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Fixed-Point Elimination in the Intuitionistic Propositional Calculus

  • Silvio Ghilardi
  • Maria João Gouveia
  • Luigi SantocanaleEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9634)

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 \(\mu \)-calculus based on intuitionistic logic is trivial, every \(\mu \)-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 \(\mu \)-formula. The axiomatization of the greatest fixed-pointis simple. The axiomatization of the least fixed-pointis more complex, in particular every monotone formula converges to its least fixed-pointby 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.

Keywords

Intuitionistic Logic Conjunctive Normal Form Propositional Variable Heyting Algebra Monotone Formula 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

1 Introduction

In [23] the author proved that, for each formula \(\phi (x)\) of the Intuitionistic Propositional Calculus, there exists a number \(n \ge 0\) such that \(\phi ^{n}(x)\)—the formula obtained from \(\phi \) by iterating n times substitution of \(\phi \) for the variable x—and \(\phi ^{n+2}(x)\) are equivalent in Intuitionistic Logic. This result has, as an immediate corollary, that a syntactically monotone formula \(\phi (x)\) converges both to its least fixed-pointand to its greatest fixed-pointin at most n steps. Using a modern notation based on \(\mu \)-calculi [3], we have \(\mu _{x}.\phi (x) = \phi ^{n}(\bot )\) and \(\nu _{x}.\phi (x) = \phi ^{n}(\top )\). These identities also show that a \(\mu \)-calculus based on Intuitionistic Logic is trivial, every \(\mu \)-formula being equivalent to a fixed-point free formula.

Ruitenberg’s work [23] leaves open how to extract or estimate the least number \(\rho (\phi )\) such that \(\phi ^{\rho (\phi )}(x) = \phi ^{\rho (\phi )+2}(x)\). Yet, our motivations stem from the theory of extremal fixed-points and \(\mu \)-calculi [3]. In principle, being able to compute or bound Ruitenberg’s number \(\rho (\phi )\) might end up in an over-approximation of the closure ordinal of \(\phi \)—the least k such that \(\mu _{x}.\phi (x) = \phi ^{k}(\bot )\). For the analogous problem with the greatest fixed-point, we shall see that the least number k such that \(\nu _{x}.\phi (x) = \phi ^{k}(\top )\) is bounded by 1, while \(\rho (\phi )\) might be arbitrarily large.

Later in [20], the author gave an independent proof that least fixed-points of monotone formulas are definable within Intuitionistic Logic. His proof relies on semantics methods and on the coding of Intuitionistic Logic into Grzegorczyk’s Logic; the proof was further refined in [21] to encompass the standard coding of Intuitionistic Logic into its modal companion, the logic S4. Curiously, no mention of greatest fixed-points appears in these works.

Another relevant source for this paper stem from the discovery that IPC has uniform interpolants [22], often named bisimulation quantifiers. Together with the deduction property of IPC, they give the category of (finitely generated) Heyting algebras—that is, the algebraic models of the Intuitionistic Propositional Calculus—a rather strong structure, axiomatized and studied in [14, 15]. It is possible to argue that in every category with similar properties the extremal fixed-points of monotone formulas are definable. This is possible by using quantified formulas analogous to the one used in [9, Sect. 3] to argue that PDL lacks the uniform interpolation property. In this paper we exploit this idea and the existential bisimulation quantifiers to characterize greatest fixed-points in the Intuitionistic Propositional Calculus.

A \(\mu \)-calculus is a prototypical kind of computational logic, obtained from a base logic or algebraic system by addition of distinct forms of iteration so to increase expressivity. This paper is part of a line of research whose goal is to understand, under a unified perspective, why alternation-depth hierarchies of \(\mu \)-calculi are degenerate or trivial. A \(\mu \)-calculus adds to an underlying logical-algebraic system formal lgfps of formula-terms whose semantic monotonicity can be witnessed at the syntactic level. When addition of extremal fixed-points is iterated, formula-terms with nested extremal fixed-points are generated. The alternation-depth hierarchy [3, Sect. 2.6] of a \(\mu \)-calculus measures the complexity of a formula-term as a function of the nesting of the different types of fixed-points, with respect to a fixed class of models. It is well known that fixed-points that are unguarded can be eliminated in the propositional modal \(\mu \)-calculus [18]. We can rephrase this fact by saying that the alternation-depth hierarchy of the \(\mu \)-calculus over distributive lattices is trivial, every \(\mu \)-term being equivalent to a fixed-point free term. A goal of [12] was to understand closely this result and to generalize it. We were able to exhibit equational classes of lattices \(\mathcal{D}_{n}\)—with \(\mathcal{D}_{0}\) the class of distributive lattices—where the extremal fixed-points can be uniformly computed by iterating a formula-term \(n+1\) times from the bottom/top of the lattice; moreover, we showed that these uniform upper bounds are optimal. The reasons for the degeneracy of the hierarchy can be ultimately found in the structural theory of lattices.

As we show in this paper, the situation is quite different when the base for the \(\mu \)-calculus is Intuitionistic Logic, with its standard models the Heyting algebras. Several ingredients contribute to the existence of a closure ordinal of each formula and to its finiteness. Among them, strongness of the monotone polynomials on Heyting algebras. This means that a monotone polynomial \(\mathfrak {f}: H \xrightarrow {\quad }H\) over a Heyting algebra H can be considered as a functor enriched over H, when H is consider as a closed category [16]. For some polynomials, existence and finiteness of the closure ordinal is a consequence of being inflating (or expanding) and, on the syntactic level, to a restriction to the use of conjunction that determines a notion of disjunctive formula. As far as the greatest fixed-pointis concerned, monotone formulas uniformly converge to it after one step. A key ingredient of the algorithm we present is creation of least fixed-points via the Rolling equation (cf. Lemma 1), a fact already used in [10]. For Intuitionistic Logic and Heyting algebras, where formula-terms can be semantically antitone (i.e. contravariant), existing greatest fixed-points create least fixed-points. The most striking difference with the case of distributive lattices (and with the case of the varieties \(\mathcal{D}_{n}\)) is the absence of a finite uniform upper bound on the closure ordinals, the rate of convergence to the least fixed-pointcrucially depending on the shape of the formula.

