A Don’t Sit on the Fence: A Static Analysis Approach to Automatic Fence Insertion

events. An abstract event represents all events with same program point, direction (write or read), and compatible memory location. An abstract event consists of first a unique identifier, then the direction specifier (W or R), and then the address specifier. In Figure 10(a), (a)Wx abstracts the events (a′)Wx1 and (a′′)Wx2 in the executions of Figure 10(b) and (c). Moreover, for example, a static event (a)W∗ would also abstract the two events, as ∗ is compatible with any memory location. We write addr(e) for the address specifier of an abstract event e. Static program order. The static program order relation pos abstracts all the (dynamic) po edges that connect two events in program order and that cannot be decomposed as a succession of po edges in this execution. We write pos (resp. pos) for the transitive (resp. reflexive-transitive) closure of this relation. We also write begin(pos) and end(pos) to denote respectively the sets of the first and last abstract events of pos. That is, if we imagine the pos relation as a directed graph, then begin(pos) contains the abstract events in pos that do not have incoming edges, and end(pos) contains the abstract events that do not have outgoing edges. Competing pairs. The external communications coe ∪ rfe ∪ fre are over-approximated by the competing pairs relation cmp. In Figure 10(a), the cmp edges (a, f), (b1, d), and (c, f) abstract in particular the fre edges (f ′, c′) and (f ′, a′), and the rfe edge (b1, d′) in Figure 10(b). We do not need to represent internal communications as they are already covered by pos . The cmp construction is similar to the first steps of static data race detection (see, e.g., [Kahlon et al. 2009, Sec. 5]), where statements involved in write-read or write-write communications are collected. As further work, we could reduce the set of competing 3We denote the function composition operator by ◦. ACM Transactions on Programming Languages and Systems, Vol. V, No. N, Article A, Publication date: January YYYY. Don’t Sit on the Fence: A Static Analysis Approach to Automatic Fence Insertion A:15 (1) assignment: lhs = rhs; i τ [lhs = rhs; i](aeg) = let Es = aeg.Es ∪ evts(lhs) ∪ evts(rhs)∪ trg(lhs) in let pos = aeg.pos∪ end(aeg.pos)× (evts(rhs) ∪ evts(lhs))∪ (evts(rhs) ∪ evts(lhs))× trg(lhs) in τ [i](Es,pos, aeg. cmp) R W τ [i] evts(lhs) ∪ evts(rhs) trg(lhs) pos pos pos (2) function call3: fun(); i τ [fun(); i] = τ [i] ◦ τ [body(fun)] τ [body(f)] τ [i] pos pos (3) guard: [guard] i1; i2 τ [ [guard] i1; i2](aeg) = let guarded = τ [i1](aeg) in let Es = aeg.Es ∪ guarded.Es in let pos = aeg.pos ∪ guarded.pos in τ [i2](Es,pos, aeg. cmp) τ [i1] τ [i2] pos pos