Fixed-point elimination in the Intuitionistic Propositional Calculus (extended version)

Silvio [e-mail] Ghilardi (1) , Maria Joao Gouveia (2) , Luigi Santocanale (3, 4)

1 UNIMI - Università degli Studi di Milano = University of Milan
2 ULISBOA - Universidade de Lisboa = University of Lisbon
3 LIS - Laboratoire d'Informatique et des Systèmes (LIS) (Marseille, Toulon)
4 LIRICA - Logique, Interaction, Raisonnement et Inférence, Complexité, Algèbre

It is a consequence of existing literature that least and greatest fixed-points of monotone polynomials on Heyting algebras—that is, the alge- braic models of the Intuitionistic Propositional Calculus—always exist, even when these algebras are not complete as lattices. The reason is that these extremal fixed-points are definable by formulas of the IPC. Consequently, the μ-calculus based on intuitionistic logic is trivial, every μ-formula being equiv- alent to a fixed-point free formula. We give in this paper an axiomatization of least and greatest fixed-points of formulas, and an algorithm to compute a fixed-point free formula equivalent to a given μ-formula. The axiomatization of the greatest fixed-point is simple. The axiomatization of the least fixed- point is more complex, in particular every monotone formula converges to its least fixed-point by Kleene’s iteration in a finite number of steps, but there is no uniform upper bound on the number of iterations. We extract, out of the algorithm, upper bounds for such n, depending on the size of the formula. For some formulas, we show that these upper bounds are polynomial and optimal.

Domaines: Logique [math.LO]; Logique en informatique [cs.LO]

Silvio [e-mail] Ghilardi, Maria Joao Gouveia, Luigi Santocanale. Fixed-point elimination in the Intuitionistic Propositional Calculus (extended version). ACM Transactions on Computational Logic, inPress, 21 (1), pp.1--37. ⟨10.1145/3359669⟩. ⟨hal-01716485⟩