diff --git a/lec-hb-vc.html b/lec-hb-vc.html new file mode 100644 index 0000000..ef2e3e3 --- /dev/null +++ b/lec-hb-vc.html @@ -0,0 +1,1806 @@ + + + + + + + + + Dynamic data race prediction - Happens-before and vector clocks + + + + + +
+

Dynamic data race prediction - Happens-before and +vector clocks

+

+Martin Sulzmann +

+
+
+

Overview

+

Data race prediction

+

We say that two read/write operations on some shared variable are +conflicting if both operations arise in different threads and +at least one of the operations is a write.

+

Dynamic analysis method

+

We consider a specific program run represented as a trace. The trace +contains the sequence of events as they took place during program +execution. Based on this trace, we carry out the analysis.

+

Exhaustive +versus efficient and approximative methods

+

To identify conflicting events that are in a race, we could check if +there is a valid reordering of the trace under which both events occur +right next to each other. In general, this requires to exhaustively +compute all reordering which is not efficient.

+

Recall the following example.

+
Trace A:
+
+     T1          T2
+
+e1.   w(x)
+e2.   acq(y)
+e3.   rel(y)
+e4.               acq(y)
+e5.               w(x)
+e6.               rel(y)
+

Via the following (valid) reordering

+
     T1          T2
+
+e4.               acq(y)
+e5.               w(x)
+e1.   w(x)
+e6.               rel(y)
+e2.   acq(y)
+e3.   rel(y)
+

we can show that events e1 and e5 are in a race.

+

Efficient methods approximate by only considering certain +reorderings.

+

Happens-before

+

The happens-before method imposes a partial order relation among +events to identify if one event happens before another event.

+

In case of of unordered events, we can assume that these events may +happen concurrently. If two conflicting operations are unordered under +the happens-before relation, then we report that these operations are in +a (data) race.

+

The happens-before (partial) order approximates the must +happen relations. Hence, we may encounter false positives.

+

In case of trace A (from above) we find that

+
e1 < e2 < e3 < e4 < e5 < e6
+

For this example, happens-before totally order all events. We fail to +report the data race.

+

Vector clocks

+

To decide if two events are (partially) ordered, +e<fe < f, +we can build the closure of all events that are ordered with respect to +the two events.

+

ESe={g∣g<e}∪{e}ES_e = \{ g \mid g < e \} \cup \{ e \}

+

ESf={g∣g<f}∪{f}ES_f = \{ g \mid g < f \} \cup \{ f \}

+

Then, we use set membership/entailment to check if +e<fe < f +holds.

+

This is not very efficient (as the closure may grow linearly in the +size of the trace).

+

Vector +clocks represent a more efficient representation for the +happens-before relation.

+

Instead of +ESESs +we build vector clocks +VeV_e +and +VfV_f. +The size of vector clocks is linear in the number of threads.

+

Deciding if +Ve<VfV_e < V_f +can be done in +O(k)O(k) +time.

+
+
+

Trace and event notation

+

We write +j#eij\#e_i +to denote event +ee +at trace position +ii +in thread +jj.

+

In plain text notation, we sometimes write j#e_i.

+

We assume the following events.

+
e ::=  acq(y)         -- acquire of lock y
+   |   rel(y)         -- release of lock y
+   |   w(x)           -- write of shared variable x
+   |   r(x)           -- read of shared variable x
+
+

Consider the trace

+

[1#w(x)1,1#acq(y)2,1#rel(y)3,2#acq(y)4,2#w(x)5,2#rel(y)6][1 \# w(x)_1, 1 \# acq(y)_2, 1 \# rel(y)_3, 2 \# acq(y)_4, 2 \# w(x)_5, 2 \# rel(y)_6]

+

and its tabular representation

+
     T1          T2
+
+1.   w(x)
+2.   acq(y)
+3.   rel(y)
+4.               acq(y)
+5.               w(x)
+6.               rel(y)
+

Instrumentation and tracing

+

We ignore the details of how to instrument programs to carry out +tracing of events. For our examples, we generally choose the tabular +notation for traces. In practice, the entire trace does not need to be +present as events can be processed online in a stream-based +fashion. A more detailed offline analysis, may get better +results if the trace in its entire form is present.

+
+
+

Trace reordering

+

To predict if two conflicting operations are in a race we could +reorder the trace. Reordering the trace means that we simply +permute the elements.

+

Consider the example trace.

+
     T1          T2
+
+1.   w(x)
+2.   acq(y)
+3.   rel(y)
+4.               acq(y)
+5.               w(x)
+6.               rel(y)
+

Here is a possible reordering.

+
     T1          T2
+
+4.               acq(y)
+1.   w(x)
+2.   acq(y)
+3.   rel(y)
+5.               w(x)
+6.               rel(y)
+

This reordering is not valid because it violates the lock +semantics. Thread T2 holds locks y. Hence, its not possible for thread +T1 to acquire lock y as well.

+

Here is another invalid reordering.

+
     T1          T2
+
+2.   acq(y)
+3.   rel(y)
+4.               acq(y)
+1.   w(x)
+5.               w(x)
+6.               rel(y)
+

The order among elements in trace has changed. This is not +allowed.

+

Valid trace reordering

+

For a reordering to be valid the following rules must hold:

+
    +

  1. The elements in the reordered trace must be part of the original +trace.

    2. +

  2. The order among elements for a given trace cannot be changed. See +The Program Order Condition.

    3. +

  3. For each release event rel(y) there must exist an earlier acquire +event acq(y) such there is no event rel(y) in between. See the Lock +Semantics Condition.

    4. +

  4. Each read must have the same last writer. See the Last +Writer Condition.

    5. +
+

A valid reordering only needs to include a subset of the events of +the original trace.

+

Consider the following (valid) reordering.

+
     T1          T2
+
+4.               acq(y)
+5.               w(x)
+1.   w(x)
+

This reordering shows that the two conflicting events are in a data +race. We only consider a prefix of the events in thread T1 and thread +T2.

+

Exhaustive data race +prediction methods

+

A “simple” data race prediction method seems to compute all possible +(valid) reordering and check if they contain any data race. Such +exhaustive methods generally do not scale to real-world +settings where resulting traces may be large and considering all +possible reorderings generally leads to an exponential blow up.

+

Approximative data +race prediction methods

+

Here, we consider efficient predictive methods. By efficient +we mean a run-time that is linear in terms of the size of the trace. +Because we favor efficiency over being exhaustive, we may compromise +completeness and soundness.

+

Complete means that all valid reorderings that exhibit some +race can be predicted. If incomplete, we refer to any not reported race +as a false negative.

+

Sound means that races reported can be observed via some +appropriate reordering of the trace. If unsound, we refer to wrongly a +classified race as a false positive.

