### FORTissimo: Automating the First-Order Theory of Rewriting

FORTissimo is about the first-order theory of rewriting, which is a decidable theory for left-linear right-ground rewrite systems in which well-known properties like confluence, normalization and termination are expressible. The decision procedure for this theory is based on tree automata techniques and a first implementation was conducted by Franziska Rapp during her master studies. The resulting tool FORT is equipped with a synthesis mode to generate rewrite systems that satisfy properties expressible in the first-order theory of rewriting.

The aim of this project is to formalize the decision procedure in the proof assistant Isabelle/HOL such that the output of FORT can be certified. Moreover, the expressiveness and the performance of FORT should be increased, and its limitations better understood. More concretely, the project has the following three main objectives:

- Formalize the basic properties of automata on
*n*-ary relations (cylindrification, projection) and ground tree transducers in Isabelle/HOL. Develop suitable certificates that can be produced by FORT and checked by the certifier obtained from the formalization in Isabelle via code generation. - Improve the performance of FORT by adopting and developing state-of-the-art tree automata techniques. Investigate methods for formula normalization in order to speed up the computation of intermediate automata. Adopt parallel programming techniques to further improve the efficiency of FORT.
- Improve the expressiveness of FORT by adding support for combinations of rewrite systems and the generation of witnesses for existentially quantified variables. Investigate to what extent properties on open terms can be simulated in FORT, and whether certain fragments of the first-order theory of rewriting are decidable for larger classes of rewrite systems.

FORTissimo will start on September 1, 2017 and run for three years.

#### Members

##### Current Members

- Aart Middeldorp (coordination)

##### Former Members

- Bertram Felgenhauer
- Jamie Hochrainer
- Johannes Koch
- Alexander Lochmann
- Fabian Mitterwallner
- T. V. H. Prathamesh
- Franziska Rapp
- Kei Shirakizawa (guest)

#### FWF project number

P30301#### Contact

aart middeldorp at uibk dot ac dot at#### Publications

Formalized Signature Extension Results for Equivalence

Alexander Lochmann, Fabian Mitterwallner, Aart Middeldorp

Proceedings of the 11th International Workshop on Confluence (IWC 2022),
pp. 42-47,
2022.

First-Order Theory of Rewriting

Alexander Lochmann, Bertram Felgenhauer

Archive of Formal Proofs 2022.

Regular Tree Relations

Alexander Lochmann, Bertram Felgenhauer, Christian Sternagel, René Thiemann, Thomas Sternagel

Archive of Formal Proofs 2021.

Formalized Signature Extension Results for Confluence, Commutation and Unique Normal Forms

Alexander Lochmann, Fabian Mitterwallner, Aart Middeldorp

Proceedings of the 10th International Workshop on Confluence (IWC 2021),
pp. 25 - 30,
2021.

Certifying Proofs in the First-Order Theory of Rewriting

Fabian Mitterwallner, Alexander Lochmann, Aart Middeldorp, Bertram Felgenhauer

27th International Conference on Tools and Algorithms for the Construction and Analysis of Systems (TACAS 2021),
LNCS 12652,
pp. 127 – 144,
2021.

A Verified Decision Procedure for the First-Order Theory of Rewriting for Linear Variable-Separated Rewrite Systems

Alexander Lochmann, Aart Middeldorp, Fabian Mitterwallner, Bertram Felgenhauer

10th ACM SIGPLAN International Conference on Certified Programs and Proofs (CPP 2021),
pp. 250 – 263,
2021.

Formalized Proofs of the Infinity and Normal Form Predicates in the First-Order Theory of Rewriting

Alexander Lochmann, Aart Middeldorp

26th International Conference on Tools and Algorithms for the Construction and Analysis of Systems,
LCNS 12079,
pp. 25 – 40,
2020.

A Verified Ground Confluence Tool for Linear Variable-Separated Rewrite Systems in Isabelle/HOL

Bertram Felgenhauer, Aart Middeldorp, T. V. H. Prathamesh, and Franziska Rapp

8th ACM SIGPLAN International Conference on Certified Programs and Proofs,
pp. 132 – 143,
2019.

Minsky Machines

Bertram Felgenhauer

Archive of Formal Proofs 2018.

Towards a Verified Decision Procedure for Confluence of Ground Term Rewrite Systems in Isabelle/HOL

Bertram Felgenhauer, Aart Middeldorp, T. V. H. Prathamesh, Franziska Rapp

Proceedings of the 7th International Workshop on Confluence (IWC 2018),
pp. 46 – 50,
2018.

FORT 2.0

Franziska Rapp, Aart Middeldorp

Proceedings of the 9th International Joint Conference on Automated Reasoning,
Lecture Notes in Artificial Intelligence 10900,
pp. 81 – 88,
2018.