > 3.

An effect, changes in some of the state variables: i := i + 1.

The entire action is atomic.

Example:

get(m): m first on q take first from q, if q now empty
and status = ? then status := true

name guard effect

Specifying At-Most-Once Messages

State: q : sequence[M] := < >
status : {true, false, ?} := true
rec_s/r : Boolean := false

“Sender Actions”			“Receiver Actions”
Name	Guard	Effect	Name	Guard	Effect
put(m)**		if ~recs then append m to q, status := ?	get(m)*	~recr, m first on q	take first from q, if q now empty and status = ? then status := true
getAck(b)*	~recs , status = b	none
crashs**		recs := true	crashr**		recr := true
recovers*	recs	recs := false	recoverr*	recr	recr := false
lose	recs or recr	delete any element of q; if it’s the last then status := false or status := false

Histories for AMO Messages

Action	q	status	Action	q	status	Action	q	status
Initially	< >	true
put(“red”)	<“red”>	?
get(“red”)	< >	true
getAck(true)	< >	true
put(“green”)	<“green”>	?
crashr, lose	<“green”>	?	crashr, lose	<“green”>	false	crashr, lose	< >	false
		?	getAck(false)	<“green”>	false	getAck(false)	< >	false
put(“blue”)	<“green”, “blue”>	?	put(“blue”)	<“green”, “blue”>	?	put(“blue”)	<“blue”>	?
get(“green”)	<“blue”>	?	get(“green”)	<“blue”>	?
get(“blue”)	< >	true
getAck(true)	< >	true

Channels

State sr : multiset[P] := {} P = I ´ M rs : multiset[P] := {}
or I ´ Bool

Name	Guard	Effect	Name	Guard	Effect
sndsr(p)		sr := sr È {p}	sndrs(p)		rs := rs È {p}
rcvsr(p)	p Î sr	sr := sr – {p}	rcvrs(p)	p Î rs	rs := rs – {p}
losesr	$ p | p Î sr	sr := sr – {p}	losers	$ p | p Î rs	rs := rs – {p}

Stable Implementation

State: mode_s : {acked, send, rec} := acked mode_r : {idle, rcvd, ack} := idle
good_s : set[ I ] := I good_r : set[ I ] := I
last_s : I last_r : I
cur : M buf : sequence[M] := <>

Name	Guard	Effect	Name	Guard		Effect
put(m)** sndsr(i, m)*	mode = acked and $ i | i Î good mode = send, i = last, m = cur	cur := m, last := i, mode := send, take i from good none	rcvsr(i, m) **		if i Î good then mode := rcvd, take i from good, last := i, append m to buf else if i = last and mode = idle then mode := ack
			get(m)*	mode = rcvd, m first on buf		take first from buf; if it’s now empty, mode := ack
rcvrs(i, –) **		if mode = send and i = last then mode := acked	sndrs(i, true)*	mode = ack, i = last		mode := idle
getAck (true)*	mode = acked	none

What Does “Implements” Mean?

Divide actions into external (marked * or **) and internal (unmarked).
External actions change external state, internal ones don’t.

An external history is a history (sequence of actions) with all the internal actions removed.

T implements S if

every external history of T is an external history of S, and

T’s liveness property implies S’s liveness property.

Abstraction Functions

Suppose we have a function f from T’s state to S’s state such that:

f takes initial states to initial states;

f maps every transition of T to a sequence of transitions of S, perhaps empty (i.e., the identity on S);

f(t) S or skip _® f(t')

t T _® t'

f maps every external action of T to a sequence containing the same external action of S and no other external actions.

f maps every internal action of T to a sequence of internal actions.

Then T implements S.
Why bother? Transitions are simpler than histories.

Proof of Stable Implementation

Invariants

(1) good_s Ç ( {last_s} È ids(sr) È ids(rs) ) = {}

(2) good_r Ç ids(rs) = {}

(3) good_r Ê good_s

(4) ((i, m) Î sr and i Î good_r) implies m = cur

Abstraction function

q = <cur> if mode_s = send and last_s Î good_r
< > otherwise
+ buf

status = ? if mode_s = send
true otherwise

Methodology for Proofs

Simplify the spec and the implementations.
Save clever encodings for later.

