geometry of trace spaces - laboratoire paul...

Geometry of Trace Spaces

Eric Goubault

CEA LIST and Ecole Polytechnique, France

joint work with Samuel Mimram, Emmanuel Haucourt, Christine Tasson, LisbethFajstrup, Martin Raussen

Inaugural Conference Labex CEMPI

27th of September 2012

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Contents of the talk

A rush tour of static analysis, and of geometric semantics ofconcurrent programs

Combinatorics/algorithmics of trace spaces

Very quick application to static analysis of programs

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Typical static analysis

Given a program, how to prove some form of correctness?(executions won’t do)

Generally: (a) to compute invariants, i.e. properties which holdtrue on any execution(b) to compute liveness properties, i.e. properties that willeventually be true (will terminate, will deadlock etc.)

Classical steps:

(1) interleaving semantics, collecting semantics for (a) (only)

(2) abstraction (Galois connections etc.)

(3) computation of least fixpoints - resolution of large systems ofequations

We are looking for a more efficient step (1) ((2) and (3) for someother talk!)

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Use of invariants

From (3) we conclude in general about the “safety” (for instanceabsence of run-time errors) of programs:

Invariants are “integral forms” of the dynamics typically (if it weredynamical systems’ theory)For (b): we can use variants (decreasing in N for instance, fortermination) for instance along the paths

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Concurrent programsshared memory style

2Q1Q 3Q 4Q 5QShared M

emory

Processes

x y z

x, y and z are locations

Not sequential programs, bad states, chaotic behavior=⇒ Need for synchronizations =⇒ Need for locks (P, V)=⇒ Interleaving semantics given by a “shuffle” of transitionsystems (or fibred product)

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Interleaving semantics

Starting with a = 1, b = 2 for instance...

T1 T2

Pa −a = a + 1 −

Va

− Pb

− b = b + 1

− Vb

Pb −b = 2 ∗ b −

Vb −− Pa

− a = 3 ∗ a

− Va

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Geometry“progress graphs” E.W.Dijkstra’68 (later V.Pratt, R. van Glabbeek’91)

T1=Pa.Pb.Vb.Va in parallel with T2=Pb.Pa.Va.Vb

Pa Pb Vb Va

Pb

Pa

Va

Vb

Forbidden

“Continuous model”: xi = local time; dark grey region=forbidden!see Algebraic Topology and Concurrency TCS 2006, L. Fajstrup, E. Goubault, M.

Raussen

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Execution pathsare continuous

T1=Pa.Pb.Vb.Va in parallel with T2=Pb.Pa.Va.Vb

Pa Pb Vb Va

Pb

Pa

Va

Vb

Traces are continuous paths increasing in each coordinate: dipaths.

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Classes of equivalent dipathsup to dihomotopy

Pa Pb Vb Va

Pb

Pa

Va

Vb

T2

T1

b=2*b

b=b−1

a=a+

1

b=b+

1

b=2a=1

T1 gets a and b before T2 => a=2 and b=4

T2 gets b and a before T1 => a=2 and b=3

Each of T1 and T2 gets a ressource=> Deadlock with a=2 and b=1

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Goal

When verifying a concurrent program,there is a priori a large number of possible interleavings to check

(exponential in the number of processes)

Many executions are equivalent:we want here to provide a minimal number of execution traces

which describe all the possible cases

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Geometric semantics

A program

Pb;x:=1;Vb;Pa;y:=2;Va | Pa;y:=3;Va

will be interpreted as a directed space:

Pb.Vb.Pa.Va

Pb Vb Pa Va

Pa.Va

Pa Va

Pb.Vb.Pa.Va | Pa.Va Forbidden regions

Pb Vb Pa Va

Pa

Va

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Models

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. Johnstoneetc., functional 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.

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More formally

Basic definitions in directed algebraic topology

Let X be a stream/d-space etc. (here we only consider apo-space, i.e. a topological space X together with a partialorder 6⊆ X × X , closed in the product topology)

p : I→ X a continuous and increasing path from po-spaceI = ([0, 1],6) (standard order) to X is a directed path

Define the path space P(X )(a, b) = {p : I→ Xmod p(0) = a, p(1) = b, p is a directed path}A dihomotopy on P(X )(a, b) is a continuous mapH : I× I → X such that Ht ∈ P(X )(a, b) for all t ∈ I .

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Trace spaces

Formally

Let X be a stream/d-space etc.

Define the trace space T(X )(a, b) to be the path spacebetween a and b modulo continuous and increasingreparametrizations (with a suitable topology...)

