Directed Algebraic Topology and Concurrency

Transcription

1 Directed Algebraic Topology and Concurrency Emmanuel Haucourt MPRI : Concurrency (2.3) Wednesday, the 4 th of January / 43

2 Locally ordered spaces Partially ordered spaces Partially ordered spaces Topology and Order, L. Nachbin, 1965 A partially ordered space (or pospace) is a topological space X together with a partial order on (the underlying set of) X such that {(a, b) X X a b} is a closed subset of X X. A pospace morphism is an order-preserving continuous map. Pospaces and their morphisms form the category PoSp. The underlying space of a pospace is Hausdorff. 2 / 43

3 Locally ordered spaces Partially ordered spaces Examples - The real line with standard topology and order. - Any subset a pospace with the induced topology and order. - The collection of compact subsets of a metric space equipped with the Hausdorff distance is a metric space. d H (K, K ) = sup { d(x, K ), d(x, K) x K; x K } d(x, K) = inf { d(x, k) k K } The induced topological space ordered by inclusion is a pospace. - Problem: there is no pospace on the circle whose collection of directed paths is { ρ(t) e iθ(t) ρ, θ : [0, r] R + increasing } 3 / 43

4 Locally ordered spaces Definition Ordered atlas Algebraic topology and concurrency, L. Fajstrup, É. Goubault, and M. Raußen, 1998 Let X be a Hausdorff space. An (ordered) chart on X is a pospace U whose underlying space is an open subset of X. An (ordered) atlas is a collection U of ordered charts on X such that: - the underlying spaces of the charts form a basis of the topology of X, and - for all U, V U for all x U V there exists W U such that x W U V and denoting by U W the relation induced by U on the underlying set of W, the restrictions of U and V to W match W. U W = W = V W Any subset of X inherits an ordered atlas from U. 4 / 43

5 Locally ordered spaces Definition Ordered atlas V x W U 5 / 43

6 Locally ordered spaces Definition Locally ordered space Two atlases on the same space are compatible when their union is still an atlas. The relation of compatibility is an equivalence relation. The union of all the atlases of a given equivalence class is still an atlas i.e. every equivalence class contains a greatest element for inclusion. A local pospace is a Hausdorff space togethter with an equivalence class of ordered atlas. 6 / 43

7 Locally ordered spaces Examples The locally ordered line Examples of equivalent atlases on R - {(I, ) I open interval of R}, - {(U, ) U open subset of R}, - {(U, U ) U open subset of R} where x U y stands for x y and [x, y] U, - {(U, U ) U open subset of R} where x U y is any extension of U. 7 / 43

8 Locally ordered spaces Examples The locally ordered circle Examples of equivalent atlases on S 1 - {(A, ) A open arc} where is the order induced by R and the restriction of the exponential map to an open subinterval of {t R e it A} of length at most 2π, - {(U, U ) U proper open subset of S 1 } where x U y means that the anticlockwise compact arc from x to y is included in U, - {(U, U ) U proper open subset of S1 } where U is any extension of the partial order U. 8 / 43

9 Locally ordered spaces Basic properties Local pospace morphisms An atlas morphism from U to V is a map f (between the underlying sets of U and V) such that for all x dom(f ) there exists an ordered chart U U and an ordered chart V V such that x U and f induces a pospace morphism from U to V (implicitly f (U) V ). e.g. t [0, 1] [2, 3] t + 2 (mod 4) [0, 1] [2, 3] Let f be an atlas morphism from U to V. If U U and V V then f is also an atlas morphism from U to V. A local pospace morphism from (X, [U] ) to (Y, [V] ) is a map from X to Y inducing an atlas morphism from U to V. A local pospace morphism defined over a locally ordered compact interval is called a directed path. 9 / 43

10 Locally ordered spaces Basic properties Pospaces as local pospaces Each pospace (X, ) can be seen as a local pospace (X, { (U, U ) U open subset of X }) The resulting functor is: - faithful - not injective on object (hence not an embedding) - not full 10 / 43

11 Locally ordered spaces Basic properties Directed loops on local pospaces A directed path δ on a local pospace X is constant iff its extremities are equal and there exists an ordered chart of some atlas of X that contains the image of δ. A vortex is a point every neighbourhood of which contains a non-constant directed loop. A local pospace has no vortex. 11 / 43

12 Locally ordered spaces Locally ordered metric graphs Ordered atlas on metric graphs Let B be the collection of open balls B of G such that - B is centred at a vertex and its radius is 1 3, or - B = {a} U for some arrow a and some open interval U ]0, 1[ of length 1 3. Let B, B B be such that B B. - If B is of the first kind, then so is B B. - If B, B are centred at v and v, we have then v = v and B B or B B 12 / 43