+

We consider here Lamport’s happens-before (HB) relation that +approximates the possible reorderings. The HB relation can be computed +efficiently but may lead to false positives and false negatives.

+
+
+

Lamport’s happens-before (HB) relation

+

Let +TT +be a trace. We define the HB relation < as the smallest +strict +partial order that satisfies the following conditions:

+

Program order

+

Let +j#ei,j#fi+n∈Tj\#e_i, j\#f_{i+n} \in T +where +n>0n>0. +Then, we find that +j#ei<j#fi+nj\#e_i < j\#f_{i+n}.

+

Critical section order

+

Let +i#rel(x)k,j#acq(x)k+n∈Ti \# rel(x)_k, j\# acq(x)_{k+n} \in T +where +i!=ji != j +and +n>0n>0. +Then, we find that +i#rel(x)k<j#acq(x)k+ni\#rel(x)_k < j\#acq(x)_{k+n}.

+ +

The HB relation does not enforce the last writer rule. More on +this later.

+

Example

+

Consider the trace

+

[1#w(x)1,1#acq(y)2,1#rel(y)3,2#acq(y)4,2#w(x)5,2#rel(y)6][1 \# w(x)_1, 1 \# acq(y)_2, 1 \# rel(y)_3, 2 \# acq(y)_4, 2 \# w(x)_5, 2 \# rel(y)_6].

+

Via program order we find that

+

1#w(x)1<1#acq(y)2<1#rel(y)3 +1 \# w(x)_1 < 1 \# acq(y)_2 < 1 \# rel(y)_3 +

+

and

+

2#acq(y)4<2#w(x)5<2#rel(y)6. +2 \# acq(y)_4 < 2 \# w(x)_5 < 2 \# rel(y)_6. +

+

Via critical section order we find that

+

1#rel(y)3<2#acq(y)4. +1 \# rel(y)_3 < 2 \# acq(y)_4. +

+

Points to note.

+

<< +is the smallest partial order that satisfies the above conditions.

+

Hence, by transitivity we can also assume that for example

+

1#w(x)1<2#w(x)5. +1 \# w(x)_1 < 2 \# w(x)_5. +

+

Happens-before data race +check

+

If for two conflicting events +ee +and +ff +we have that neither +e<fe < f +nor +f<ef < e, +then we say that +(e,f)(e,f) +is a HB data race pair.

+

The argument is that if +e<fe < f +nor +f<ef < e +we are free to reorder the trace such that +ee +and +ff +appear right next to each other (in some reordered trace).

+

Note. If +(e,f)(e,f) +is a HB data race pair then so is +(f,e)(f,e). +In such a situation, we consider +(e,f)(e,f) +and +(f,e)(f,e) +as two distinct representative for the same data race. When reporting +(and counting) HB data races we only consider a specific +representative.

+
+
+

Event sets

+

Consider event +ee. +We denote by +ESeES_e +the set of events that happen-before +ee. +We assume that +ee +is also included in +ESeES_e. +Hence, +ESe={f∣f<e}∪{e}ES_e = \{ f \mid f < e \} \cup \{ e \}.

+

Example - Trace +annotated with event sets

+

We write ei to denote the event at trace position +i.

+
     T1          T2            ES
+
+1.   w(x)                     ES_e1= {e1}
+2.   acq(y)                   ES_e2= {e1,e2}
+3.   rel(y)                   ES_e3 = {e1,e2,e3}
+4.               acq(y)       ES_e4 = ES_e3 cup {e4} = {e1,e2,e3,e4}
+5.               w(x)         ES_e5 = {e1,e2,e3,e4,e5}
+6.               rel(y)       ES_e6 = {e1,e2,e3,e4,e5,e6}
+

In the above, we write cup to denote set union +∪\cup.

+

Observations:

+

To enforce the critical section order we simply need to add the event +set +ESrel(y)ES_{rel(y)} +of some release event to the event set +ESacq(y)ES_{acq(y)} +of some later acquire event.

+

To enforce the program order, we keep accumulating events within one +thread (in essence, building the transitive closure).

+

To decide if +e<fe < f +we can check for +ESe⊂ESfES_e \subset ES_f.

+

Consider two conflicting events +ee +and +ff +where +ee +appears before +ff +in the trace. To decide if +ee +and +ff +are in a race, we check for +e∈ESfe \in ES_f. +If yes, then there’s no race (because +e<fe < f). +Otherwise, there’s a race.

+

Set-based race predictor

+

We compute event sets by processing each event in the trace (in the +order events are recorded).

+

We maintain the following state variables.

+
D(t)
+
+  Each thread t maintains the set of events that happen before when processing events.
+
+
+R(x)
+
+  Most recent set of concurrent reads.
+
+W(x)
+
+  Most recent write.
+
+
+Rel(y)
+
+  Critical sections are ordered as they appear in the trace.
+  Rel(y) records the event set of the most recent release event on lock y.
+

All of the above are sets of events and assumed to be initially +empty. The exception is W(x). We assume that initially W(x) is +undefined.

+

For each event we call its processing function. We write +e@operation to denote that event e will be +processed by operation.

+
e@acq(t,y) {
+   D(t) = D(t) cup Rel(y) cup { e }
+}
+
+e@rel(t,y) {
+   D(t) = D(t) cup { e }
+   Rel(y) = D(t)
+
+e@fork(t1,t2) {
+  D(t1) = D(t1) cup { e }
+  D(t2) = D(t1)
+}
+
+e@join(t1,t2) {
+  D(t1) = D(t1) cup D(t2) cup { e }
+  }
+
+e@write(t,x) {
+  If W(x) exists and W(x) not in D(t)
+  then write-write race (W(x),e)
+
+  For each r in R(x),
+  if r not in D(t)
+  then read-write race  (r,e)
+
+  D(t) = D(t) cup { e }
+  W(x) = e
+}
+
+e@read(t,x) {
+  If W(x) exists and W(x) not in D(t)
+  then write-read race (W(x),e)
+
+  R(x) = {e} cup {r' | r' in R(x), r' \not\in D(t) }
+  D(t) = D(t) cup { e }
+}
+

Examples

+

Race not detected

+
    T0            T1
+
+1.  fork(T1)
+2.  acq(y)
+3.  wr(x)
+4.  rel(y)
+5.                acq(y)
+6.                rel(y)
+7.                wr(x)
+

The HB relation does not reorder critical sections. Therefore, we +report there is no race. This is a false negative because there is a +valid reordering to shows the two writes on x are in a race.

+

Here’s the annotated trace with the information computed by the +set-based race predictor.