As emphasized in [19] for the propositional modal \(\mu \)-calculus, once a formula is known to be equivalent to some other formula of smaller complexity, we should also be able to effectively compute this second formula. Thus, the fact that the alternation hierarchy is trivial for \(\mu \)-calculi based on the IPC should not be the end of the story. The main contribution of this paper is to achieve an effective transformation of an intuitionisitc \(\mu \)-formula into an equivalent fixed-point free intuitionisitc formula. The size of the formula might exhibit an exponential grow during this transformation. Yet, this is mainly due, as usual, to the need of precompiling a formula into an equivalent one in some kind of conjunctive normal form. We might use sharing in substitutions—or introduce the appropriate formalism for approximants to least fixed-points—so that, if we are given an already precompiled formula, then its least fixed-pointw.r.t. the variable x is only polynomially bigger than the original formula. For these formulas, we instantiate this claim by explicitly giving a way of computing \(f(\phi )\) such that \(\mu _{x}.\phi (x) = \phi ^{f(\phi )}(\bot )\), so that \(f(\phi )\) is an upper bound to the closure ordinal of \(\phi \). In some cases we are able to show that \(f(\phi )\) is optimal, by exhibiting some formula \(\phi (x)\) such that \(\phi ^{f(\phi ) -1}(\bot ) <\mu _{x}.\phi (x) \).

The paper is structured as follows. We recall in Sect. 2 some elementary facts from fixed-pointtheory. In Sect. 3 we recall the Intuitionistic Propositional Calculus and introduce the Intuitionistic Propositional \(\mu \)-Calculus. In Sect. 4 we argue that monotone polynomials are strong and exhibit the interactions between least fixed-points and strong functions. In Sect. 5 we use the existential bisimulation quantifier to argue that monotone polynomials converge to their greatest fixed-pointin one step. Section 6 is the core of our paper, where we show ho to eliminate a least fixed-pointfrom a formula. Together with the result in the previous Section, this leads to a procedure to eliminate off the fixed-points from a \(\mathbf{IPC }_{\mu }\) formula. Finally, in Sect. 7, we show how upper bounds to closure ordinals can be extracted from the procedure elimination of the least fixed-points. In Sect. 8 we present our final remarks.

2 Notation and Elementary Concepts

Let P and Q be posets. A function \(f : P \xrightarrow {\quad }Q\) is monotone if \(x \le y\) implies \(f(x) \le f(y)\), for each \(x,y \in P\). If \(f : P \xrightarrow {\quad }P\) is a monotone endofunction, then \(x \in P\) is a prefixed-point of f if \(f(x) \le x\); we denote by \(\texttt {Pre}_{f}\) the set of prefixed points of f. Whenever \(\texttt {Pre}_{f}\) has a least element, we denote it by \(\mu .f\). Therefore, \(\mu .f\) denotes the least prefixed-point of f, whenever it exists. If \(\mu .f\) exists, then it is a fixed-pointof f, necessarily the least one. The notions of least prefixed-point and of least fixed-pointcoincide on complete lattices or when the least fixed-pointis computed by iterating from the bottom of a lattice; for our purposes they are interchangeable, so we shall abuse of language and refer to \(\mu .f\) as the least fixed-pointof f. Dually (and abusing again of language), the greatest fixed-point of f shall be denoted by \(\nu .f\).

Let us mention few elementary facts from fixed-pointtheory.

Lemma 1

Let PQ be posets, \(f : P \xrightarrow {\quad }Q\) and \(g : Q \xrightarrow {\quad }P\) be monotone functions. If \(\mu .(g \circ f)\) exists, then \(\mu .(f \circ g)\) exists as well and is equal to \(f(\mu .(g \circ f))\).

As we do not work in complete lattices (so we are not ensured that least fixed-points exist) we express the above statement via the equality

where the colon emphasizes existence: if the least fixed-pointin the expression on the right exists, then this expression is the least fixed-pointof \(f\circ g\). Analogous notations will be used later. We endow the product of two posets P and Q with the coordinatewise ordering. Therefore a function \(f : P \times Q \xrightarrow {\quad }R\) is monotone if, as a function of two variables, it is monotone in each variable. To deal with least fixed-points of functions of many variables, we use the standard notation: for example, if \(f : P\times P\xrightarrow {\quad }P\) is the monotone function f(xy), then, for a fixed \(p \in P\), \(\mu _{x}.f(x,p)\) denotes the least fixed-pointof f(xp). Let us recall that the correspondence \(p \mapsto \mu _{x}.f(x,p)\)—noted \(\mu _{x}.f(x,y)\)—is again monotone.

Lemma 2

If P is a poset and \(f : P\times P \xrightarrow {\quad }P\) is a monotone mapping, then

Lemma 3

If P and Q are posets and \(\langle f,g\rangle : P\times Q \xrightarrow {\quad }P\times Q\) is a monotone function, then \(\mu .\langle f,g \rangle := \langle \mu _{1},\mu _{2}\rangle \), where

3 The Intuitionistic Propositional \(\mu \)-Calculus

Formulas of the Intuitionistic Propositional Calculus are generated according to the following grammar:
$$\begin{aligned} \phi&\;\;\Rightarrow \;\;x \mid \top \mid \phi \wedge \phi \mid \bot \mid \phi \vee \phi \mid \phi \rightarrow \phi \, , \end{aligned}$$
(1)
where x ranges over a countable set \(\mathbb {X}_{}\) of propositional variables. For the IPC, the formulation of the consequence relation \(\vdash _\mathbf{LJ }\) (relating a set of formulas to a formula) goes back to Gentzen’s work on the system LJ [13]. It is well known that Intuitionistic Logic is sound and complete w.r.t. the class of its algebraic models, the Heyting algebras.

Definition 1

A Heyting algebraH is a bounded lattice (with least element \(\bot \) and greatest element \(\top \)) equipped with a binary operation \(\rightarrow \) such that the following equations hold in H:
$$\begin{aligned} x \wedge (x \rightarrow y)&= x \wedge y\,,&x \wedge (y \rightarrow x)&= x\,, \\ \nonumber x \rightarrow x&= \top \,,&x \rightarrow (y \wedge z)&= (x \rightarrow y) \wedge (x \rightarrow z)\,. \end{aligned}$$
(2)

We can define on any Heyting algebra a partial order by saying that \(x \le y\) holds when \(x \vee y = y\). We identify formulas of the IPC with terms of the theory of Heyting algebras, constructed therefore from variables and using the signature \(\langle \top ,\wedge , \bot ,\vee , \rightarrow \rangle \). For \(\phi \) such a formula-term, H a Heyting algebra, and \(v : \mathbb {X}_{}\xrightarrow {\quad }H\) a valuation of the propositional variables in H, let us write \(\llbracket \phi \rrbracket _{v}\) for the result of evaluating the formula in H, starting from the variables. The soundness and completeness theorem of the IPC over Heyting algebras—see e.g. [6]—can then be stated as follows: if\(\varGamma \)is a finite set of formula-terms and\(\phi \)is a formula-term, then\(\varGamma \vdash _\mathbf{LJ }\phi \)holds if and only if\(\bigwedge _{\gamma \in \varGamma } \llbracket \gamma \rrbracket _{v} \le \llbracket \phi \rrbracket _{v}\)holds, in every Heyting algebra Hand for every valuation of the propositional variables\(v : \mathbb {X}_{}\xrightarrow {\quad }H\). Given this theorem, we shall often abuse of notation and write \(\le \) in place of \(\vdash _\mathbf{LJ }\), and the equality symbol \(=\) to denote logical equivalence of formulas.

