Projective relative unification through duality

Quentin Gougeon (1, 2) , Philippe Balbiani (1, 3)

1 IRIT-LILaC - Logique, Interaction, Langue et Calcul
2 UT3 - Université Toulouse III - Paul Sabatier
3 CNRS - Centre National de la Recherche Scientifique

Unification problems can be formulated and investigated in an algebraic setting, by identifying substitutions to modal algebra homomorphisms. This opens the door to applications of the notorious duality between Heyting or modal algebras and descriptive frames. Through substantial use of this correspondence, we give a necessary and sufficient condition for formulas to be projective. A close inspection of this characterization will motivate a generalization of standard unification, which we dub relative unification. Applying this result to a number of different logics, we then obtain new proofs of their projective-or non-projective-character. Aside from reproving known results, we show that the projective extensions of K5 are exactly the extensions of K45. This resolves the open question of whether K5 is projective.

Domains: Computer Science [cs]

Quentin Gougeon, Philippe Balbiani. Projective relative unification through duality. Journal of Logic and Computation, 2023, pp.1--21. ⟨10.1093/logcom/exad058⟩. ⟨hal-04307724⟩