Fixed-point Elimination in the Intuitionistic Propositional Calculus

Abstract

It is a consequence of existing literature that least and greatest fixed-points of monotone polynomials on Heyting algebras-that is, the algebraic 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 equivalent 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 fixedpoint 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.

Free Heyting algebra endomorphisms: Ruitenburg’s Theorem and beyond

“…the implicational degree and the number of propositional variables of a formula. The syntactic computations in [13] for fixpoints convergence also yield tighter bounds.…”

“…An interesting consequence of this result is that least (and greatest) fixpoints of monotonic formulae are definable in (IP C) [19,18,13]: this is because the sequence (1) becomes increasing when evaluated on ⊥/x (if A is monotonic in x), so that the period is decreased to 1. Thus the index of the sequence becomes a finite upper bound for the fixpoint approximations convergence: in fact we have, ⊢ IP C A N (⊥/x) → A N +1 (⊥/x) and ⊢ IP C A N +1 (⊥/x) → A N +2 (⊥/x) by the monotonicity of A, yielding ⊢ IP C A N (⊥/x) ↔ A N +1 (⊥/x) by (2).…”