Valid Trace Reorderings

In the literature referred to as Correct Trace Reordering.

Two requirements:

Source trace $S$ is well-formed.
Reordered trace $R$ is still well-formed and satisfies program-order and last-writer conditions.

Well-formedness

We have seen Lock-semantics condition:

Lock-1: if $P = P_1 acq(y) P_2 acq(y) P_3$ , then there exists $rel(y)$ in $P_2$ .

Lock-2: if $P = P_1 rel(y) P_2$ , then there exists $acq(y)$ such that $P_1 = P'_1 acq(y) P''_1$ and there is no $rel(y)$ in $P''_1$ . That is, we have $P = P'_1 acq(y) P''_1 rel(y) P_2$ .
Additionally, we have Fork-join condition:

Fork-1: for each thread $T \neq T1$ (not the main thread), there exists at most one $fork(T)$ in $P$ , and there is no $fork(T1)$ in $P$ .

Fork-2: for each event $e$ in thread $T \neq T1$ , there exists $fork(T)$ in thread $T' \neq T$ and $fork(T) <P e$ .

Join: for each $join(T)$ in thread $T'$ we have $T \neq T'$ and, for each event $e$ in $T$ , $e <P join(T)$ .

We often omit $fork$ events, assuming all threads are initially forked by the main thread.

HB additionally requires:

If $fork(T) \in P$ , then, for each $e \in P \downarrow T$ , $fork(T) < e$ .

Recall thread projection: $P \downarrow T$ is the list of events in thread $T$ contained in trace $P$ .
If $join(T) \in P$ , then, for each $e \in P \downarrow T$ , $e < join(T)$ .

Program-order and Last-writer Conditions

We have seen:

Program-order condition:

The condition requires, for each thread $T$ , that $S \downarrow T = R \downarrow T$ .
Last-writer condition:

For each read $r(x)$ in $R$ , the last-writer of $r(x)$ is the same in $S$ and $R$ .

Examples

HB, ESs, and Set-based Race Predictor.

Example 1

     T1            T2            ES
              
e1.   fork(T2)                    
e2.   wr(x)                       
e3.   rd(x)                       
e4.                 rd(x)         
e5.                 wr(x)

HB: $1 < 2 < 3$ and $1 < 4 < 5$ .

     T1            T2            ES            
                                               
e1.   fork(T2)                    {1}              
e2.   wr(x)                       {1,2}              
e3.   rd(x)                       {1,2,3}              
e4.                 rd(x)         {1,4}
e5.                 wr(x)         {1,4,5}

Example 2

     T1            T2            ES            
                                               
e1.   fork(T2)                                  
e2.   wr(x)                                     
e3.   wr(y)                                     
e4.                 rd(y)                       
e5.                 wr(x)

HB: $1<2<3$ and $1,4,5$ .

     T1            T2            ES                       
                                                            
e1.   fork(T2)                    {1}         D(T1) = {1} ; D(T2) = {1}        
e2.   wr(x)                       {1,2}       D(T1) = {1,2} ; W(x) = 2         
e3.   wr(y)                       {1,2,3}     D(T1) = {1,2,3} ; W(y) = 3         
e4.                 rd(y)         {1,4}       W(y) notin D(T2) ; R(y) = {4} ; D(T2) = {1,4}        
e5.                 wr(x)         {1,4,5}     W(x) notin D(T2) ; D(T2) = {1,4,5} ; W(x) = 5

Example 3

     T1            T2          ES             
                                            
e1.   fork(T2)                                 
e2.   wr(a)                                    
e3.   acq(x)                                   
e4.   wr(b)                                    
e5.   rel(x)                                   
e6.   wr(c)                                 
e7.                 acq(x)                     
e8.                 rd(a)                      
e9.                 rd(b)                      
e10.                rel(x)                     
e11.                wr(c)

HB: $1<2<3<4<5<6$ , $1<7,8,9,10,11$ , and $5<7$ .

