Deterministic simulation of a NFA with k symbol lookahead fileDeterministic simulation of a NFA with k‐symbol lookahead SOFSEM 2007 Bala Ravikumar, California State University (joint

Deterministic simulation of a NFA with k‐symbol lookahead

SOFSEM 2007

Bala Ravikumar, California State University

(joint work with Nic Santean, University of Waterloo)

Overview

• Definitions: DFA, NFA and lookahead DFA

• Motivation: automated e‐service composition and delegation

• The NFA delegation model, examples, properties

• Characterization of existence of delegator

• Complexity of finding a delegator for unambiguous and general NFA

• An algorithm for general NFA delegation

• Conclusion, further work

DFA – Deterministic Finite Automaton

( )0

~

, , , ,

:

A Q s F

Q Q

δ

δ

= ∑

× ∑ ⎯⎯→

0s 1s

0 1

0

1

( ) ( ){ } ( )2is odd 0 1 1L A w w ∗∗= ∈Σ = +

0 1 1 0 1

NFA – Nondeterministic Finite Automaton

0s 1s

1 0 1 0 1( )( )

0, , , ,

:

M Q s F

Q P Q

δ

δ

= ∑

× ∑ →

( ) ( ) ( ) ( )0 1 1 0 00 1 0 1L A ∗ ∗ ∗= + +

12s 3s1

0

0

0,1 0,1

0

k‐symbols Lookahead DFA

0 1 0 0 1 0 1 0( )0

~

, , , ,

: k

D Q s F

Q Q

δ

δ ≤

= ∑

× ∑ ⎯⎯→

1 010 2 3

4

100

001

101

5

0010

0

3‐symbol lookahead


0 1 0 0 1 0 1 0

1 010 2 3

4

100

001

101

5

0010

0

( )0

~

, , , ,

: k

D Q s F

Q Q

δ

δ ≤

= ∑

× ∑ ⎯⎯→


0 1 0 0 1 0 1 0

1 010 2 3

4

100

001

101

5

0010

0

( )0

~

, , , ,

: k

D Q s F

Q Q

δ

δ ≤

= ∑

× ∑ ⎯⎯→


0 1 0 0 1 0 1 0

1 010 2 3

4

100

001

101

5

0010

0

( )0

~

, , , ,

: k

D Q s F

Q Q

δ

δ ≤

= ∑

× ∑ ⎯⎯→


0 1 0 0 1 0 1 0

1 010 2 3

4

100

001

101

5

0010

0

( )0

~

, , , ,

: k

D Q s F

Q Q

δ

δ ≤

= ∑

× ∑ ⎯⎯→


0 1 0 0 1 0 1 0

1 010 2 3

4

100

001

101

5

0010

0

( )0

~

, , , ,

: k

D Q s F

Q Q

δ

δ ≤

= ∑

× ∑ ⎯⎯→


0 1 0 0 1 0 1 0

1 010 2 3

4

100

001

101

5

0010

0

( )0

~

, , , ,

: k

D Q s F

Q Q

δ

δ ≤

= ∑

× ∑ ⎯⎯→


0 1 0 0 1 0 1 0

1 010 2 3

4

100

001

101

5

0010

0

( )0

~

, , , ,

: k

D Q s F

Q Q

δ

δ ≤

= ∑

× ∑ ⎯⎯→

E‐Services Composition and Delegation

e-Service 1

e-Service j

e-Service r

...

1 2 3 task t ask tastask k ...i i ii + + +

server...Based on the "status" of every e-service, the server

dispatches the next task to e-service .

Each e-service may be modeled by an activity automaton ,and every task by a symbol from a finite alph

i

i j

i Aabet.

E‐Services Composition and Delegation

e-Service 1

e-Service j

e-Service r

...

1 2 3 task task task task ...i i i i+ + +

server...Based on the "status" of every e-service, the server

dispatches the next task to e-service .

Each e-service may be modeled by an activity automaton ,and every task by a symbol from a finite alph

i

i j

i Aabet.

