How to be an effective profiler: A 15-745 Project

value v ::= n constant | πi function parameter | v1 ⊕ v2 operation | ld(πH , v) memory load | P ? v1 : v2 conditional value | > unknown abstract proposition P ::= v1 = v2 | v1 < v2 | . . . abstract heap H ::= πH heap parameter | [v1 7→ v2]H assignment | > unknown heapheap H ::= πH heap parameter | [v1 7→ v2]H assignment | > unknown heap Figure 2: Interpretation lattice We now give a formal description of a simplified version of the abstract interpretation used in our effect analysis. In this simplified account, we ignore external system calls, function return results, and the intricacies of L3’s fat pointers—we give some discussion of those issues in later sections. For our purposes in this section, abstract interpretation produces for each function call just the heap update performed on that trace, in terms of the function parameters and input heap. The interpretation lattice is given in Figure 2. It is mostly straightforward, but the analysis of pointers deserves some discussion. The state of the heap at the beginning of a function call is represented by the abstract parameter πH , and a memory load of location v from that heap is denoted by ld(πH , v). Assignments are dealt with by treating the heap as a function—a load of address v in the heap [v1 7→ v2]H , evaluates to v2 if v = v1, or else to the value determined by the load in H . Evaluation of a trace τ on an abstract heap H produces a new heap H ′ assuming that conditions φ are met. This is represented by the judgment τ,H ⇓ φ ⊃ H ′, which is given in Figure 3. Note that on a function call, we generalize the function parameters and input heap to be π1, . . . , πn and πH respectively, so as to obtain the most general profile information for the callee. Given this general profile, we perform a substitution to obtain the effects in terms of the caller’s parameters and heap. 3.3 Simplifying effect profiles via theorem proving The conditions generated along an abstract execution in Figure 3 are often redundant in the sense that they are tautologous given the preceding conditions. For example, suppose that executing the first iteration of a loop generates the condition 2 ∗π1 < π2, and the next iteration generates π1 < π2—the second condition is strictly redundant. Similarly, symbolic heaps may be overly complicated, if they contain conditional values P ? v1 : v2 where P is always valid given the preconditions.