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

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 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 other. Lisbeth Fajstrup (AAU) Cut-o ff theorems for deadlocks and serializability July 2013 2 / 28

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 ::= P a | V a | p . p | p | p | p + p | p ∗ P a is a request to access the resource a , if granted acces, lock it. V a 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-o ff theorems for deadlocks and serializability July 2013 3 / 28

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. T 1 = T 2 = Pa . Pb . Va . Pa . Vb . Va κ ⌘ 1 T 2 Va Vb A A Pa B Va Pb A A Pa T 1 Pa Pb Va Pa Vb Va Lisbeth Fajstrup (AAU) Cut-o ff theorems for deadlocks and serializability July 2013 4 / 28

An execution is a directed path T 2 Va Vb A A Pa B Va Pb A A Pa T 1 Pa Pb Va Pa Vb Va Lisbeth Fajstrup (AAU) Cut-o ff theorems for deadlocks and serializability July 2013 5 / 28

Deadlock - no directed paths leave the point T 2 Va Vb A A Pa B Va Pb A A Pa T 1 Pa Pb Va Pa Vb Va Lisbeth Fajstrup (AAU) Cut-o ff theorems for deadlocks and serializability July 2013 6 / 28

Definition A state x = ( x 1 , . . . , x n ) 2 T 1 ⇥ T 2 ⇥ · · · ⇥ 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 x i = > or x i = Pr ( i ) and κ ( r ( i )) other threads hold a lock on r ( i ) at x Lisbeth Fajstrup (AAU) Cut-o ff theorems for deadlocks and serializability July 2013 7 / 28

A cut-o ff 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 = Σ r ∈ R κ ( r ). Lisbeth Fajstrup (AAU) Cut-o ff theorems for deadlocks and serializability July 2013 8 / 28

The bound M is sharp. Theorem 2 Given R = { r 1 , . . . , r k } with capacity κ : R ! N , the thread T = Pr 1 Pr 2 Vr 1 Pr 3 Vr 2 . . . Pr k Vr k − 1 Pr 1 Vr k Vr 1 satisfies: There is a deadlock in T M (and hence for all n � M ) where M = Σ r ∈ R κ ( r ). There are no deadlocks in T n for n  M Lisbeth Fajstrup (AAU) Cut-o ff theorems for deadlocks and serializability July 2013 9 / 28

The bound M is sharp. Theorem 2 Given R = { r 1 , . . . , r k } with capacity κ : R ! N , the thread T = Pr 1 Pr 2 Vr 1 Pr 3 Vr 2 . . . Pr k Vr k − 1 Pr 1 Vr k Vr 1 satisfies: There is a deadlock in T M (and hence for all n � M ) where M = Σ r ∈ R κ ( r ). There are no deadlocks in T n for n  M Proof: The deadlock is x = ( x 1 , . . . , x 1 , x 2 , . . . , x 2 , . . . , x k . . . x k ) where for i 6 = 1, x i = Pr i and x i is repeated κ ( r i − 1 ) times. x 1 is the last Pr 1 and is repeated κ ( r k ) 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-o ff theorems for deadlocks and serializability July 2013 9 / 28

Example. When κ ⌘ 1 x = ( x 1 , x 2 , . . . , x k ) and M = k : T 1 requests r 1 and has a lock on r k Ti requests r i and has a lock on r i +1 There are no deadlocks in T n for n < M : If y is a deadlock, then there is a y i 1 = Pr ( i 1) and another y i 2 with a lock on r ( i 1), so y i 2 = Pr ( i 1 + 1) (or the last P ( r 1) if r ( i 1) = r k . Hence, y is a permutation of ( x , > , > , . . . , > ). Similarly for general κ . Still need to see that x is reachable. Lisbeth Fajstrup (AAU) Cut-o ff theorems for deadlocks and serializability July 2013 10 / 28

Example T = PaPbVaPcVbPaVcVa , κ ⌘ 1. T 3 has a deadlock at (6 , 2 , 4) T 2 Va Vc A A Pa C Vb Pc B Va Pb A A Pa T 1 Pa Pb Va Pc Vb Pa Vc Va Lisbeth Fajstrup (AAU) Cut-o ff theorems for deadlocks and serializability July 2013 11 / 28

x = ( x 1 , . . . , x 1 , x 2 , . . . , x 2 , . . . , x k . . . x k ) is reachable T = Pr 1 Pr 2 Vr 1 Pr 3 Vr 2 . . . Pr k Vr k − 1 Pr 1 Vr k Vr 1 a dipath from 0 to x is composed of the pieces: γ 0 : 0 ! ( x 1 , 0 ) serially - one coordinate at a time. OBS, now hold κ ( r k ) locks on r k Lisbeth Fajstrup (AAU) Cut-o ff theorems for deadlocks and serializability July 2013 12 / 28

