A Geometric Approach
to the Problem of
Unique Decomposition of Processes
I
Thibaut Balabonski & Emmanuel Haucourt
Geometric Semantics
PPS
Université Paris Diderot
LIST
CEA
Definition of
Independence
Decomposition
Theorem
CONCUR’10: September 3
Implementation
T. Balabonski & E. Haucourt
Geometric Decomposition of Processes — CONCUR’10: September 3 — 1/22
Concurrent programming
Background
• A set of concurrent processes
• Some shared resources
I
Geometric Semantics
Definition of
Independence
Decomposition
Theorem
Tools for peaceful interaction
• Semaphores
• Synchronisation barriers
• Communications
• ...
Implementation
T. Balabonski & E. Haucourt
Geometric Decomposition of Processes — CONCUR’10: September 3 — 2/22
A toy language for semaphores
Two instructions
Pa Take one occurrence of a.
Va Release a.
I
Geometric Semantics
Each semaphore a is characterized by its arity.
Definition of
Independence
Decomposition
Theorem
Implementation
T. Balabonski & E. Haucourt
Geometric Decomposition of Processes — CONCUR’10: September 3 — 3/22
Nontrivial independence
k
k
k
I
Geometric Semantics
Definition of
Independence
a, b
c
π1
π2
π3
π4
= Pa.Pc.Vc.Va
= Pa.Pc.Vc.Va
= Pb.Pc.Vc.Vb
= Pb.Pc.Vc.Vb
mutual exclusion
two occurrences
Decomposition
Theorem
Implementation
T. Balabonski & E. Haucourt
Geometric Decomposition of Processes — CONCUR’10: September 3 — 4/22
Nontrivial independence
k
k
k
I
Geometric Semantics
Definition of
Independence
a, b
c
π1
π2
π3
π4
= Pa.Pc.Vc.Va
= Pa.Pc.Vc.Va
= Pb.Pc.Vc.Vb
= Pb.Pc.Vc.Vb
mutual exclusion
two occurrences
Decomposition
Theorem
Implementation
T. Balabonski & E. Haucourt
Geometric Decomposition of Processes — CONCUR’10: September 3 — 4/22
Nontrivial independence
k
k
k
I
Geometric Semantics
Definition of
Independence
a, b
c
π1
π2
π3
π4
= Pa.
= Pa.
= Pb.
= Pb.
Va
Va
Vb
Vb
mutual exclusion
two occurrences
Decomposition
Theorem
Implementation
T. Balabonski & E. Haucourt
Geometric Decomposition of Processes — CONCUR’10: September 3 — 4/22
Roadmap
1 A Geometric Semantics for Concurrence
2 Semantic Definition of Independence
3 A General Decomposition Theorem
4 Implementation
T. Balabonski & E. Haucourt
Geometric Decomposition of Processes — CONCUR’10: September 3 — 5/22
A Geometric Semantics for Concurrence
Interpreting programs by Euclidean spaces.
Semantic Definition of Independence
A General Decomposition Theorem
Implementation
T. Balabonski & E. Haucourt
Geometric Decomposition of Processes — CONCUR’10: September 3 — 6/22
One process, one dimension
Pa.Pb.Vb.Va
IGeometric Semantics
Definition of
Independence
·
Pa
·
Pb
·
Vb
·
Va
·
Decomposition
Theorem
Implementation
T. Balabonski & E. Haucourt
Geometric Decomposition of Processes — CONCUR’10: September 3 — 6/22
Two processes, two dimensions ?
Pa.Pb.Vb.Va k Pb.Pa.Va.Vb
·
·
·
·
·
·
·
·
·
·
·
·
·
·
·
·
·
·
·
·
Vb
IGeometric Semantics
Va
Pa
Definition of
Independence
Pb
·
Pa
·
Pb
·
Vb
·
Va
·
Decomposition
Theorem
Implementation
T. Balabonski & E. Haucourt
Geometric Decomposition of Processes — CONCUR’10: September 3 — 7/22
Two processes, two dimensions ?