13 Locally ordered spaces Locally ordered metric graphs Ordered open stars An element B of B centred at v of radius r 1 is the disjoint union of {v} together with 3 - {a} ]0, r[ for each arrow a such that - a = v - {a} ]1 r, 1[ for each arrow a such that + a = v The partial order on B is characterized by the following constraints: - each branch {a} ]1 r, 1[ and {a} ]0, r[ inherits its order from R - {v} {a} ]0, r[ for each arrow a such that - a = v - {a} ]1 r, 1[ {v} for each arrow a such that + a = v We have B B B B B and B B B = B B = B B B The metric graph of G thus becomes a local pospace. The locally ordered metric graph construction is functorial. 13 / 43

14 Category theory Cartesian product in Set A B := { (a, b) a A and b B } There exist two mappings π A and π B π A : A B (a, b) A XXXXXXπ B : A B B a (a, b) b such that for all sets X the following map is a bijection Set[X, A B] Set[X, A] Set[X, B] h ( π A h, π B h ) 14 / 43

15 Category theory Cartesian product in a category C The object c is the Cartesian product (in C) of a and b when there exist two morphisms π a : c a and π b : c b such that for all objects x of C the following map is a bijection C[x, c] C[x, a] C[x, b] h ( π a h, π b h ) When such an object c exists we write c = a b 15 / 43

16 Category theory Cartesian product in the category of graphs (Grph) A s t V t A s = V A A s s V V t t The Cartesian product in Grph is deduced form the Cartesian product in Set 16 / 43

17 Category theory Examples of Cartesian products - The product of (X, Ω X ) and (Y, Ω Y ) in Top is given by X Y together with unions of subsets of the form U V with U Ω X and V Ω Y. It is the least topology making the projections continuous. - The product of (X, X ) and (Y, Y ) in Pos is given by X Y and the partial order defined by (x, y) (x, y ) when x X x and y Y y. It is the greatest partial order such that the projection are poset morphisms. - The product of (X, X ) and (Y, Y ) in PoSp is given by X Y and the product order X Y. - The product of (X, [U] ) and (Y, [V] ) in Lpo is X Y together with the collection of ordered charts U V with U U and V V. - The product of (X, d X ) and (Y, d Y ) in Met emb does not exist. - The product of (X, d X ) and (Y, d Y ) in Met ctr is given by X Y together with d ( (x, y), (x, y ) ) = max{d X (x, x ), d Y (y, y )}. - The product of (X, d X ) and (Y, d Y ) in Met top can also be given by X Y together with the Euclidean product d ( (x, y), (x, y ) ) = dx 2 (x, x ) + dy 2 (y, y ) - Categories of models of algebraic theories. 17 / 43

18 Category theory Infinite Cartesian product The product of a family (A i ) i I of objects of a category C, when it exists, is an object A i i together with projections π Aj : A i i A j such that the next mapping is a bijection. C(X, A i ) i h C(X, A i ) i ( π Ai h ) Infinite products of directed circle does not exist in Lpo. 18 / 43

19 Turning the discrete models into continuous ones Canonical partition G : A + - V G = V A ]0, 1[ G 1 G n = ( V 1 A 1 ]0, 1[ ) ( V n A n ]0, 1[ ) G 1 G n = points p of G1,..., Gn {p} ]0, 1[ dim(p 1,...,p n) where p = (p 1,..., p n), p i V i A i, and dim p = # { i {1,..., n} p i A i } B p = {p} ]0, 1[ dim(p 1,...,p n) is called a canonical block The collection of canonical blocks forms the canonical partition of G 1 G n. 19 / 43

20 Turning the discrete models into continuous ones Justifying de definition of discrete directed paths Let B p and B p be canonical blocks. If there exists a directed path starting in B p, ending in B p, and whose image is contained in B p B p then one of the following facts is satisfied: - for all i {1,..., n}, p i = p i or p i is the source of the arrow p i, or - for all i {1,..., n}, p i = p i or p i is the target of the arrow p i. 20 / 43

21 Turning the discrete models into continuous ones Discretization and lifting - Given a directed path γ on the local pospace G 1 G n we have a finite partition I 0 < < I N of dom(γ) such that for all k {0,..., N}, there exists a (necessarily unique) point p k such that γ(i k ) B p k. - The sequence p 0,..., p N is a directed path on (G 1,..., G n), it is called the discretization of γ and denoted by D(γ). - Given a directed path δ on (G 1,..., G n) there exists a directed path γ on G 1 G n whose discretization is δ, such a directed path γ is said to be a lifting of δ. 21 / 43

