Geometric Quantifier Elimination Heuristics for Automatically Generating Octagonal and Max-plus Invariants

Geometric heuristics for the quantifier elimination approach presented by Kapur (2004) are investigated to automatically derive loop invariants expressing weakly relational numerical properties (such as l≤x≤h or l≤±x ±y≤h) for imperative programs. Such properties have been successfully used to analyze commercial software consisting of hundreds of thousands of lines of code (using for example, the Astree tool based on abstract interpretation framework proposed by Cousot and his group). The main attraction of the proposed approach is its much lower complexity in contrast to the abstract interpretation approach (O(n2) in contrast to O(n4), where n is the number of variables) with the ability to still generate invariants of comparable strength. This approach has been generalized to consider disjunctive invariants of the similar form, expressed using maximum function (such as max (x+a,y+b,z+c,d)≤max (x+e,y+f,z+g,h)), thus enabling automatic generation of a subclass of disjunctive invariants for imperative programs as well.