x = ( x 1 , . . . , x 1 , x 2 , . . . , x 2 , . . . , x k . . . x k ) is reachable T = Pr 1 Pr 2 Vr 1 Pr 3 Vr 2 . . . Pr k Vr k − 1 Pr 1 Vr k Vr 1 a dipath from 0 to x is composed of the pieces: γ 0 : 0 ! ( x 1 , 0 ) serially - one coordinate at a time. OBS, now hold κ ( r k ) locks on r k γ 1 : ( x 1 , 0 ) ! ( x 1 , 0 , x k ) one coordinate at a time. Now also κ ( r k − 1 ) locks on r k − 1 Lisbeth Fajstrup (AAU) Cut-o ff theorems for deadlocks and serializability July 2013 12 / 28

x = ( x 1 , . . . , x 1 , x 2 , . . . , x 2 , . . . , x k . . . x k ) is reachable T = Pr 1 Pr 2 Vr 1 Pr 3 Vr 2 . . . Pr k Vr k − 1 Pr 1 Vr k Vr 1 a dipath from 0 to x is composed of the pieces: γ 0 : 0 ! ( x 1 , 0 ) serially - one coordinate at a time. OBS, now hold κ ( r k ) locks on r k γ 1 : ( x 1 , 0 ) ! ( x 1 , 0 , x k ) one coordinate at a time. Now also κ ( r k − 1 ) locks on r k − 1 γ j : ( x 1 , 0 , x k − j + 2 , . . . , x k ) ! ( x 1 , 0 , x k − j + 1 , . . . , x k ). Now κ ( r i ) locks on r k − j , . . . , r k . For j = 2 , . . . , k � 1. Lisbeth Fajstrup (AAU) Cut-o ff theorems for deadlocks and serializability July 2013 12 / 28

x = ( x 1 , . . . , x 1 , x 2 , . . . , x 2 , . . . , x k . . . x k ) is reachable T = Pr 1 Pr 2 Vr 1 Pr 3 Vr 2 . . . Pr k Vr k − 1 Pr 1 Vr k Vr 1 a dipath from 0 to x is composed of the pieces: γ 0 : 0 ! ( x 1 , 0 ) serially - one coordinate at a time. OBS, now hold κ ( r k ) locks on r k γ 1 : ( x 1 , 0 ) ! ( x 1 , 0 , x k ) one coordinate at a time. Now also κ ( r k − 1 ) locks on r k − 1 γ j : ( x 1 , 0 , x k − j + 2 , . . . , x k ) ! ( x 1 , 0 , x k − j + 1 , . . . , x k ). Now κ ( r i ) locks on r k − j , . . . , r k . For j = 2 , . . . , k � 1. Need: No resource is locked above its capacity along γ . Let ρ i ( y ) be the number of locks held on r i at y 2 T M . Lisbeth Fajstrup (AAU) Cut-o ff theorems for deadlocks and serializability July 2013 12 / 28

x = ( x 1 , . . . , x 1 , x 2 , . . . , x 2 , . . . , x k . . . x k ) is reachable T = Pr 1 Pr 2 Vr 1 Pr 3 Vr 2 . . . Pr k Vr k − 1 Pr 1 Vr k Vr 1 a dipath from 0 to x is composed of the pieces: γ 0 : 0 ! ( x 1 , 0 ) serially - one coordinate at a time. OBS, now hold κ ( r k ) locks on r k γ 1 : ( x 1 , 0 ) ! ( x 1 , 0 , x k ) one coordinate at a time. Now also κ ( r k − 1 ) locks on r k − 1 γ j : ( x 1 , 0 , x k − j + 2 , . . . , x k ) ! ( x 1 , 0 , x k − j + 1 , . . . , x k ). Now κ ( r i ) locks on r k − j , . . . , r k . For j = 2 , . . . , k � 1. Need: No resource is locked above its capacity along γ . Let ρ i ( y ) be the number of locks held on r i at y 2 T M . For γ 0 : ρ i ( γ 0 ( t ))  1 for i 6 = k ρ k ( γ 0 ( t ))  κ ( r k ). ρ i ( γ 1 ( t ))  1 for i 6 = k � 1 , k . ρ k ( γ 1 ( t )) = k . ρ k − 1 ( γ 1 ( t ))  κ ( r k − 1 ). ρ i ( γ j ( t ))  1 for i  k � j + 1. ρ i ( γ j ( t )) = κ ( r i ) for i � k � j + 2. ρ k − j +1 ( γ j ( t ))  κ ( r k − j +1 ). Lisbeth Fajstrup (AAU) Cut-o ff theorems for deadlocks and serializability July 2013 12 / 28

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 = Σ r ∈ R κ ( r ). Prove: If there is a deadlock in T n , then there is a deadlock in T M . Lisbeth Fajstrup (AAU) Cut-o ff theorems for deadlocks and serializability July 2013 13 / 28

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 = Σ r ∈ R κ ( r ). Prove: If there is a deadlock in T n , then there is a deadlock in T M . For n < M : Let x = ( x 1 , . . . , x n ) be a (reachable) deadlock in T n . I.e. x i = Pr j ( i ) and ρ j ( i ) ( x ) = κ ( r j ( i ) ) or x i = > . x = ( x 1 , . . . , x n , > , . . . , > ) 2 T M is a (reachable) deadlock in T M , Then ˜ since ρ i ( ˜ x ) = ρ i ( x ) for all i Lisbeth Fajstrup (AAU) Cut-o ff theorems for deadlocks and serializability July 2013 13 / 28