Cut-o ff theorems for deadlocks and serializability Lisbeth Fajstrup - - PowerPoint PPT Presentation

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Cut-o ff theorems for deadlocks and serializability Lisbeth Fajstrup - - PowerPoint PPT Presentation

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Cut-o ff theorems for deadlocks and serializability Lisbeth Fajstrup Department of Mathematics Aalborg University ACAT in Bremen, 2013 Lisbeth Fajstrup (AAU) Cut-o ff theorems for deadlocks and serializability July 2013 1 / 28 Cut-O ff

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SLIDE 1

Cut-off theorems for deadlocks and serializability

Lisbeth Fajstrup

Department of Mathematics Aalborg University

ACAT in Bremen, 2013

Lisbeth Fajstrup (AAU) Cut-off theorems for deadlocks and serializability July 2013 1 / 28

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Cut-Off Theorems

General

Property P holds for all n 2 N if and only P holds for n  M Think of n as a dimension. Here: Given thread T, T n means n copies of T run in parallel. The property P is deadlock free in one theorem and serializable in the

  • ther.

Lisbeth Fajstrup (AAU) Cut-off theorems for deadlocks and serializability July 2013 2 / 28

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SLIDE 3

PV -programs - controlling concurrency through locks

A set of shared resources R - memory, printers,... A capacity function κ : R ! N. PV programs p are defined by the grammar p ::= Pa | Va | p.p | p|p | p + p | p∗ Pa is a request to access the resource a, if granted acces, lock it. Va releases the resource. A program without the parallel operator is a thread. Resource r may be accessed by at most κ(r) threads at a time. At a final point, the program has released all resources. At an initial point, no resources are locked

Lisbeth Fajstrup (AAU) Cut-off theorems for deadlocks and serializability July 2013 3 / 28

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SLIDE 4

Geometrically

One thread is represented by a graph. n threads in parallel are represented by a product of n graphs with “holes” where a resource is locked above its capacity. T1 = T2 = Pa.Pb.Va.Pa.Vb.Va κ ⌘ 1 Pa Pb Va Pa Vb Va Pa Pb Va Pa Vb Va A A A A B T1 T2

Lisbeth Fajstrup (AAU) Cut-off theorems for deadlocks and serializability July 2013 4 / 28

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SLIDE 5

An execution is a directed path

Pa Pb Va Pa Vb Va Pa Pb Va Pa Vb Va A A A A B T1 T2

Lisbeth Fajstrup (AAU) Cut-off theorems for deadlocks and serializability July 2013 5 / 28

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Deadlock - no directed paths leave the point

Pa Pb Va Pa Vb Va Pa Pb Va Pa Vb Va A A A A B T1 T2

Lisbeth Fajstrup (AAU) Cut-off theorems for deadlocks and serializability July 2013 6 / 28

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SLIDE 7

Definition

A state x = (x1, . . . , xn) 2 T1 ⇥ T2 ⇥ · · · ⇥ Tn is a deadlock if The only dipaths starting in x are constant. x is reachable: There is a dipath from 0 to x x 6= > - not all Ti have finished If x is a deadlock, then xi = > or xi = Pr(i) and κ(r(i)) other threads hold a lock on r(i) at x

Lisbeth Fajstrup (AAU) Cut-off theorems for deadlocks and serializability July 2013 7 / 28

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SLIDE 8

A cut-off theorem for deadlocks

Theorem 1

Let T be a PV thread accessing resources R with capacity κ : R ! N. Let T n be n copies of T run in parallel. T n is deadlock free for all n if and only if T M is deadlock free, where M = Σr∈Rκ(r).

Lisbeth Fajstrup (AAU) Cut-off theorems for deadlocks and serializability July 2013 8 / 28

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The bound M is sharp.

Theorem 2

Given R = {r1, . . . , rk} with capacity κ : R ! N, the thread T = Pr1Pr2Vr1Pr3Vr2 . . . PrkVrk−1Pr1VrkVr1 satisfies: There is a deadlock in T M (and hence for all n M) where M = Σr∈Rκ(r). There are no deadlocks in T n for n  M

Lisbeth Fajstrup (AAU) Cut-off theorems for deadlocks and serializability July 2013 9 / 28

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The bound M is sharp.

