Untyping Typed Algebraic Structures and Colouring Proof Nets of Cyclic Linear Logic

Damien Pous (1)

1 SARDES - System architecture for reflective distributed computing environments

Algebraic structures sometimes need to be typed. For example, matrices over real numbers form a ring, but multiplication is a only a partial operation: dimensions have to agree. Therefore, a natural way to look at the ring of matrices algebraically is to consider ``typed rings''. We prove several ``untyping'' theorems: in some algebras (semirings, Kleene algebras, residuated lattices, involutive residuated lattices), typed equations can be derived from the underlying untyped equations. As a consequence, the corresponding untyped decision procedures can be extended for free to the typed settings. Some of these theorems are obtained via a detour through fragments of cyclic linear logic.

Mots clés: algebra; typed algebra; kleene algebra; residuated lattice; gentzen proof system; cyclic linear logic; decision procedures; coq proof assistant; partial algebra

Domaines: Mathématique discrète [cs.DM]; Logique en informatique [cs.LO]; Catégories et ensembles [math.CT]

Damien Pous. Untyping Typed Algebraic Structures and Colouring Proof Nets of Cyclic Linear Logic. 2010. ⟨hal-00421158v2⟩