+
   T0            T1
+
+1.  fork(T1)
+2.  acq(y)                      D(T0) = [0:fork(T1)_1,0:acq(y)_2]
+3.  wr(x)                       W(x) = undefined
+4.  rel(y)                      Rel(y) = [0:fork(T1)_1,0:acq(y)_2,0:wr(x)_3,0:rel(y)_4]
+5.                acq(y)        D(T1) = [0:fork(T1)_1,0:acq(y)_2,0:wr(x)_3,0:rel(y)_4,1:acq(y)_5]
+6.                rel(y)        Rel(y) = [0:fork(T1)_1,0:acq(y)_2,0:wr(x)_3,0:rel(y)_4,1:acq(y)_5,1:rel(y)_6]
+7.                wr(x)         W(x) = 0:wr(x)_3
+ +

Race detected

+
    T0            T1
+
+1.  fork(T1)
+2.  acq(y)
+3.  wr(x)
+4.  rel(y)
+5.                wr(x)
+6.                acq(y)
+7.                rel(y)
+

Under the HB relation, the two writes on x are not ordered. Hence, we +report that they are in a race.

+

Here’s the annotated trace.

+
    T0            T1
+
+1.  fork(T1)
+2.  acq(y)                      D(T0) = [0:fork(T1)_1,0:acq(y)_2]
+3.  wr(x)                       W(x) = undefined
+4.  rel(y)                      Rel(y) = [0:fork(T1)_1,0:acq(y)_2,0:wr(x)_3,0:rel(y)_4]
+5.                wr(x)         W(x) = 0:wr(x)_3
+6.                acq(y)        D(T1) = [0:fork(T1)_1,0:acq(y)_2,0:wr(x)_3,0:rel(y)_4,1:wr(x)_5,1:acq(y)_6]
+7.                rel(y)        Rel(y) = [0:fork(T1)_1,0:acq(y)_2,0:wr(x)_3,0:rel(y)_4,1:wr(x)_5,1:acq(y)_6,1:rel(y)_7]
+

Earlier race not detected

+
    T0            T1
+
+1.  fork(T1)
+2.  acq(y)
+3.  wr(x)
+4.  wr(x)
+5.  rel(y)
+6.                wr(x)
+7.                acq(y)
+8.                rel(y)
+

The write at trace position 3 and the write at trace position 6 are +in a race. However, we only maintain the most recent write. Hence, we +only report that the write at trace position 4 is in a race with the +write at trace position 6.

+

Here’s the annotated trace.

+
    T0            T1
+
+1.  fork(T1)
+2.  acq(y)                      D(T0) = [0:fork(T1)_1,0:acq(y)_2]
+3.  wr(x)                       W(x) = undefined
+4.  wr(x)                       W(x) = 0:wr(x)_3
+5.  rel(y)                      Rel(y) = [0:fork(T1)_1,0:acq(y)_2,0:wr(x)_3,0:wr(x)_4,0:rel(y)_5]
+6.                wr(x)         W(x) = 0:wr(x)_4
+7.                acq(y)        D(T1) = [0:fork(T1)_1,0:acq(y)_2,0:wr(x)_3,0:wr(x)_4,0:rel(y)_5,1:wr(x)_6,1:acq(y)_7]
+8.                rel(y)        Rel(y) = [0:fork(T1)_1,0:acq(y)_2,0:wr(x)_3,0:wr(x)_4,0:rel(y)_5,1:wr(x)_6,1:acq(y)_7,1:rel(y)_8]
+

Read-write races

+
    T0            T1            T2
+
+1.  wr(x)
+2.  fork(T1)
+3.  fork(T2)
+4.  rd(x)
+5.                rd(x)
+6.                              acq(y)
+7.                              wr(x)
+8.                              rel(y)
+

We find that the two reads are in a race with the write.

+

Here’s the annotated trace.

+
    T0            T1            T2
+
+1.  wr(x)                                     W(x) = undefined
+2.  fork(T1)
+3.  fork(T2)
+4.  rd(x)                                     W(x) = 0:wr(x)_1
+5.                rd(x)                       W(x) = 0:wr(x)_1
+6.                              acq(y)        D(T2) = [0:wr(x)_1,0:fork(T1)_2,0:fork(T2)_3,2:acq(y)_6]
+7.                              wr(x)         W(x) = 0:wr(x)_1  R(x) = [0:rd(x)_4,1:rd(x)_5]
+8.                              rel(y)        Rel(y) = [0:wr(x)_1,0:fork(T1)_2,0:fork(T2)_3,2:acq(y)_6,2:wr(x)_7,2:rel(y)_8]
+

Note that when processing the write in T2, we find that D(T2) +contains the earlier write.

+

Can we find a +more compact representation for D(t)?

+

The size of D(t) may grow linearly in the size of the trace.

+

To check for a race we check if some element is in D(t).

+

If there are n events, this means set-based race prediction requires +O(n*n) time.

+
+
+

From event sets to timestamps

+

Timestamps

+

For each thread we maintain a timestamp.

+

We represent a timestamp as a natural number.

+

Each time we process an event we increase the thread’s timestamp.

+

Initially, the timestamp for each thread is 1.

+

Example - Trace +annotated with timestamps

+
     T1          T2      TS_T1       TS_T2
+
+1.   w(x)                  2
+2.   acq(y)                3
+3.   rel(y)                4
+4.               acq(y)               2
+5.               w(x)                 3
+6.               rel(y)               4
+

Representing +events via its thread id and timestamp

+

Let +ee +be an event in thread +ii +and +jj +its timestamp.

+

Then, we can uniquely identify +ee +via +ii +and +jj.

+

We write +i#ji \# j +to represent event +ee. +In the literature, +i#ji \# j +is referred to as an epoch.

+

Example - Trace annotated +with epochs

+
     T1          T2           Epochs
+
+1.   w(x)                     1#1
+2.   acq(y)                   1#2
+3.   rel(y)                   1#3
+4.               acq(y)       2#1
+5.               w(x)         2#2
+6.               rel(y)       2#3
+

We use the timestamp “before” processing the event. We could also use +the timestamp “after” (but this needs to be done consistently).

+

From event sets to sets of +epoch

+

Instead of events sets we record the set of epochs.

+

We group together epochs belonging to the same thread. For example, +in case of +{1#1,1#2}\{ 1 \# 1, 1 \# 2 \} +we write +{1#{1,2}}.\{ 1 \# \{ 1, 2\} \}.

+

Example - Trace +annotated with sets of epochs

+
     T1          T2            Sets of epochs
+
+1.   w(x)                     {1#1}
+2.   acq(y)                   {1#{1,2}}
+3.   rel(y)                   {1#{1,2,3}}
+4.               acq(y)       {1#{1,2,3}, 2#1}
+5.               w(x)         {1#{1,2,3}, 2#{1,2}}
+6.               rel(y)       {1#{1,2,3}, 2#{1,2,3}}
+
+
+

From sets of timestamps to vector clocks

+

Insight:

+

For each thread only keep most recent timestamp.