22 Turning the discrete models into continuous ones Example of discretization 22 / 43

23 Turning the discrete models into continuous ones Admissible directed paths and execution traces on G 1 G n The sequence of multi-instructions of a directed path γ on G 1 G n is that of its discretization of D(γ). A directed path on G 1 G n is admissible (resp. an execution trace) iff so is its discretization. The action of a directed path γ on G 1 G n on the right of a state σ is that of its discretization of D(γ). 23 / 43

24 Turning the discrete models into continuous ones Potential function on G 1 G n If the program under consideration is conservative, then we have the potential function F : G 1 G n S {multisets over {1,..., n}} The function F is constant on each canonical block. 24 / 43

25 Turning the discrete models into continuous ones Geometric model The forbidden region is B p forbidden points p of (G 1,..., G n) The directed continuous model is G 1 G n \ {forbidden region} 25 / 43

26 The motivating theorem Geometric models are sound and complete - Any directed path on a continuous model is admissible. - Conversely, for each admissible path on a continuous model which meets a forbidden point, there exists a directed path which avoids them and such that both directed paths induce the same sequence of multi-instructions. 26 / 43

27 The motivating theorem Trade off More mathematics for more properties? - Both discrete and geometric models are sound and complete. - The continuous models satisfy extra properties that are naturally expressed in terms of metrics. 27 / 43

28 The motivating theorem The geometric model of a conservative program The continuous model X of a conservative program whose running processes are G 1,..., G n is a sub-local pospace of G 1 G n. The continuous model X inherits a distance d X from the distances d Gi of the metric graphs G i d X (p, p ) = max { d Gi (p i, p i ) i {1,..., n}} The distance d X is in accordance with the fact that the execution time of the simultaneous execution of many processes is the longest execution time among that of the processes considered individually. 28 / 43

29 The motivating theorem Uniform distance between directed paths Given a compact Hausdorff space K and a metric space (X, d X ), the set of continuous maps from K to X can be equipped with the uniform distance d(f, g) = max{d X (f (k), g(k)) k K}. We consider the case where K = [0, r] is the domain of definition of a directed path and (X, d X ) is the geometric model of a conservative program. 29 / 43

30 The motivating theorem The main theorem Let B p and B p be canonical blocks of the geometric model X of a conservative program. Let dx [0,r] (B p, B p ) be the set of directed paths on X whose sources and targets lie in B p and B p respectively. Let γ be an element of dx [0,r] (B p, B p ). There exists an open ball Ω of dx [0,r] (B p, B p ), centred in γ, such that all the elements of Ω induce the same action on valuations. Moreover, if γ is an execution trace, then so are all the elements of Ω. 30 / 43

31 The motivating theorem Desynchronization one of the artifact used in the proof y:=2 P 2 γ 2 γ 3 γ 1 y:=2 x:=1 γ 3 P 1 y:=2 x:=1 γ 2 P 1 P 2 P 2 x:=1 P 1 x:=1 γ 1 P 1 y:=2 P 2 31 / 43

32 Directed homotopy between directed paths Standard homotopy of paths Let γ and δ be two paths on X defined over the segment [0, r] A homotopy from γ to δ is a continuous map h from [0, r] [0, q] to X such that - The mappings h(0, ) : s [0, q] h(0, s) and h(r, ) : s [0, q] h(r, s) are constant - The mappings h(, 0) : t [0, r] h(t, 0) and h(, q) : s [0, r] h(t, q) are γ and δ As a consequence we have γ(0) = δ(0) and γ(r) = δ(r). 32 / 43

33 Directed homotopy between directed paths Uniform distance and Curryfication Suppose that X is a metric space. For all compact Hausdorff space K, the homset Top(K, X ) with the (topology induced by the) uniform distance is denoted by X K The Curryfication ˆ ( ) induces a homeomorphism from X [0,r] [0,q] to ( X [0,r]) [0,q] (h : [0, r] [0, q] X ) (ĥ : [0, q] X [0,r] ) 33 / 43

34 Directed homotopy between directed paths The two faces of homotopies h is a continuous map from [0, r] [0, q] to X i.e. h Top [ [0, r] [0, q], X ] but is also a path from γ to δ in the space X [0,r] i.e. h Top [ [0, q], X [0,r]] γ [0, q] h x h y δ [0, r] The second point of view leads us to introduce the following notation γ x δ h y 34 / 43