We aim at studying extremal fixed-points on Heyting algebras. To this end, we formalize the Intuitionistic Propositional \(\mu \)-Calculus.

An occurrence of a variable x is positive in a formula-term \(\phi \) if, in the syntax tree of \(\phi \), the path from the root to the leaf labeled by this variable contains an even number of nodes labeled by subformulas \(\psi _{1} \rightarrow \psi _{2}\) immediately followed by a node labeled by the subformula \(\psi _{1}\). If, on this path the number of those nodes is odd, then we say that this occurrence of x is negative in \(\phi \). A variable x is positive in a formula \(\phi \) if each occurrence of x is positive in \(\phi \). A variable x is negative in a formula \(\phi \) if each occurrence of x is negative in \(\phi \). If we add to the previous grammar (1) the following productions:
$$\begin{aligned} \phi \;\;\Rightarrow \;\;&\mu _{x}.\phi \,,&\phi \;\;\Rightarrow \;\;&\nu _{x}.\phi \,, \end{aligned}$$
subject to the restriction that x is positive in \(\phi \), we obtain then a grammar for the formulas of \(\mathbf{IPC }_{\mu }\), the Intuitionistic Propositional \(\mu \)-Calculus. The semantics of these formulas is the expected one. Let \(\phi \) be a formula of \(\mathbf{IPC }_{\mu }\), and let x be positive in \(\phi \). Let us denote by \(\mathbb {X}_{\phi }\) the set of variables having an occurrence in \(\phi \). If \(v : \mathbb {X}_{\phi } \setminus \{\, x\,\}\xrightarrow {\quad }H\) is a valuation of all the variables of \(\phi \) but x in a complete Heyting algebra, then the function \(\llbracket \phi \rrbracket _{v}\), defined by
$$\begin{aligned} h&\mapsto \llbracket \phi \rrbracket _{v,h/x}\,, \end{aligned}$$
is monotone, so \(\mu _{x}.\phi \) (resp., \(\nu _{x}.\phi \)) is to be evaluated over the least fixed-point(resp., the greatest fixed-point) of this function. A sequent calculus for \(\mathbf{IPC }_{\mu }\) is presented in [7, Sect. 2].

Let us say that a formula \(\phi \) of \(\mathbf{IPC }_{\mu }\) is fixed-point free if it is a formula of IPC, that is, it does not contain either of the symbols \(\mu ,\nu \).

Proposition 1

Every formula \(\phi \) of \(\mathbf{IPC }_{\mu }\) is equivalent to a fixed-point free formula \(\phi '\).

Proof

Clearly, the statement holds if we can show that it holds whenever \(\phi = \mu _{x}.\psi \) or \(\phi = \nu _{x}.\psi \), where \(\psi \) is a fixed-point free formula. For a natural number \(n \ge 0\), let \(\psi ^{n}(x)\) denote the formula obtained by substituting x for \(\psi \)n times. Ruitenburg [23] proves that, for each intuitionisitic formula \(\psi \), there exists a number \(n \ge 0\) such that the formulas \(\psi ^{n}(x)\) and \(\psi ^{n + 2}(x)\) are equivalent. If x is positive in \(\psi \), then instantiating x with \(\bot \), leads to the equivalence \(\psi ^{n+1}(\bot ) \equiv \psi ^{n}(\bot )\), exhibiting \(\psi ^{n}(\bot )\) as the least fixed-pointof \(\psi \). Similarly, \(\psi ^{n}(\top )\) is the greatest fixed-pointof \(\psi \).    \(\square \)

While it is an obvious step to derive the previous Proposition from Ruitenburg’s result, there has been no attempt (as far as we know) to compute an upper bound on \(n \ge 0\) such that \(\psi ^{n}(x)\) and \(\psi ^{n + 2}(x)\) are equivalent. Nor is such an n necessarily a tight upper bound for convergence of a formula to its least or greatest fixed-point.

4 Strong Monotone Functions and Fixed-Points

If H is a Heyting algebra and \(f : H \xrightarrow {\quad }H\) is a monotone function, then we say that f is strong if
The interplay between fixed-points and this class of functions has already been emphasized, mainly in the context of categorical proof-theory and semantics of functional programming languages with inductive data types [7, 8].

Lemma 4

A monotone \(f : H \xrightarrow {\quad }H\) is strong if and only if any of the following equivalent conditions holds in H:
$$\begin{aligned} x \wedge f(y)&\le f(x \wedge y)\,, \end{aligned}$$
(3)
$$\begin{aligned} f(x \rightarrow y)&\le x \rightarrow f(y)\,, \end{aligned}$$
(4)
$$\begin{aligned} x \rightarrow y&\le f(x) \rightarrow f(y)\,. \end{aligned}$$
(5)

The proof of these equivalences is usual in categorical algebra [17] and therefore it is omitted here.

Definition 2

Let H be a Heyting algebra. We say that a function \(f : H \xrightarrow {\quad }H\) is monotone polynomial if there exist a formula \(\phi \) of the IPC, a variable x positive in \(\phi \), and a valuation \(\mathbf {v} : \mathbb {X}_{\phi } \setminus \{\, x\,\} \xrightarrow {\quad }H\) such that, for each \(h \in H\), we have \(f(h) = \llbracket \phi \rrbracket _{\mathbf {v},h/x}\).

Proposition 2

Every monotone polynomial \(\mathfrak {f}\) on a Heyting algebra is strong.

Proof

Recall that the replacement Lemma holds in the IPC: \(z \leftrightarrow w \vdash _\mathbf{LJ }\phi (z) \leftrightarrow \phi (w)\). Substituting x for z and \(x \wedge y\) for w, and considering that \(x \rightarrow y \vdash _\mathbf{LJ }x \leftrightarrow (x \wedge y) \), we derive that \(x \rightarrow y \vdash _\mathbf{LJ }\phi (x) \leftrightarrow \phi (x \wedge y)\). Assuming that u is positive in \(\phi (u)\), we have \( \phi (x) \leftrightarrow \phi (x \wedge y) \vdash _\mathbf{LJ }\phi (x) \rightarrow \phi (x \wedge y) \vdash _\mathbf{LJ }\phi (x) \rightarrow \phi (y)\), whence \(x \rightarrow y \vdash _\mathbf{LJ }\phi (x) \rightarrow \phi (y)\). The last relation immediately implies that Eq. (4) from Lemma 4 holds, when \(\mathfrak {f}\) is a monotone polynomial.    \(\square \)

It can be shown that the relation \(\mathfrak {f}(x) \wedge y = \mathfrak {f}(x\wedge y) \wedge y\) holds (for any xy and) for any polynomial on a Heyting algebra. The analogous remark for Boolean algebras is credited to Peirce, in view of the iteration rule for existential graphs of type Alpha, see [11].