+

For example, in case of +{1#{1,2}}\{ 1 \# \{ 1, 2\} \} +we write +{1#2}\{ 1 \# 2 \}

+

Example - +Trace annotated with most recent timestamps

+
     T1          T2            Sets of most recent timestamps
+
+1.   w(x)                     {1#1}
+2.   acq(y)                   {1#2}
+3.   rel(y)                   {1#3}
+4.               acq(y)       {1#3, 2#1}
+5.               w(x)         {1#3, 2#2}
+6.               rel(y)       {1#3, 2#3}
+

Vector clocks

+

Instead of a set use a list (vector).

+
V  ::=  [i1,...,in]    -- vector clock with n time stamps
+

The entry associated to each thread is identified via its position in +that list.

+

If there’s no entry for a thread, we assume the timestamp 0.

+

Example - Trace +annotated with vector clocks

+
     T1          T2            Vector clocks
+
+1.   w(x)                     [1,0]
+2.   acq(y)                   [2,0]
+3.   rel(y)                   [3,0]
+4.               acq(y)       [3,1]
+5.               w(x)         [3,2]
+6.               rel(y)       [3,3]
+

Mapping event sets to +vector clocks

+

Based on the above, each event set can be represented as the set +({1#E1,...,n#En}(\{1 \# E_1,..., n \# E_n\} +where sets +EjE_j +are of the form +{1,...,k}\{1,...,k\}.

+

We define a mapping +Φ\Phi +from event sets to vector clocks as follows.

+

Φ({1#E1,...,n#En})=[max(E1),...,max(En)]\Phi(\{1 \# E_1,..., n \# E_n\}) = [max(E_1),...,max(E_n)]

+

Properties

+
    +

  1. De⊂DfD_e \subset D_f +iff +ϕ(De)<ϕ(Df)\phi(D_e) < \phi(D_f)

    2. +

  2. ϕ(De∪Df)=sync(ϕ(De),ϕ(Df))\phi(D_e \cup D_f) = sync(\phi(D_e),\phi(D_f))

    3. +

  3. Let +e,fe, f +be two events where +ee +appears before +ff +and +e=i#ke = i\# k. +Then, +e∉Dfe \not\in D_f +iff +k>Φ(Df)(i)k > \Phi(D_f)(i).

    4. +
+

We define +<< +and +syncsync +for vector clocks as follows.

+
Synchronize two vector clocks by taking the larger time stamp
+
+     sync([i1,...,in],[j1,...,jn]) = [max(i1,j1), ..., max(in,jn)]
+
+Order among vector clocks
+
+      [i1,...,in]  < [j1,...,jn]
+
+if ik<=jk for all k=1...n and there exists k such that ik<jk.
+
+
+

Vector clock based race detector a la FastTrack

+

Further vector clock operations.

+
Lookup of time stamp
+
+   [i1,...,ik,...,in](k) = ik
+
+
+Increment the time stamp of thread i
+
+    inc([k1,...,ki-1,ki,ki+1,...,kn],i) = [k1,...,ki-1,ki+1,ki+1,...,kn]
+

We maintain the following state variables.

+
Th(i)
+
+ Vector clock of thread i
+
+
+R(x)
+
+   Vector clock for reads we have seen
+
+W(x)
+
+  Epoch of most recent write
+
+
+Rel(y)
+
+  Vector clock of the most recent release on lock y.
+

Initially, the timestamps in R(x), W(x) and Rel(y) are all set to +zero.

+

In Th(i), all time stamps are set to zero but the time stamp of the +entry i which is set to one.

+

Event processing is as follows.

+
acq(t,y) {
+  Th(t) = sync(Th(t), Rel(y))
+  inc(Th(t),t)
+}
+
+rel(t,y) {
+   Rel(y) = Th(t)
+   inc(Th(t),t)
+}
+
+fork(t1,t2) {
+  Th(t2) = Th(t1)
+  inc(Th(t2),t2)
+  inc(Th(t1),t1)
+}
+
+join(t1,t2) {
+  Th(t1) = sync(Th(t1),Th(t2))
+  inc(Th(t1),t1)
+  }
+
+write(t,x) {
+  If not (R(x) < Th(t))
+  then write in a race with a read
+
+  If W(x) exists
+  then let j#k = W(x)                    -- means W(x) equals j#k
+       if k > Th(t)(j)
+       then write in a race with a write
+
+  W(x) = t#Th(t)(t)
+  inc(Th(t),t)
+}
+
+read(t,x) {
+  If W(x) exists
+  then let j#k = W(x)
+       if k > Th(t)(j)
+       then read in a race with a write
+  R(x) = sync(Th(t), R(x))
+  inc(Th(t),t)
+}
+

Points note note.

+

We include fork and join events.

+

FastTrack (like the HB relation) does not enforce the “last writer” +rule. This is in line with Go’s/Java’s “weak” memory semantics.

+

The last-write order is enforced for atomic variables. Atomic +variables are protected by a lock. As we order critical sections based +on their textual occurrence, we get the last-write order for atomic +variables for free.

+
+
+

FastTrack Examples (from before)

+

The following examples are automatically generated. There is a slight +change in naming convention. Instead of lock variable “x” we write “L1”. +Similarly, we write “V0” for some shared variable “y”.

+

Race not detected

+

Consider the following trace.

+
   T0            T1
+e1. fork(T1)
+e2. acq(L1)
+e3. wr(V2)
+e4. rel(L1)
+e5.               acq(L1)
+e6.               rel(L1)
+e7.               wr(V2)
+

We apply the FastTrack algorithm on the above trace. For each event +we annotate its vector clock before and after processing the event.

+
   T0                            T1
+e1. [1,0]_fork(T1)_[2,0]
+e2. [2,0]_acq(L1)_[3,0]
+e3. [3,0]_wr(V2)_[4,0]
+e4. [4,0]_rel(L1)_[5,0]
+e5.                               [1,1]_acq(L1)_[4,2]
+e6.                               [4,2]_rel(L1)_[4,3]
+e7.                               [4,3]_wr(V2)_[4,4]
+

Here are the individual processing steps in detail.