Theorem 2

Given R = {r1, . . . , rk} with capacity κ : R ! N, the thread T = Pr1Pr2Vr1Pr3Vr2 . . . PrkVrk−1Pr1VrkVr1 satisfies: There is a deadlock in T M (and hence for all n M) where M = Σr∈Rκ(r). There are no deadlocks in T n for n  M Proof: The deadlock is x = (x1, . . . , x1, x2, . . . , x2, . . . , xk . . . xk) where for i 6= 1, xi = Pri and xi is repeated κ(ri−1) times. x1 is the last Pr1 and is repeated κ(rk) times. All resources r are locked κ(r) times. Hence x is a deadlock and 6= >. (Need to check reachability and no deadlocks in lower dimensions)

Lisbeth Fajstrup (AAU) Cut-off theorems for deadlocks and serializability July 2013 9 / 28

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  • Example. When κ ⌘ 1

x = (x1, x2, . . . , xk) and M = k: T1 requests r1 and has a lock on rk Ti requests ri and has a lock on ri+1 There are no deadlocks in T n for n < M: If y is a deadlock, then there is a yi1 = Pr(i1) and another yi2 with a lock on r(i1), so yi2 = Pr(i1 + 1) (or the last P(r1) if r(i1) = rk. Hence, y is a permutation of (x, >, >, . . . , >). Similarly for general κ. Still need to see that x is reachable.

Lisbeth Fajstrup (AAU) Cut-off theorems for deadlocks and serializability July 2013 10 / 28

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SLIDE 12

Example

T = PaPbVaPcVbPaVcVa, κ ⌘ 1. T 3 has a deadlock at (6, 2, 4) Pa Pb Va Pc Vb Pa Vc Va Pa Pb Va Pc Vb Pa Vc Va A A A A B C T1 T2

Lisbeth Fajstrup (AAU) Cut-off theorems for deadlocks and serializability July 2013 11 / 28

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x = (x1, . . . , x1, x2, . . . , x2, . . . , xk . . . xk) is reachable

T = Pr1Pr2Vr1Pr3Vr2 . . . PrkVrk−1Pr1VrkVr1 a dipath from 0 to x is composed of the pieces: γ0 : 0 ! (x1, 0) serially - one coordinate at a time. OBS, now hold κ(rk) locks on rk

Lisbeth Fajstrup (AAU) Cut-off theorems for deadlocks and serializability July 2013 12 / 28

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x = (x1, . . . , x1, x2, . . . , x2, . . . , xk . . . xk) is reachable

T = Pr1Pr2Vr1Pr3Vr2 . . . PrkVrk−1Pr1VrkVr1 a dipath from 0 to x is composed of the pieces: γ0 : 0 ! (x1, 0) serially - one coordinate at a time. OBS, now hold κ(rk) locks on rk γ1 : (x1, 0) ! (x1, 0, xk) one coordinate at a time. Now also κ(rk−1) locks on rk−1

Lisbeth Fajstrup (AAU) Cut-off theorems for deadlocks and serializability July 2013 12 / 28

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x = (x1, . . . , x1, x2, . . . , x2, . . . , xk . . . xk) is reachable

T = Pr1Pr2Vr1Pr3Vr2 . . . PrkVrk−1Pr1VrkVr1 a dipath from 0 to x is composed of the pieces: γ0 : 0 ! (x1, 0) serially - one coordinate at a time. OBS, now hold κ(rk) locks on rk γ1 : (x1, 0) ! (x1, 0, xk) one coordinate at a time. Now also κ(rk−1) locks on rk−1 γj : (x1, 0, xk−j+2, . . . , xk) ! (x1, 0, xk−j+1, . . . , xk). Now κ(ri) locks

  • n rk−j, . . . , rk.

For j = 2, . . . , k 1.

Lisbeth Fajstrup (AAU) Cut-off theorems for deadlocks and serializability July 2013 12 / 28

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SLIDE 16

x = (x1, . . . , x1, x2, . . . , x2, . . . , xk . . . xk) is reachable

T = Pr1Pr2Vr1Pr3Vr2 . . . PrkVrk−1Pr1VrkVr1 a dipath from 0 to x is composed of the pieces: γ0 : 0 ! (x1, 0) serially - one coordinate at a time. OBS, now hold κ(rk) locks on rk γ1 : (x1, 0) ! (x1, 0, xk) one coordinate at a time. Now also κ(rk−1) locks on rk−1 γj : (x1, 0, xk−j+2, . . . , xk) ! (x1, 0, xk−j+1, . . . , xk). Now κ(ri) locks

  • n rk−j, . . . , rk.

