Interprocedural Dataflow Analysis, Part III: Newtonian Program Analysis Via Tensor Product CS701

In this lecture, we summarize the interprocedural-dataflow-analysis method of Esparza et al., which generalizes Newton’s method from a numerical-analysis algorithm for finding roots of real-valued functions to a method for finding fixed-points of systems of equations over semirings. As in its real-valued counterpart, each iteration of the method of Esparza et al. solves a simpler “linearized” problem. We then move on to a recent method of Reps et al., which attempts to provide an improved technique for solving the linearized problems produced during successive rounds of Newton’s method for semirings. 1 Newtonian Program Analysis (NPA) In previous lectures, we described how interprocedural dataflow-analysis problems could be formalized as path problems in which there are two operators, extend (denoted by ⊗) and combine (denoted by ⊕). Technically, we are dealing with a semiring, which can be defined as follows: Definition 1.1 A semiring S = (D,⊕,⊗, 0, 1) consists of a set of elements D equipped with two binary operations: combine (⊕) and extend (⊗). ⊕ and ⊗ are associative, and have identity elements 0 and 1, respectively. ⊕ is commutative, and ⊗ distributes over ⊕. (A semiring is sometimes called a weight domain, in which case elements are called weights.) If fact, we need the semiring to have some additional properties—chiefly so that it makes sense to define a Kleene-star operator in the following way: ∗ : D → D is the operation a = ⊕ i∈N a, where a denotes the i term in the sequence a0 = 1 and ai+1 = a⊗ a. See App. A for the technical details. Our focus is on semirings in which ⊕ is idempotent (i.e., for all a ∈ D, a⊕ a = a). In an idempotent semiring, the order on elements is defined by a ⊑ b iff a⊕ b = b. (Idempotence would be expected in the context of dataflow analysis because an idempotent semiring is a join semilattice (D,⊕) in which the join operation is ⊕.) A semiring is commutative if for all a, b ∈ D, a⊗ b = b⊗ a. We work with non-commutative semirings, and henceforth use the term “semiring”—and symbol S—to mean an idempotent, noncommutative, (ω-continuous) semiring . To simplify notation, we sometimes abbreviate a⊗ b as ab, and we assume the following precedences for operators: ∗ > ⊗ > ⊕. We also sometimes use a ∈ S rather than a ∈ D.