Schedules are dihomotopy classes of dipaths (π0(T(X )))

We wish to study the homotopy type of T(X )(a, b), which isrelated to “schedules of execution”

There is a homotopy equivalence between T(X )(a, b) and acertain prodsimplicial complex, which can be calculatedcombinatorially, on our simple spaces...

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Schedulings

A scheduling is the homotopy class of a path.

Pb Vb Pa Va

Pa

Va

We want to compute a path in every scheduling.Pb;x:=1;Vb;Pa;y:=2;Va | Pa;y:=3;Va is scheduled here in

three different ways:

y:=3;x:=1;y:=2 x:=1;y:=3;y:=2 x:=1;y:=2;y:=3(x , y) = (1, 2) (x , y) = (1, 2) (x , y) = (1, 3)

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Examples of geometric semantics

To each program p we associate a d-space (Hp, bp, ep):

Pa.Va|Pa.Va Pa.Pb.Vb.Va|Pb.Pa.Va.Vb Pa.(Va.Pa)∗|Pa.Va

bp

ep

bp

ep

bp

ep

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Examples of geometric semantics

Pa.Va|Pa.Va|Pa.Va Pa.Va|Pa.Va|Pa.Va

(κa = 2) (κa = 1)

t0

t1

t2

t0

t1

t2

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Determining traces can be intricate!

Px.Py.Pz.Vx.Pw.Vz.Vy.Vw | Pu.Pv.Px.Vu.Pz.Vv.Vx.Vz Py.Pw.Vy.Pu.Vw.Pv.Vu.Vv

LipskyE. Goubault

From now on...no loops for a start

Suppose given a program with n threads

p = p0|p1| . . . |pn−1

Under mild assumptions, the geometric semantics is of the form

Gp = ~I n \l−1⋃i=0

R i ; R i =n−1∏j=0

]x ij , y

ij [

Example Pa.Va.Pb.Vb|Pb.Vb.Pa.Va

t0

t1

0

1

x00 y0

0 x10 y1

0

x11

y11

x01

y01

a b

b

a

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Determining trace spaces

The trace space between two points b and e is the space ofdirected paths modulo reparameterization, suitably topologized...

To determine it (its connected components?), the main (naive?)idea is to extend the forbidden cubes downwards in variousdirections and look whether there is a path from b to e.

t0

t1

t0

t1

t0

t1

By combining those information, we will be able to compute tracesmodulo homotopy.

The directions in which to extend the holes will be coded byboolean matrices M.

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The index poset

Ml ,n: boolean matrices with l rows and n columns.

XM : space obtained by extendingfor every (i , j) such that M(i , j) = 1

the forbidden cube i downwardsin every direction other than j

0

1

t0

t1

0

1

t0

t1

0

1

t0

t1

(1 01 0

) (0 11 0

) (1 00 1

)alive alive dead

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The index poset

Ml ,n: boolean matrices with l rows and n columns.

XM :

0

1

t0

t1

0

1

t0

t1

0

1

t0

t1

(1 01 0

) (0 11 0

) (1 00 1

)alive alive dead

Ψ :Ml ,n → {0, 1}:Ψ(M) = 0 if there is a path b → e: M is alive

Ψ(M) = 1 if there is no path b → e: M is dead

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The index poset

Pa.Va.Pb.Vb | Pa.Va.Pb.Vb | Pa.Va.Pb.Vb

(κa = 2, κb = 2)

0

1

t0

t1

t2

t0

t1

t2

t0

t1

t2

t0

t1

t2

(0 0 00 0 0

) (1 0 00 0 1

) (0 0 11 0 0

) (0 0 01 1 1

)alive alive alive dead

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The index poset

Alive and dead?

Important matrices are

the dead poset D(X ) = {M ∈MCl ,n / Ψ(M) = 1}.

the index poset C(X ) = {M ∈MRl ,n / Ψ(M) = 0} (the alive

matrices).

consider the entrywise ordering (0 < 1) on matrices.

General results:

D(X ) C(X ) trace spaces, up to homotopy equivalence

(hence at least homotopy classes of traces...)

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The dead poset

Proposition

A matrix M ∈MCl ,n is in D(X ) iff it satisfies

∀(i , j) ∈ [0 : l [×[0 : n[, M(i , j) = 1 ⇒ x ij < min

i ′∈R(M)y i ′j

where R(M): indexes of non-null rows of M.

Example

t0

t1

0

1

x00 x1

0 y00 y1

0

x01

y01

x11

y11

M =

(0 11 0

)x01 = 1 < 2 = min(y0

1 , y11 )

x10 = 2 < 3 = min(y0

0 , y10 )

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The index poset

Proposition

A matrix M is in C(X ) iff for every N ∈ D(X ), N 66 M.