Composition and Delegation Models

( ) ( ) ( ) ( )1

1 2

: represents valid sequences of atomic tasks (e.g. a web session): activity automaton for a specific e-service

; , ..., is if

com

posabl

#

e

#...#

i

r

r

AA i

A A A

L A L A L A L A⊆

(e.g. # )

This is a necessary condition to ensure that the web applicationis feasible. Question: is it sufficient?

xaayby aab xyy∈

Composition and Delegation Models

( ) ( ) ( ) ( )1

1 2

: represents valid sequences of atomic tasks (e.g. a web session): activity automaton for a specific e-service

; , ..., is if

com

# #...#

posab

le

i

r

r

AA i

A A A

L A L A L A L A⊆

(e.g. # )

This is a necessary condition to ensure that the web applicationis feasible. Question: is it sufficient?

xaayby aab xyy∈

Composition and Delegation

1 2 3 4 5

1

22

1 1

2

1

:

:

: ..

.

: rr r

q

p

p

q q q q

p p

p

p

A

p

A

A

A

′ ′′

′

′

2 1 2 3 1 21

4 1 2 5, 1 2

1 2 ( , , ,..., ) ( , , ,..., ) ( , , ,..., ) ( , ,..( , , ,..

,)

).,

.r r r

r r

q p p p q p p pq p p p q p p

q p pp

p ′ ′′

′′ ′ ′′ ′ ′

( )1 #...# rA A A∩

1 2 3 4 a a a a

Composability and Delegation

31 2 4

2 2

5

11 1 1

2

:

:

: ...

:

r rr

A q q

A p p

p

p

q q q

p

p

p

A

A

′′

′

′

′

1 1 2 2 1 2 3 1 2

4 1 2 5, 1 2

( , , ,..., ) ( , , ,.

(.., )

, , ,..., ) ( , ,. ( , , ,..., )

.., ) r

r r

r r q p p pq p p

q p p p q p p pp q p p p

′′

′′ ′ ′′ ′ ′

′

1a

1a

1a

21 3 4 a a a a

( )1 #...# rA A A∩


2 3

1 1 1

41

2 2

1

5

2

:

:

: .

..

: r rr

A q q

A p p

p

p

q

p

pA

q

pA

q

′ ′′

′

′

2 1 2 3 1 21

4 1 2 5, 1

1

2

2( , , ,..., )

( , , ,..., )

( , , ,..., )( , , ,..., ) ( , ,..., )

r r r

r r

qq p p p q p

q p p p p qp

p pp p pp′′ ′ ′′ ′ ′

′ ′′

2a1a

1a 2a

1a2a

1 32 4 a a aa

( )1 #...# rA A A∩


1 2

1 1 1

3 4

1

22

5

2

:

:

: ...

:

r r r

A qq q

A p p

A

A

q

p

p

p

p

q

p

′′

′

′

′

3 1 2

4 1

1 1 2 2 1 2

5, 1 22

( , , ,..., ) (( , , ,.

( , ,...,.., )

,( , , ,..., )

,

)

,..., )r r

r

r

r

q p p pq

q p p p qq p

ppp p p

p pp

′′

′′ ′′ ′ ′′

′

2a

3a

1a

1a 2a

3a

1a2a 3a

3a

31 2 4 aa a a

( )1 #...# rA A A∩

Nondeterministic Delegator

1 2 4 5

1

2 2

3

1 1 1

2

:

:

: ...

:

r r r

A q q

p

A p p

q q q

A p

p

p

A p

′ ′′

′

′

1 1 2 2 1

4 1 2 5, 1 2

2 3 1 2( , , ,..., ) ( , , ,..., ) ( , , ,..., ) ( , , ,..., ) ( , ,..., )

r r

r

r

rqq p p p q p p p

p p p q pq

pp p p

p′′ ′ ′′ ′ ′

′ ′′

2a

3a

4a

1a

1a 2a

3a 4a

1a2a 3a 4a

3a

1 2 3 4 a a aa

( )1 #...# rA A A∩

k‐Lookahead Delegation Model

( ) ( )

( )

0

~0

1

a

, , , , : the composite NFA #...#

How can we use M determi

delegator for is an equivalent -symbol lookaheadDFA , , , , , with : verify

nistically ?

