Scalable Analysis of Linear Systems using Mathematical Programming Sriram Sankaranarayanan, Henny B. Sipma, and Zohar Manna ⋆ Computer Science Department Stanford University Stanford, CA 94305-9045 {srirams,sipma,zm}@theory.stanford.edu Abstract. We present a method for generating linear invariants for large systems. The method performs forward propagation in an abstract domain consisting of arbitrary polyhedra of a predefined fixed shape. The basic operations on the domain like abstraction, intersection, join and inclusion tests are all posed as linear optimization queries, which can be solved efficiently by existing LP solvers. The number and dimension- ality of the LP queries are polynomial in the program dimensionality, size and the number of target invariants. The method generalizes sim- ilar analyses in the interval, octagon, and octahedra domains, without resorting to polyhedral manipulations. We demonstrate the performance of our method on some benchmark programs. 1 Introduction Static analysis is one of the central challenges in computer science, and increas- ingly, in other disciplines such as computational biology. Static analysis seeks to discover invariant relationships between the variables of a system that hold on every execution of the system. In computer science, knowledge of these relation- ships is invaluable for verification and optimization of systems; in computational biology this knowledge may lead to better understanding of the system’s dynam- ics. Linear invariant generation, the discovery of linear relationships between vari- ables, has a long history, starting with Karr [9], and cast in the general framework of abstract interpretation by Cousot and Cousot [6]. The most general form of linear invariant generation is polyhedral analysis. The analysis is performed in the abstract domain of all the linear inequalities over all the system variables [7]. Although impressive results have been achieved in this domain, its applicability is severely limited by its worst-case exponential time and space complexity. This has led to the investigation of more restricted domains which seek to trade off ⋆ This research was supported in part by NSF grants CCR-01-21403, CCR-02-20134 and CCR-02-09237, by ARO grant DAAD19-01-1-0723, by ARPA/AF contracts F33615-00-C-1693 and F33615-99-C-3014, by NAVY/ONR contract N00014-03-1- 0939, and by the Siebel Graduate Fellowship.