Proposition 3

If \(\mathfrak {f}\) is a strong monotone function on H and \(a \in H\), then
$$\begin{aligned} \mu .a \rightarrow \mathfrak {f}&:= a \rightarrow \mu .\mathfrak {f}\,,&\mu .a \wedge \mathfrak {f}&:= a \wedge \mu .\mathfrak {f}\,. \end{aligned}$$
(6)

Proof

Let us argue first that first equation holds. To this end, let us set \({\mathfrak {f}^{a}}(x) =_{\texttt {def}}a \rightarrow \mathfrak {f}(x)\). From \(\mathfrak {f}\le {\mathfrak {f}^{a}}\) we have \(\texttt {Pre}_{{\mathfrak {f}^{a}}} \subseteq \texttt {Pre}_{\mathfrak {f}}\). Thus, if \(p \in \texttt {Pre}_{{\mathfrak {f}^{a}}}\), then \(\mu _{x}.\mathfrak {f}(x) = \mathfrak {f}(\mu _{x}.\mathfrak {f}(x)) \le \mathfrak {f}(p)\) and \(a \rightarrow \mu .\mathfrak {f}\le a \rightarrow \mathfrak {f}(p) = {\mathfrak {f}^{a}}(p) \le p\). That is, \(a \rightarrow \mu .\mathfrak {f}\) is below any element of \(\texttt {Pre}_{{\mathfrak {f}^{a}}}\). To obtain the proposition, we need to argue that \(a \rightarrow \mu .\mathfrak {f}\) belongs to \(\texttt {Pre}_{{\mathfrak {f}^{a}}}\). To this end, we notice that \(\{\, a \rightarrow p \mid p \in \texttt {Pre}_{\mathfrak {f}}\,\} \subseteq \texttt {Pre}_{{\mathfrak {f}^{a}}}\), since if \(\mathfrak {f}(p) \le p\), then \({\mathfrak {f}^{a}}(a \rightarrow p) = a \rightarrow \mathfrak {f}(a \rightarrow p) \le a \rightarrow \mathfrak {f}(p) \le a \rightarrow p\), where we used the fact that \(\mathfrak {f}\) is strong, thus (4) holds.

Let us come now to the second equation, for which we set \({\mathfrak {f}_{a}}(x) =_{\texttt {def}}a \wedge \mathfrak {f}(x)\). Suppose \(a \wedge \mathfrak {f}(p) \le p\), so \(\mathfrak {f}(p) \le a \rightarrow p\). Then \(\mathfrak {f}(a \rightarrow p) \le a \rightarrow \mathfrak {f}(p) \le a \rightarrow p\), using (4), whence \(\mu .\mathfrak {f}\le a \rightarrow p\) and \(a \wedge \mu .\mathfrak {f}\le p\). Thus we are left to argue that \(a \wedge \mu .\mathfrak {f}\) is a prefixed-point of \({\mathfrak {f}_{a}}\). Yet, this is true for an arbitrary prefixed-point p of \(\mathfrak {f}\): \(a \wedge \mathfrak {f}(a \wedge p) \le a \wedge \mathfrak {f}(p) \le a \wedge p\).    \(\square \)

Corollary 1

For each \(n \ge 1\) and each collection \(\mathfrak {f}_{i}\), \(i = 1,\ldots ,n\) of monotone polynomials, we have the following distribution law:
$$\begin{aligned} \mu _{x}. \bigwedge _{i = 1,\ldots ,n} \mathfrak {f}_{i}(x)&:= \bigwedge _{i = 1,\ldots ,n} \mu _{x}.\mathfrak {f}_{i}(x)\,. \end{aligned}$$
(7)

Proof

For \(n = 1\) there is nothing to prove. We suppose therefore that the statement holds for every collection of size \(n \ge 1\), and prove it holds for a collection of size \(n+1\). We have

   \(\square \)

The elimination of greatest fixed-points is easy for strong monotone functions (we are thankful to the referee for pointing out the following fact, which greatly simplifies our original argument):

Proposition 4

If \(\mathfrak {f}: L \xrightarrow {\quad }L\) is any strong monotone function on a bounded lattice L, then \(\mathfrak {f}^{2}(\top ) = \mathfrak {f}(\top )\). Thus \(\mathfrak {f}(\top )\) is the greatest fixed-pointof \(\mathfrak {f}\).

Proof

Indeed, we have \(\mathfrak {f}(\top ) = \mathfrak {f}(\top ) \wedge \mathfrak {f}(\top ) \le \mathfrak {f}(\mathfrak {f}(\top ) \wedge \top ) = \mathfrak {f}^{2}(\top )\).    \(\square \)

5 A Digression on Fixpoints and Bisimulation Quantifiers

The connection between extremal fixed-points and bisimulation quantifiers, as emphasized in [9], was a main motivation to tackle this research. Although in the end our computations are independant on that, we nevertheless want to have a closer look to the topic (the content of this section is not needed afterwards).

It was discovered in [22] that IPC has the uniform interpolation property. As made clear from the title of [22], this property amounts to an internal existential and universal quantification. This result was further refined in [15] to show that any morphism between finitely generated Heyting algebras has a left and a right adjoint. We shall be interested in Heyting algebras H[x] of polynomials with coefficients from H, and to (the left and right adjoints to) the inclusion of H into H[x]. The algebra of polynomials H[x] is formally defined as the coproduct (in the category of Heyting algebras) of H with the free Heyting algebra on one generator. The universal property gives that if \(h_{0} \in H\), then there exists a unique morphism \(\llbracket \cdot \rrbracket _{h_{0}/x} : H[x] \xrightarrow {\quad }H\) such that \(\llbracket x\rrbracket _{h_{0}/x} = h_{0}\) and \(\llbracket h\rrbracket _{h_{0}/x} = h\), for each \(h \in H\). Thus, for \(\mathfrak {f}\in H[x]\) and \(h \in H\), we can define \(\mathfrak {f}(h) = \llbracket \mathfrak {f}\rrbracket _{h/x}\). It follows from [15] that if H is finitely generated, then the inclusion \(i_{x} : H \xrightarrow {\quad }H[x]\) has both adjoints \(\exists _{x} ,\forall _{x} : H[x] \xrightarrow {\quad }H\), with \(\exists _{x} \dashv i_{x} \dashv \forall _{x}\). In particular, we shall use the unit relation for \(\exists _{x}\):
$$\begin{aligned} \mathfrak {f}&\le i_{x}(\exists _{x}(\mathfrak {f})) \,,\qquad {\text {for all}}\, \mathfrak {f}\in H[x]\,. \end{aligned}$$
Identifying \(h \in H\) with \(i_{x}(h) \in H[x]\), we can read the above inequality as \(\mathfrak {f}\le \exists _{x}.\mathfrak {f}\). We can identify a monotone polynomial, as defined in Definition 2, as an element \(\mathfrak {f}\in H[x]\) such that \(\llbracket \mathfrak {f}\rrbracket _{h_{0}/x} \le \llbracket \mathfrak {f}\rrbracket _{h_{1}/x}\) whenever \(h_{0} \le h_{1}\).