Pa.Pb.Vb.Va k Pb.Pa.Va.Vb
·
·
·
·
·
·
·
·
·
·
·
·
·
·
·
·
·
·
·
·
Vb
IGeometric Semantics
Va
Pa
Definition of
Independence
Pb
·
Pa
·
Pb
·
Vb
·
Va
·
Decomposition
Theorem
Implementation
T. Balabonski & E. Haucourt
Geometric Decomposition of Processes — CONCUR’10: September 3 — 7/22
Euclidean spaces with holes
Pa.Pb.Vb.Va k Pb.Pa.Va.Vb
Vb
IGeometric Semantics
Va
Pa
Pb
Definition of
Independence
Decomposition
Theorem
Pa Pb Vb Va
Implementation
T. Balabonski & E. Haucourt
Geometric Decomposition of Processes — CONCUR’10: September 3 — 8/22
Euclidean spaces with holes
Pa.Pb.Vb.Va k Pb.Pa.Va.Vb
Vb
IGeometric Semantics
Va
Fa
Pa
Pb
Definition of
Independence
Decomposition
Theorem
Pa Pb Vb Va
Implementation
T. Balabonski & E. Haucourt
Geometric Decomposition of Processes — CONCUR’10: September 3 — 8/22
Euclidean spaces with holes
Pa.Pb.Vb.Va k Pb.Pa.Va.Vb
Vb
IGeometric Semantics
Va
Fb
Pa
Pb
Definition of
Independence
Decomposition
Theorem
Pa Pb Vb Va
Implementation
T. Balabonski & E. Haucourt
Geometric Decomposition of Processes — CONCUR’10: September 3 — 8/22
Euclidean spaces with holes
Pa.Pb.Vb.Va k Pb.Pa.Va.Vb
Vb
IGeometric Semantics
Va
F
Pa
Pb
Definition of
Independence
Decomposition
Theorem
Pa Pb Vb Va
Implementation
T. Balabonski & E. Haucourt
Geometric Decomposition of Processes — CONCUR’10: September 3 — 8/22
A 3D example
π1 = Pa.Pc.Vc.Va
k π2 = Pa.Pc.Vc.Va
k π3∗ =
Pc.Vc
IGeometric Semantics
Fa
Fc
Definition of
Independence
Decomposition
Theorem
Implementation
T. Balabonski & E. Haucourt
Geometric Decomposition of Processes — CONCUR’10: September 3 — 9/22
A Geometric Semantics for Concurrence
Semantic Definition of Independence
Cartesian product and factorization are the key.
A General Decomposition Theorem
Implementation
T. Balabonski & E. Haucourt
Geometric Decomposition of Processes — CONCUR’10: September 3 — 10/22
Independence of programs
Geometric Semantics
Σ1 independent from Σ2 :
Σ2 has no effect on the execution of Σ1 .
IDefinition of
Independence
Semantic definition
Σ1 and Σ2 are independent
iff
J Σ1 k Σ2 K = J Σ1 K × J Σ2 K
Decomposition
Theorem
Implementation
T. Balabonski & E. Haucourt
Geometric Decomposition of Processes — CONCUR’10: September 3 — 10/22
Independent programs: an example
Geometric Semantics
Pa.Pc.Vc.Va
k Pa.Pc.Vc.Va
Pc.Vc
IDefinition of
Independence
Decomposition
Theorem
Pa.Pc.Vc.Va
k Pa.Pc.Vc.Va
k
Pc.Vc
Implementation
T. Balabonski & E. Haucourt
Geometric Decomposition of Processes — CONCUR’10: September 3 — 11/22
Algebraic notions
Irreducibles
A decomposition
always exists.
Primes
Any decomposition
is unique.
Prime element x: if x|y1 ...yn then there is i such that x|yi
T. Balabonski & E. Haucourt
Geometric Decomposition of Processes — CONCUR’10: September 3 — 12/22
A Geometric Semantics for Concurrence
Semantic Definition of Independence
A General Decomposition Theorem
Homogeneous tuples of coordinates,
up to permutation.
Implementation
T. Balabonski & E. Haucourt
Geometric Decomposition of Processes — CONCUR’10: September 3 — 13/22
Homogeneous sets of words
Geometric Semantics
Definition of
Independence
N-dimensional geometric objects
• Point: N-uple of coordinates.
• Box: N-uple of intervals.
• Gist: word of length N. (on a given alphabet)
IDecomposition
Theorem
Implementation
T. Balabonski & E. Haucourt
Geometric Decomposition of Processes — CONCUR’10: September 3 — 13/22
Homogeneous sets of words
Geometric Semantics
Definition of
Independence
N-dimensional geometric objects
• Point: N-uple of coordinates.
