Invariant and Type Inference for Matrices

Abstract

Abstract. We present a loop property generation method for loops iterating over multi-dimensional arrays. When used on matrices, our method is able to infer their shapes (also called types), such as upper-triangular, diagonal, etc. To generate loop properties, we first transform a nested loop iterating over a multidimensional array into an equivalent collection of unnested loops. Then, we infer quantified loop invariants for each unnested loop using a generalization of a recurrence-based invariant generation technique. These loop invariants give us conditions on matrices from which we can derive matrix types automatically using theorem provers. Invariant generation is implemented in the software package Aligator and types are derived by theorem provers and SMT solvers, including Vampire and Z3. When run on the Java matrix package JAMA, our tool was able to infer automatically all matrix types describing the matrix shapes guaranteed by JAMA's API.

Inferring Loop Invariants Using Postconditions

“…Soundness and completeness are achieved by leveraging the decidability of the underlying mathematical domains they represent; this implies that the extension of these techniques to new classes of properties is often limited by undecidability. In fact, state-of-the-art static techniques can mostly infer invariants in the form of "well-behaving" mathematical domains such as linear inequalities [11,9], polynomials [38,37], restricted properties of arrays [7,5,20], and linear arithmetic with uninterpreted functions [1]. Loop invariants in these forms are extremely useful but rarely sufficient to prove full functional correctness of programs.…”

Loop invariants

“…Soundness and completeness are achieved by leveraging the decidability of the underlying mathematical domains they represent; this implies that the extension of these techniques to new classes of properties is often limited by undecidability. State-of-the-art static techniques can infer invariants in the form of mathematical domains such as linear inequalities [Cousot and Halbwachs 1978;Colón et al 2003], polynomials [Sankaranarayanan et al 2004;Rodríguez-Carbonell and Kapur 2007], restricted properties of arrays [Bradley et al 2006;Bozga et al 2009;Henzinger et al 2010], and linear arithmetic with uninterpreted functions [Beyer et al 2007]. …”

Automatically inferring loop invariants via algorithmic learning

“…A loop property generation method for loops iterating over multi-dimensional arrays is introduced in Henzinger et al (2010). For inferring range predicates, Jhala and McMillan (2007) described a framework that uses infeasible counterexample paths.…”