Proposition 5

If \(\mathfrak {f}\) is a monotone polynomial on a finitely generated Heyting algebra, then
$$\begin{aligned} \nu .\mathfrak {f}&:= \exists _{x}.( x \wedge (x \rightarrow \mathfrak {f}(x)))\,. \end{aligned}$$
(8)

Proof

By the unit relation \(x \wedge x \rightarrow \mathfrak {f}(x) \le \exists _{x}.(x \wedge x \rightarrow \mathfrak {f}(x))\). Recall that evaluation at \(p \in H\) is a Heyting algebra morphism, thus it is monotone. Therefore, if \(p \in H\) is a postfixed-point of \(\mathfrak {f}\), then by evaluating the previous inequality at p, we have
$$\begin{aligned} p&= p\wedge p \rightarrow \mathfrak {f}(p) \le \exists _{x}.(x \wedge x \rightarrow \mathfrak {f}(x))\,, \end{aligned}$$
so that \(\exists _{x}.(x \wedge x \rightarrow \mathfrak {f}(x))\) is greater than any postfixed-point of \(\mathfrak {f}\). Let us show that \(\exists _{x}.(x \wedge x \rightarrow \mathfrak {f}(x))\) is also a postfixed-point. To this end, it will be enough to argue that \(x \wedge x \rightarrow \mathfrak {f}(x) \le \mathfrak {f}(\exists _{x}.(x \wedge x \rightarrow \mathfrak {f}(x)))\) in H[x]. We compute as follows:

In a similar fashion, we can prove that if \(\mathfrak {f}\) is a monotone polynomial on a finitely generated Heyting algebra, then \(\mu .\mathfrak {f}:= \forall _{x}.((\mathfrak {f}(x) \rightarrow x) \rightarrow x)\). As an application, we give an alternative proof of Proposition 4:

Corollary 2

If \(\mathfrak {f}\) is a monotone polynomial on a Heyting algebra H, then
$$\begin{aligned} \nu .\mathfrak {f}&:= \mathfrak {f}(\top )\,. \end{aligned}$$
(9)

Proof

It is easy to see that if \(\mathfrak {f}\) is a monotone polynomial on a finitely generated Heyting algebra, then \(\exists _{x}.\mathfrak {f}= \mathfrak {f}(\top )\). Thus we have
$$\begin{aligned} \nu .\mathfrak {f}&= \exists _{x}.( x \wedge (x \rightarrow \mathfrak {f}(x))) = \exists _{x}.( x \wedge \mathfrak {f}(x)) = \top \wedge \mathfrak {f}(\top ) = \mathfrak {f}(\top )\,. \end{aligned}$$
Therefore, if \(\phi \) is a formula-term whose variables are among set \(x,y_{1},\ldots ,y_{n}\), then the equation \(\phi ^{2}(\top ) = \phi (\top )\) holds in the free Heyting algebra on the set \(\{\, y_{1},\ldots ,y_{n}\,\}\). Consequently, the equation \(\mathfrak {f}(\top ) = \mathfrak {f}^2(\top )\) holds in H, making \(\mathfrak {f}(\top )\) into the greatest fixed-pointof \(\mathfrak {f}\).    \(\square \)

6 The Elimination Procedure

In this Section we present our main result, a procedure that both axiomatizes and eliminates least fixed-points of the form \(\mu _{x}.\phi (x)\) with \(\phi \) fixed-point free. Together with the axiomatization of greatest fixed-points given in Sect. 5, the procedure can be extended to a procedure to construct a fixed-point free formula \(\psi \) equivalent to a given formula \(\chi \) of the \(\mathbf{IPC }_{\mu }\).

Definition 3

An occurrence of the variable x is strongly positive in a formula-term \(\phi \) if there is no subformula \(\psi \) of \(\phi \) of the form \(\psi _{0} \rightarrow \psi _{1}\) such that x is located in \(\psi _{0}\). A formula-term \(\phi \) is strongly positive in the variable x if every occurrence of x is strongly positive in \(\phi \). An occurrence of a variable x is weakly negative in a formula-term \(\phi \) if it is not strongly positive. A formula-term \(\phi \) is weakly negative in the variable x if every occurrence of x is weakly negative in \(\phi \).

Observe that a variable might be neither strongly positive nor weakly negative in a formula-term. A second key concept for the elimination is the following notion of disjunctive formula.

Definition 4

The set of formula-terms that are disjunctive in the variablex is generated by the following grammar:
$$\begin{aligned} \phi&\;\;\Rightarrow \;\;x \mid \beta \vee \phi \mid \phi \vee \beta \mid \alpha \rightarrow \phi \mid \phi \vee \phi \,, \end{aligned}$$
(10)
where \(\alpha \) and \(\beta \) are formulas with no occurrence of the variable x. A formula-term \(\phi \) is in normal form (w.r.t. x) if it is a conjunction of formula-terms \(\phi _{i}\), \(i \in I\), so that each \(\phi _{i}\) either does not contain the variable x, or it is disjunctive in x.

Notice that disjunctive formula-terms are strongly positive in x. Due to Eq. (2) and since the usual distributive laws hold in Heyting algebras, we have the following Lemma.

Lemma 5

Every strongly positive formula-term is equivalent to a formula-term in normal form.

In order to compute the least fixed-point\(\mu _{x}.\phi \), we take the following steps:
  1. 1.

    We rename all the weakly negative occurrences of x in \(\phi \) to a fresh variable y, so \(\phi (x) = \psi (x,x/y)\) with \(\psi \) strongly positive in x and weakly negative in y.

     
  2. 2.

    We compute a normal form of \(\psi (x,y)\), so this formula is equivalent to a conjunction \(\bigwedge _{i \in I} \psi _{i}(x,y)\) with each \(\psi _{i}\) disjunctive in x or not containing the variable x.

     
  3. 3.

