A mathematical model for concurrent parallel computing
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1 A mathematical model for concurrent parallel computing Cleveland State University p/ November 11, University of Oregon
2 The end of Moore s Law?
3 Example: Multi-core processors Intel AMD
4 Example: Internet database
5 Classical non-parallel computing Process processor private resources A process with its own private resources
6 Concurrent parallel computing Process 1 Process 2... Process N Several processes with shared resources
7 A mathematical model Concurrent systems can be modeled by spaces in which only some certain paths are allowed. Definition A directed space is a topological space X with a distinguished set of paths, γ : [0,1] X, called directed paths. Example
8 A concurrent system Example 2 processes using 2 shared resources a and b which can only be used by one process at a time Notation Px - a process locks resource x Vx - a process releases resource x Program The first process: Pa Pb Vb Va The second process: Pb Pa Va Vb
9 The Swiss flag Example Vb Va Pa Pb Pa Pb Vb Va
10 The Swiss flag Example Vb Va Pa Pb Pa Pb Vb Va
11 The Swiss flag Example Vb Va Pa Pb Pa Pb Vb Va Problem: The state space is infinite.
12 The essential schedules of the Swiss flag Example Vb Va Pa Pb Pa Pb Vb Va This is a subspace of the Swiss flag.
13 Goal Develop a framework for concurrency where equivalences are accounted for. We would like equivalences that allow a piece-by-piece analysis. This will make the analysis of large programs tractable.
14 Goal Develop a framework for concurrency where equivalences are accounted for. We would like equivalences that allow a piece-by-piece analysis. This will make the analysis of large programs tractable. Idea Use algebraic topology.
15 Undirected equivalences Definition Given continuous maps f,g : B C, B g C B f
16 Undirected equivalences Definition Given continuous maps f,g : B C, I B H g C B f
17 Undirected equivalences Definition Given continuous maps f,g : B C, I B H g C B f a homotopy between f and g is a continuous map H : B I C restricting to f and g. This is an equivalence relation. Write H : f g.
18 Undirected equivalences Definition Spaces B,C are homotopy equivalent if there are maps f : B C : g such that g f Id B and f g Id C. B f C g
19 Directed spaces Definition A directed space is a topological space X with a distinguised set of directed paths. A directed map is a continuous map f : X Y between directed spaces such that if γ : [0,1] X is a directed paths in X, then f (γ) is a directed path in Y. Remark Subspaces, products, and quotients of directed spaces inherit a directed space structure.
20 Directed equivalences Definition A homotopy between directed maps f,g : B C is a directed map H : B I C restricting to f and g. Write H : f g. I B H g C B f
21 Directed equivalences Definition Write f g if there is a chain of dihomotopies f f 1 f 2... f n g. Po-Spaces B,C are dihomotopy equivalent if there are dimaps f : B C : g such that g f Id B and f g Id C.
22 The fundamental category Definition The fundamental category π 1 (X) has objects: the points in X morphisms: homotopy classes of directed paths
23 The fundamental category Definition The fundamental category π 1 (X) has objects: the points in X morphisms: homotopy classes of directed paths Problem The fundamental category is enormous.
24 Full subcats fund.bip.graph Retracts Extremal models Full subcategories of the fundamental category Plan We would like to derive a small category from the fundamental category that still contains useful information. Definition Given A X, let π 1 (X,A) have objects: points in A morphisms: homotopy classes of paths in X
25 The fundamental bipartite graph Full subcats fund.bip.graph Retracts Extremal models Definition For (X,dX) write x y if there exists a dipath γ with γ(0) = 1 and γ(1) = y. This gives X a preorder.
26 The fundamental bipartite graph Full subcats fund.bip.graph Retracts Extremal models Definition For (X,dX) write x y if there exists a dipath γ with γ(0) = 1 and γ(1) = y. This gives X a preorder. Definition Let Min(X) = {a X a a = a = a}. Let Max(X) = {b X b b = b = b }. Definition (B) The fundamental bipartite graph of X is π 1 (X,Min(X) MaxX).
27 Full subcats fund.bip.graph Retracts Extremal models Example of the fundamental bipartite graph b a b a
28 Future retracts Full subcats fund.bip.graph Retracts Extremal models Definition A future retract of π 1 (X) moves each x X along a directed path in X to a point x + which has the same future. c a A! b x x + [γ x ] x y P + : π 1 (X) π 1 (X,A)
29 Past retracts Full subcats fund.bip.graph Retracts Extremal models Definition A past retract of π 1 (X) moves each x X backwards along a directed path in X to a point x which has the same past. b! x [γ x ] x a A a P : π 1 (X) π 1 (X,A)
30 Extremal models Full subcats fund.bip.graph Retracts Extremal models Definition (B) An extremal model is a chain of future retracts and past retracts π 1 (X) P+ 1 π 1 (X,X 1 ) P 2 π 1 (X,X 2 ) P P± n π 1 (X,A), such that Min(X) Max(X) A.
31 Extremal models Full subcats fund.bip.graph Retracts Extremal models Definition (B) An extremal model is a chain of future retracts and past retracts π 1 (X) P+ 1 π 1 (X,X 1 ) P 2 π 1 (X,X 2 ) P P± n π 1 (X,A), such that Min(X) Max(X) A. Proposition (B) An extremal model induces an injection of fundamental bipartite graphs. Theorem (B) If X is a compact and is a partial order, then extremal models induce an isomorphism of fundamental bipartite graphs.
32 Examples of extremal models Full subcats fund.bip.graph Retracts Extremal models b b b a a a
33 Full subcats fund.bip.graph Retracts Extremal models An extremal model for the Swiss flag d d d c b c b c b a a a
34 Examples of extremal models Full subcats fund.bip.graph Retracts Extremal models c c c b b b a a a
35 An extremal model of S 1 Full subcats fund.bip.graph Retracts Extremal models Let x S 1. There is a future retract P + : π 1 ( S 1 ) π 1 ( S 1,x) = (N,+). It is a minimal extremal model.
36 Piecewise analysis Full subcats fund.bip.graph Retracts Extremal models Theorem (B) Extremal models can be constructed in a piecewise manner.
37 Full subcats fund.bip.graph Retracts Extremal models Van Kampen for extremal models example
38 Full subcats fund.bip.graph Retracts Extremal models Van Kampen for extremal models example
39 Full subcats fund.bip.graph Retracts Extremal models Van Kampen for extremal models example
40 Summary Summary Directed spaces provide a good mathematical model for concurrent parallel computing. Using directed homotopies one can hope to cope with the state space explosion. The homotopy classes of directed paths assemble into the fundamental category. Minimal extremal models provide a way to generalize the fundamental group to directed spaces. Extremal models can be constructed in a piecewise manner.
41 Applications Summary L. Fajstrup, E. Goubault, and M. Raussen (1998) used geometry and directed topology to give an algorithm for detecting deadlocks, unsafe regions and inaccessible regions for po-spaces such as the Swiss flag, in any dimension. E. Goubault and E. Haucourt (2005) reduced the fundamental category to components to develop a static analyzer (ALCOOL) of concurrent parallel programs.
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