k

r

k M

Q s F

kQ s

A

F Q

M A

Q

A

δ δ

δ

≤

= ∩

−

′ ′∑ ×Σ ⎯⎯

∑

→

( ) ( ) ( )1 1 1

ing:

, ... : , ... , , ki iq a a Q q a a q a i kδ δ≤ ′∀ ∈ ×Σ ∈ ≤


( ) ( )

( )

10

0~

, , , , : the composite NFA #...#

How can we use M deterministically ?

a for is an equivalent -symbol lookaheadDFA , , , ,

delegato, with : ve i

rr fy

r

k

Q s F

kQ s F

M A A A

M kQ Q

δ

δ δ ≤

= ∩

′ ′ ×Σ ⎯

∑

−

⎯→∑

( ) ( ) ( )1 1 1

ing:

, ... : , ... , , ki iq a a Q q a a q a i kδ δ≤ ′∀ ∈ ×Σ ∈ ≤


q

1q

1a ...

1a

jq

tq

...

1a

( ) ( ) ( )1 1 1, ... : , ... ,ki iq a a Q q a a q aδ δ≤ ′∀ ∈ ×Σ ∈

Interpretation:

q

1q

1a

1a

( )1, ...j iq q a aδ ′=

tq

1 2... ia a a

an NFA a delegator fo r k MM −

( )1,q aδ

⎫⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎭

Example: an unambiguous NFA

1

2

4 5

3

a

b

a

a

a

a

( ) ( )( )a b ab aa∗ ∗+

Example: a 2‐delegator

1

2

4 5

3

a

b

a

a

aa

aba

a

b

a

,b

,a,a

( ) ( )( )a b ab aa∗ ∗+

This unambiguous NFA has a 2‐delegator.

Example: an unambiguous NFA

1

2

4 5

3

a

a

a

a

a

a

( )a+

this unambigous NFA has no -delegator, .0k k∀ >

Example: no delegators

1

2

4 5

3

a

a

a

a

a

a

( )a+

This unambigous NFA has no -delegator, .0k k∀ >

Example: an ambiguous NFA

...0 1kp −2p

1p

1kq −2q ...

kp

kq

0 0

0 0 0

0

1( )10 0 1k− ∗ ∗+

Example: a k‐delegator, no (<k)‐delegator

...10 ,0k k− 1kp −2p

1p

1kq −2q ...

kp

kq

1 20 ,0 ,0k k k− −

10 1k−2 2 20 1,0 1k k− −

10 ,0 ,...,0k k−

11 ,1 ,...,1k k−

( )10 0 1k− ∗ ∗+

...

...

( )

( )

This NFA has a -delegator but no 1 - deleg

Note: -delegation

ators.

1 -delegationk k

k k

⇒ +

−

Example: an ambiguous NFA

0s 1s12s 3s1

0

0

0,1 0,1

( ) ( ) ( )0 1 1 0 00 1 0 1∗ ∗ ∗+ +

Example: no delegators

0s 1s12s 3s1

0

0

0,1 0,1

( ) ( ) ( )0 1 1 0 00 1 0 1∗ ∗ ∗+ +

-1 k-1 1idea: ta

This NFA has n

ke 10 0

o -delegator, for any

1 and

0

10 0 101

.

k k k

k k

+

>

Why Study NFA Delegation

• E‐Service Composition – given some specifications and a set of e‐services, implement an application which follows the specifications.

• Deterministic simulation of NFA: alternative to classical subset construction to simulate an NFA by a DFA. Linear blow‐up vs. exponential blow‐up.

• A theoretical metric for nondeterminism: if an NFA M has a 3‐delegator and M’ does not have a k‐delegator for, say k < 10, then M’ is “more nondeterministic” than M.

