## Section:
New Results2>
### Non linear analysis: fast inference of polynomial invariants3>

Participants : Thomas Jensen, David Cachera, Arnaud Jobin.

The problem of automatically inferring non-linear (polynomial) invariants of programs is still a challenge in program verification. A central observation in existing work on generating polynomial invariants is that n-ary relations between variables that can be described as the zeroes of a set of polynomials, correspond to a lattice of polynomials ideals. Such ideals are finitely generated , and all the approaches proposed so far in the literature rely on Gröbner base computations for computing ideal intersection or inclusion, or analysing the effects of polynomial assignments to variables. Computing Gröbner bases however slows down considerably the overall analysis.

We have proposed an abstract interpretation based method for inferring polynomial invariants that entirely avoids computing Gröbner bases. The method is precise and efficient, and is obtained without restricting the expressiveness of the polynomial programming language. Our analysis handles a general polynomial structured programming language that includes if and while constructs where branching conditions are both polynomial equalities and disequalities. Our analysis uses a form of weakest precondition calculus for showing that a polynomial relation holds at the end of a program. We show that this backward approach, which was already observed to be well adapted to polynomial disequality guards can be extended to equality guards by using parameterized polynomial division.

Based on this anlysis, we have designed a constraint-based algorithm for inferring polynomial invariants. Such constraint-based techniques (rather than iteration) when dealing with loops means that it becomes feasible to analyse conditionals precisely, using parameterized polynomial division. A salient feature of this analysis, which distinguishes it from previous analyses, is that it does not require the use of Gröbner base computations. We have implemented this algorithm in Maple and our benchmarks show that our analyzer can successfully infer invariants on a sizeable set of examples, while performing two orders of magnitude faster than other existing implementations [19] .