On the geometric nature of mutual exclusion Eric Goubault & Samuel Mimram Work in progress CEA LIST and Ecole Polytechnique, France ACAT Conference, Bremen 15th of July 2013 E. Goubault & S. Mimram
Contents of the talk Mostly trying to wrap things up... Motivation: study of concurrent systems and their schedules; talk influenced by: The surprising efficiency of classical mutual exclusion models in the analysis of concurrent systems Geometric group theory, and the ATMCS 2004 talk by R. Ghrist, work on trace spaces by M. Raussen, universal dicoverings of L. Fajstrup, our paper in GETCO 2010 etc. In the particular case of mutual exclusion Corresponds exactly to the NPC cubical complexes CAT(0) cubical complexes are prime event structures Classical unfoldings of prime event structures/safe Petri nets are universal dicoverings of NPC cubical complexes As a potential application: “directed” H 1 in the NPC case E. Goubault & S. Mimram
Original motivation of this work Processes Q Q Q Q Q 1 2 3 4 5 Shared Memory x y z x, y and z are locations Not sequential programs, bad states, chaotic behavior = ⇒ Need for synchronizations = ⇒ Need for locks: Py, Vy (binary and counting semaphores = ⇒ Interleaving semantics given by a “shuffle” of transition systems (or fibred product) - Can we “do” better? E. Goubault & S. Mimram
Original motivation of this work P b . V b . P a . V a | P a . V a Pb Vb Pa Va Va Va Va Va Pb Vb Pa Pa Pa Pa Pb Vb Pa Va Not sequential programs, bad states, chaotic behavior = ⇒ Need for synchronizations = ⇒ Need for locks: Py, Vy (binary and counting semaphores ⇒ Interleaving semantics given by a “shuffle” of transition systems = (or fibred product) - Can we “do” better? E. Goubault & S. Mimram
Original motivation of this work P b . V b . P a . V a | P a . V a Pb Vb Pa Va Va Va Va Va Pb Vb Pa Pa Pa Pa Pb Vb Pa Va Not sequential programs, bad states, chaotic behavior = ⇒ Need for synchronizations = ⇒ Need for locks: Py, Vy (binary and counting semaphores ⇒ Interleaving semantics given by a “shuffle” of transition systems = (or fibred product) - Can we “do” better? E. Goubault & S. Mimram
Models On the “geometry” side Cubical sets (pre-existing the field of course!) Po-spaces (i.e. topological space with closed partial order), introduced first in other fields (domain theory P. Johnstone etc., functionnal analysis L. Nachbin etc.) local po-spaces (atlas of po-spaces - L. Fajstrup, E. Goubault, M. Raussen) d-spaces (M. Grandis) Flows (P. Gaucher) Streams (S. Krishnan) etc. More classical models in computer science Transition systems, prime event structures, Petri nets etc. E. Goubault & S. Mimram
Geometric semantics A program P b ;x:=1; V b ; P a ;y:=2; V a | P a ;y:=3; V a will be interpreted as a directed space : P b . V b . P a . V a P b V b P a V a P a . V a P a V a P b . V b . P a . V a | P a . V a Forbidden regions V a P a P b V b P a V a E. Goubault & S. Mimram
Geometric semantics A program P b ;x:=1; V b ; P a ;y:=2; V a | P a ;y:=3; V a will be interpreted as a directed space : P b . V b . P a . V a P b V b P a V a P a . V a P a V a P b . V b . P a . V a | P a . V a Forbidden regions V a P a P b V b P a V a E. Goubault & S. Mimram
Geometric semantics A program P b ;x:=1; V b ; P a ;y:=2; V a | P a ;y:=3; V a will be interpreted as a directed space : P b . V b . P a . V a P b V b P a V a P a . V a P a V a P b . V b . P a . V a | P a . V a Forbidden regions V a P a P b V b P a V a E. Goubault & S. Mimram
Schedules of executions directed homotopies A directed homotopy (with fixed extremities) between directed paths f and g φ : f → g : − → I → Y is a directed map φ : X × − → I → Y such that: for all x ∈ [ 0 , 1 ] , φ ( x , 0 ) = f ( x ) , φ ( x , 1 ) = g ( x ) for all t ∈ [ 0 , 1 ] , φ ( 0 , t ) = f ( 0 ) , φ ( 1 , t ) = f ( 1 ) (in particular, for all t ∈ [ 0 , 1 ] , φ ( ., t ) is a directed path) Schedules are directed paths up to directed homotopies. How to compute them? (fundamental category − → π 1 ( X ) , category of components, trace spaces... still quite computationally demanding!) E. Goubault & S. Mimram
Examples of geometric semantics To each program p we associate a d-space ( H p , b p , e p ) : P a . V a | P a . V a P a . P b . V b . V a | P b . P a . V a . V b P a . ( V a . P a ) ∗ | P a . V a e p e p e p b p b p b p E. Goubault & S. Mimram
Examples of geometric semantics P a . V a | P a . V a | P a . V a P a . V a | P a . V a | P a . V a ( κ a = 2 ) ( κ a = 1 ) t 1 t 1 t 2 t 2 t 0 t 0 E. Goubault & S. Mimram