35 Directed homotopy between directed paths Directed homotopy on a locally ordered space Let γ, δ Lpo([0, r], X ) such that - γ = - δ and + γ = + δ. - A directed homotopy from γ to δ is a homotopy of paths h : [0, r] [0, q] X that induces a local pospace morphism. γ h δ - A weakly directed homotopy from γ to δ is a homotopy of paths h : [0, r] [0, q] X whose intermediate paths h(, s), for s [0, q], are directed. - Any directed homotopy is a weakly directed homotopy. The converse is false. 35 / 43

36 Directed homotopy between directed paths Theorem Two weakly dihomotopic paths on the geometric model of a conservative program induce the same action on valuations. Moreover, if one of them is an execution trace, then so is the other. 36 / 43

37 Directed homotopy between directed paths Proof By a standard result from general topology and the Curryfication of h, namely is a continuous path on dx [0,r] (p, p ). ĥ : s [0, q] (t [0, r] h(t, s) X ) The image of ĥ is thus compact, so we cover it with open balls given by the main theorem of geometric models. By the Lebesgue number theorem there exists a real number ε > 0 such that s s ε implies that ĥ(s) and ĥ(s ) belong to the same open ball from the covering. The conclusion follows considering the sequence ĥ(0), ĥ(ε), ĥ(2ε), ĥ(3ε),, ĥ(nε), ĥ(q) where n is the greatest natural number such that nε q. 37 / 43

38 Directed homotopy between directed paths Concatenation of homotopies vertical composition Let g : [0, r] [0, q ] X and h : [0, r] [0, q] X be homotopies from γ to ξ and from ξ to δ. The mapping h g : [0, r] [0, q + q ] X defined by h g(t, s) = { g(t, s) if 0 s q h(t, s q) if q s q + q is a homotopy from γ to δ. γ g ξ h δ γ h g δ If g and h are (weakly) directed homotopies, then so is their concatenation h g. 38 / 43

39 Directed homotopy between directed paths Homotopy and dihomotopy relations An anti-directed homotopy from γ to δ is a homotopy of paths h : [0, r] [0, q] X such that (t, s) h(t, q s) is a directed homotopy from δ to γ. An elementary homotopy between γ to δ is a homotopy of paths h : [0, r] [0, q] X obtained as a finite concatenation of directed homotopies and anti-directed homotopies. Two paths γ and γ are said to be homotopic when there exists a homotopy between them. We have the equivalence relation h between paths on a topological space. They are said to be dihomotopic when there exists an elementary homotopy between them. We have the equivalence relation d between directed paths on a locally ordered space. They are said to be weakly dihomotopic when there exists a weakly directed homotopy between them. We have the equivalence relation w between directed paths on a locally ordered space. 39 / 43

40 Directed homotopy between directed paths An important remark If h is a homotopy from γ to γ on the topological space X and f : X Y is a continuous map, then f h is a homotopy from f γ to f γ on the topological space Y. If h is a (weakly) directed homotopy from γ to γ on the local pospace space X and f : X Y is a local pospace morphism, then f h is a (weakly) directed homotopy from f γ to f γ on the local pospace space Y. If γ, γ : [0, r] X are ((weakly) di)homotopic, then so are f γ, f γ : [0, r] Y. 40 / 43

41 Directed homotopy between directed paths Reparametrization An increasing and surjective map θ : [0, r] [0, r] is called a reparametrization. The mapping h : (t, s) [0, r] [0, 1] θ(t) + s (max(t, θ(t)) θ(t)) [0, r] is a directed homotopy from θ to max(id [0,r], θ). If γ : [0, r] X is a directed path on the local pospace X, then γ h is a directed homotopy from γ θ to γ max(id [0,r], θ) Therefore γ and γ θ are dihomotopic. 41 / 43

42 Directed homotopy between directed paths Images of directed paths on a pospace Theorem The image of a nonconstant directed path on a pospace is isomorphic to [0, 1]. Corollary Two directed paths on a posapce having the same image are dihomotopic. proof: Suppose that im(γ) = im(γ ). φ : [0, r] im(γ) a pospace isomorphism. φ 1 γ and φ 1 γ are reparametrization. We have h an elementary homotopy from φ 1 γ to φ 1 γ. Hence φ h is an elementary homotopy from γ and γ. 42 / 43

43 Directed homotopy between directed paths Programs with mutex only a result by É. Goubault and S. Mimram Let X be the geometric model of a conservative program whose semaphores have arity 1 (mutex), then two directed paths on X are dihomotopic if and only if they are homotopic. 43 / 43