A Quantifier-Elimination Based Heuristic for Automatically Generating Inductive Assertions for Programs

Abstract

A method using quantifier-elimination is proposed for automatically generating program invariants/inductive assertions. Given a program, inductive assertions, hypothesized as parameterized formulas in a theory, are associated with program locations. Parameters in inductive assertions are discovered by generating constraints on parameters by ensuring that an inductive assertion is indeed preserved by all execution paths leading to the associated location of the program. The method can be used to discover loop invariants–properties of variables that remain invariant at the entry of a loop. The parameterized formula can be successively refined by considering execution paths one by one; heuristics can be developed for determining the order in which the paths are considered. Initialization of program variables as well as the precondition and postcondition, if available, can also be used to further refine the hypothesized invariant. The method does not depend on the availability of the precondition and postcondition of a program. Constraints on parameters generated in this way are solved for possible values of parameters. If no solution is possible, this means that an invariant of the hypothesized form is not likely to exist for the loop under the assumptions/approximations made to generate the associated verification condition. Otherwise, if the parametric constraints are solvable, then under certain conditions on methods for generating these constraints, the strongest possible invariant of the hypothesized form can be generated from most general solutions of the parametric constraints. The approach is illustrated using the logical languages of conjunction of polynomial equations as well as Presburger arithmetic for expressing assertions.