Termination Detection

Part 1

Goal

• Study the development of a protocol for termination detection with the help of invariants.

Termination Detection

• Rules:– A process is either active or passive– An active process can become passive at any time– A passive process can become active only if it

receives an computation message– Only active processes can send computation

messages. All processes can receive them– Any process can send control messages, I.e.,

messages sent for detecting termination

• A system is said to be terminated if– All processes are passive– No computation messages are in transit

• Reminder: We distinguish between computation messages and messages sent for detecting termination. Any process can send and receive them. These messages do not change the status of a process

Application

• A solution for termination detection allows one to ensure that all tasks in a system are indeed complete, even though the tasks may create additional tasks that are run at other processors in the system

Observation

• Termination detection is a stable property– Once true, it remains true forever

• Detecting such properties is important for many problems including– Garbage collection– Deadlock detection

We will consider two algorithms

• Based on the idea of diffusion

• Based on the idea of global snapshot

– We will study these aspects later.

Approach 1: Dijkstra Scholten

• Assumptions– Initially one process is active– No failures, lost messages etc.

• Each process j maintains a variable P.j that is its parent in the tree– At root, P.root = root– Initially for all other processes,

• P.j = NULL

Predicate in Invariant (1)

• The set of active processes form a tree– True in the initial state– Ensure that this remains true during

computation

When a Process becomes active

• Consider the case when j changes from Passive to Active– It must be the case that j received a

computation message from some process, say k

• P.j = k• Become active

Action (1)

P.j = NULL j receives a message from k

P.j = k, j becomes active

When a Process Becomes Passive

• Consider the case when j changes from Active to Passive – It must be the case that j has no children

Action (2)

P.j = NULL j receives a message from k

P.j = k, j becomes active

j is active j wants to become passive j has no children

j becomes passive, P.j = NULL

Alternatives

• J is active• J is passive and for all children of j

– Set their parent to P.j– Or set their parent to root

• There are some paeprs with this idea. But we will skip them here

Problem?

• Does not deal with messages.

Predicate in Invariant (2)

• The set of active processes form a tree

• If j is passive then all messages it sent have been received– True initially– Preserve this predicate during computation

Action (3)

• Maintain a variable oc.j that denotes the number of messages that j has sent and are not yet acknowledged

Action (2) corrected

P.j = NULL j receives a message from k

P.j = k, j becomes active

j is active j wants to become passive j has no children oc.j = 0

j becomes passive, P.j = NULL

• The actions on previous slide can be used to implement termination detection.

• Consider second actionj is active j wants to become passive j has no children

oc.j = 0

j becomes passive, P.j = NULL

• Is it possible to drop ` j has no children’ from the guard?

Answer

• We could if we guarantee that – oc.j = 0 j has no children– Same as

• j has children oc.j > 0• Could be achieved if the child does not respond

to at least one of parent’s message (first one?)

• Thus, checking oc.j is 0 sufficient

Action (3)

P.j = NULL j receives a message from k

P.j = k, j becomes active (Don’t send ack to this message)

j is active j wants to become passive oc.j = 0

j becomes passive, P.j = NULL; send ack to parent

j is active j receives a message from k

Send ack to k

Other simple actions for maintaining oc.j

Summarizing Approach 1

• Goal– Active processes form a rooted tree

• If process k activates j then j sets its parent to k

– If a process is passive, all messages it sent have been received

• Acknowledge each message (at some time)

– A process becomes passive only when all its children are passive; in other words, force a process to wait for its children to become passive.

• This is achieved if the children do not send an acknowledgment for the first message received from the parent until they become passive.