For j = 2, . . . , k 1. Need: No resource is locked above its capacity along γ. Let ρi(y) be the number of locks held on ri at y 2 T M.

Lisbeth Fajstrup (AAU) Cut-off theorems for deadlocks and serializability July 2013 12 / 28

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x = (x1, . . . , x1, x2, . . . , x2, . . . , xk . . . xk) is reachable

T = Pr1Pr2Vr1Pr3Vr2 . . . PrkVrk−1Pr1VrkVr1 a dipath from 0 to x is composed of the pieces: γ0 : 0 ! (x1, 0) serially - one coordinate at a time. OBS, now hold κ(rk) locks on rk γ1 : (x1, 0) ! (x1, 0, xk) one coordinate at a time. Now also κ(rk−1) locks on rk−1 γj : (x1, 0, xk−j+2, . . . , xk) ! (x1, 0, xk−j+1, . . . , xk). Now κ(ri) locks

  • n rk−j, . . . , rk.

For j = 2, . . . , k 1. Need: No resource is locked above its capacity along γ. Let ρi(y) be the number of locks held on ri at y 2 T M. For γ0: ρi(γ0(t))  1 for i 6= k ρk(γ0(t))  κ(rk). ρi(γ1(t))  1 for i 6= k 1, k. ρk(γ1(t)) = k. ρk−1(γ1(t))  κ(rk−1). ρi(γj(t))  1 for i  k j + 1. ρi(γj(t)) = κ(ri) for i k j + 2. ρk−j+1(γj(t))  κ(rk−j+1).

Lisbeth Fajstrup (AAU) Cut-off theorems for deadlocks and serializability July 2013 12 / 28

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Proof of theorem 1

Theorem 1

Let T be a PV thread accessing resources R with capacity κ : R ! N. Let T n be n copies of T run in parallel. T n is deadlock free for all n if and only if T M is deadlock free, where M = Σr∈Rκ(r). Prove: If there is a deadlock in T n, then there is a deadlock in T M.

Lisbeth Fajstrup (AAU) Cut-off theorems for deadlocks and serializability July 2013 13 / 28

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Proof of theorem 1

Theorem 1

Let T be a PV thread accessing resources R with capacity κ : R ! N. Let T n be n copies of T run in parallel. T n is deadlock free for all n if and only if T M is deadlock free, where M = Σr∈Rκ(r). Prove: If there is a deadlock in T n, then there is a deadlock in T M. For n < M: Let x = (x1, . . . , xn) be a (reachable) deadlock in T n. I.e. xi = Prj(i) and ρj(i)(x) = κ(rj(i)) or xi = >. Then ˜ x = (x1, . . . , xn, >, . . . , >) 2 T M is a (reachable) deadlock in T M, since ρi(˜ x) = ρi(x) for all i

Lisbeth Fajstrup (AAU) Cut-off theorems for deadlocks and serializability July 2013 13 / 28

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Proof of theorem 1

Theorem 1

Let T be a PV thread accessing resources R with capacity κ : R ! N. Let T n be n copies of T run in parallel. T n is deadlock free for all n if and only if T M is deadlock free, where M = Σr∈Rκ(r). Prove: If there is a deadlock in T n, then there is a deadlock in T M. For n < M: Let x = (x1, . . . , xn) be a (reachable) deadlock in T n. I.e. xi = Prj(i) and ρj(i)(x) = κ(rj(i)) or xi = >. Then ˜ x = (x1, . . . , xn, >, . . . , >) 2 T M is a (reachable) deadlock in T M, since ρi(˜ x) = ρi(x) for all i ˜ x is reachable: There is a serial path to (0, >, . . . , >). Compose with (γ(t), >, . . . , >), where γ(t) : 0 ! x 2 T n

Lisbeth Fajstrup (AAU) Cut-off theorems for deadlocks and serializability July 2013 13 / 28

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Proof of Theorem 1. n > M

x = (x1, . . . , xn) is a (reachable) deadlock in T n. Construct the directed wait for graph G(x) = (V , E). V = {x1, . . . , xn} There is an edge E(xi, xk) if xi = Prj and ρj(xk) = 1.