Language vs. Machine Property

Is delegation a language property or a machine

1'st Question:

(there exist infinitely many NFA for a given regular l

prope

angu

rty?

age)

Delegation as a Language Property

( )( )( )

( )( )( )

for 0

0 for

- regular language

1. is if has a d

weakly delegable

strongly delegab

elegator

1. is if has a delegato

ler

NFA L k

k NFA L

L

LM k M k

Lk M M k

>

>

∀ ∃ −

∃ ∀ −

Delegation as a Language Property

.

is strongly deleg. is weakly deleg. is finite

It is more interesting to see delegabilit

.

y a

Theorem

L L L

Consequence

⇐ ⇒⇔ ⇔

s a machine property.

Complexity of Finding a k‐Delegator

( )

( )

1

2

Problems:

P is an integer not part of the input. : an NFA : if has a , otherwise P : an NFA and an integer

delega

tor

kinput Moutput YES M NO

input M

k

k

−

( )3

: if has a , otherwise P : an NFA : if has a , otherwis

delegator

delegator e

output YES M NO

input Moutput YES M NO

k −

State Blindness

q

p

1a

...

1a

p′ 2... ka a v

2... ka a v

( ) ( )( ) ( )

1 2

1 1

2 2

1 2 1 2

is ... -blind if

, , , :

, ... , , ...

a state is -blind if there exists ... such that it is ... -blind

k

k k

k k

q a a a

p q a p q a v

p a a v F p a a v F

k a a a a a a

δ δ

δ δ

∗′∀ ∈ ∃ ∈ ∈ ∑

′∉ ∈

[ ]1 2... ka a a

State Blindness

q

p

1a

...

1a

p′ 2... ka a v

2... ka a v

( ) ( )( ) ( )

1 2

1 1

2 2

1 2 1 2

is if

, , , :

, ... , , ...

a state is -blind if there exists ... suc

... -blin

h that it is ... -blin

d

d

k

k

k

k k

q

p q a p q a v

p a a v F p a a v F

k a a a a a a

a a a

δ δ

δ φ δ φ

∗′∀ ∈ ∃ ∈ ∈ ∑

′∩ = ∩ ≠

[ ]1 2... ka a a

State‐blindness is computable

Recall that a string w is q‐blind if it is not possible in state q to find a deterministic choice given w as the look‐ahead.

Define Bq = { w | w is q‐blind }

Theorem: Let M be a DFA with n states, and over an alphabet of size m, and let q be a state of M. The language Bqis regular, and there is a DFA with at most (4n + 1)m that accepts Bq.

A Characterization for Unambiguous NFA

.

An unambiguous NFA has a -delegator iff none ofits states are -blind.

Consequently, an unambiguous NFA has a delegatoriff all its states have finite blindness.

Theorem

kk

Problem 1 for unambiguous NFA’s

Theorem: Let k be a fixed integer. There is a polynomial time algorithm that given an unambiguous NFA M determines if M has a k‐delegator.

Proof (sketch) The problem can be reduced to the problem of containment problem for unambiguous NFA’s.

From a result of Hunt and Stearns, this problem is known to be solvable in polynomial time.

Delegation for Unambiguous NFA

( ) ( ) ( )1 2 3

( ).

unambiguous NFA P co-NP PSPACE

Theorem s

P P P

The proof of co‐NP and PSPACE upper‐bounds for P2 and P3 are similar to that of Problem 1.

Arbitrary NFA

All 3 problems are significantly harder for general NFA:

- a delegator for a (trim) unambiguous NFA must use all its states: local properties (blindness) have a global impact;

- containment and equivalence problems are decidable in polynomial time for unamb. NFA.

Arbitrary NFA: Forbidden Words

( )

1 2

1 2

1

is ... -forbidden if one of the following two conditionsis satisfied recursively (intuitive def.) :

1. is ... -blind; 2. for every state , there exists such

k

k

p

q a a a

q a a ap q a bδ∈ ∈ ∑

2 that is ... -forbidden.k pp a a b

q

p

1a

...

1a

r 2... k ra a b

2... k pa a b[ ]1 2... ka a a

Arbitrary NFA: Forbidden Words

( )

1

1 2

1

2

(intuitive def.) is if one of the following two conditions

is satisfied recursively :

... -forbidden

blind 1. is ... - ;

2. for every state , there exists such

k

p

kq

q a a ap q

a

a b

a a

δ∈ ∈ ∑

2 that is . fo.. - .rbiddenk pp a a b

q

p

1a

...

