Generalizing the Template Polyhedral Domain

Abstract

Template polyhedra generalize weakly relational domains by specifying arbitrary fixed linear expressions on the left-hand sides of inequalities and undetermined constants on the right. The domain operations required for analysis over template polyhedra can be computed in polynomial time using linear programming. In this paper, we introduce the generalized template polyhedral domain that extends template polyhedra using fixed left-hand side expressions with bilinear forms involving program variables and unknown parameters to the right. We prove that the domain operations over generalized templates can be defined as the best possible abstractions of the corresponding polyhedral domain operations. The resulting analysis can straddle the entire space of linear relation analysis starting from the template domain to the full polyhedral domain. We show that analysis in the generalized template domain can be performed by dualizing the join, post-condition and widening operations. We also investigate the special case of template polyhedra wherein each bilinear form has at most two parameters. For this domain, we use the special properties of two dimensional polyhedra and techniques from fractional linear programming to derive domain operations that can be implemented in polynomial time over the number of variables in the program and the size of the polyhedra. We present applications of generalized template polyhedra to strengthen previously obtained invariants by converting them into templates. We describe an experimental evaluation of an implementation over several benchmark systems.