    Strongly positive elimination. For each \(i \in I\): if x has an occurrence in \(\psi _{i}\), we compute then a formula \(\psi '_{i}\) equivalent to the least fixed-point\(\mu _{x}.\psi _{i}(x,y)\) and observe that \(\psi '_{i}\) is weakly negative in y; otherwise, we let \(\psi '_{i} = \psi _{i}\).

     
  4. 4.

    Weakly negative elimination. The formula \(\bigwedge _{i \in I} \psi '_{i}(y)\) is weakly negative in y; we compute a formula \(\chi \) equivalent to \(\mu _{y}.\bigwedge _{i} \psi '_{i}(y)\) and return it.

     
The correction of the procedure relies on the following chain of equivalences:

6.1 Strongly positive Elimination

We tackle here the problem of computing the least fixed-point\(\mu _{x}.\phi \) of a formula-term \(\phi \) which is disjunctive in x. Recall that the formulas \(\alpha \) and \(\beta \) appearing in a parse tree as leaves—according to the grammar (10)—do not contain the variable x. We call such a formula \(\alpha \) a head subformula of \(\phi \), and such a \(\beta \) a side subformula of \(\phi \), and thus we put:
$$\begin{aligned} \mathsf {Head}(\phi )&=_{\texttt {def}}\{\, \alpha \mid \alpha \text { is a head subformula of } \phi \,\} \,,\\ \mathsf {Side}(\phi )&=_{\texttt {def}}\{\, \beta \mid \beta \text { is a side subformula of } \phi \,\} \,. \end{aligned}$$
Recall that a monotone function \(f : P \xrightarrow {\quad }P\) is inflating if \(x \le f(x)\).

Lemma 6

The interpretation of a strongly positive disjunctive formula \(\phi \) as a function of x is inflating.

The key observation needed to prove Proposition 6 is the following Lemma on monotone inflating functions. In the statement of the Lemma we assume that P is a join-semilattice, and that \(f \vee g\) is the pointwise join of the two functions f and g.

Lemma 7

If \(f,g:P \xrightarrow {\quad }P\) are monotone inflating functions, then \(\texttt {Pre}_{f \vee g} = \texttt {Pre}_{f \circ g}\). Consequently, for any monotone function \(h : P \xrightarrow {\quad }P\), we have
$$\begin{aligned} \mu .(\,f \vee g \vee h\,)&:=: \mu .(\,(f \circ g) \vee h\,)\,. \end{aligned}$$
(11)

Proof

Observe firstly that \(\texttt {Pre}_{f \vee g}= \texttt {Pre}_{f} \cap \texttt {Pre}_{g}\). If \(p \in \texttt {Pre}_{f \circ g}\), then \(f(p) \le f(g(p)) \le p\) and \(g(p) \le f(g(p)) \le p\), showing that \(p \in \texttt {Pre}_{f \vee g}\). Conversely, if \(p \in \texttt {Pre}_{f \vee g}\), then p is a fixed point of both f and g, since these functions are inflating. It follows that \(f(g(p)) = f(p) = p\), showing \(p \in \texttt {Pre}_{f \circ g}\).

We have argued that \(\texttt {Pre}_{f \vee g}\) coincides with \(\texttt {Pre}_{f \circ g}\); this implies that \(\texttt {Pre}_{(f \circ g) \vee h} = \texttt {Pre}_{f \vee g \vee h}\) and, from this equality, Eq. (11) immediately follows.    \(\square \)

To ease reading of the next Proposition and of its proof, let us put
$$\begin{aligned} \left[ \alpha \right] \phi&=_{\texttt {def}}\alpha \rightarrow \phi \,. \end{aligned}$$

Proposition 6

If \(\phi \) is a disjunctive formula-term, then
$$\begin{aligned} \mu .\phi&= \left[ \bigwedge _{\alpha \in \mathsf {Head}(\phi )} \alpha \right] (\bigvee _{\beta \in \mathsf {Side}(\phi )} \beta )\,. \end{aligned}$$
(12)

Proof

For \(\psi ,\chi \) formula-terms, let us write \(\psi \sim \chi \) when \(\mu .\psi = \mu .\chi \). We say that a disjunctive formula \(\psi \) is reduced (w.r.t. \(\phi \)) if either it is x, or it is of the form \(\beta \vee x\) (or \(x \vee \beta \)) for some \(\beta \in \mathsf {Side}(\phi )\), or of the form \(\left[ \alpha \right] x\) for some \(\alpha \in \mathsf {Head}(\phi )\). A set \(\varPhi \) of disjunctive formulas is reduced if every formula in \(\varPhi \) is reduced.

We shall compute a reduced set of disjunctive formulas \(\varPhi _{k}\) such that \(\phi \sim \bigvee \varPhi _{k}\). Thus let \(\varPhi _{0} = \{\, \phi \,\}\). If \(\varPhi _{i}\) is not reduced, then there is \(\phi _{0} \in \varPhi _{i}\) which is not reduced, thus of the form (a) \(\beta \vee \psi \) (or \(\psi \vee \beta \)) with \(\psi \ne x\), or (b) \(\left[ \alpha \right] \psi \) with \(\psi \ne x\), or (c) \(\psi _{1} \vee \psi _{2}\). According to the case (\(\ell \)), with \(\ell \in \{\, a,b,c\,\}\), we let \(\varPhi _{i + 1}\) be \((\varPhi _{i} \setminus \{\, \phi _{0}\,\}) \cup \varPsi _{\ell }\) where \(\varPsi _{\ell }\) is as follows:
$$\begin{aligned} \varPsi _{a}&= \{\, \beta \vee x, \psi \,\},&\varPsi _{b}&= \{\, \left[ \alpha \right] x, \psi \,\},&\varPsi _{c}&= \{\, \psi _{1},\psi _{2}\,\}\,. \end{aligned}$$
By Lemma 7, we have \(\bigvee \varPhi _{i} \sim \bigvee \varPhi _{i+1}\). Morever, for some \(k \ge 0\), \(\varPhi _{k}\) is reduced and \(\varPhi _{k} \subseteq \{\, \left[ \alpha \right] x \mid \alpha \in \mathsf {Head}(\phi )\,\} \cup \{\, \beta \vee x \mid \beta \in \mathsf {Side}(\phi )\,\} \cup \{\, x\,\}\). Consequently
$$\begin{aligned} \mu _{x}.\phi (x)&= \mu _{x}.\bigvee \varPhi _{k} \le \mu _{x}.(x \vee \bigvee _{\alpha \in \mathsf {Head}(\phi )} \left[ \alpha \right] x \vee \bigvee _{\beta \in \mathsf {Side}(\phi )} \beta \vee x )\,. \end{aligned}$$
(13)
On the other hand, if \(\alpha \in \mathsf {Head}(\phi )\), then \(\phi (x) = \psi _{1}(x,\left[ \alpha \right] \psi _{2}(x))\) for some disjunctive formulas \(\psi _{1}\) and \(\psi _{2}\), so
$$\begin{aligned} \left[ \alpha \right] x&\le \left[ \alpha \right] \psi _{2}(x) \le \psi _{1}(x,\left[ \alpha \right] \psi _{2}(x)) = \phi (x) \end{aligned}$$
and, similarly, \(\beta \vee x \le \phi (x)\), whenever \(\beta \in \mathsf {Side}(\phi )\). It follows that
$$\begin{aligned} x \vee \bigvee _{\alpha \in \mathsf {Head}(\phi )} \left[ \alpha \right] x \vee \bigvee _{\beta \in \mathsf {Side}(\phi )} \beta \vee x&\le \phi (x)\,, \end{aligned}$$
whence, by taking the least fixed-pointin both sides of the above inequality, we derive equality in (13). Finally, in order to obtain (12), we compute as follows:

   \(\square \)

6.2 Weakly Negative Elimination

If \(\phi \) is weakly negative in x then we can write
$$\begin{aligned} \phi (x)&= \psi _{0}(\psi _{1}(x), \ldots ,\psi _{n}(x))\,, \end{aligned}$$
(14)
for formula-terms \(\psi _{0}(y_{1},\ldots ,y_{n})\) and \(\psi _{i}(x)\), \(i = 1,\ldots ,n\), such that: (a) all the variables \(y_{i}\) are negative in \(\psi _{0}\); (b) for \(i = 1,\ldots ,n\), x is negative \(\psi _{i}\).

Proposition 7

Let \(\langle \nu _{1},\ldots ,\nu _{n}\rangle \) be a collection of formula-terms denoting the greatest solution of the system of equations \(\{\, y_{i} = \psi _{i}(\psi _{0}(y_{1},\ldots ,y_{n})) \mid i = 1,\ldots ,n \,\}\). Then \(\psi _{0}(\nu _{1},\ldots ,\nu _{n})\) is a formula equivalent to \(\mu _{x}.\phi (x)\).

Proof

Let \(v : \mathbb {X}_{}\setminus \{\, x,y_{1},\ldots ,y_{n}\,\} \xrightarrow {\quad }H\) be a partial valuation into an Heyting algebra H, put \(\mathfrak {f}_{0} = \llbracket \psi _{0}\rrbracket _{v}\) and, for \(i = 1,\ldots ,n\), \(\mathfrak {f}_{i} = \llbracket \psi _{i}\rrbracket _{v}\). Then \(\mathfrak {f}_{0}\) is a monotone function from \( [ H^{op}]^{n}\) to H. Here \(H^{op}\) is the poset with the same elements as H but with the opposite ordering relation. Similarly, for \(1 \le i \le n\), \(\mathfrak {f}_{i} : H \xrightarrow {\quad }H^{op}\). If we let \(\bar{\mathfrak {f}} = \langle \mathfrak {f}_{i} \mid i = 1,\ldots ,n \rangle \circ \mathfrak {f}_{0}\), then \(\bar{\mathfrak {f}} :{[H^{op}]}^{n} \xrightarrow {\quad }{[H^{op}]}^{n}\). We exploit next the fact that \((\cdot )^{op}\) is a functor, so that \(f^{op} : P^{op} \xrightarrow {\quad }Q^{op}\) is the same monotone function as f, but considered as having distinct domain and codomain. Then, using (Roll), we can write
$$\begin{aligned} \mu . (\,\mathfrak {f}_{0} \circ \langle \mathfrak {f}_{i} \mid i = 1,\ldots ,n \rangle \,)&= \mathfrak {f}_{0}(\, \langle \mathfrak {f}_{i} \mid i = 1,\ldots ,n \rangle \circ \mathfrak {f}_{0}\,)\nonumber \\&= \mathfrak {f}_{0}(\,\mu .\bar{\mathfrak {f}}\,) = \mathfrak {f}_{0}(\,\nu .\bar{f}^{op}\,)\,, \end{aligned}$$
(15)
since the least fixed-pointof f in \(P^{op}\) is the greatest fixed-pointof \(f^{op}\) in P. That is, if we consider the function \(\langle \mathfrak {f}_{i} \mid i = 1,\ldots ,n \rangle \circ \mathfrak {f}_{0}\) as sending a tuple of elements of H (as opposite to \(H^{op}\)) to another such a tuple, then Eq. (15) proves that a formula denoting the least fixed-pointof \(\phi \) is constructible out of formulas for the greatest solution of the system mentioned in the statement of the Proposition.    \(\square \)

As far as computing the greatest solution of the system mentioned in the Proposition, this can be achieved by using the Bekic elimination principle, see Lemma 3. This principle implies that solutions of systems can be constructed from solutions of linear systems, i.e. from usual parametrized fixed-points. In our case, as witnessed by Eq. (9), these parametrized greatest fixed-points are computed by substituting \(\top \) for the fixed-pointvariable. In the next Section we shall give a more explicit description, by means of approximants, of the least fixed-pointof a weakly negative formula \(\phi \).

7 Upper Bounds on Closure Ordinals

Recall that Ruitenburg’s result [23] implies that a monotone formula converges to its (parametrized) least fixed-pointby iterating the formula n times from \(\bot \), for some \(n \ge 0\). That is, we can always substitute \(\mu _{x}.\phi (x)\) for some equivalent \(\phi ^{n}(\bot )\). We show, in this Section, how to extract, from the procedure just seen, upper bounds for such a number n.

Proposition 8

If \(\phi \) is a disjunctive formula and n is the cardinality of the set \(\mathsf {Head}(\phi )\), then
$$\begin{aligned} \mu _{x}.\phi (x)&= \phi ^{n +1}(\bot )\,. \end{aligned}$$
(16)

Proof

We have seen, in the proof of Propositon 6, that \(\left[ \alpha \right] x \le \phi (x)\) for any \(\alpha \in \mathsf {Head}(\phi )\) and, similarly, \(\beta \vee x \le \phi (x)\) for any \(\beta \in \mathsf {Side}(\phi )\). Thus we have
$$\begin{aligned} \bigvee _{\beta \in \mathsf {Side}(\phi )} \beta&= \bigvee _{\beta \in \mathsf {Side}(\phi )} \beta \vee \bot \le \phi (\bot )\,. \end{aligned}$$
Let \(\mathsf {Head}(\phi ) = \{\, \alpha _{1},\ldots ,\alpha _{n}\,\}\). Supposing that \(\left[ \alpha _{i} \right] \ldots \left[ \alpha _{1} \right] (\bigvee _{\beta \in \mathsf {Side}(\phi )} \beta ) \le \phi ^{i +1}(\bot )\), then
$$\begin{aligned} \left[ \alpha _{i+1} \right] \left[ \alpha _{i} \right] \ldots \left[ \alpha _{1} \right] (\bigvee _{\beta \in \mathsf {Side}(\phi )} \beta ) \le \left[ \alpha _{i+1} \right] (\phi ^{i +1}(\bot )) \le \phi (\phi ^{i +1}(\bot )) = \phi ^{i +2}(\bot )\,. \end{aligned}$$
Whence
   \(\square \)

