Automated generation of BSP Automata Thibaut Tachon Ga´etan Hains Supervisor in Huawei
&
Fr´ed´eric Loulergue Academic supervisor
Frebrary 15th, 2017
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Content Introduction Definitions BSP Language BSP Regular Expressions BSP Automaton Applications Query Language Parallel recognition of RE From BSPRE to BSPA Desynchronization Br¨ uggemann-Klein Synchronization Conclusion 2/28
Regular language Regular Expressions r
::= | | | | |
Finite Automata
r r0 r + r0 r∗ a ∅
A = (Q, Σ, δ, I , F ) Q Σ δ I F
State set Alphabet Transitions Initial states Final states
Br¨uggemann-Klein r = (a + b)ba∗
b
=⇒
0
2
a
a b b
3
a
4
1
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Bulk-Synchronous Parallel model • Superstep sequence • Local determinism → global determinism • Neither deadlocks nor data races • Simple cost model
Async computations
Communications Sync
local comp. 0 local comp. 1
Next Superstep comp. 0
...
comp. 1
...
comp. 2
...
comp. 3
...
Barr. local comp. 2 local comp. 3
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BSP Language Regular language Symbol : Word : Language :
Type Σ Σ∗ P Σ∗
Example a abccd {a, aab, ab}
BSP words and language Word vector : BSP word : BSP Language :
p processors
Type
Example
(Σ∗ )p
hab, a, bbbi hab, aihbb, abai {hab, ai, hab, baiha, bbbi}
((Σ∗ )p )∗ P ((Σ∗ )p )∗
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BSP Regular Expressions Grammar R ::= R R 0 | R + R 0 | R ∗ | ∅ | | hr 0 , . . . , r p−1 i
Language r
L(r )
r0
L(r 0 )
r L(r ) · 0 r + r L(r ) ∪ L(r 0 ) ∞
r∗
∪Li (r ) i=0
a ∅
{a} {} {}
R R0
L(R)
R R + R0
L(R) · L(R 0 ) L(R) ∪ L(R 0 )
R∗
∪Li (R) i=0
∞
hr0 , . . . , rp−1 i L(r0 ) × · · · × L(rp−1 ) {} ∅ {} 6/28
BSP Automata : Definition
A = ({Q i }i∈[p] , Σ, {δ i }i∈[p] , {I i }i∈[p] , {F i }i∈[p] , ∆) • p sized automata vector • Common alphabet • Synchronization function
Delta
Notation
• ∆ : Qp → Qp
• p processors
• Q p = (Q 0 × ... × Q p−1 )
• [p] = {0 . . . p − 1}
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BSP Automata: Example (p = 2) c 0
a
1
b
b
b
2 Δ
a 0'
3
2'
3'
b
5 4 a
d d
7
6 Δ 4'
5'
a
1'
8/28
BSP Automata: Example (p = 2) c 0
a
1
b
b
b
2 Δ
a 0'
3
2'
3'
b
5 4 a
d d
7
6 Δ 4'
5'
a
1'
Legend
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BSP Automata: Example (p = 2) c 0
a
1
b
b
b
2
b
Δ
a 0'
3
2'
5 4 a
3'
d d
7
6 Δ 4'
5'
a
1'
Legend
Delta ∆ = { h2, 20 i → h3, 30 i , h6, 40 i → h7, 50 i}
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BSP Automata: Example (p = 2) c 0
a
1
b
b
b
2
b
Δ
a 0'
3
2'
5 4 a
3'
d d
7
6 Δ 4'
5'
a
1'
Legend
Delta ∆ = { h2, 20 i → h3, 30 i , h6, 40 i → h7, 50 i}
Accept : hab, aihbd, ai ?
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BSP Automata : Recognition hq00 , . . . , q0p−1 i
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BSP Automata : Recognition hq00 , . . . , q0p−1 i
input 6=
Apply ∆
yes
yes
(Apply δ ∗ )p
∆ arg
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BSP Automata : Recognition hq00 , . . . , q0p−1 i
input 6=
Apply ∆
yes
yes
(Apply δ ∗ )p
∆ arg
no Reject
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BSP Automata : Recognition hq00 , . . . , q0p−1 i
input 6=
yes
(Apply δ ∗ )p
no Apply ∆
All final yes
Accept
yes
∆ arg
no no Reject
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BSP Automata: Example (p = 2) c 0
a
1
b
b
b
2
b
Δ
a 0'
3
2'
5 4 a
3'
d d
7
6 Δ 4'
5'
a
1'
Legend
Delta ∆ = { h2, 20 i → h3, 30 i , h6, 40 i → h7, 50 i}
Accept : hab, aihbd, ai ?
