Use trace algebra to formalize the YAPI model

Use trace algebra to formalize the YAPI modelUse trace algebra to formalize the YAPI model

EE290N Spring2002

Alessandro Pinto

Mentors: Roberto Passerone

Jerry Burch

OutlineOutline

References

Introduction to YAPI

Implication of select

Introduction to Trace Algebra

Traces to model YAPI

Buffer model

Conservative Approximation

Example

Conclusion

ReferencesReferences

K.A. Vissers et. al. “YAPI: Application Modeling for Signal Processing System”, DAC00

J. Burch, R. Passerone, ASV “Overcoming Heterophobia: Modeling Concurrency in Heterogeneous Systems”

J. Burch, R. Passerone, ASV “Modeling Techniques in Design-by-Refinement Methodologies”

E.A. Lee, T.M. Parks “Dataflow Process Networks”

G. Kahn, “The Semantics of a Simple Language for Parallel Programming”

G. Kahn, D.B. MacQueen, “Corutines and Networks of Parallel Processes”

The YAPI Model 1The YAPI Model 1

KPN (blocking read, non-blocking write)

Non-determinism

i1

i2

oread(i1,i1_int,1) ;read(i2,i2_int,1);o_int=in1_int+in2_int ;write(o,o_int,1) ;

),...,,,( 2211 kk npnpnpselects

The YAPI Model 2The YAPI Model 2

Selection can be done on input and output

Read,Write on unbounded FIFOs

Read,Write on bounded FIFOs

YAPI

TTL

Implication of selectImplication of select

continuitymonotonicity determinacy

i1=[1], i2=[]

F(i1,i2) = [1]

i1=[1,1], i2=[2]

F(i1,i2) = [2,1]o

if (select(i1,1,i2,1) == 1) read(i1,o_int,1);else read(i2,o_int,1);write(o,o_int,1);

i1

i2

read(i,i_int,1);if (select(o1,1,o2,1) == 1) write(o1,i_int,1);else write(o2,i1_int,1);

i

o1

o2

i=[1]

F(i) = ([1],[])

i=[1,3]

F(i) = ([1],[3]), ([3],[1]), ([1,3],[]) …

Trace Algebra 1Trace Algebra 1

Concurrency Algebra• Set of agents• Parallel composition of agents• Projection of agents• Renaming of agents

Trace structure algebra

A

Model of individual behaviors

Model of agents

Trace algebra

C

Trace Algebra• Set of traces• Projection of traces• Renaming of traces

Source: R. Passerone

I1=[1,3,4,-1]

I2=[0,3,5,2]

O=[1,6,9,1]

i1 oi2 +


Alphabet

Set of traces over alphabet A

Renaming y = rename(r) (x) where x is a trace and r is a renaming function

Projection y = proj(B)(x) where x is a trace and BA

Trace structure T=(,P) where is the signature and P B(A)

A set of axioms must be verified !!!!!!!

WA)(AB



A’


A

Trace algebra

C

Trace algebra

C’

Homomorphismh

u linv

“Abstract” Domain

“Detailed” Domain

Let Tspec and Timpl be trace structures in A. Then

if u( Timpl ) l( Tspec ) then Timpl Tspec

Derive

Source: R. Passerone

Implication of select: Trace Algebra ViewImplication of select: Trace Algebra View Monotonicity is captured by projection

proj(I)(x) proj(I)(y) ⊑ proj(O)(x) proj(O)(y)⊑

For the previous example two traces belongs to the same process:

]1[

][2

]1[1

o

i

i

]1,2[

]2[2

]1,1[1

o

i

i

First step: Set of TracesFirst step: Set of Traces

Which set of traces should we choose?

Streams

Synchronous

DE

VAAB )(

)()( VANAB

)()( VAAB

Streams 1Streams 1

No timing information

KPN can be expressed on the same set of traces Continuity is a constraint on the trace structures

In this sense YAPI processes can form more trace structures then KPN

23,10

0,1,6,4,32

3,1,5,4,11

o

i

i

KPN

YAPI

Streams 2Streams 2

Operation on traces Renaming

Projection

Operation on trace structures

rfxrrename

VAfx

))((

:

BfxBproj

VAfx

|))((

:

)''))(''('))('(|{''||' PxAprojPxAprojxTTT

23,10

0,1,6,4,32

3,1,5,4,11

o

i

i

23,10

0,1,6,4,34

3,1,5,4,13

aux

i

i

23,10

0,1,6,4,32

3,1,5,4,11

o

i

i

0,1,6,4,32

3,1,5,4,11

i

i

A more detailed domain 1A more detailed domain 1

We want to capture two important things

Relative arrival time

Bounded FIFOs

Synchronous Domain

)()( VAAB V

A more detailed domain 2A more detailed domain 2

Renaming

Projection

rfxrrename

VAfx

))((

:

BfxBproj

VAfx

|))((

:

o

if (select(i1,1,i2,1,p,1)==3)read(p,op,1);

read(i1,i1_int,1);read(i2,i2_int,1);o_int=op(i1_int,i2_int);write(o,o_int,1);

i1

i2

p

3,2,11,8,4

,,,,

0,1,6,4,32

3,1,5,4,11

o

subsump

i

i

3,2,1,8,,4

,,,,,

0,,1,6,4,32

3,1,5,4,,11

o

subsump

i

i

Two possible traces

BuffersBuffers

Buffers are modeled as processes

The strict version of a sequence a is

We indicate with ak the sequence up to instant k

Definition of a buffer Trace structure TB=((I,O),P)

}][|][{ iaiaas

sssk

sk OIthbufferlengOlenIlenkxP )()(0,0|

Buffer ConstraintBuffer Constraint

3,2,11,8,4

,,,,

0,1,6,4,32

3,1,5,4,11

o

subsump

i

i

3,2,1,8,,,,4

,,,,,,,

0,,1,6,4,,,32

3,1,5,4,,,,11

o

subsump

i

i

0,1,6,4,32

3,1,5,4,11

o

o

o

i1

i2

p

o1

o2

Gen Buf1

Buf2

AddSub

Two possible traces of AddSub

T=Gen||Buf1||Buf2||AddSub

Bufferlength =2

Bufferlength =1

3,2,11,8,4

,,,,

o

subsump

3,2,1,8,,,,4

,,,,,,,

o

subsump

P

P

3,2,11,8,4

,,,,

o

subsump

3,2,1,8,,,,4

,,,,,,,

o

subsump

P

P

Conservative Approximation 1Conservative Approximation 1

Synch

Nondet KPN

Approximation Homomorphism

h(x)

VAAB )(

)()( VAAB


h(P)

P

h is in general many to one so this is an upper bound

))(,( Phu

Detailed Domain

Abstract Domain


h(P)

P

B(A) - P

h(B(A) – P)

h(P) - h(B(A) – P)

)))(()(,( PABhPhl

Detailed Domain

Abstract Domain

From Synch to ND-KPN 1From Synch to ND-KPN 1

Definition of the homomorphism

sVAxh )(

3,2,1,8,,4

,,,,,

0,,1,6,4,32

3,1,5,4,,11

o

subsump

i

i

3,2,1,8,4

,

0,1,6,4,32

3,1,5,4,11

o

subsump

i

ih(x)

From Synch to ND-KPN 2From Synch to ND-KPN 2

Let’s consider a process: If p is true add the inputs, if p is false subtract them

lu

]3,4[

],[

]5,3[2

]2,1[1

o

falsetruep

i

i

]3,4,[

],,[

]5,3,[2

]2,,1[1

o

falsetruep

i

i

]3,4,,[

],,,,[

]5,3,,[2

]2,1,,[1

o

falsetruep

i

i

h(x) h(x)

P ))(( PAB

Applications of the ApproximationApplications of the Approximation

Verification Problem

Design Problem

TTTAprojTTTAproj luu )||)(()())(||)()(( 2121

)())(||)()(()||)(( 2121 TTTAprojTTTAproj invinvinv

Example 1Example 1

Norm

n

iixxNorm

1

)(

2

n

iix

1

=(I,O) =(I,O) =(I,O)

Bufferlength=n

Example 1Example 1

2

n

iix

1

=(I,O) =(I,O) =(I,O)

Bufferlength=n

otherwiseIifkIkOVOIfP ,][][|)),(:( 2

Example 1Example 1

2

n

iix

1

=(I,O) =(I,O) =(I,O)

Bufferlength=n


otherwiseIifkIkOVOIfP ,][][|)),(:(

Example 1Example 1

2

n

iix

1

=(I,O) =(I,O) =(I,O)

Bufferlength=n


otherwiseIifkIkOVOIfP ,][][|)),(:(

otherwisekOkInkOnkkIVOIfPn

j

][,][][][|)),(:(1

Example 2Example 2

One possible execution

2

n

iix

1

]64,,16,,9,9,1[

]8,,4,,3,3,1[

o

i

]99,,,,,,,,,,,[

]64,16,9,9,1,,,,,,,[

o

i

]99,,,,,,,,,,,[

]99,,,,,,,,,,,[

o

i

Bufferlength=n

Example 3Example 3

2

n

iix

1

h(x)

2][][|),(: kIkOVOIfP

][][|),(: kIkOVOIfP

1

0

]*[][|),(:n

j

jnkIkOVOIfP

id

Example 4Example 4

One possible execution

2

n

iix

1

]64,16,9,9,1[

]8,4,3,3,1[

o

i

]99[

]64,16,9,9,1[

o

i

]99[

]99[

o

i

Example 4Example 4

Design problem

2

n

iix

1

2

n

iix

1

ConclusionConclusion

The YAPI model was a good motivation for this project

Trace algebra has been applied to describe the denotational semantics of the model at different level of abstraction

A Conservative approximation between Synchronous model and ND-KPN allows abstraction from one model to another for verification and design

Use trace algebra to formalize the YAPI model

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