An Infinitary Treatment of Full Mu-Calculus

Abstract

We explore the proof theory of the modal $\mu$-calculus with converse, aka the ‘full $\mu$-calculus’. Building on nested sequent calculi for tense logics and infinitary proof theory of fixed point logics, a cut-free sound and complete proof system for full $\mu$-calculus is proposed. As a corollary of our framework, we also obtain a direct proof of the regular model property for the logic: every satisfiable formula has a tree model with finitely many distinct subtrees. To obtain the results we appeal to the basic theory of well-quasi-orderings in the spirit of Kozen’s proof of the finite model property for $\mu$-calculus without converse.

Appendices

A Well-Annotated Formulæ

We begin by making more precise the definition of well-annotated $\kappa$-formulæ, and the properties that this class satisfy.

Fix $\kappa \le \omega _1$ and let $\triangleleft$ denote the subsumption ordering on $\mathsf {Var}$, where $x \triangleleft y$ reads as x subsumes y. We assume $\triangleleft$ is a strict partial order on $\mathsf {Var}$ which is downwards linear. Recall that we consider $\triangleleft$ fixed and that all formulæ respect $\triangleleft$. Hence, if $\mu y \phi$ is a formula with x free, then $x \triangleleft y$.

An $\kappa$ -assignment is a partial function from $\mathsf {Var}$ into ordinals $\mathord <\kappa$ whose domain is linearly ordered by $\triangleleft$. is the set of $\kappa$-assignments and we let denote the domain of . It proves convenient to occasionally treat assignments as total functions $o :\mathsf {Var}\rightarrow \kappa + 1$, and set . Given and $x \in \mathsf {Var}$, $o_{\triangleleft x}$ denotes the restriction of o to the variables subsuming x:

$$ o_{\triangleleft x} (y) = {\left\{ \begin{array}{ll} o(y), &{}\text {if }y \triangleleft x, \\ \kappa ,&{}\text {otherwise.} \end{array}\right. } $$

For and , $\phi ^o$ is the $\kappa$-formula generated as follows.

That is, $\nu$-quantifiers in $\phi ^o$ are approximated by their value under o (which is no approximation if the variable is outside the domain) except for variables occurring within the scope of a variable lower in the subsumption ordering. The significance of constraining to be linearly ordered will become apparent shortly when we consider a quasi-ordering of .

Example 1

Suppose $x \triangleleft y$ and $o(x) = \alpha$ and $o(y)=\beta$, with $\alpha ,\beta <\kappa$. Let $\phi$ be a formula without quantifiers containing both x and y free. Then $(\nu y \nu x \phi )^o = \nu ^\beta y \nu ^\alpha \phi$, whereas $(\nu x \nu y \phi )^o = \nu ^\alpha x \nu y \phi$. The requirement that is linear means that if $((\nu x \phi ) \vee (\nu z \psi ))^o = (\nu ^\alpha x \phi ' ) \vee (\nu ^\gamma z \chi ' )$ then either one of $\alpha$ and $\gamma$ is $\kappa$, or x and z are comparable in $\triangleleft$.

Definition 10

The image of a well-formed formula under a $\kappa$-assignment is well-annotated. We let be the set of well-annotated $\kappa$-formulæ.

Recall that substitution is well-defined for well-formed formulæ.

Lemma 6

If $\varrho$ is well-named then every formula in $\mathbb {SC}_\kappa (\varrho )$ is well-annotated.

Proof

Suppose $\phi = (\nu x \, \psi )^o = \nu ^{\alpha } x\, \psi ^{o_{\triangleleft x}} \in \mathbb {SC}_\kappa (\varrho )$. Then for each $\beta <\alpha$, we have $\phi ' = \psi ^{o_{\triangleleft x}} ( x/ \nu ^{\beta } x\, \psi ^{o_{\triangleleft x}} ) \in \mathbb {SC}_\kappa (\varrho )$ by the closure condition and we require to show that $\phi '$ is well-annotated. Assume $o(x) < \kappa$ (otherwise the result is immediate) and let $o'$ be the assignment with domain determined by $o'(y) = o(y)$ for $y \triangleleft x$ and $o'(x) = \beta$. Given the fact that $\varrho$ is well-named, x does not appear bound in $\psi$, whence it is easy to check that $\phi ' = \psi (x/ \nu x\, \psi ) ^{o'}$.

The other closure conditions are straightforward.

As defined, $\kappa$-assignments do not uniquely determine the formulæ in . Each determines an obvious equivalence relation on , given by $o \sim _\psi o'$ iff $\psi ^o = \psi ^{o'}$. However, for each there exists a unique $\kappa$-assignment o with smallest domain such that $\phi = \psi ^{o}$, where $\psi = \phi ^-$ is the template of $\phi$. We call this assignment the ordinal assignment of $\phi$ and denote it $o_\phi$.