The upper bound given in (16) is optimal: if we let \(\phi _{n}(x) =_{\texttt {def}}b \vee \bigvee _{i=1,\ldots ,n} a_{i} \rightarrow x\) and consider the Heyting algebra of downsets of \(\langle P(\{\, 1,\ldots ,n\,\}),\subseteq \rangle \), then, interpreting b as \(\{\, \emptyset \,\}\) and \(a_{i}\) as \(\{\, s \subseteq \{\, 1,\ldots ,n\,\} \mid i \not \in s\,\}\), \(\phi _{n}\) converges exactly after \(n + 1\) steps.

In order to tackle convergence of weakly negative formulas, we mention some general statements, where we assume that all the posets have a least element.

Lemma 8

Convergence forRoll). Let \(f : P \xrightarrow {\quad }Q\) and \(g: Q \xrightarrow {\quad }P\) be monotone functions. If \(\mu . (f \circ g) = (f \circ g)^{n}(\bot )\), then \(\mu . (g \circ f) = (g \circ f)^{n+1}(\bot )\).

Lemma 9

Convergence forDiag. Let \(f : P \times P \xrightarrow {\quad }P\) be a monotone function. For each \(p \in P\), put \(g_{p}(y) = f(p,x)\) and \(h(x) = \mu _{y}.g_{x}(y)\). Suppose that, for each \(p \in P\), \(h(p) = \mu _{y}.f(p,y) =g_{p}^{n} (\bot )\) and that \(\mu _{x}.h(x) = h^{m}(\bot )\). Then \(\mu _{x}.f(x,x) = f^{nm}(\bot ,\bot )\).

For our purposes, the following Lemma provides more accurate upper bounds than Lemma 9.

Lemma 10

Let \(f,g : H \xrightarrow {\quad }H\) be strong monotone mappings. If \(\mu .f = f^{n}(\bot )\) and \(\mu .g = g^{m}(\bot )\), then \(\mu .f \wedge g = (f \wedge g)^{n +m -1}(\bot )\).

For the Bekic property we have a similar statement, bounding convergence of \(\langle f,g\rangle \) by \((n + 1)(m+1) -1\), with m and n being bounds on convergence of \(\mu _{y}.g(x,y)\) and \(\mu _{x}.f(x,\mu _{y}.g(x,y))\), respectively. While in general this bound is optimal, the relevant observation is, for our purposes, the following Lemma.

Lemma 11

Let \(\{ x_{i} = f_{i}(x_{1},\ldots ,x_{k}) \mid i = 1,\ldots ,k \}\) be a monotone system of equations P on some poset with least element \(\bot \). Suppose that all the functions generated under substitution from \(\{\, f_{1},\ldots ,f_{k} \,\} \cup \{\, \bot \,\}\) converge to their parametrized least fixed-pointin one step. Then the least solution of this system of equations is obtained by iterating k times \(\langle f_{1},\ldots ,f_{k} \rangle \) from \((\bot ,\ldots ,\bot ) \in P^{k}\).

Proposition 9

Let \(\phi (x)\) be a weakly negative formula, so that we have a decomposition of the form (14). Then \(\phi (x)\) converges at its least fixed-pointin at most \(n + 1\) steps.

Proof

Applying Lemma 11, we have
$$\begin{aligned} \nu .(\langle \psi _{i} \mid i = 1,\ldots ,n \rangle \circ \psi _{0})&= (\langle \psi _{i} \mid i = 1,\ldots ,n \rangle \circ \psi _{0})^{n}(\top )\,. \end{aligned}$$
(17)
Considering that
we can use (17) and Lemma 8 to deduce that

   \(\square \)

It is possible to combine Propositions 8 and 9 with Lemma 8 to obtain upper bounds for all formulas. Yet, mainly due to the exponential blow-up in computing an equivalent normal-form of a given formula, that is, step 2 of the procedure described in Sect. 6, these bounds turn out to be exponential functions of the size of the formula. It is possible on the other hand to pinpoint fragments of the \(\mathbf{IPC }_{\mu }\) for which we still have polynomial bounds. For example, if we define a formula-term to be weakly disjunctive if it is generated by the grammar (10), with the difference that we allow x to have weakly negative occurrences in \(\alpha \) and \(\beta \), then bounds are polynomials of order 2.

8 Conclusions

As mentioned in the Introduction, a main motivation for the research described in this paper was to provide in-depth answers to the question of why alternation-depth hierarchies in \(\mu \)-calculi collapse or are trivial. Until now, the authors dealt with trivial alternation-depth hierarchies. The tools and ideas so far developed still need to be tested when a hierarchy does not completely collapse at its lowest level. In particular, and given the closeness of Intuitionistic Logic with Modal Logic based on transitive frames, it becomes appealing to investigate further connections with existing work on the subject [1, 2, 10, 24].

Compared to other works, such as [20, 21], we definitely took an algebraic and constructive approach to the problem of showing definability of least fixed-points within the IPC. Witnessing the fruitfulness of our approach, the algebra made the goal of computing upper bounds of closure ordinals of the monotone functions denoted by intuitionisitc formulas an accessible task. Let us notice on the way that our work leads to an obvious decision procedure, based on any decision procedure for IPC, for the Intuitionistic Propositional \(\mu \)-Calculus. This logic, already studied on the side of proof theory and of game semantics [7], should also be of interest in verification, for example when transition systems come with some ordering and upward or downward closed properties are defined by \(\mu \)-formulas, see [5].

Overall, we believe that understanding extremal fixed-points and more in general fixed-points in an intuitionisitc setting—where sparse but surprising results are known, see for [4] example—is still in quest for an elementary but solid theory to be developed. The present paper is a contribution toward this goal.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  • Silvio Ghilardi
    • 1
  • Maria João Gouveia
    • 2
  • Luigi Santocanale
    • 3
    Email author
  1. 1.Dipartimento di MatematicaUniversità degli Studi di MilanoMilanItaly
  2. 2.CEMAT-CIÊNCIASUniversidade de LisboaLisboaPortugal
  3. 3.LIF, CNRS UMR 7279Aix-Marseille UniversitéMarseilleFrance

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