Weakly endochronous systems

Dumitru Potop-Butucaru

IRISA, France

Joint work with A. Benveniste and B. Caillaud

Outline

• Introduction (the distribution problem)

• Weakly endochronous systems

• Applications– Weak isochrony– Main result

• Conclusion

module M: input A,B,R;relation A#R,B#R;abort loop await A || await B endwhen Rend module

0 2

3

1

RR

R

A,BAB

BA

A

B

BA

R

inputSynchrony

– Global clock, reactions– Can test the absence– Composition:

• synchronized product

(unification on reactions)

0 2

3

1

RR

R

A,BAB

BA

A

B

BA

R

inputAsynchrony

– No global clock– No reaction, no absence– Composition:

• unification of FIFO histories

0 2

3

1

RR

R

A,BAB

BA

A

B

BA

R

input

0 2

3

1

RR

R

A,BAB

BA

executive

A

B

BA

R

Good executions

BA

R

BA

Asynchrony– No global clock– No reaction, no absence– Composition:

• unification of FIFO histories

AB

BA

R

input

0 2

3

1

RR

R

A,BAB

BA

executive

R

Bad executions

Asynchrony– No global clock– No reaction, no absence– Composition:

• unification of FIFO histories

A

R

Model

• Components = synchronous automatons

no causality• Synchronous composition (automaton product)

Σ1Σ2

• Desynchronization operator

• Asynchronous composition (on traces)

43

2

1

34

21

δ

1321

13321a b c d a b c

Model

• Correct resynchronization (Benveniste):

(L(Σ1Σ2)) = (L(Σ1))||(L(Σ2))

• Intuition:

– the asynchronous observations do not

change when changing FIFO size and delay

• Condition on infinite traces…

Previous work

• Latency-insensitivity (Carloni et al., 1999)– Hardware-directed (handle long wires)– Use all channels in each reaction

(transmit values)• Monoclock – not efficient• Difficult to re-configure the communication scheme

– Extension: Singh and Theobald, 2003

• Endo/Isochrony (Benveniste et al., 2000)

Endo/Isochrony

• An endochronous component decides itself how to read its inputs– No event-driven, no real concurrency, determinism!– Comparable to Kahn processes (but no causality)– Executive = state machine and blocking reads

• A pair of components is isochronous if no incorrect synchronization takes place

• Compositionality problem– Endochrony is not compositional– Difficult to generalize isochrony to >2 components

AB

BA

R

input

0 2

3

1

RR

R

A,BAB

BA

executive

R

Bad executions

A

R

AB

BA

R

input

Event-executive

0

0

1

0

0

1

0 2

3

1

R,E=1

A,D=0

B,E=0

B,E=0

A,D=0

A,B,D=E=0

R,D=E=1

R,D=1

What I want– Non-determinism

(Event-driven execution)

Weak endochrony

• Independent reactions =non-overlapping, fully commuting

• Weak endochrony = allow online re-synchronization in any environment without restricting concurrency between independent reactions:– Choice between non-independent reactions is

visible on a channel– Causal ordering of reactions is visible on a

channel

• Non-independent reactions share a common support variable with different value

• Non-independent causally ordered reactions share a common variable

0 2

3

1

R,E=1

A,D=0

B,E=0

B,E=0

A,D=0

A,B,D=E=0

R,D=E=1

R,D=1

Weak endochrony

1. Determinism: ss1, ss2r1=r2

2. Commutation: if r1,r2 do not share present values

• ss1, ss2s’:ss’

• ss1s2s’:ss2

3. Decomposition: if ss1, ss2 and r1, r2 do not share present variables with different values

Weak endochrony axiomsr1 r2

r1 r2 r1r2

r1 r2 r2

s’:ss’

r1 r2

r1r2

s1

s2

r1\r2

r2\r1

• All reactions are composed of atoms

• Disjoint atoms in a state fully commute and can be freely combined

0 2

3

1

R,E=1

A,D=0

B,E=0

B,E=0

A,D=0

A,B,D=E=0

R,D=E=1

R,D=1

Properties

State 0 atoms:(A,D=0),(B,E=0),(R,D=E=1)