1a

r 2... k ra a b

2... k pa a b[ ]1 2... ka a a

Arbitrary NFA: Delegation Characterization

( )( )

0

0

0

Notation: is the set of all forbidden words for .

.

An NFA , , , , has a -delegator iff

.

Consequently, an NFA has a delegator iff is finite.

q

q k M

q

F q

Theorem

M Q q F k

F pref L

F

δ

φ

= Σ

∩ =

Delegation for Unambiguous NFA

( ) ( ) ( )1 2 3

( ).

general NFA PSPACE-complete PSPACE-hard ?

Theorem s

P P P

Wrap‐up

• NFA delegation is a finite‐state model used in web service applications, task scheduling, NFA simulation, measure of nondeterminism, etc.

• NFA delegation is a machine property and its computational complexity is machine dependent:

( ) ( ) ( )1 2 3

unambiguous P co-NP PSPACE

general PSPACE-c

P P P

1(previous work: for

omplete PSPACE-h

1, in the ge

ard

neral case : EXPTIME

)

?

k P=

Direction for future workMain open problem:

• Investigate complexity matters for other families of NFA. For example, study NFA that are a shuffle product of DFA.

• Is delegation decidable for arbitrary NFA? Study the nature of .

Other Questions:

• The complexity result for Problem 1 (general case) was proven for a 4‐letter alphabet. Can the PSPACE‐completeness proof be extended to smaller alphabets?

• Is problem 2 complete for co‐NP for unambiguous NFA? Is problem 3 complete for PSPACE for unambiguous NFA?

0qF

References

Service‐oriented computing:

‐ R. Hull and J. Su. Tools for Design of Composite Web Services. In SIGMOD 2004, pp. 958‐961 (2004)

‐ M. Mecella and G. D. Giacomo. Service Composition: Technologies, Methods and Tools for Synthesis and Orchestration of Composite Services and Processes. Tutorial in ICSOC 2004.

‐ D. Berardi, D. Calvanese, G. D. Giacomo, M. Lenzerini, and M. Mecella. Automatic Composition of e‐Services that Export their Behaviour. In ICSOC 2003, LNCS 2910, pp. 43‐58, 2003.

Finite state models for e‐services:

‐ C. E. Gerede, O. H. Ibarra, B. Ravikumar, and J. Su. On‐line and Ad‐hoc Minimum Cost Delegation in e‐Service Composition. In IEEE SCC, pp. 103‐112, 2005.

‐ Z. Dang, O. H. Ibarra, and J. Su. Composability of Infinite‐State Activity Automata. In ISAAC 2004, LNCS 3341, pp. 377‐388.

‐ C. E. Gerede, R. Hull, O. H. Ibarra, and J. Su. Automated Composition of e‐Services: Lookaheads. In ISOC 2004, pp. 252‐262.

Design Methodology ‐ Example

d

d d

p

P

1: downloads and processes files sequentiallyeS 2: downloads 2 files at a time

and processes them in paralleleS

S - application specification:download and process, havingno more than 2 files in the systemat any time

d d

P

p p

2 2

2

1 1

1

3

3

Design Methodology – Step 1

1 2construct :eS eS#

[ ]1,1

[ ]2,1

[ ]1,2

d

d

p

[ ]2,2

[ ]1,3

d

d

[ ]2,3

p

d

dp

P

P

1 2it represents the shuffle behavior of and eS eS


1 2construct :S eS eS∩ #

[ ]1,1,1

[ ]2,2,1

[ ]2,1,2

2d

1d1p

[ ]3,2,2

[ ]3,1,3

2d

1d

1p

2P

it represents the nondeterministic behavior of the application ( )S


find a delegator :

2d d

1d p

1 1,p d p

2d P

1d p

1p d

2 2,P P d

it represents the design of S

d d p d P d p

1eS

2eS

Deterministic simulation of a NFA with k symbol lookahead fileDeterministic simulation of a NFA with k‐symbol lookahead SOFSEM 2007 Bala Ravikumar, California State University (joint

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