• Box: N-uple of intervals.
• Gist: word of length N. (on a given alphabet)
IDecomposition
Theorem
Homogeneous sets of words
Sets whose elements have the same length.
Implementation
T. Balabonski & E. Haucourt
Geometric Decomposition of Processes — CONCUR’10: September 3 — 13/22
Product of sets
Geometric Semantics
Definition of
Independence
Cartesian product
Concatenation of all pairs.
IDecomposition
Theorem
×
=
With homogeneous sets:
dim(S1 × S2 ) = dim(S1 ) + dim(S2 )
Implementation
T. Balabonski & E. Haucourt
Geometric Decomposition of Processes — CONCUR’10: September 3 — 14/22
Permutation of coordinates
Geometric Semantics
Gap
Definition of
Independence
Parallel composition should be commutative,
Cartesian product is not.
IDecomposition
Theorem
Permutation
of coordinates
(group action)
Homogeneous monoid: quotient by permutation.
Implementation
T. Balabonski & E. Haucourt
Geometric Decomposition of Processes — CONCUR’10: September 3 — 15/22
Factorization result
Geometric Semantics
Definition of
Independence
Equivalence Theorem
On the homogeneous monoid,
Irreducibility = Primality
IDecomposition
Theorem
Factorization Corollary
Each element of the homogeneous monoid has a
unique finest decomposition.
Implementation
T. Balabonski & E. Haucourt
Geometric Decomposition of Processes — CONCUR’10: September 3 — 16/22
A Geometric Semantics for Concurrence
Semantic Definition of Independence
A General Decomposition Theorem
Implementation
The constructive side of the factorization result.
T. Balabonski & E. Haucourt
Geometric Decomposition of Processes — CONCUR’10: September 3 — 17/22
A Galois connection
γ(F) :=
Families of n-cubes
S
F
Subsets of Rn
α(A) := {maximal cubes of A}
T. Balabonski & E. Haucourt
Geometric Decomposition of Processes — CONCUR’10: September 3 — 17/22
Cubical Areas
Subsets of Rn with finitely many maximal cubes.
T. Balabonski & E. Haucourt
Geometric Decomposition of Processes — CONCUR’10: September 3 — 18/22
The ALCOOL analyzer: features
Geometric Semantics
Definition of
Independence
Decomposition
Theorem
IImplementation
•
Based on the structure of cubical areas.
•
Recognizes an extension of the PV language.
•
Detects deadlocks, overflows, unreachable states...
•
Factorizes concurrent programs.
•
Modelling of loops in progress.
T. Balabonski & E. Haucourt
Geometric Decomposition of Processes — CONCUR’10: September 3 — 19/22
Algorithm: An Example
Geometric Semantics
Definition of
Independence
Decomposition
Theorem
IImplementation
T. Balabonski & E. Haucourt
k
k
k
π1
π2
π3
π4
= Pa.Pc.Vc.Va
= Pb.Pc.Vc.Vb
= Pa.Pc.Vc.Va
= Pb.Pc.Vc.Vb
Geometric Decomposition of Processes — CONCUR’10: September 3 — 20/22
Algorithm: The State Space
Geometric Semantics
Definition of
Independence
Decomposition
Theorem