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Model Regular language Symbol : Word : Language :
Type Σ Σ∗ P Σ∗
model an instruction a program trace all possible traces of a program
BSP words and language Word vector : BSP word : BSP Language :
p processors
Type
model
(Σ∗ )p
one superstep a BSP program trace all possible program traces
((Σ∗ )p )∗ P ((Σ∗ )p )∗
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Query language Which target • BSP program trace • BSP code (well structured)
DSL for queries • based on BSPRE • computed by a BSPA
Limitations • regular expressions expressiveness • output is a boolean
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Parallel recognition of RE
RE
Σ∗
DFA Bool
13/28
Parallel recognition of RE RE
DFA
Σ∗
Bool
BSPRE
(Σ∗ )p
∗
BSPA Bool 13/28
Parallel recognition of RE RE
Σ∗
DFA Bool
RE
Σ∗
BSPRE
(Σ∗ )p
∗
BSPA Bool 13/28
From RE to BSPRE String research r = σ ∗ string σ ∗
where r ∈ RE σ = ∀σi ∈ Σ, (σ0 + σ1 + . . . )
Distribution of input Block
P1
P2
P3
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Block distribution
P1
P2
P3
R ∈ BSPRE
h σ∗ , σ∗ , σ ∗ string σ ∗ i R = +h σ∗ , σ ∗ string σ ∗ , σ∗ i + h σ ∗ string σ ∗ , σ∗ , σ∗ i ?
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Block distribution s t r i ng
P1
P2
P3
R ∈ BSPRE
h σ∗ , σ∗ , σ ∗ string σ ∗ i R = +h σ∗ , σ ∗ string σ ∗ , σ∗ i + h σ ∗ string σ ∗ , ∗ ∗ σ , σ i ?
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Block distribution
s t r i ng
P1
P2
P3
R ∈ BSPRE
h σ∗ , σ∗ , σ ∗ string σ ∗ i R = +h σ∗ , σ ∗ string σ ∗ , σ∗ i + h σ ∗ string σ ∗ , σ∗ , σ∗ i ?
15/28
Block distribution
s t r i ng
P1
P2
P3
R ∈ BSPRE
h σ∗ , σ∗ , σ ∗ string σ ∗ i R = +h σ∗ , σ ∗ string σ ∗ , σ∗ i + h σ ∗ string σ ∗ , σ∗ , σ∗ i ?
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Block distribution
P1
R=
h +h +h
P2
σ∗ σ∗ σ∗
, , ,
P3
σ∗ σ∗ σ∗ .. .
, σ ∗ string σ ∗ i s , tring σ∗ i st , ring σ∗ i
+h σ∗ , σ ∗ string σ ∗ , .. . ∗ ∗ + h σ string σ , σ∗ ,
σ∗
i
σ∗
i 15/28
Block distribution s t r i ng
P1
R=
h +h + h
P2
σ∗ σ∗ σ∗
, , ,
P3
σ∗ σ∗ σ∗ .. .
, σ ∗ string σ ∗ i s , tring σ∗ i st , ring σ∗ i
+h σ∗ , σ ∗ string σ ∗ , .. . ∗ ∗ + h σ string σ , σ∗ ,
σ∗
i
σ∗
i 15/28
Block distribution s t r i ng
P1
R=
h +h + h
P2
σ∗ σ∗ σ∗
, , ,
P3
σ∗ σ∗ σ∗ .. .
, σ ∗ string σ ∗ i s , tring σ∗ i st , ring σ∗ i
+h σ∗ , σ ∗ string σ ∗ , .. . ∗ ∗ + h σ string σ , σ∗ ,
σ∗
i
σ∗
i 15/28
Block distribution let m = |string |
R=
h +h +h
σ∗ σ∗ σ∗
, , ,
σ∗ σ∗ σ∗ .. .