+
****
+Step 1: Processing event fork(T1)in thread T0
+BEFORE
+Thread VC = [1,0]
+AFTER
+Thread VC = [2,0]
+****
+Step 2: Processing event acq(L1)in thread T0
+BEFORE
+Thread VC = [2,0]
+Rel[L1] = [0,0]
+AFTER
+Thread VC = [3,0]
+Rel[L1] = [0,0]
+****
+Step 3: Processing event wr(V2)in thread T0
+BEFORE
+Thread VC = [3,0]
+R[V2] = [0,0]
+W[V2] = undefined
+AFTER
+Thread VC = [4,0]
+R[V2] = [0,0]
+W[V2] = 0#3
+****
+Step 4: Processing event rel(L1)in thread T0
+BEFORE
+Thread VC = [4,0]
+Rel[L1] = [0,0]
+AFTER
+Thread VC = [5,0]
+Rel[L1] = [4,0]
+****
+Step 5: Processing event acq(L1)in thread T1
+BEFORE
+Thread VC = [1,1]
+Rel[L1] = [4,0]
+AFTER
+Thread VC = [4,2]
+Rel[L1] = [4,0]
+****
+Step 6: Processing event rel(L1)in thread T1
+BEFORE
+Thread VC = [4,2]
+Rel[L1] = [4,0]
+AFTER
+Thread VC = [4,3]
+Rel[L1] = [4,2]
+****
+Step 7: Processing event wr(V2)in thread T1
+BEFORE
+Thread VC = [4,3]
+R[V2] = [0,0]
+W[V2] = 0#3
+AFTER
+Thread VC = [4,4]
+R[V2] = [0,0]
+W[V2] = 1#3
+

Race detected

+
   T0            T1
+e1. fork(T1)
+e2. acq(L1)
+e3. wr(V2)
+e4. rel(L1)
+e5.               wr(V2)
+e6.               acq(L1)
+e7.               rel(L1)
+

In case of a race detected, we annotate the triggering event.

+
   T0                            T1
+e1. [1,0]_fork(T1)_[2,0]
+e2. [2,0]_acq(L1)_[3,0]
+e3. [3,0]_wr(V2)_[4,0]
+e4. [4,0]_rel(L1)_[5,0]
+e5.                               [1,1]_wr(V2)_[1,2]   WW
+e6.                               [1,2]_acq(L1)_[4,3]
+e7.                               [4,3]_rel(L1)_[4,4]
+

The annotation WW indicates that write event +e4 is in a race with some earlier write.

+

For brevity, we omit the detailed processing steps.

+

Earlier race not detected

+
   T0            T1
+e1. fork(T1)
+e2. acq(L1)
+e3. wr(V2)
+e4. wr(V2)
+e5. rel(L1)
+e6.               wr(V2)
+e7.               acq(L1)
+e8.               rel(L1)
+

Event e6 is in a race with e4 and +e3. However, FastTrack only maintains the “last” write. +Hence, FastTrack only reports the race among e6 and +e4.

+

Here is the annotated trace.

+
   T0                            T1
+e1. [1,0]_fork(T1)_[2,0]
+e2. [2,0]_acq(L1)_[3,0]
+e3. [4,0]_wr(V2)_[5,0]
+e4. [4,0]_wr(V2)_[5,0]
+e5. [5,0]_rel(L1)_[6,0]
+e6.                               [1,1]_wr(V2)_[1,2]   WW
+e7.                               [1,2]_acq(L1)_[5,3]
+e8.                               [5,3]_rel(L1)_[5,4]
+

Read-write races

+

The following trace contains reads as well as writes.

+
   T0            T1            T2
+e1. wr(V2)
+e2. fork(T1)
+e3. fork(T2)
+e4. rd(V2)
+e5.               rd(V2)
+e6.                             acq(L1)
+e7.                             wr(V2)
+e8.                             rel(L1)
+

FastTrack reports that the write e7 is in a race with a +read. See the annotation RW below.

+
   T0                            T1                            T2
+e1. [1,0,0]_wr(V2)_[2,0,0]
+e2. [2,0,0]_fork(T1)_[3,0,0]
+e3. [3,0,0]_fork(T2)_[4,0,0]
+e4. [4,0,0]_rd(V2)_[5,0,0]
+e5.                               [2,1,0]_rd(V2)_[2,2,0]
+e6.                                                             [3,0,1]_acq(L1)_[3,0,2]
+e7.                                                             [3,0,2]_wr(V2)_[3,0,3]   RW
+e8.                                                             [3,0,3]_rel(L1)_[3,0,4]
+

There are two read-write races: (e4,e7) and +(e5,e7). In our FastTrack implementation, we +only report the triggering event.

+

Here is a variant of the above example where we find write-write and +write-read and read-write races.

+
   T0            T1            T2
+e1. fork(T1)
+e2. fork(T2)
+e3. wr(V2)
+e4. rd(V2)
+e5.               rd(V2)
+e6.                             acq(L1)
+e7.                             wr(V2)
+e8.                             rel(L1)
+

Here is the annotated trace.

+
   T0                            T1                            T2
+e1. [1,0,0]_fork(T1)_[2,0,0]
+e2. [2,0,0]_fork(T2)_[3,0,0]
+e3. [3,0,0]_wr(V2)_[4,0,0]
+e4. [4,0,0]_rd(V2)_[5,0,0]
+e5.                               [1,1,0]_rd(V2)_[1,2,0]  WR
+e6.                                                             [2,0,1]_acq(L1)_[2,0,2]
+e7.                                                             [2,0,2]_wr(V2)_[2,0,3]   RW WW
+e8.                                                             [2,0,3]_rel(L1)_[2,0,4]
+
+
+

FastTrack Implementation in Go

+

Go playground

+

Check out main and the examples.