     T1            T2          ES                        
                                                         
e1.   fork(T2)                  {1}                     D(T1) = {1} ; D(T2) = {1}        
e2.   wr(a)                     {1,2}                   D(T1) = {1,2} ; W(a) = 2          
e3.   acq(x)                    {1,2,3}                 D(T1) = {1,2,3}            
e4.   wr(b)                     {1,2,3,4}               D(T1) = {1,2,3,4} ; W(b) = 4              
e5.   rel(x)                    {1,2,3,4,5}             D(T1) = {1,2,3,4,5} ; Rel(x) = {1,2,3,4,5}                
e6.   wr(c)                     {1,2,3,4,5,6}           D(T1) = {1,2,3,4,5,6} ; W(c) = 6               
e7.                 acq(x)      {1,2,3,4,5,7}           D(T2) = {1,2,3,4,5,7}     
e8.                 rd(a)       {1,2,3,4,5,7,8}         R(a) = {8} ; D(T2) = {1,2,3,4,5,7,8}       
e9.                 rd(b)       {1,2,3,4,5,7,8,9}       R(b) = {9} ; D(T2) = {1,2,3,4,5,7,8,9}         
e10.                rel(x)      {1,2,3,4,5,7,8,9,10}    D(T2) = {1,2,3,4,5,7,8,9,10} ; Rel(x) = {1,2,3,4,5,7,8,9,10}            
e11.                wr(c)       {1,2,3,4,5,7,8,9,10,11} W(c) notin D(T2) ; D(T2) = {1,2,3,4,5,7,8,9,10,11} ; W(c) = 11

Example 4

     T1            T2                        
              
e1.   fork(T2)               
e2.   acq(x)                 
e3.   wr(a)                  
e4.   rel(x)                 
e5.                 acq(x)   
e6.                 rd(a)    
e7.                 rel(x)

HB: $1<2<3<4$ , $1<5<6<7$ , and $4<5$ .

     T1            T2                        
              
e1.   fork(T2)               D(T1) = {1} ; D(T2) = {1}                   
e2.   acq(x)                 D(T1) = {1,2}     
e3.   wr(a)                  D(T1) = {1,2,3} ; W(a) = 3     
e4.   rel(x)                 D(T1) = {1,2,3,4} ; Rel(x) = {1,2,3,4}     
e5.                 acq(x)   D(T2) = {1,2,3,4,5}     
e6.                 rd(a)    D(T2) = {1,2,3,4,5,6} ; W(a) = 6     
e7.                 rel(x)   D(T2) = {1,2,3,4,5,6,7} ; Rel(x) = {1,2,3,4,5,6,7}

Example 5

     T1            T2            
              
e1.   fork(T2)                    
e2.   acq(l)                      
e3.   wr(x)                       
e4.   rel(l)                      
e5.                 acq(l)        
e6.                 wr(x)         
e7.                 rel(l)        
e8.                 rd(x)

HB: $1<2<3<4$ , $1<5<6<7<8$ , and $4<5$ .

Correct reordering with a data race:

     T0            T1            
              
e1.   fork(T1)                    
e5.                 acq(l)        
e6.                 wr(x)         
e7.                 rel(l)        
e2.   acq(l)                      
e8.                 rd(x)         
e3.   wr(x)                       
e4.   rel(l)

Example 6

     T1            T2            T3            
              
e1.   fork(T3)                                  
e2.   acq(l)                                    
e3.   fork(T2)                                  
e4.                 wr(x)                       
e5.   join(T2)                                  
e6.   rel(l)                                    
e7.                               acq(l)        
e8.                               wr(x)         
e9.                               rel(l)        
e10.                              rd(x)

HB: $1<2<3<5<6$ , $3<4<5$ , $1<7<8<9<10$ , and $6<7$ .

Correct reordering with a data race: ~~~ T0 T1 T2

e1. fork(T2)
e7. acq(l)
e8. wr(x)
e9. rel(l)
e2. acq(l)
e3. fork(T1)
e10. rd(x)
e4. wr(x)
e5. join(T1)
e6. rel(l)
~~~

Dynamic data race prediction (Tutorial)

Summary