IImplementation
T. Balabonski & E. Haucourt
[0, 1[
[0, 1[
[0, 1[
[0, 1[
[4, ∞[
[4, ∞[
[4, ∞[
[4, ∞[
[0, ∞[
[0, ∞[
[0, ∞[
[0, ∞[
[0, ∞[
[0, ∞[
[0, ∞[
[0, ∞[
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
[0, 1[
[4, ∞[
[0, ∞[
[0, ∞[
[0, 1[
[4, ∞[
[0, ∞[
[0, ∞[
[0, 1[
[0, 1[
[4, ∞[
[4, ∞[
[0, ∞[
[0, ∞[
[0, ∞[
[0, ∞[
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
[0, ∞[
[0, ∞[
[0, ∞[
[0, ∞[
[0, ∞[
[0, ∞[
[0, ∞[
[0, ∞[
[0, 1[
[4, ∞[
[0, 1[
[4, ∞[
[0, 1[
[0, 1[
[4, ∞[
[4, ∞[
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
[0, ∞[
[0, ∞[
[0, 1[
[4, ∞[
[0, ∞[
[0, ∞[
[0, 1[
[4, ∞[
[0, ∞[
[0, ∞[
[0, ∞[
[0, ∞[
[0, 1[
[4, ∞[
[0, 1[
[4, ∞[
Geometric Decomposition of Processes — CONCUR’10: September 3 — 20/22
Algorithm: testing {2, 3}
T. Balabonski & E. Haucourt
[0, 1[
[0, 1[
[4, ∞[
[4, ∞[
×
[0, ∞[
×
[0, ∞[
×
[0, 1[
[4, ∞[
[0, 1[
[4, ∞[
[0, ∞[
[0, ∞[
×
[0, ∞[
×
[0, 1[
×
[0, 1[
[4, ∞[
[0, ∞[
[0, ∞[
×
[0, ∞[
×
[4, ∞[
×
[0, 1[
[4, ∞[
[0, 1[
[4, ∞[
×
[0, 1[
×
[0, ∞[
×
[0, ∞[
[0, ∞[
[0, 1[
[4, ∞[
×
[4, ∞[
×
[0, ∞[
×
[0, ∞[
[0, ∞[
[0, ∞[
[0, ∞[
[0, ∞[
[0, ∞[
×
×
×
×
[0, 1[
[0, 1[
[4, ∞[
[4, ∞[
×
×
×
×
[0, 1[
[4, ∞[
[0, 1[
[4, ∞[
×
×
×
×
[0, ∞[
[0, ∞[
[0, ∞[
[0, ∞[
Geometric Decomposition of Processes — CONCUR’10: September 3 — 20/22
Algorithm: testing {2, 4}
[0, 1[
[4, ∞[
[0, ∞[
[0, ∞[
×
[0, 1[
[0, 1[
[4, ∞[
[0, ∞[
[0, ∞[
×
[4, ∞[
[0, 1[
[4, ∞[
[0, ∞[
[0, ∞[
[0, 1[
[4, ∞[
[0, ∞[
[0, ∞[
T. Balabonski & E. Haucourt
×
×
[0, ∞[
[0, ∞[
×
[0, ∞[
[0, ∞[
[0, 1[
[4, ∞[
×
[0, ∞[
×
[0, ∞[
[0, ∞[
[0, 1[
[4, ∞[
×
[0, ∞[
×
[0, ∞[
[0, ∞[
[0, 1[
[4, ∞[
×
[0, 1[
×
[0, ∞[
[0, ∞[
[0, 1[
[4, ∞[
×
[4, ∞[
Geometric Decomposition of Processes — CONCUR’10: September 3 — 20/22
Some tests
Example
6 philosophers
7 philosophers
8 philosophers
9 philosophers
10 philosophers
Example
Σ2,2
Σ2,2,2
Σ3,3
Σ2,2,2,2
Σ4,4
Σ3,3,3
Σ4,5
Σ5,5
Time
0.1
0.1
0.13
0.13
1
1.5
6.1
50
T. Balabonski & E. Haucourt
Time
0.2
0.7
3.5
21
152
Decomp.
No
No
No
No
No
Time is given in seconds.
Decomposition
{1, 3}{2, 4}
{1, 4}{2, 5}{3, 6}
{1, 3, 5}{2, 4, 6}
{1, 5}{2, 6}{3, 7}{4, 8}
{1, 3, 5, 7}{2, 4, 6, 8}
{1, 4, 7}{2, 5, 8}{3, 6, 9}
{1, 3, 5, 7}{2, 4, 6, 8, 9}
{1, 3, 5, 7, 9}{2, 4, 6, 8, 10}
Example
Σ02,2
Σ02,2,2
Σ03,3
Σ02,2,2,2
Σ04,4
Σ03,3,3
Σ04,5
Σ05,5
Time
0.1
0.3
0.52
7.1
33
293
327
2875
Decomp.
No
No
No
No
No
No
No
No
Geometric Decomposition of Processes — CONCUR’10: September 3 — 21/22
Conclusion
Geometric Semantics
Definition of
Independence
Decomposition
Theorem
Implementation
•
Geometry is cool.
•
Cubical areas can model several
mainstream concurrent primitives.
•
“Torical” areas give support for loops.
•
Relation to algebraic topology and directed
homotopy.
I
T. Balabonski & E. Haucourt
Geometric Decomposition of Processes — CONCUR’10: September 3 — 22/22