+
package main
+
+// FastTrack
+
+import "fmt"
+import "strconv"
+import "strings"
+
+
+
+// Example traces.
+
+// Traces are assumed to be well-formed.
+// For example, if we fork(ti) we assume that thread ti exists.
+
+// Race not detected
+func trace_1() []Event {
+    t0 := mainThread()
+    t1 := nextThread(t0)
+    x := 1
+    z := 2
+    return []Event{
+        fork(t0, t1),
+        acq(t0, x),
+        wr(t0, z),
+        rel(t0, x),
+        acq(t1, x),
+        rel(t1, x),
+        wr(t1, z)}
+}
+
+// Race detected
+func trace_2() []Event {
+    t0 := mainThread()
+    t1 := nextThread(t0)
+    x := 1
+    z := 2
+    return []Event{
+        fork(t0, t1),
+        acq(t0, x),
+        wr(t0, z),
+        rel(t0, x),
+        wr(t1, z),
+        acq(t1, x),
+        rel(t1, x)}
+
+}
+
+// Earlier race not detected
+func trace_2b() []Event {
+    t0 := mainThread()
+    t1 := nextThread(t0)
+    x := 1
+    z := 2
+    return []Event{
+        fork(t0, t1),
+        acq(t0, x),
+        wr(t0, z),
+        wr(t0, z),
+        rel(t0, x),
+        wr(t1, z),
+        acq(t1, x),
+        rel(t1, x)}
+
+}
+
+// Read-write races
+func trace_3() []Event {
+    t0 := mainThread()
+    t1 := nextThread(t0)
+    t2 := nextThread(t1)
+    x := 1
+    z := 2
+    return []Event{
+        wr(t0, z),
+        fork(t0, t1),
+        fork(t0, t2),
+        rd(t0, z),
+        rd(t1, z),
+        acq(t2, x),
+        wr(t2, z),
+        rel(t2, x)}
+
+}
+
+// Write-write and wwrite-read and read-write races
+func trace_3b() []Event {
+    t0 := mainThread()
+    t1 := nextThread(t0)
+    t2 := nextThread(t1)
+    x := 1
+    z := 2
+    return []Event{
+        fork(t0, t1),
+        fork(t0, t2),
+        wr(t0, z),
+        rd(t0, z),
+        rd(t1, z),
+        acq(t2, x),
+        wr(t2, z),
+        rel(t2, x)}
+
+}
+
+func main() {
+
+    //  fmt.Printf("\n%s\n", displayTrace(trace_1()))
+    //  run(trace_1(), true)
+
+    //  fmt.Printf("\n%s\n", displayTrace(trace_2()))
+    //  run(trace_2(), true)
+
+    //  fmt.Printf("\n%s\n", displayTrace(trace_2b()))
+    //  run(trace_2b(), true)
+
+    //  fmt.Printf("\n%s\n", displayTrace(trace_3()))
+    //  run(trace_3(), true)
+
+    //      fmt.Printf("\n%s\n", displayTrace(trace_3b()))
+    //      run(trace_3b(), true)
+
+}
+
+
+
+
+///////////////////////////////////////////////////////////////
+// Helpers
+
+func max(x, y int) int {
+    if x < y {
+        return y
+    }
+    return x
+}
+
+func debug(s string) {
+    fmt.Printf("%s", s)
+
+}
+
+///////////////////////////////////////////////////////////////
+// Events
+
+type EvtKind int
+
+const (
+    AcquireEvt EvtKind = 0
+    ReleaseEvt EvtKind = 1
+    WriteEvt   EvtKind = 2
+    ReadEvt    EvtKind = 3
+    ForkEvt    EvtKind = 4
+    JoinEvt    EvtKind = 5
+)
+
+type Event struct {
+    k_  EvtKind
+    id_ int
+    a_  int
+}
+
+func (e Event) thread() int   { return e.id_ }
+func (e Event) kind() EvtKind { return e.k_ }
+func (e Event) arg() int      { return e.a_ }
+
+// Some convenience functions.
+func rd(i int, a int) Event {
+    return Event{ReadEvt, i, a}
+}
+
+func wr(i int, a int) Event {
+    return Event{WriteEvt, i, a}
+}
+
+func acq(i int, a int) Event {
+    return Event{AcquireEvt, i, a}
+}
+
+func rel(i int, a int) Event {
+    return Event{ReleaseEvt, i, a}
+}
+
+func fork(i int, a int) Event {
+    return Event{ForkEvt, i, a}
+}
+
+func join(i int, a int) Event {
+    return Event{JoinEvt, i, a}
+}
+
+// Trace assumptions.
+// Initial thread starts with 0. Threads have ids in ascending order.
+
+func trace_info(es []Event) (int, map[int]bool, map[int]bool) {
+    max_tid := 0
+    vars := map[int]bool{}
+    locks := map[int]bool{}
+    for _, e := range es {
+        max_tid = max(max_tid, e.thread())
+        if e.kind() == WriteEvt || e.kind() == ReadEvt {
+            vars[e.arg()] = true
+        }
+        if e.kind() == AcquireEvt { // For each rel(x) there must exists some acq(x)
+            locks[e.arg()] = true
+        }
+    }
+    return max_tid, vars, locks
+}
+
+func mainThread() int      { return 0 }
+func nextThread(i int) int { return i + 1 }
+
+const (
+    ROW_OFFSET = 14
+)
+
+// Omit thread + loc info.
+func displayEvtSimple(e Event, i int) string {
+    var s string
+    arg := strconv.Itoa(e.arg())
+    switch {
+    case e.kind() == AcquireEvt:
+        s = "acq(L" + arg + ")" // L(ock)
+    case e.kind() == ReleaseEvt:
+        s = "rel(L" + arg + ")"
+    case e.kind() == WriteEvt:
+        s = "wr(V" + arg + ")" // V(ar)
+    case e.kind() == ReadEvt:
+        s = "rd(V" + arg + ")"
+    case e.kind() == ForkEvt:
+        s = "fork(T" + arg + ")" // T(hread)
+    case e.kind() == JoinEvt:
+        s = "join(T" + arg + ")"
+    }
+    return s
+}
+
+func colOffset(i int, row_offset int) string {
+    n := (int)(i)
+    return strings.Repeat(" ", n*row_offset)
+}
+
+// Thread ids fully cover [0..n]
+func displayTraceHelper(es []Event, row_offset int, displayEvt func(e Event, i int) string) string {
+    s := ""
+    m := 0
+    for i, e := range es {
+        row := "\n" + "e" + strconv.Itoa(i+1) + ". " + colOffset(e.thread(), row_offset) + displayEvt(e, i)
+        s = s + row
+        m = max(m, e.thread())
+    }
+    // Add column headings.
+    heading := "   "
+    for i := 0; i <= m; i++ {
+        heading += "T" + strconv.Itoa(i) + strings.Repeat(" ", row_offset-2)
+    }
+    s = heading + s
+
+    return s
+}
+
+func displayTrace(es []Event) string {
+    return displayTraceHelper(es, ROW_OFFSET, displayEvtSimple)
+}
+
+func displayTraceWithVC(es []Event, pre map[Event]VC, post map[Event]VC, races map[Event]string) string {
+    displayEvt := func(e Event, i int) string {
+        out := pre[e].display() + "_" + displayEvtSimple(e, i) + "_" + post[e].display()
+        info, exists := races[e]
+        if exists {
+            out = out + "  " + info
+
+        }
+        return out
+
+    }
+    return displayTraceHelper(es, 30, displayEvt)
+}
+
+///////////////////////////////////////////////////////////////
+// Vector Clocks
+
+type VC []int
+
+type Epoch struct {
+    tid        int
+    time_stamp int
+}
+
+func (e Epoch) display() string {
+    return strconv.Itoa(e.tid) + "#" + strconv.Itoa(e.time_stamp)
+
+}
+
