René Thiemann

We report on the new ARI infrastructure that supports tools and competitions in term rewriting. It offers ARI-COPS, a database for confluence problems and competition results, and ARIWeb, a convenient web interface for tools that participate in the annual confluence competition. These are built on top of the new ARI format for rewrite systems, a format converter, certifiers for competition results, and a duplicate checker.

In this paper, we present an Isabelle/HOL formalization of co-rewrite pairs for non-reachability analysis in term rewriting. In particular, we formalize polynomial interpretations over negative integers as well as the weighted path order (WPO) and its variant co-WPO. With this formalization, the verified certifier CeTA is now able to check such non-reachability proofs, including those for non-reachability problems of a database where existing tools fail to provide certified proofs.

By means of a simple reduction from Hilbert's 10th problem we prove the somewhat surprising result that termination of one-rule rewrite systems by a linear interpretation in the natural numbers is undecidable. The very same reduction also shows the undecidability of termination of one-rule rewrite systems using the Knuth-Bendix order with subterm coefficients. The linear termination problem remains undecidable for one-rule rewrite systems that can be shown terminating by a (non-linear) polynomial interpretation. We further show the undecidability of the problem whether a one-rule rewrite system can be shown terminating by a polynomial interpretation with rational or real coefficients. Several of our results have been formally verified in the Isabelle/HOL proof assistant.

This work presents formal correctness proofs in Isabelle/HOL of algorithms to transform a matrix into Smith normal form, a canonical matrix form, in a general setting: the algorithms are written in an abstract form and parameterized by very few simple operations. We formally show their soundness provided the operations exist and satisfy some conditions, which always hold on Euclidean domains. We also provide a formal proof on some results about the generality of such algorithms as well as the uniqueness of the Smith normal form. Since Isabelle/HOL does not feature dependent types, the development is carried out by switching conveniently between two different existing libraries by means of the lifting and transfer package and the use of local type definitions, a sound extension to HOL.

AProVE is a powerful termination prover for various programming languages, including a termination analysis method for imperative programs specified in the LLVM intermediate representation (IR). The method internally works in three steps: first, it transforms LLVM IR code into a symbolic execution graph; second, the graph is translated into an integer transition system; finally, termination of the transition sys- tem is proved by the back end of AProVE. Since AProVE is unverified software, our aim is to increase its reliability by certifying the generated proofs. To this end, we require formal semantics of all program representations, i.e., for LLVM IR, for symbolic execution graphs and for inte- ger transition systems. As the latter is already available, we define the former ones. We note that our semantics for LLVM IR use arithmetic with unbounded integers. We further verify the first and the second step of AProVE’s termination method, including verified algorithms to check concrete proofs. Since the third step can already be certified, we obtain a complete formally verified method for certifying AProVE’s termination proofs of LLVM IR programs. The whole formalization has been done in Isabelle/HOL and our certifier is available as a Haskell program via code generation.

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.

The LLL basis reduction algorithm was the first polynomial-time algorithm to compute a reduced basis of a given lattice, and hence also a short vector in the lattice. It approximates an NP-hard problem where the approximation quality solely depends on the dimension of the lattice, but not the lattice itself. The algorithm has applications in number theory, computer algebra and cryptography. In this paper, we provide an implementation of the LLL algorithm. Both its soundness and its polynomial running-time have been verified using Isabelle/HOL. Our implementation is nearly as fast as an implementation in a commercial computer algebra system, and its efficiency can be further increased by connecting it with fast untrusted lattice reduction algorithms and certifying their output. We additionally integrate one application of LLL, namely a verified factorization algorithm for univariate integer polynomials which runs in polynomial time.