Concurrency theory

Concurrency theory

Exercise sheet 10

Sebastian Muskalla, Prakash Saivasan Winter term 2017/18

Out: January 10 Due: January 16

Submit your solutions until Tuesday, January 16, during the lecture.

Exercise 1: Bounded round reachability

Describe the general case for the bounded round TSO-reachability problem that was described in the lecture. LetPbe a parallel program withn∈Nthreads and a boundk∈Non the number of rounds that each thread can make. Explain how to construct a programP^′such that for each program counterpcinPand its equivalent program counterpc^′inP^′, the following holds.

pcis TSO-reachable inPiffpc^′is SC-reachable inP^′.

Note: You do not have to give a formal construction. It is sufficient to list the additional global variables needed, explain their meaning and how they are used byP^′.

Exercise 2: Trace robustness strictly implies reachability robustness Prove the following Lemma from the lecture.

a) If Tr_TSO( P)

= Tr_SC( P)

for some program, then Reach_TSO(P) = Reach_SC(P).

Here, Reach_TSO(P) ={

pccf₀→^∗_TSO (pc,val,buf) withbuf(i) =εfor alli}

and Reach_SC(P) is ob- tained by restricting the definition to computations in which each issue (STORE) is followed by the store (UPDATE).

b) The reverse implication does not hold.

Remark: Relations

Recall the following basic definitions forrelations.

LetNbe a set and let⩽⊆N×Nbe a relation.

Recall thatNisreflexiveifx ⩽ xfor allx∈ N. It isantisymmetricifx⩽ yandy⩽ ximplyx =y (for allx,y ∈ N). It istransitiveifx ⩽ yandy ⩽ zimplyx ⩽ z(for allx,y,z ∈ N). If all three properties hold, we call⩽apartial order.

A partial order is calledtotal(or linear) if any two elements are comparable, i.e.

∀x,y∈N: x⩽yory⩽x.

We let⩽^∗denote the reflexive-transitive closure of⩽, the smallest subset ofN×Nthat contains

⩽and is reflexive and transitive.

We may see (N,⩽) as a directed graph. We call⩽acyclicif this graph does not contain a non- trivial cyclex₀⩽x₁⩽. . .⩽x_m ⩽x₀. (Cycles of the shapex₀ ⩽x₀are trivial.)

Exercise 3: Relations

LetNbe afiniteset and let⩽⊆N×Nbe a relation.

a) Explain how to construct⩽^∗from⩽within a finite number of steps.

b) Prove that⩽^∗is a partial order (i.e. antisymmetric) if and only if⩽is acyclic.

c) Now assume that⩽po is some partial order. Prove that there is a total order⩽to ⊆ N ×N containing⩽po, i.e.⩽po⊆⩽to.

d) (Bonus exercise, not graded.) Do b) and c) still hold ifNis infinite?

Exercise 4: Shasha and Snir

Prove the Lemma by Shasha and Snir:

A trace Tr( τ)

∈Tr_TSO( P)

is in Tr_SC( P)

if and only if its happens-before relation→_hbis acyclic.

Proceed as follows:

a) Show that for traces of SC computations,→_hbis necessarily acyclic.

b) Show how from a trace with acyclic→_hb, one can construct an SC computationτ^′with Tr( τ^′)

= Tr( τ)

. Hint:Use Exercise 3.