+func (v VC) display() string {
+    var s string
+    s = "["
+    for index := 0; index < len(v)-1; index++ {
+        s = s + strconv.Itoa(v[index]) + ","
+
+    }
+    if len(v) > 0 {
+        s = s + strconv.Itoa(v[len(v)-1]) + "]"
+    }
+    return s
+
+}
+
+func copyVC(v VC) VC {
+    new := make(VC, len(v))
+    copy(new, v)
+    return new
+}
+
+func sync(v VC, v2 VC) VC {
+    l := max(len(v), len(v2))
+
+    v3 := make([]int, l)
+
+    for tid, _ := range v3 {
+
+        v3[tid] = max(v[tid], v2[tid])
+    }
+
+    return v3
+}
+
+func (v VC) happensBefore(v2 VC) bool {
+    b := false
+    for tid := 0; tid < max(len(v), len(v2)); tid++ {
+        if v[tid] > v2[tid] {
+            return false
+        } else if v[tid] < v2[tid] {
+            b = true
+        }
+
+    }
+    return b
+}
+
+func (e Epoch) happensBefore(v VC) bool {
+    return e.time_stamp <= v[e.tid]
+
+}
+
+///////////////////////////////////////////////////////////////
+// Vector Clocks
+
+type FT struct {
+    Th    map[int]VC
+    pre   map[Event]VC
+    post  map[Event]VC
+    Rel   map[int]VC
+    W     map[int]Epoch
+    R     map[int]VC
+    races map[Event]string
+    i     int
+}
+
+func run(es []Event, full_info bool) {
+
+    var ft FT
+
+    ft.Th = make(map[int]VC)
+    ft.pre = make(map[Event]VC)
+    ft.post = make(map[Event]VC)
+    ft.Rel = make(map[int]VC)
+    ft.W = make(map[int]Epoch)
+    ft.R = make(map[int]VC)
+    ft.races = make(map[Event]string)
+
+    n, vars, locks := trace_info(es)
+    n = n + 1
+
+    // Initial vector clock of T0
+    ft.Th[0] = make([]int, n)
+    for j := 0; j < n; j++ {
+        ft.Th[0][j] = 0
+    }
+    ft.Th[0][0] = 1
+
+    for x, _ := range vars {
+        ft.R[x] = make([]int, n) // all entries are zero
+    }
+
+    for y, _ := range locks {
+        ft.Rel[y] = make([]int, n) // all entries are zero
+    }
+
+    exec := func(e Event) {
+        switch {
+        case e.kind() == AcquireEvt:
+            ft.acquire(e)
+        case e.kind() == ReleaseEvt:
+            ft.release(e)
+        case e.kind() == ForkEvt:
+            ft.fork(e)
+        case e.kind() == JoinEvt:
+            ft.join(e)
+        case e.kind() == WriteEvt:
+            ft.write(e)
+        case e.kind() == ReadEvt:
+            ft.read(e)
+        }
+
+    }
+
+    infoBefore := func(i int, e Event) {
+
+        if full_info {
+
+            x := strconv.Itoa(e.arg())
+
+                out := "\n****\nStep " + strconv.Itoa(i) + ": Processing event " + displayEvtSimple(e, i) + "in thread T" + strconv.Itoa(e.thread())
+            out = out + "\nBEFORE"
+            out = out + "\nThread VC = " + ft.Th[e.thread()].display()
+
+            if e.kind() == AcquireEvt || e.kind() == ReleaseEvt {
+                out = out + "\nRel[L" + x + "] = " + ft.Rel[e.arg()].display()
+
+            }
+
+            if e.kind() == WriteEvt || e.kind() == ReadEvt {
+                out = out + "\nR[V" + x + "] = " + ft.R[e.arg()].display()
+
+                ep, exists := ft.W[e.arg()]
+                if exists {
+                    out = out + "\nW[V" + x + "] = " + ep.display()
+                } else {
+                    out = out + "\nW[V" + x + "] = undefined"
+                }
+
+            }
+
+            fmt.Printf("%s", out)
+        }
+
+    }
+
+    infoAfter := func(e Event) {
+
+        if full_info {
+
+            x := strconv.Itoa(e.arg())
+
+            out := "\nAFTER"
+            out = out + "\nThread VC = " + ft.Th[e.thread()].display()
+
+            if e.kind() == AcquireEvt || e.kind() == ReleaseEvt {
+                out = out + "\nRel[L" + x + "] = " + ft.Rel[e.arg()].display()
+
+            }
+
+            if e.kind() == WriteEvt || e.kind() == ReadEvt {
+                out = out + "\nR[V" + x + "] = " + ft.R[e.arg()].display()
+
+                ep, exists := ft.W[e.arg()]
+                if exists {
+                    out = out + "\nW[V" + x + "] = " + ep.display()
+                } else {
+                    out = out + "\nW[V" + x + "] = undefined"
+                }
+
+            }
+
+            fmt.Printf("%s", out)
+        }
+
+    }
+
+    for i, e := range es {
+        infoBefore(i+1, e)
+        ft.pre[e] = copyVC(ft.Th[e.thread()])
+        exec(e)
+        infoAfter(e)
+        ft.post[e] = copyVC(ft.Th[e.thread()])
+    }
+
+    fmt.Printf("\n%s\n", displayTraceWithVC(es, ft.pre, ft.post, ft.races))
+
+}
+
+// We only report the event that triggered the race.
+func (ft *FT) logWR(e Event) {
+    ft.races[e] = "WR" // "write-read race"
+}
+
+// A write might be in a race with a write and a read, so must add any race message.
+func (ft *FT) logRW(e Event) {
+    _, exists := ft.races[e]
+    if !exists {
+        ft.races[e] = ""
+    }
+    ft.races[e] = ft.races[e] + " " + "RW" // "read-write race"
+
+}
+
+func (ft *FT) logWW(e Event) {
+    _, exists := ft.races[e]
+    if !exists {
+        ft.races[e] = ""
+    }
+    ft.races[e] = ft.races[e] + " " + "WW" // "write-write race"
+}
+
+func (ft *FT) inc(t int) {
+    ft.Th[t][t] = ft.Th[t][t] + 1
+
+}
+
+func (ft *FT) acquire(e Event) {
+    t := e.thread()
+    y := e.arg()
+    vc, ok := ft.Rel[y]
+    if ok {
+        ft.Th[t] = sync(ft.Th[t], vc)
+    }
+    ft.inc(t)
+
+}
+
+func (ft *FT) release(e Event) {
+    t := e.thread()
+    y := e.arg()
+    ft.Rel[y] = copyVC(ft.Th[t])
+    ft.inc(t)
+}
+
+func (ft *FT) fork(e Event) {
+    t1 := e.thread()
+    t2 := e.arg()
+    ft.Th[t2] = copyVC(ft.Th[t1])
+    ft.inc(t1)
+    ft.inc(t2)
+}
+
+func (ft *FT) join(e Event) {
+    t1 := e.thread()
+    t2 := e.arg()
+    ft.Th[t1] = sync(ft.Th[t1], ft.Th[t2])
+    ft.inc(t1)
+
+}
+
+func (ft *FT) write(e Event) {
+    t := e.thread()
+    x := e.arg()
+
+    if !ft.R[x].happensBefore(ft.Th[t]) {
+        ft.logRW(e)
+    }
+
+    ep, exists := ft.W[x]
+    if exists {
+        if !ep.happensBefore(ft.Th[t]) {
+            ft.logWW(e)
+        }
+
+    }
+
+    ft.W[x] = Epoch{tid: t, time_stamp: ft.Th[t][t]}
+    ft.inc(t)
+
+}
+
+func (ft *FT) read(e Event) {
+    t := e.thread()
+    x := e.arg()
+    ep, exists := ft.W[x]
+    if exists {
+        if !ep.happensBefore(ft.Th[t]) {
+            ft.logWR(e)
+        }
+
+        ft.R[x] = sync(ft.Th[t], ft.R[x])
+
+    }
+    ft.inc(t)
+
+}
+
+
+