, σ ∗ string σ ∗ i s , tring σ∗ i st , ring σ∗ i
+h σ∗ , σ ∗ string σ ∗ , .. . ∗ ∗ + h σ string σ , σ∗ ,
σ∗
i
σ∗
i
Possibilities number (p − 1) × m + 1 15/28
Parallel recognition of RE
RE
Σ∗
BSPRE
(Σ∗ )p
∗
BSPA Bool
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From BSPRE to BSPA
Algorithm global scheme
(Σ, p) bspre
(Σ, p) bspa
Desynchronization
(Σ ∪ {;t }) re
p
Synchronization
(BK : re → nfa)
p
(Σ ∪ {;t }) nfa
p
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Desynchronization
(Σ, p) bspre → (Σ ∪ {;t }) re
p
Pre-processing • Vectors annotated with their position • Depth First Search of syntactic tree • Vector number : S
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Desynchronization
(Σ, p) bspre → (Σ ∪ {;t }) re
p
Pre-processing • Vectors annotated with their position • Depth First Search of syntactic tree • Vector number : S
Annotation example a, bb, (a+b)∗ , h a+b i+h a∗ i h aa i
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Desynchronization
(Σ, p) bspre → (Σ ∪ {;t }) re
p
Pre-processing • Vectors annotated with their position • Depth First Search of syntactic tree • Vector number : S
Annotation example
=⇒
a, bb, (a+b)∗ , h a+b i+h a∗ i h aa i a, bb, (a+b)∗ , h a+b i0 +h a∗ i1 h aa i2
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Desynchronization
(Σ, p) bspre → (Σ ∪ {;t }) re
p
Auxiliary recursive function dsni (F ·BSP G ) = dsni (F ) · dsni (G ) dsni (F +BSP G ) = dsni (F ) + dsni (G ) dsni (F ∗BSP ) = dsni (F )∗ dsni (hr0 , . . . , rp−1 it ) = ri · ;t dsni (BSP ) = dsni (∅BSP ) = ∅
Main function Dsn(R) = hdsn0 (R), . . . , dsnp−1 (R)i 19/28
Desynchronization
(Σ, p) bspre → (Σ ∪ {;t }) re
p
Auxiliary recursive function dsni (F ·BSP G ) = dsni (F ) · dsni (G ) dsni (F +BSP G ) = dsni (F ) + dsni (G ) dsni (F ∗BSP ) = dsni (F )∗ dsni (hr0 , . . . , rp−1 it ) = ri · ;t dsni (BSP ) = dsni (∅BSP ) = ∅
Main function 0
Dsn(R) = hdsn (R), . . . , dsn
p−1
(R)i
Σdsync = Σ∪
S
;t
t∈[S] 19/28
From BSPRE to BSPA
Algorithm global scheme
(Σ, p) bspre
(Σ, p) bspa
Desynchronization
(Σ ∪ {;t }) re
p
Synchronization
(BK : re → nfa)
p
(Σ ∪ {;t }) nfa
p
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Br¨uggemann-Klein
(Σ ∪ {;t }) re → (Σ ∪ {;t }) nfa
Principles • Improvement of Glushkov O(n3 ) • Pre-processing : Star Normal From
Features • Time complexity : O(n2 ) • Space complexity : O(n) • No -transitions. i ∀i ∈ [p], Ai = (Q i , Σdsync , δdsync , Ii, Fi)
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From BSPRE to BSPA
Algorithm global scheme
(Σ, p) bspre
(Σ, p) bspa
Desynchronization
(Σ ∪ {;t }) re
p
Synchronization
(BK : re → nfa)
p
(Σ ∪ {;t }) nfa
p
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Synchronization : Auxiliary functions For each ;t
...
1
...
2
...
3 4
;t ;t ;t
5
...
6
...
7
...
;t
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Synchronization : Auxiliary functions For each ;t
...
1
...
2
...
3 4
;t ;t ;t
5
...
6
...
7
...
;t
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Synchronization : Auxiliary functions For each ;t
... ...
...
1 × 2 × 3 4
;t ;t ;t
5
...
6
...
7
...
;t
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Synchronization : Auxiliary functions For each ;t arg t : state vector set ... ...
...