Remark

N 66 M: there exists (i , j) s.t. N(i , j) = 1 and M(i , j) = 0.

Remark

Since C(X ) is downward closed it will be enough to compute theset Cmax(X ) of maximal alive matrices.

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Connected components

M ∧ N: pointwise min of M and N

Definition

Two matrices M and N are connected when M ∧ N does notcontain any null row.

Proposition

The connected components of C(X ) are in bijection withhomotopy classes of traces b → e in X .

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Combinatorics of traces modulo homotopy

First dead matrix:

0

1

t0

t11 1

0 0

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Second dead matrix:

0

1

t0

t10 0

1 1

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Third and last (minimal) deadmatrix:

0

1

t0

t1 1 0

0 1

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First (maximal) alive matrix:

t0

t10 1

0 1

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Second alive matrix:

t0

t11 0

1 0

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Third (and last) maximal alivematrix:

t0

t1 0 1

1 0

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Combinatorics

Hypergraph transversal

An hypergraph H = (V ,E ) consists of a set V of vertices anda set E of edges, where an edge is a subset of V

A transversal T of H is a subset of V such that T ∩ e 6= ∅ forevery edge e ∈ E .

D(X ) ⇒ hypergraph H:

vertices: [0 : l [×[0 : n[

hyperedges: {(i , j) / D(i , j) = 1} (D is a matrix in D(X ))

The sets {(i , j) / M(i , j) = 0}, where M is a maximal matrixof C(X ), correspond to minimal transversals/hitting sets(wrt inclusion order) of H.

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Minimal transversal hypergraph

...is itself an hypergraph (same vertices, but hitting sets ashyper-edges):

. .

.

M2

.

M3 M1

t0

t1

t0

t1

t0

t1

(1 01 0

) (0 10 1

) (0 11 0

)M1 M2 M3

and they are all disconnected.(linked with Herlihy/Rajsbaum protocol complex...)

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The prodsimplicial structure of trace spaces

A prod-simplicial space is just a space made up of simplices,and products of simplices, glued together along their faces(natural generalization of cubical and simplicial sets)

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A prod-simplicial space is just a space made up of simplices,and products of simplices, glued together along their faces(natural generalization of cubical and simplicial sets)

Example:

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Each matrix of C represents a prod-simplex, product of onen-simplex per line, n=number of 1 per line minus 1...

Recall:

t0

t1

M3 =

(0 11 0

)product of 2 0-simplices = point!

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D(X )(0, 1) = {(111)}C(X )(0, 1) = {(110), (101), (011)}

(1 1 0)

(1 0 1)(0 1 1)

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C(X )(0, 1) = {(110), (101), (011)}and common faces are meet ofmatrices

(0,1,0)(1 1 0)

(1 0 0)

(1 0 1)

(0 0 1)

(0 1 1)

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C(X )(0, 1) = {(110), (101), (011)}connected, not simply-connected(reflecting the fact thatπ2(X ) = Z)

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A more intricate example

D(X )(0, 1) =

{(1 1 10 0 0

),

(0 0 01 1 1

)}t0

t1

t2

C(X )(0, 1) ={(1 1 01 1 0

),

(1 0 11 1 0

), . . .

}

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C(X )(0, 1) ={(1 1 01 1 0

),

(1 0 11 1 0

), . . .

}

(1 1 0)

(1 0 1)(0 1 1)

× (1 1 0)

(1 0 1)(0 1 1)

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C(X )(0, 1) ={(1 1 01 1 0

),

(1 0 11 1 0

), . . .

}

(π1 is Z× Z)

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And now, loops?

We consider programs of the form

p ::= p∗0 |p∗1 | . . . |p∗n−1

constituted of n threads pi which are looping.

We write X for the geometric semantics of theprogram p0|p1| . . . |pn−1, with X = ~I n \

⋃l−1i=0 R i with R i =]x i , y i [.

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Delooping

This is a particular case of delooping (here, just deloop once inevery direction)

Given a n-dimensional vector m whose components (mj)06j<n arein N, we write pm for the m-delooping of p, i.e. the program

pm00 |p

m11 | . . . |p

mn−1

n−1

and X m for the associated geometric semantics.

The trace space (at least its connected components) of all X m canbe generated fairly easily given what we did before...