Lisbeth Fajstrup (AAU) Cut-off theorems for deadlocks and serializability July 2013 14 / 28

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Proof of Theorem 1. n > M

x = (x1, . . . , xn) is a (reachable) deadlock in T n. Construct the directed wait for graph G(x) = (V , E). V = {x1, . . . , xn} There is an edge E(xi, xk) if xi = Prj and ρj(xk) = 1. Properties of G(x) If xi = >, then the vertex xi is isolated. If xi = Prj, then the vertex xi has κ(rj) outgoing edges. There are circuits in G(x): Start a walk at a non-isolated vertex. If xj is the target of an edge, then xj 6= >, so the walk continues. G(x) is finite, so there is a circuit L.

Lisbeth Fajstrup (AAU) Cut-off theorems for deadlocks and serializability July 2013 14 / 28

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Proof of Theorem 1. n > M

x = (x1, . . . , xn) is a (reachable) deadlock in T n. Construct the directed wait for graph G(x) = (V , E). V = {x1, . . . , xn} There is an edge E(xi, xk) if xi = Prj and ρj(xk) = 1. Properties of G(x) If xi = >, then the vertex xi is isolated. If xi = Prj, then the vertex xi has κ(rj) outgoing edges. There are circuits in G(x): Start a walk at a non-isolated vertex. If xj is the target of an edge, then xj 6= >, so the walk continues. G(x) is finite, so there is a circuit L. Let " L be the vertices reachable from L. Let ˜ x = (xi1, xi2, . . . , xim) where xij are all the vertices in " L

Lisbeth Fajstrup (AAU) Cut-off theorems for deadlocks and serializability July 2013 14 / 28

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˜ x = (xi1, xi2, . . . , xim) where xij are all the vertices in " L, future of a circuit in the wait-for-graph. Claim:

1 ˜

x is a (reachable) deadlock.

2 m  M

Reachability: Let γ : 0 ! x in T n. The restriction to Ti1, . . . Tim is a dipath 0 ! ˜ x

Lisbeth Fajstrup (AAU) Cut-off theorems for deadlocks and serializability July 2013 15 / 28

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The restricted Wait for Graph - future of the loop

xi1 xi2 xi3 xi4 xi5 xil κ(r(il)) locks κ(r(i1)) locks κ(r(i1)) edges xij 6= >, so xij = P(r(ij)). In G(x), xij = P(r(ij)) had κ(r(ij)) outgoing

  • edges. It still has in G(˜

x), since " L contains all targets. Hence, ˜ x is a deadlock.

Lisbeth Fajstrup (AAU) Cut-off theorems for deadlocks and serializability July 2013 16 / 28

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m  M

In G(˜ x), all the m vertices are targets. Hence, they hold a lock on a resource. The maximal number of locks at an allowed state is M = Σr∈Rκ(r) Hence, there are at most M vertices in G(˜ x) (Less, if some xi hold a lock on more than one resource.)

Lisbeth Fajstrup (AAU) Cut-off theorems for deadlocks and serializability July 2013 17 / 28

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Now to something different: Serializability

Pa Pb Va Pa Vb Va Pa Pb Va Pa Vb Va A A A A B T1 T2 Two executions up to dihomotopy. Equivalent to the serial executions T1.T2 and T2.T1

Lisbeth Fajstrup (AAU) Cut-off theorems for deadlocks and serializability July 2013 18 / 28

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Serializability

T = PaPbVaPcVbPaVcVa is not serializable. Pa Pb Va Pc Vb Pa Vc Va Pa Pb Va Pc Vb Pa Vc Va A A A A B C T1 T2 4 Executions up to dihomotopy. Two serial executions (green). Two non-serializable executions (red).

Lisbeth Fajstrup (AAU) Cut-off theorems for deadlocks and serializability July 2013 19 / 28

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Serializability - a cut-off theorem

Definition

T1|T2 . . . |Tn is serializable if all execution paths are dihomotopic to a serial execution Ti1.Ti2....Tin.

Theorem 3

Let T be a PV -thread accessing resources R all of capacity 1. Let T n be n copies of T run in parallel. Then T n is serializable if and only is T 2 is serializable.