1 × 2 × 3 4
;t ;t ;t
5
...
6
...
7
...
;t
arg t
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Synchronization : Auxiliary functions For each ;t
... ...
...
1 × 2 × 3 4 arg t
;t ;t ;t
5 × 6 × 7
...
arg t : state vector set res t : state vector set
... ...
;t
res t
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Synchronization : Auxiliary functions For each ;t
... ...
...
1 × 2 × 3 4 arg t
;t ;t ;t
5 × 6 × 7
arg t : state vector set res t : state vector set
... ... ...
∆=
[
[
#» q → #» q0
#» q 0 ∈res t q ∈arg t #»
;t
res t
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Synchronization : Auxiliary functions For each ;t
... ...
...
1 × 2 × 3 4 arg t
;t ;t ;t
5 × 6 × 7
arg t : state vector set res t : state vector set
... ... ...
∆=
[ [ t∈[S]
[
#» q → #» q0
#» q 0 ∈res t q ∈arg t #»
;t
res t
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Synchronization : Auxiliary functions For each ;t
... ...
...
1 × 2 × 3 4 arg t
;t ;t ;t
5 × 6 × 7
arg t : state vector set res t : state vector set
... ... ...
∆=
[ [ t∈[S]
[
#» q → #» q0
#» q 0 ∈res t q ∈arg t #»
;t
res t
∆ = { h1, 2, 3i → h5, 6, 7i , h1, 2, 4i → h5, 6, 7i , ... }
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Synchronization : Auxiliary functions Input vectors of ∆ src t (δdsync ) =
n o q | (q, ;t ) → q 0 ∈ δdsync
p−1 0 arg t ({δ i }i∈[p] ) = src t (δdsync ) × · · · × src t (δdsync )
Output vectors of ∆ dst t (δdsync ) =
n o q 0 | (q, ;t ) → q 0 ∈ δdsync
p−1 0 res t ({δ i }i∈[p] ) = dst t (δdsync ) × · · · × dst t (δdsync )
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Synchronization
(Σ ∪ {;t }) nfa
p
→ (Σ, p) bspa
i Syn( ({Q i }i∈[p] , Σdsync , {δdsync }i∈[p] , {I i }i∈[p] , {F i }i∈[p] ))
= ({Q i }i∈[p] , Σ, {δ i }i∈[p] , {I i }i∈[p] , {F i }i∈[p] , ∆) where ∆=
[ [ t∈[S]
#» q ∈arg t
i δ i = δdsync \
[
[
#» q → #» q0
#» q 0 ∈res t
i (q, ;t ) → q 0 ∈ δdsync
t∈[S]
Σ = Σdsync \
S
;t
t∈[S] 25/28
Example
BSPRE (a+b)a, h ab i0
(+b)b, ab, h ba i1 +h a i2
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Example
BSPRE (a+b)a, h ab i0
(+b)b, ab, h ba i1 +h a i2
Desynchronization
h
(a+b) a ;0 (+b) b ;1)+(a b ;2) ,
a b ;0 (b a ;1)+(a ;2)
i
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Example BSPRE
h
(a+b)a, i0 ab
h
(+b)b, ab, i1 +h a i2 ba
Desynchronization
h
(a+b) a ;0 (+b) b ;1)+(a b ;2) ,
a b ;0 (b a ;1)+(a ;2)
i
BK a b 0
2
a
a a
3
;0
4
b
a
1'
5
b
b
2'
;0
3'
a b
7' 4'
9
;2
10
6
;1
7
;1
6'
b
b
1
0'
8
;2 a
8' 5'
26/28
Example Desynchronized automata 8
a 2
b 0
a
a a
3
;0
4
b
5
a
1'
b
2'
;0
3'
a b
7' 4'
9
;2
10
6
;1
7
;1
6'
b
b
1
0'
b
;2 a
8' 5'
Synchronization b b 0
2
a 1
0'
a
1'
5
b
b
6
a a b
4
3 Δ 2'
3'
a b
8 4' a
b a
7 Δ 6'
9 5' 7'
Δ
10 8'
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Conclusion
What is done • Software for RE and Automata transformation • BSPRE → BSPA
What is being done • Determinize and minimize BSPA • Automatize RE → BSPRE • Implement BSPA recognition
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