The Interprocedural Coincidence Theorem

We present an interprocedural generalization of the well-known (intraprocedural) Coincidence Theorem of Kam and Ullman, which provides a suucient condition for the equivalence of the meet over all paths (MOP) solution and the maximal xed point (MFP) solution to a data ow analysis problem. This generalization covers arbitrary imperative programs with recursive procedures, global and local variables, and formal value parameters. In the absence of procedures, it reduces to the classical intraprocedural version. In particular, our stack-based approach generalizes the coincidence theorems of Barth and Sharir/Pnueli for the same setup, which do not properly deal with local variables of recursive procedures. 1 Motivation Data ow analysis is a classical method for the static analysis of programs that supports the generation of eecient object code by \optimizing" compilers (cf. He, MJ]). For imperative languages, it provides information about the program states that may occur at some given program points during execution. Theoretically well-founded are data ow analyses that are based on abstract interpretation (cf. CC1]). The point of this approach is to replace the \full" semantics by a simpler more abstract version, which is tailored to deal with a speciic problem. Usually, the abstract semantics is speciied by a local semantic functional, which gives abstract meaning to every program statement in terms of a transformation function from a lattice C into itself. The elements of C express the data ow information of interest. The (global) abstract semantics then results from one of the following two globalization strategies; the \operational" meet over all paths (MOP) strategy, and the \denotational" maximal xed point (MFP) strategy 1 in the sense of Kam and Ullman KU]. 2 The MOP-strategy directly mimics possible program executions: it \meets" (intersects) all information corresponding to program paths reaching the program point under consideration. This speciies the optimal result of a globalization but is in general not eeective. The MFP-strategy iteratively approximates the greatest solution of a system of equations that express consistency between preconditions and post-conditions that are given in terms of data ow information: the precondition of a statement must be implied by each of the post-conditions of the predecessors, and the post-condition must be implied by the result of transforming the precondition according to the (abstract) meaning of the statement. In general, this leads to a suboptimal but algorithmic description. 1 These are the strategies that lead to the MOP-solution and MFP-solution, respectively. 2 The operational and …