Lisbeth Fajstrup (AAU) Cut-off theorems for deadlocks and serializability July 2013 20 / 28

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In general studying pairwise interaction is not enough

Example

Let T1 = PcVcPaVa, T2 = PcVcPbVb, T3 = PaVaPbVb, Each pair Ti, Tj shares one resource. T1 T2 T3

Lisbeth Fajstrup (AAU) Cut-off theorems for deadlocks and serializability July 2013 21 / 28

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Schedules as in Raussens Trace algorithm

When κ = 1 All conflict n-rectangles are ⇥n

k=1Ik, Ik = I except for two directions

Ii =]ai, bi[, Ij =]aj, bj[ One choice at a given such n-rectangle “above or below” in the ij-plane - i waits or j waits.

Lisbeth Fajstrup (AAU) Cut-off theorems for deadlocks and serializability July 2013 22 / 28

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T = PaVa, T 3

T T T 3! serial executions, pairwise inequivalent.

Lisbeth Fajstrup (AAU) Cut-off theorems for deadlocks and serializability July 2013 23 / 28

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SLIDE 33

T 2 serializability

If choice at one rectangle implies choice at all other rectangles - below all or above all rectangles Equivalently: Only schedules J = ({1}, {1}, . . . , {1}) and J = ({2}, {2}, . . . , {2}) are allowed. Equivalently: The closure of the forbidden area under adding unreachable and unsafe areas is connected. OBS: Two phase locking is not required. PaPbVaPcVb PdVcVd Pa Pb Va Pc Vb Pd Vc Vd A D B C T1 T2 Pa Pb Va Pa Vb Va Pa Pb Va Pa Vb Va A A A A B T1 T2

Lisbeth Fajstrup (AAU) Cut-off theorems for deadlocks and serializability July 2013 24 / 28

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T 2 serializability

If choice at one rectangle implies choice at all other rectangles - below all or above all rectangles Equivalently: Only schedules J = ({1}, {1}, . . . , {1}) and J = ({2}, {2}, . . . , {2}) are allowed. Equivalently: The closure of the forbidden area under adding unreachable and unsafe areas is connected. OBS: Two phase locking is not required. PaPbVaPcVb PdVcVd Pa Pb Va Pc Vb Pd Vc Vd A D B C T1 T2 A A A A B Pd Pd Pd Vd Vd Vd Pd Pd Pd Vd Vd Vd D D D D D D D D D T1 T2

Lisbeth Fajstrup (AAU) Cut-off theorems for deadlocks and serializability July 2013 24 / 28

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SLIDE 35

T n

If T is non-trivial, all the n! serial executions are non-equivalent, since They have different schedules wrt. the n·(n−1)

2

n-rectangles induced by just one PV -interval. Equivalently: Their projections to at least one of the T 2 are non-equivalent.

Lisbeth Fajstrup (AAU) Cut-off theorems for deadlocks and serializability July 2013 25 / 28

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Proof of Theorem 3

Suppose T 2 is serializable, then

1 Let ]ar, br[ correspond to a lock Pr, Vr. There are n! schedules for

the rectangles {⇥n

k=1Ik, Ik =]ar, br[ for k = i, j, i < j 2 [1 : n]} -

corresponding to the serial executions.

2 Fix i, j There are two schedules - i last for all or j last for all the

rectangles {I ⇥ I ⇥ · · · ⇥]ar

s, br s[⇥I · · · ⇥]ar t, br t[⇥I ⇥ · · · ⇥ I} where

]ar

s, br s[⇥]ar t, br t[ are forbidden rectangles in T 2

3 Choose one of the n! schedules for 1); that fixes all schedules in 2).

Hence, there are n! inequivalent schedules.

4 These are all realized by serial executions. Lisbeth Fajstrup (AAU) Cut-off theorems for deadlocks and serializability July 2013 26 / 28

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Serializability for κ > 1

There are non-serializable executions. All serial executions are equivalent (all boundary 2-cells are allowed) The scheduling algorithm calculates components of the trace space. More than one component , non-serializable.

Lisbeth Fajstrup (AAU) Cut-off theorems for deadlocks and serializability July 2013 27 / 28

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More to do

Serializability for κ > 1 - guess: There is a homological obstruction . This needs to be described specifically for the symmetric case. Cut-off for other properties - linearizability?

Lisbeth Fajstrup (AAU) Cut-off theorems for deadlocks and serializability July 2013 28 / 28