Make a “working spec” that’s easier to handle:
It implements the actual spec.
It has as much non-determinism as possible.
All the prophecy is between it and the actual spec.

actual_¬ implements working_¬ implements implemen-

spec spec tation

Find the abstraction function. The rest is automatic.

Give names to important functions of your state variables.

To design an implementation, first invent the guards you need,
then figure out how to implement them.

History Variables

If you add a variable h to the state space such that

If s is an old initial state then there’s an h such that (s, h) is initial;

If (s, h) ® (s', h') then s ® s';

If s ® s' then for any h there’s an h' such that (s, h) ® (s', h')

then the new state machine has the same histories as the old one.

Predicting Non-Determinism

Suppose we add mode := acked to crash_s.

Consider the sequence put(“red”), snd, crash_s, put(“blue”), snd.

Now we have sr = {(1, “red”), (2, “blue”)}. We need an ordering on identifiers to order these packets and maintain fifo delivery. On rcv_sr(i, m) the receiver must remove all identifiers ≤ i from good_r.

But now “red” is lost if (2, “blue”) is received first. If we use the obvious abstraction function

q = the m’s from {(i, m) Î sr È (last_s, cur)| i Î good_r} sorted by i,

this loss doesn’t happen between crash_s and recover_s, as allowed by the spec, but later at the rcv.

We give a working spec that makes this delay explicit.

Delayed-Decision Specification

State: q : sequence[(M, Flag)] := < > Flag = {OK, ?}
status : ({true, false, ?}, Flag) := true
rec_s/r : Boolean := false

Name	Guard	Effect	Name	Guard	Effect
put(m)**		if ~recs then add (m, OK) to q, status := (?, OK)	get(m)*	~recr, (m, –) first on q	take first from q, if q now empty and status = (?, f) status := (true, f)
getAck(b)*	~recs , status = (b, –)	none
mark	recs or recr	in some element of q or in status, set flag := ?	unmark		in some element of q or in status, set flag := OK
drop	true	delete some element of q with flag := ?, and if it’s the last, status := (false, OK) or if status = (–, ?), status := (false, OK)

Prophecy Variables

If you add a variable p to the state space such that

If s is an old initial state then there’s a p such that (s, p) is initial;

If (s, p) ® (s', p') then s ® s';

If s ® s' then for any h' there exists an h such that (s, h) ® (s', h')

If s is an old state, there’s a p such that (s, p) is a new state.

then the new state machine has the same histories as the old one.

If T implements S, you can always add history and prophecy variables to T and then find an abstraction function to S.

Prophecy for Delayed-Decision

Extend Flag to include a lost component drawn from the set {OK, lost}.

The lost component prophesies whether drop will attack or not.

The abstraction function is

q_S = the first components of elements of q_DP that are not lost

status_S = the first component of status_DP if it is not lost, else false.

Delayed-Decision with Prophecy DP

State: q : sequence[(M, Flag)] := < > Flag = ({OK, ?},
status : ({true, false, ?}, Flag) := (true, OK²⁾ {OK, lost})
rec_s/r : Boolean := false

Name	Guard	Effect	Name	Guard	Effect
put(m)**		if ~recs then add (m,OK^2) to q, status := (?, OK^2)	get(m)*	~recr, (m, –) first on q and not lost	take first from q, if q now empty and status = (?, f) status := (true, f)
getAck(b)*	~recs , status = (b, –), not lost	none
mark	recs or recr, $ x Î {OK, lost}	in some element of q or in status, set flag := (?, x). If last of q is lost, set status flag (?, lost)	unmark		in some element of q or in status that isn’t lost, set flag := OK
drop	true	delete some lost element of q or if status is lost, status := (false, OK^2)