On the period of sequences (<i>A<sup>n</sup>(p)</i>) in intuitionistic propositional calculus

Ruitenburg, Wim

doi:10.2307/2274142

Cited by 13 publications

(18 citation statements)

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“…This corollary also follows from Ruitenburg's theorem (Ruitenburg, 1984). Namely, for a formula ϕ(p, q 1 , .…”

Section: Switches Of Truth Valuesmentioning

confidence: 71%

Definable fixed points in modal and temporal logics : A survey

Mardaev¹

2007

Journal of Applied Non-Classical Logics

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The paper presents a survey of author's results on definable fixed points in modal, temporal, and intuitionistic propositional logics. The well-known Fixed Point Theorem considers the modalized case, but here we investigate the positive case. We give a classification of fixed point theorems, describe some classes of models with definable least fixed points of positive operators, special positive operators, and give some examples of undefinable least fixed points. Some other interesting phenomena are discovered -definability by formulas that do not preserve positivity of parameters and definability by finite sets of formulas. We also consider negative operators, graded modalities, construct undefinable inflationary fixed points, and put some problems. . Modal logic in Russia Let us first recall some standard definitions. Modal propositional formulas are constructed from propositional variables p, q, r . . . and the constant ⊥ (falsity) using the binary connectives ∧, ∨, and the unary connectives ¬ and . We introduce the following abbreviations:A modal Kripke frame W, R consists of a non-empty set W with a binary relation R. A Kripke model based on a frame W, R is a triple W, R, v , where v is a valuation function assigning a subset of W (the value) to every propositional variable. This function is extended to formulas in the usual way as follows: the value of the constant ⊥ is always the empty set; the connectives ¬, ∧, ∨ respectively correspond to the complement, the intersection, and the union; and ♦ correspond to the following operations on sets:A = {x | ∀y (xRy ⇒ y ∈ A)}, ♦A = {x | ∃y (xRy ∧ y ∈ A)}.

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“…This corollary also follows from Ruitenburg's theorem (Ruitenburg, 1984). Namely, for a formula ϕ(p, q 1 , .…”

Section: Switches Of Truth Valuesmentioning

confidence: 71%

Definable fixed points in modal and temporal logics : A survey

Mardaev¹

2007

Journal of Applied Non-Classical Logics

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“…Remark 46. The upper bound given in Theorem 45 appears to be orthogonal to bound implicit in Ruitenburg's paper [30]. In the bound ρ(φ) ≤ 2n+ 2, the size n of φ is at least the number of implication subformulas of φ.…”

mentioning

confidence: 89%

“…By φ n we denote the iterated substitution of x in φ for φ, defined by induction by φ 0 = def x and φ n+1 = def φ[φ n /x]. We let ρ(φ) be the least non-negative integer n such that the relation φ n+2 = φ n holds; ρ(φ) is defined for any formula φ of the Intuitionistic Propositional Calculus, by [30], and moreover cl(φ) ≤ ρ(φ). A fine analysis of Ruitenburg's work shows that ρ(φ) ≤ 2n + 2, where n counts the implication subformulas and the propositional variables in φ.…”

Section: Ruitenburg's Numbers For Strongly Positive Formulasmentioning

confidence: 99%

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Fixed-point Elimination in the Intuitionistic Propositional Calculus

Ghilardi

Gouveia

Santocanale

2019

ACM Trans. Comput. Logic

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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.

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“…In [23] the author proved that, for each formula φ(x) of the Intuitionistic Propositional Calculus, there exists a number n ≥ 0 such that φ n (x)-the formula obtained from φ by iterating n times substitution of φ for the variable x-and φ n+2 (x) are equivalent in Intuitionistic Logic. This result has, as an immediate corollary, that a syntactically monotone formula φ(x) converges both to its least fixed-point and to its greatest fixed-point in at most n steps.…”

Section: Introductionmentioning

confidence: 99%

Fixed-Point Elimination in the Intuitionistic Propositional Calculus

Ghilardi

Gouveia

Santocanale

2016

Lecture Notes in Computer Science

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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.

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On the period of sequences (Aⁿ(p)) in intuitionistic propositional calculus

Cited by 13 publications

References 1 publication

Definable fixed points in modal and temporal logics : A survey

Definable fixed points in modal and temporal logics : A survey

Fixed-point Elimination in the Intuitionistic Propositional Calculus

Fixed-Point Elimination in the Intuitionistic Propositional Calculus

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On the period of sequences (An(p)) in intuitionistic propositional calculus

Cited by 13 publications

References 1 publication

Definable fixed points in modal and temporal logics : A survey

Definable fixed points in modal and temporal logics : A survey

Fixed-point Elimination in the Intuitionistic Propositional Calculus

Fixed-Point Elimination in the Intuitionistic Propositional Calculus

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On the period of sequences (Aⁿ(p)) in intuitionistic propositional calculus