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How to extend previous work, on an example

Consider (Pa.Va)∗ | (Pa.Va)∗:

The (1,1)-delooping:

0

t0

t1

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How to extend previous work, on an example

Consider (Pa.Va)∗ | (Pa.Va)∗:

The (2,2)-delooping:

t0

t1

etc.

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Trace spaces of deloopings

The dead matrix for the (1,1)-delooping is:

0

t0

t1

M0 = ( 1 1 )

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Trace spaces of deloopings

Hence the maximal alive matrices for the (1,1)-delooping are:

0

t0

t1

0

t0

t1

M0 = ( 0 1 ) M1 = ( 1 0 )

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Now...

Test all products of letters M0, M1 along coordinates 0 (t0) and 1(t1):

t0

t1

M0 ∗0 M0

t0

t1

M0 ∗0 M1

t0

t1

M1 ∗0 M1

t0

t1

M1 ∗0 M0 dead!

(see ESOP 2012)

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Automaton of dipaths mod dihomotopy

By symmetry we obtain an automaton of compositions:

M0

0,1

��0 ++

M11kk

1,0

��

(means we can write M0 ∗0 M1, but not M0 ∗1 M1 etc.)

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Relations!

We may have extra relations among the words in this automaton!Consider all possible concatenations of 2 letters on t0 and on t1:

?

t0

t1?

t0

t1?

t0

t1?

t0

t1

(M0 ?M0 M0

) (M0 ?M0 M1

) (M0 ?M1 M1

) (M1 ?M1 M1

)(M0 ∗0 M0) ∗1 (M0∗0?) (M0 ∗0 M1) ∗1 (M0∗0?) (M1 ∗0 M1) ∗1 (M0∗0?) (M1 ∗0 M1) ∗1 (M1∗0?)

Only the two in the middle give rise to commutations.

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Relations

?

t0

t1

?

t0

t1

(M0 ?M0 M1

) (M0 ?M1 M1

)= (M0 ∗1 M0) ∗0 (M1∗1?) (M1 ∗0 M1) ∗1 (M0∗0?)(M0 ∗0 M1) ∗1 (M0∗0?) = (M1 ∗1 M0) ∗0 (M1∗1?)

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Finally:

So the dipaths modulo dihomotopy are given by the wordsaccepted by the automaton:

M0

0,1

��0 ++

M11kk

1,0

��

modulo the relations:

M0 0 // M1 1 // M0 = M0 1 // M0 0 // M0

M0 0 // M1 1 // M1 = M0 1 // M0 0 // M1

M1 0 // M1 1 // M0 = M1 1 // M0 0 // M0

M1 0 // M1 1 // M1 = M1 1 // M0 0 // M1

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Application to static analysis

Suppose we want to analyze program

(Pa.a = a− 1.Va)∗ | (Pa.a = a/2.Va)∗

What are the possible sets of values reached, for a, starting witha ∈ [0, 1]?

Write (A0: set of possible values for a at the end of dipath M0, A1:set of possible values for a at the end of dipath M1):

M0 : [a← a− 1]

0,1

��0 ..

M1 : [a← a2 ]

1nn

1,0

{A0 ∪ = A0 − 1 ∪ A1

A1 ∪ = A12 ∪ A0

(plus relations) as some form of CFG for our program...theninterpret this as a least-fixed point equations over the lattice ofintervals...

Finds invariant a∞ =]−∞, 1]

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Conclusion and future work

Lots of experiments (implementation in “ALCOOL”) and lotsof mathematics to be investigated yet on trace spaces...

Applications: possibility/impossibility results for fault-tolerantprotocols (a la Herlihy/Rajsbaum)

Extension to randomized algorithms: random simplicial sets!Application: possibility of consensus...

Logical interpretation?

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Logical/type theoretical interpretation

The po-space could be interpreted as a type X . Terms p are pathsfrom beginning to end: we note ` p : X . A term H with` H : IdX (p, q) is a (di-)homotopy between paths p, q (` p, q : X ).

0

1

q

t0

t1

0

1

p

t0

t1

(A0

A1

)=

(1 01 0

) (B0

A1

)=

(0 11 0

)alive i.e. p ` A0 ∧ A1 alive i.e. q ` B0 ∧ A1

(in the style of Awodey, Voevodsky - although naively so, we wouldneed to encode cofibrations A0 → X etc. fully in the types!)

E. Goubault

Thanks for your attention!

http://acat.lix.polytechnique.frhttp://www.appliedtopology.org/

http://labex-digicosme.fr/

E. Goubault

http://acat.lix.polytechnique.fr

http://www.appliedtopology.org/

http://labex-digicosme.fr/

geometry of trace spaces - laboratoire paul...

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