Untyping Typed Algebraic Structures

Damien Pous (1)

1 SARDES - System architecture for reflective distributed computing environments

Algebraic structures sometimes need to be typed. For example, matrices over a ring form a ring, but the product is a only a partial operation: dimensions have to agree. Therefore, an easy way to look at matrices algebraically is to consider ``typed rings''. We prove some ``untyping'' theorems: in some algebras (semirings, Kleene algebras, residuated monoids), types can be reconstructed from valid untyped equalities. As a consequence, the corresponding untyped decision procedures can be extended to the typed setting.

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

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

Damien Pous. Untyping Typed Algebraic Structures. 2009. ⟨hal-00421158v1⟩