State 1 atoms:(B,E=0),(R,E=1)

• Mazurkiewicz traces over atoms– The letters are the atoms, disjoint atoms are

independent– Normal form

• trace equivalence upto commutation of atoms• take at each reaction the maximal reaction (unique)

– Confluence for a given history

• Tetris-like heaps (gravity => normal form)• Generalization of endochrony, latency-

insensitivity

Related models

Consequences

• Accept concurrency between independent behaviors– non-determinism– support event-driven implementations

• Compositionality:Σ1,Σ2 weakly endochronous

Σ1Σ2 weakly endochronous

Simpler correctness criterion

(s1,s2)RSS(Σ1Σ2),iΣi(si),i=1,2: 1||2 exists

1,2 are both void traces

or

Σ1Σ2 can perform a common transition

• Still not an online correctness criterion!

Weak isochrony

(s1,s2)RSS(Σ1Σ2), ai atom in Σi(si), i=1,2, Ai reaction in Σi(si) such that

• A1||A2 maximal• ai Ai,i=1,2

i reaction in Σi(si), i=1,2 such that• ai i Ai,i=1,21||2 exists

Correct desynchronization

Theorem

If Σ1,Σ2 weakly endochronous, (Σ1,Σ2) weakly isochronous then:

(L(Σ1Σ2)) = (L(Σ1))||(L(Σ2))

Methodology

1. Make components weakly endochronous by deciding which signals are missing

2. Incrementally build communication protocols that insure the weak isochrony

3. Remove the useless variables created at point (1)

Conclusion

• Weak endochrony/isochrony – Compositional– Supports the development of communication

protocols that minimize communication– Not yet effective (not efficiently)– No causality

• Future:– Define appropriate algorithms– Extend with causality

Weak isochrony

(s1,s2)RSS(Σ1Σ2), ai atom in Σi(si), i=1,2, Ai reaction in Σi(si) such that

• A1||A2 maximal• ai Ai,i=1,2

i reaction in Σi(si), i=1,2 such that• ai i Ai,i=1,21||2 exists

Synchrony and asynchrony

• Composition:– Synchronous composition

– Asynchronous composition

Σ1 Σ2

1

2 2

1

Σ1||Σ2:

Σ1Σ2:

Σ1 Σ2

1

2 2

1

rejected

accepted

Desynchronization

• Stuttering-invariant synchronous specification, distributed architecture (FIFOs)

• Two aspects:– Correct re-synchronization

– Causality, full asynchrony

Σ1 Σ2

1

2 2

1FIFO

FIFOWRONG

Correct resynchronization

• Model – labeled synchronous transition systems – synchronous, n-place bidirectional buffers – no causality

• Correctness criterion: – the traces accepted with empty buffers do not

change from n=0 to n=

Σ1 Σ2

1

2 2

1Bn

Bn

Correct resynchronization

• Equivalent formulation (Benveniste):

(L(Σ1Σ2)) = (L(Σ1))||(L(Σ2))

= desynchronization operator

|| = composition of asynchronous traces (equality on interface union)

01112

11021

δ

Previous work

• Latency-insensitivity (Carloni et al., 1999)– Hardware-directed (handle long wires)– A component reads/emits all its inputs, or none – Monoclock– Endochrony-like extensions (Singh and

Theobald, 2003)

• Endo/Isochrony (Benveniste et al., 2000)

Endo/Isochrony

• An endochronous component decides itself how to read its inputs Kahn!!!– Determinism, no real concurrency

• A pair of components is isochronous if non-contradictory reactions can be fully unified

• Compositionality problem– Endochrony is not compositional– Difficult to generalize isochrony to >2

components

Weakly endochronous systems

• Allow online re-synchronization in any environment without restricting concurrency:– Each computation choice is visible on an

interface variable

– Causally ordered reactions share a common variable 1

2

10

20

2

1

21

10

or

after

Weakly endochronous systems

• Example

• Not weakly endochronous.

