Inference of Polynomial Invariants for Imperative Programs: A Farewell to Gröbner Bases

We propose a static analysis for computing polynomial invariants for imperative programs. The analysis is derived from an abstract interpretation of a backwards semantics, and computes pre-conditions for equalities like g = 0 to hold at the end of execution. A distinguishing feature of the technique is that it computes polynomial loop invariants without resorting to Gröbner base computations. The analysis uses remainder computations over parameterized polynomials in order to handle conditionals and loops efficiently. The algorithm can analyse and find a large majority of loop invariants reported previously in the literature, and executes significantly faster than implementations using Gröbner bases.