Here: cubical complexes Geometric realization Maps pre-cubical sets to a gluing of hyper-cubes [ 0 , 1 ] n along their faces (with quotient topology) (directed path structure agreeing with partial order on each hypercube) E. Goubault & S. Mimram
Here: cubical complexes Produces cubical complexes Topological spaces X that are particular cellular complexes whose building blocks (cells) are n -cubes, i.e. [ 0 , 1 ] n i : [ 0 , 1 ] n → X In particular we have, as data, maps c n identifying the i th cell of dimension n in X (homeomorphism from the interior of [ 0 , 1 ] n onto X plus some extra properties for the boundaries) (directed path structure agreeing with partial order on each hypercube) E. Goubault & S. Mimram
Adding geodesic metric structure Idea: identify directed paths with shortest paths... Undirected path, “longer length” than the directed one Or on π 1 ... f 5 f 4 f 3 f 1 f − 1 2 f 1 f − 1 f 3 f 4 f 5 is a “longer word” in π 1 (seen as generated by a 2 semi-group of “directed” generators) When can we give a meaning to this? E. Goubault & S. Mimram
Adding geodesic metric structure d 2 and d ∞ metrics on a (arc connected) cubical complex X Each of the cell c n i (( 0 , 1 ) n ) inherits from the d k (here k = 2 or ∞ ) metrics by: � � ( c n i ) − 1 ( x ) , ( c n i ) − 1 ( y ) d k ( x , y ) = d k For any two points x , y in X , � d k ( x , y ) = inf d k ( γ ( t k ) , γ ( t k + 1 )) γ ∈ Γ i = 0 , n γ − 1 where Γ is the set of paths from x to y and ( γ ( t k ) , γ ( t k + 1 )) belongs to the same k -cell of X (hence d k ( γ ( t k ) , γ ( t k + 1 )) is well-defined) E. Goubault & S. Mimram
Arc-length of a curve in a d ∞ -space Arc-length Let c : [ a , b ] → X a curve in X , define its arc-length as: n − 1 � l ( c ) = sup d ( c ( t i ) , c ( t i + 1 )) a = t 0 � t 1 � ... � t n = b i = 0 (the supremum is taken over all n and all possible partitions) Rectifiability All paths in a finitely generated d ∞ -space have finite length. b t 3 t 2 t 4 a t 1 E. Goubault & S. Mimram
Geodesics and local geodesics Local geodesic A map c : [ a , b ] → X is a local geodesic is for all t ∈ [ a , b ] , there exists ǫ > 0 such that d ( c ( t ′ ) , c ( t ′′ )) = | t ′ − t ′′ | for all t ′ , t ′′ ∈ [ a , b ] with | t − t ′ | + | t − t ′′ | � ǫ Geodesic A map c : [ 0 , l ] → X is a geodesic (from c ( a ) to c ( b ) ) if d ( c ( t ) , c ( t ′ )) = | t − t ′ | for all t , t ′ ∈ [ 0 , l ] (in particular l = l ( c ) = d ( x , y ) ). E. Goubault & S. Mimram
Cubical sets and geodesics Geodesic metric space Such metrics give connected cubical complexes the structure of a geodesic metric space, i.e. any two points can be linked by a geodesic (complete in case it is a finite cubical complexes - Bridson & Haefliger 1999) Extension to more general d-spaces? Open question... E. Goubault & S. Mimram
d ∞ geodesics (start to end) Example: 2 unit 2-cells “Left-greedy” [Ghrist] - arc length 2 E. Goubault & S. Mimram
d ∞ geodesics (start to end) Example: 2 unit 2-cells More of them (still arc length 2 for d ∞ !): E. Goubault & S. Mimram
d ∞ geodesics (start to end) Example: 2 unit 2-cells More of them (still arc length 2 for d ∞ !): E. Goubault & S. Mimram
d ∞ geodesics (start to end) Example: 2 unit 2-cells More of them (still arc length 2 for d ∞ !): E. Goubault & S. Mimram
d ∞ geodesics (start to end) Example: 2 unit 2-cells More of them (still arc length 2 for d ∞ !): Of course, a unique d 2 geodesic! √ ( d 2 arc length is 5 > 2) E. Goubault & S. Mimram
CAT(0) and NPC CAT(0) geometry X is CAT(0) iff all triangles are no fatter than in the planar case X has non-positive curvature (NPC) iff X is locally CAT ( 0 ) . E. Goubault & S. Mimram
Combinatorial condition Gromov’s condition Property: a d ∞ -space is NPC iff its link at every vertex is a flag complex (“if the red edges look like a k -simplex, then there is really a k -simplex linking them”) Implies: empty squares are OK, but not empty cubes nor empty hypercubes of dimension � 3 E. Goubault & S. Mimram
NPC is exactly mutual exclusion models P a . V a | P a . V a | P a . V a P a . V a | P a . V a | P a . V a ( κ a = 2 ) ( κ a = 1 ) t 1 t 1 t 2 t 2 t 0 t 0 not NPC NPC In the right-hand side case, we know that the trace space (M. Raussen) is discrete, − → π 1 is “combinatorial”...can we find this with the geodesic metric approach? E. Goubault & S. Mimram
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