0 2

3

1

R

module M: input A,B,R;relation A#R,B#R;abort loop await A || await B endwhen Rend module

RR

A,BAB

BA

Weakly endochronous systems

• Example

0 2

3

1

R,E=1

A,D=0

B,E=0

B,E=0

A,D=0

A,B,D=E=0

R,D=E=1

R,D=1

module M: input A,B,R;relation A#R,B#R;abort loop await A || await B endwhen Rend module

Weakly endochronous systems

• All reactions are composed of atoms

• Disjoint atoms in a state fully commute and can be freely combined

• A reaction can be decomposed into its atoms

0 2

3

1

R,E=1

A,D=0

B,E=0

B,E=0

A,D=0

A,B,D=E=0

R,D=E=1

R,D=1State 0 atoms:

(A,D=0),(B,E=0),(R,D=E=1)

State 1 atoms:(B,E=0),(R,E=1)

Related models

• Mazurkiewicz traces over atoms– The letters are the atoms, disjoint atoms are

independent– Normal form

• trace equivalence upto commutation of atoms• take at each reaction the maximal reaction (unique)

– Confluence for a given history

• Tetris-like heaps (gravity => normal form)

• Generalization of endochrony

Consequences

• Accept concurrency between independent behaviors– non-determinism– support event-driven implementations

• Normal form

• Compositionality:Σ1,Σ2 weakly endochronous

Σ1Σ2 weakly endochronous

Simpler correctness criterion

(s1,s2)RSS(Σ1Σ2),iΣi(si),i=1,2: 1||2 exists

1,2 are both void traces

or

Σ1Σ2 can perform a common transition

• Weak endochrony online correctness criterion!

Weak isochrony

(s1,s2)RSS(Σ1Σ2), ai atom in Σi(si), i=1,2, Ai reaction in Σi(si) such that

• A1||A2 maximal• ai Ai,i=1,2

i reaction in Σi(si), i=1,2 such that• ai i Ai,i=1,21||2 exists

Weak isochrony

(s1,s2)RSS(Σ1Σ2), ai atom in Σi(si), i=1,2, Ai reaction in Σi(si) such that

• A1||A2 maximal• ai Ai,i=1,2

i reaction in Σi(si), i=1,2 such that• ai i Ai,i=1,21||2 exists

Correct desynchronization

Theorem

If Σ1,Σ2 weakly endochronous, (Σ1,Σ2) weakly isochronous then:

(L(Σ1Σ2)) = (L(Σ1))||(L(Σ2))

Conclusion

• Weak endochrony/isochrony – Compositional– Supports the development of communication

protocols that minimize communication– Not yet effective (not efficiently)– No causality

• Future:– Define appropriate algorithms– Extend with causality

Methodology

1. Make components weakly endochronous by deciding which signals are missing

2. Incrementally build communication protocols that insure the weak isochrony

3. Remove the useless variables created at point (1)

Weakly endochronous systems

• Composition– Weak endochrony is compositional– Weak endochrony insures the equivalence

between C1 and C2– Re-synchrhronize a atome pres– Confluence, unique etat pour une histoire

donnee

Correct resynchronization

• Correctness criterion, revisited:

L0(Σ1Σ2B0B0) = L0(Σ1Σ2BB)

• Equivalent formulation (Benveniste):

(L(Σ1Σ2)) = (L(Σ1))||(L(Σ2)) = desynchronization operator

|| = composition of asynchronous traces (equality on interface union)

01112

11021

δ

Weakly endochronous systems

• Example

module M: input A,B,R;output E,F:integer;relation A#R,B#R;abort await immediate A;emit E(0)|| await immediate B;emit F(0)when R;emit E(1);emit F(1)end module

emit A;emit C;await immediate

await immediate C;emit R;

FIFO

FIFO

FIFO

A

R

C

AFIFO

Weakly endochronous systems

• Examplemodule M: input A,B,R;relation A#R,B#R;abort await immediate A|| await immediate Bwhen Rend module

emit A;emit R

FIFO

FIFOR

AB

Weakly endochronous systems

• Example

emit A;assert ?D=1;emit R;assert ?D=0;

module M:input A,B,R;output D,E:int;relation A#R,B#R;abort await immediate A;emit D(1)|| await immediate B;emit E(1)when R do emit D(0);emit E(0) endend module

A

BR

D