We can thus give the formal definition of the quasi-order $\sqsubseteq$ introduced immediately prior to Definition 9. This starts with a quasi-order $\le$ on $\kappa$-assignments, defined by $o \le \hat{o}$ iff and for every maximal chain the sequence $(o(x_0), \ldots , o(x_n))$ is lexicographically prior to $(\hat{o}(x_0), \ldots , \hat{o}(x_n))$.

Lemma 7

is a well-quasi-order. Moreover, there exists k such that for every set with there exists $o,\hat{o} \in X$ s.t. $o < \hat{o}$.

Proof

Transitivity of $\le$ is established by induction along $\triangleleft$ in $\mathsf {Var}$. So, $\le$ is a quasi-order. Moreover, this quasi-order is a well-order on sets of $\kappa$-assignments with the same domain since it reduces to the lexicographic ordering on $\kappa ^k$ for some k (as domains are linearly ordered by $\triangleleft$). Since $\mathsf {Var}$ is a finite set, both claims follow.    $\square$

Definition 11

Fix and for $\phi ,\psi \in \mathbb {SC}_\kappa (\varrho )$ define $\phi \sqsubseteq \psi$ iff $\phi ^- = \psi ^-$ and $o_\phi \le o_\psi$.

This relation is well-defined because of Lemma 6, which implies that every formula in the strong closure of an formula is well-annotated and, hence, has a defined ordinal assignment.

We consider it instructive to note that there is another natural quasi-order sitting strictly between Kozen’s $\preccurlyeq$ and our $\sqsubseteq$, obtained by dropping the restriction of linearity of annotated quantifiers but otherwise applying the lexicographic ordering in $\sqsubseteq$. This too is a wqo, but does not satisfy the second part of Lemma 4.

B Omitted Proofs

We now present some missing arguments from the main text. We begin with Lemma 4 as this result follows directly from our work on ordinal assignments:

Lemma 4. $(\mathbb {SC}_\kappa (\varrho ),\sqsubseteq )$ is a wqo. Moreover, there exists k such that for every $X \subseteq \mathbb {SC}_\kappa (\varrho )$, .

Proof

We need only remark that the quasi-order $(\mathbb {SC}_\kappa (\varrho ),\sqsubseteq )$ can be expressed as the disjoint union of finitely many copies of , one for each formula in $\mathbb {FL}(\varrho )$, an operation that preserves wqo-ness. The second claim follows from this fact and Lemma 7.

Lemma 1(2): For all and contexts $\varGamma []$, .

Proof

Induction on $\phi =\phi (x_1 , \ldots , x_k)$ shows the inference

$$\begin{aligned} \frac{{ \varGamma [ \psi _1 , \chi _1 ] }\quad { \cdots }\quad { \varGamma [ \psi _k , \chi _k ] }}{{ \varGamma [ \overline{\phi }( \psi _1 , \ldots , \psi _k ) , \phi ( \chi _1 , \ldots , \chi _k ) ] }} \end{aligned}$$

is admissible in and . For the case $\phi = \nu y \phi _0$, we have a derivation of $\varGamma [ \overline{\phi }, \nu ^0 y \phi _0 ]$ by $\nu {.}0$, and from $\varGamma [ \overline{\phi }, \nu ^\alpha y \phi _0 ]$ we derive $\varGamma [ \overline{\phi }, \nu ^{\alpha +1} y \phi _0 ]$ via the induction hypothesis and inferences $\mu$ and $\nu {.}(\alpha +1)$. Thus transfinite induction shows that $\varGamma [ \overline{\phi }, \nu ^\alpha y \phi _0 ]$ is derivable for every $\alpha < \kappa$, whence $\varGamma [ \overline{\phi }, \phi ]$ results.

Theorem 4. The proof of this theorem ends with a statement of the following equivalence:

$$ \forall X , Y \subseteq \mathbb {SC}_\kappa (\varrho ) : X \sqsubseteq ^* Y \quad \text {iff}\quad X^* \le _{\text {pw}} Y^* $$

On first appearance this result appears non-trivial. However, it is an easy consequence of the following result relating finite sets of ordinals, the verification of which is straightforward.

Lemma 8

Given a non-empty finite set of ordinals A, let denote the unique sequence such that . Fix a principal ordinal $\kappa$ and let $A , B \subset \kappa$ be non-empty finite sets of the same cardinality. There exists such that $B = \{ f(\alpha ) \mid \alpha \in A \}$ iff $A^* \le _{\text {pw}} B^*$.