PROALG · Proof-theoretic methods in algebra

The goal of this project is to develop and exploit proof-theoretic methods for ordered algebraic structures. Traditionally, algebra and proof theory represent two distinct approaches within logic: the former concerned with semantic meaning and structures, the latter with syntactic and algorithmic aspects. In many intriguing cases, however, methods from one field have been essential to obtaining proofs in the other. In particular, proof-theoretic techniques have been used to establish important results for classes of algebras in the framework of residuated lattices. This includes both algebras for a wide range of non-classical logics investigated across mathematics, computer science, philosophy, and linguistics, and also important examples from algebra such as lattice-ordered groups. In recent years, researchers from many countries have begun to explore connections between these two fields more closely, providing algebraic interpretations of proof-theoretic methods, and vice versa. The time is now ripe to clarify and exploit these connections.The concrete objectives of the project are: (A) to define uniform proof systems for classes of algebras and logic (such as e.g., lattice-ordered groups or cancellative residuated lattices) not covered by known frameworks, (B) to use proof systems to establish new decidability and complexity results, (C) to investigate relationships between the algebraic property of amalgamation and the logical property of interpolation and use proof systems to settle open problems, (D) to use proof systems to establish correspondences between algebraic properties and admissible rules. The main challenge and originality of the project will be to combine new insights and techniques from algebra and proof theory to tackle these goals.