Actions

• Passive Active– If j is passive and receives a computation

message from k then• P.j = k• Become active

Actions

• Active Active– If j is active and receives a computation

message from l• Send an acknowledgment

Actions

• Message send– If j wants to send message (it must be

active)• oc.j ++ • (Number of outstanding acknowledgments is

increased)

• Acknowledgement receive– oc.j = oc.j – 1

Actions

• Active passive– If j wants to be passive and oc.j = 0

• Send an acknowledgment to P.j (Observe that the first message from parent was not immediately acknowledged)

• Become passive• If j is the root then declare termination

Diffusing Computation

• Crucial for various applications• General outline

– root(?) sends the diffusion message– Every node that receives the diffusion message for the first

time forwards it to its neighbors• First node from which diffusion is received is called parent

– All subsequent diffusion messages are acknowledged– Upon receiving acknowledgements form all neighbors, a

node completes the diffusion and sends acknowledgment to parent

– When root completes the diffusion, the diffusion computation is complete

Termination Detection II

Approach

• Arrange processes in a (hypothetical) ring– The ring is used only for the sake of

termination detection– Any process can communicate with any

other process

Approach

• Each process maintains a variable c.jc.j = number of messages sent by j

–

number of messages received by j

• Initially, c values are all 0

Action (1)

When j wants to sent a message

c.j := c.j + 1

 

When j receives a message

c.j := c.j – 1

 

Observation

• Number of messages in transit =

• Detect the value of

• Ensure that when c.j is read j is passive

n

j

jc0

.

n

j

jc0

.

Action (2)

• Send a token along the ring to compute

When 0 sends a token (token is sent only when the previous token is received and process 0 is passive)

token.sum = c.0 Forwarding the token by process j, j <> 0 (token

is sent only when j is passive)token.sum := token.sum + c.j

n

j

jc0

.

Remark

• Observe that the token is taking a snapshot of the system – The global snapshot consists of local

snapshot of every process

Invariant (1)

• P1 =

(token is between k and k+1

((token.sum = c.0 + c.1 + c.2 + … + c.k) /\

(processes 0..k are passive))

• If P1 were true and the toekn is between n and 0 then token.sum would capture the sum of c values and all processes would be passive

Problem with P1

• After a token is sent by k, some process in the range 0..k may receive a message, thereby violating P1

• To deal with this, we have two options– Strengthen P1 so that such a message

cannot be received– Weaken P1 so that the invariant contains

states that are reached due to such messages

• We need to follow this approach

Question

• What can be said of states reached due to violation of P1

Invariant (2)

• P1 \/ P2, where

P2 =

(token is between k and k+1

(token.sum + c.(k+1) + c.(k+2) … + c.n > 0)

If we start from a state where P1 is true and P1 becomes false then in that state P2 is true

Problem with P1 \/ P2

• Consider the case where the token is between k and k+1

• Some process, j, j < k, is active

• j sends a message to a process l, l > k– In this scenario, we want to make sure that

l invalidates the token circulation so that process 0 ignores token.sum

Introduce a color for process

When j receives a messagec.j := c.j – 1;

color.j := purple // Addition to previous action

Invariant (3)

• P1 \/ P2 \/ P3, where

P3 = (token is between k and k+1

l : l > k : color.l = purple)

Problem

• Consider the case where l is purple and intends to ensure that the token circulation is invalidated– What happens if l forwards the token– All predicates P1, P2, P3 can be false.

• Also, the color needs to be changed back to yellow so that the token circulation will be eventually valid

Solution

• Introduce a color.token– Initially yellow

Forwarding the token by process j, j <> 0 (remember: token is sent only when j is passive)

token.sum := token.sum + c.j

if (color.j = purple)

color.token = purple

color.j = yellow

else

// Preserve the token color that you received.

// Basically, do nothing.

Invariant (4)

• P1 \/ P2 \/ P3 \/ P4, where

P4 = color.token = purple

When Should color.token be reset?

• At process 0?– Is P1 \/ P2 \/ P3 \/ P4 violated in such

circumstances?

• If no, which of these predicates is guaranteed to be true

When is Termination Detected?

• Token returns to 0

• color.token = yellow, and

• token.sum = 0

Can we Deduce Termination from Invariant?

• How?