Verifying a Solver for Linear Mixed Integer Arithmetic in Isabelle/HOL

We implement a decision procedure for linear mixed integer arithmetic and formally verify its soundness in Isabelle/HOL. We further integrate this procedure into one application, namely into , a formally verified certifier to check untrusted termination proofs. This checking involves assertions of unsatisfiability of linear integer inequalities; previously, only a sufficient criterion for such checks was supported. To verify the soundness of the decision procedure, we first formalize the proof that every satisfiable set of linear integer inequalities also has a small solution, and give explicit upper bounds. To this end we mechanize several important theorems on linear programming, including statements on integrality and bounds. The procedure itself is then implemented as a branch-and-bound algorithm, and is available in several languages via Isabelle’s code generator. It internally relies upon an adapted version of an existing verified incremental simplex algorithm.

Extending Non-Termination Proof Techniques to Asynchronously Communicating Concurrent Programs

Currently, there are no approaches known that allow for non-termination proofs of concurrent programs which account for asynchronous communication via FIFO message queues. Those programs may be written in high-level languages such as Java or Promela. We present a first approach to prove non-termination for such programs. In addition to integers, the programs that we consider may contain queues as data structures. We present a representation of queues and the operations on them in the domain of integers, and generate invariants that help us prove non-termination of selected control flow loops using a theorem proving approach. We illustrate this approach by applying a prototype tool implementation to a number of case studies.