An Algorithm Inspired by Constraint Solvers to Infer Inductive Invariants in Numeric Programs

Abstract

Abstract. This paper addresses the problem of proving a given invariance property ϕ of a loop in a numeric program, by inferring automatically a stronger inductive invariant ψ. The algorithm we present is based on both abstract interpretation and constraint solving. As in abstract interpretation, it computes the effect of a loop using a numeric abstract domain. As in constraint satisfaction, it works from "above"-interactively splitting and tightening a collection of abstract elements until an inductive invariant is found. Our experiments show that the algorithm can find non-linear inductive invariants that cannot normally be obtained using intervals (or octagons), even when classic techniques for increasing abstract-interpretation precision are employed-such as increasing and decreasing iterations with extrapolation, partitioning, and disjunctive completion. The advantage of our work is that because the algorithm uses standard abstract domains, it sidesteps the need to develop complex, non-standard domains specialized for solving a particular problem.

Synthesizing Invariants by Solving Solvable Loops

“…The tool is available at [7]. [16,24] Let us now detail the work performed by Pilat over the example of figure 6 (taken from [16]). First, our tool generates the shape of the invariant, i.e.…”

“…Synthesis of inequality invariants has become a growing field [16,22], for example in linear filters analysis and automatic verification in general as it provides good knowledge of the variables bounds when computing floating point operations. Abstract interpretation [6] with widening operators allows good approximation of loops with the desired format.…”

Lifting CDCL to Template-Based Abstract Domains for Program Verification

“…SMPP enumerates program paths using a SAT formula, which are then verified using abstract interpretation. The work of [22] proposes an algorithm inspired by constraint solvers for inferring disjunctive invariants using intervals. The lifting of CDCL to first-order theories is proposed in [11,21,23].…”

Counterexample- and Simulation-Guided Floating-Point Loop Invariant Synthesis

“…Some previous techniques for loop invariant synthesis for numerical programs require a target property to be given [19,29,38,40,43]; in most cases this is a set of unsafe states that should be proven to be unreachable. However, for floatingpoint loops where the goal is to compute as tight invariants as possible, specifying unsafe states essentially amounts to finding the invariant itself.…”

“…We thus require nonlinear loop invariants expressed as polynomial inequalities to handle many numerical loops. However, existing techniques each have limitations, as they require templates to be given by the user [1]; are limited to linear loops only [36]; do not always produce invariants that satisfy the precondition [33]; or require a target range in order to produce tight invariants [29].…”