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

Abstract

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.

Study of Isabelle/HOL on Formal Algorithm Analysis and Code Generation

“…Actually, as far as we know, there is only some less correlated partial results. This is surprising because correctness of key algorithms is essential for the guarantee of good behavior of software and there has been a trend these days of verifying algorithms through formal methods such as model checking, mechanical theorem proving [6] [7] [8].…”

“…It can further extend to executional code by using certified compilers such as CompCert [9]. Comparing to verifying the binary code directly, using such a correctnessassurence-extending tool chain will save considerable effort [8]. Nevertheless, the generated code's efficiency should also be taken into account.…”