Systematic design of program analysis frameworks

Semantic analysis of programs is essential in optimizing compilers and program verification systems. It encompasses data flow analysis, data type determination, generation of approximate invariant assertions, etc. Several recent papers (among others Cousot & Cousot[77a], Graham & Wegman[76], Kam & Ullman[76], Kildall[73], Rosen[78], Tarjan[76], Wegbreit[75]) have introduced abstract approaches to program analysis which are tantamount to the use of a program analysis framework (A,t,ã) where A is a lattice of (approximate) assertions, t is an (approximate) predicate transformer and ã is an often implicit function specifying the meaning of the elements of A. This paper is devoted to the systematic and correct design of program analysis frameworks with respect to a formal semantics. Preliminary definitions are given in Section 2 concerning the merge over all paths and (least) fixpoint program-wide analysis methods. In Section 3 we briefly define the (forward and backward) deductive semantics of programs which is later used as a formal basis in order to prove the correctness of the approximate program analysis frameworks. Section 4 very shortly recall the main elements of the lattice theoretic approach to approximate semantic analysis of programs. The design of a space of approximate assertions A is studied in Section 5. We first justify the very reasonable assumption that A must be chosen such that the exact invariant assertions of any program must have an upper approximation in A and that the approximate analysis of any program must be performed using a deterministic process. These assumptions are shown to imply that A is a Moore family, that the approximation operator (wich defines the least upper approximation of any assertion) is an upper closure operator and that A is necessarily a complete lattice. We next show that the connection between a space of approximate assertions and a computer representation is naturally made using a pair of isotone adjoined functions. This type of connection between two complete lattices is related to Galois connections thus making available classical mathematical results. Additional results are proved, they hold when no two approximate assertions have the same meaning. In Section 6 we study and examplify various methods which can be used in order to define a space of approximate assertions or equivalently an approximation function. They include the characterization of the least Moore family containing an arbitrary set of assertions, the construction of the least closure operator greater than or equal to an arbitrary approximation function, the definition of closure operators by composition, the definition of a space of approximate assertions by means of a complete join congruence relation or by means of a family of principal ideals. Section 7 is dedicated to the design of the approximate predicate transformer induced by a space of approximate assertions. First we look for a reasonable definition of the correctness of approximate predicate transformers and show that a local correctness condition can be given which has to be verified for every type of elementary statement. This local correctness condition ensures that the (merge over all paths or fixpoint) global analysis of any program is correct. Since isotony is not required for approximate predicate transformers to be correct it is shown that non-isotone program analysis frameworks are manageable although it is later argued that the isotony hypothesis is natural. We next show that among all possible approximate predicate transformers which can be used with a given space of approximate assertions there exists a best one which provides the maximum information relative to a program-wide analysis method. The best approximate predicate transformer induced by a space of approximate assertions turns out to be isotone. Some interesting consequences of the existence of a best predicate transformer are examined. One is that we have in hand a formal specification of the programs which have to be written in order to implement a program analysis framework once a representation of the space of approximate assertions has been chosen. Examples are given, including ones where the semantics of programs is formalized using Hoare[78]'s sets of traces. In Section 8 we show that a hierarchy of approximate analyses can be defined according to the fineness of the approximations specified by a program analysis framework. Some elements of the hierarchy are shortly exhibited and related to the relevant literature. In Section 9 we consider global program analysis methods. The distinction between "distributive" and "non-distributive" program analysis frameworks is studied. It is shown that when the best approximate predicate transformer is considered the coincidence or not of the merge over all paths and least fixpoint global analyses of programs is a consequence of the choice of the space of approximate assertions. It is shown that the space of approximate assertions can always be refined so that the merge over all paths analysis of a program can be defined by means of a least fixpoint of isotone equations. Section 10 is devoted to the combination of program analysis frameworks. We study and examplify how to perform the "sum", "product" and "power" of program analysis frameworks. It is shown that combined analyses lead to more accurate information than the conjunction of the corresponding separate analyses but this can only be achieved by a new design of the approximate predicate transformer induced by the combined program analysis frameworks.