Summary

+

False negatives

+

The HB method has false negatives. This is due to the fact +that the textual order among critical sections is preserved.

+

Consider the following trace.

+
     T1          T2
+
+1.   w(x)
+2.   acq(y)
+3.   rel(y)
+4.               acq(y)
+5.               w(x)
+6.               rel(y)
+

We derive the following HB relations.

+

The program order condition implies the following ordering +relations:

+

w(x)1<acq(y)2<rel(y)3w(x)_1 < acq(y)_2 < rel(y)_3

+

and

+

acq(y)4<w(x)5<rel(y)6acq(y)_4 < w(x)_5 < rel(y)_6.

+

The critical section order condition implies

+

rel(y)3<acq(y)4rel(y)_3 < acq(y)_4.

+

Based on the above we can conclude that

+

w(x)1<w(x)5w(x)_1 < w(x)_5.

+

How? From above we find that

+

w(x)1<acq(y)2<rel(y)3w(x)_1 < acq(y)_2 < rel(y)_3 +and +rel(y)3<acq(y)4rel(y)_3 < acq(y)_4.

+

By transitivity we find that

+

w(x)1<acq(y)4w(x)_1 < acq(y)_4.

+

In combination with +acq(y)4<w(x)5<rel(y)6acq(y)_4 < w(x)_5 < rel(y)_6 +and transitivity we find that +w(x)1<w(x)5w(x)_1 < w(x)_5.

+

Because of +w(x)1<w(x)5w(x)_1 < w(x)_5, +the HB method concludes that there is no data race because the +conflicting events +w(x)1w(x)_1 +and +w(x)5w(x)_5 +are ordered. Hence, the HB conclude that there is no data race.

+

This is a false negative. Consider the following reordering.

+
     T1          T2
+
+4.               acq(y)
+5.               w(x)
+1.   w(x)
+

We execute first parts of thread T2. Thus, we find that the two +conflicting writes are in a data race.

+

False positives

+

The first race reported by the HB method is an “actual” race. +However, subsequent races may be false positives.

+

Consider the program

+
func example3() {
+    x := 1
+    y := 1
+
+    // Thread T1
+    go func() {
+        x = 2
+        y = 2
+    }()
+
+    // Thread T2 = Main Thread
+    if y == 2 {
+        x = 3
+    }
+
+}
+

We consider a specific execution run that yields the following +trace.

+
Trace I:
+
+     T1            T2
+
+1.   w(x)
+2.   w(y)
+3.                 r(y)
+4.                 w(x)
+

We encounter a write-read race because +w(y)2w(y)_2 +and +r(y)3r(y)_3 +appear right next to each other in the trace.

+

It seems that there is also a HB write-write data race. The HB +relations derived from the above trace are as follows:

+

w(x)1<w(y)2w(x)_1 < w(y)_2 +and +r(y)3<w(x)4r(y)_3 < w(x)_4.

+

Hence, +w(x)1w(x)_1 +and +w(x)4w(x)_4 +are unordered. Hence, we find the write-write data race +(w(x)4,w(x)1)(w(x)_4, w(x)_1).

+

We reorder the above trace (while maintaining the program order HB +relations). For the reordered trace we keep the original trace +positions.

+
Trace II:
+
+     T1            T2
+
+3.                 r(y)
+4.                 w(x)
+1.   w(x)
+2.   w(y)
+

In the reordered trace II, the two writes on x appear right next to +each other. Is there a program run that yields the above reordered +trace? No!

+

In the reordered trace II, we violate the write-read dependency +between +w(y)2w(y)_2 +and +r(y)3r(y)_3. +w(y)2w(y)_2 +is the last write that takes place before +r(y)3r(y)_3. +The read value +yy +influences the control flow. See the above program where we only enter +the if-statement if +w(y)2w(y)_2 +takes place (and sets +yy +to 2).

+

We conclude that the HB relation does not take into account +write-read dependencies and therefore HB data races may not correspond +to actual data traces.

+

We say may not because based on the trace alone we cannot +decide if the write-read dependency actually affects the control +flow.

+

For example, trace I could be the result of a program run where we +assume the following program.

+
func example4() {
+    var tmp int
+    x := 1
+    y := 1
+
+    // Thread T1
+    go func() {
+        x = 2
+        y = 2 // WRITE
+    }()
+
+    // Thread T2 = Main Thread
+    tmp = y // READ
+    x = 3
+
+}
+

There is also a write-read dependency, see locations marked WRITE and +READ. However, the read value does not influence the control flow. +Hence, for the above program trace II would be a valid reordering of +trace I.

+

Weak memory models

+

There are further reasons why we can argue that there is no false +positive. Consider the original trace.

+
Trace I:
+
+     T1            T2
+
+1.   w(x)
+2.   w(y)
+3.                 r(y)
+4.                 w(x)
+

The HB relation applies the program order condition. If we consider a +specific thread, then events are executed one after the other.

+

On today’s modern computer architecture such rules no longer apply. +We may execute events “out of order” if they do not interfere.

+

For example, we can argume that the read event r(y) and +the write event w(x) in thread T2 do not interfere with +each other (as they operate on distinct variables). Hence, we may +process w(x) before r(y)! There are also +compiler optimizations where we speculatively execute w(x) +before r(y). Hence, we can argue that the data race among +the writes on x is not a false positive.

+

Further reading

+

What Happens-After the First +Race?

+

Shows that the “first” race reported by Lamport’s happens-before +relation is sound.

+

ThreadSanitizer

+

C/C++ implementation of Lamport’s happens-before relation (to analyze +C/C++).

+

Go’s data race +detector

+

Based on ThreadSanitizer.

+
+ +