Uniform interpolation via nested sequents and hypersequents

Uniform interpolation is stronger than Craig interpolation and provides a simulation of quantifiers in a logic. Similar to Craig interpolation, uniform interpolation is useful in computer science, for example, in quantifier elimination procedures [11] or in knowledge representation to perform tasks such as forgetting irrelevant information in descriptive logics [16]. This shows the practical value of uniform interpolation. The goal of this paper is to expand the reach of proof-theoretic method of proving uniform interpolation.

A propositional (modal) logic ${\sf L}$ admits the Craig interpolation property (CIP) if for any formulas $\varphi$ and $\psi$ such that $\vdash_{\sf L}\varphi\rightarrow\psi$, there is an interpolant $\theta$ containing only atomic propositions that occur in both $\varphi$ and $\psi$ such that $\vdash_{\sf L}\varphi\rightarrow\theta$ and $\vdash_{\sf L}\theta\rightarrow\psi$. One could say that the purpose of the interpolant is to state the reason $\psi$ follows from $\varphi$ by using the common language of the two. Logic ${\sf L}$ has the uniform interpolation property (UIP) if for each formula $\varphi$ and each atomic proposition $p$ there are uniform interpolants $\exists p\varphi$ and $\forall p\varphi$ containing only atomic propositions occurring in $\varphi$ except for $p$ such that for all formulas $\psi$ not containing $p$:

It is well known that this property is stronger than Craig interpolation. Indeed, by computing uniform interpolants consecutively, it is possible to remove a given set of atomic propositions and construct a formula that would uniformly serve as a Craig interpolant for a fixed $\varphi$ and all $\psi$ with a given common language.

Analytic sequent calculi can be used to prove the CIP constructively. For the UIP, terminating cut-free sequent calculi play a similar role. Whereas for the CIP the syntactic proofs are often straightforward, the case of the UIP is much more complicated. Pitts provided a first syntactic proof of this kind, establishing the UIP for ${\sf IPC}$ [24]. Bílková successfully adjusted the method to (re)prove the UIP for several modal logics including ${\sf K},{\sf T}$, and ${\sf GL}$ [2]. Iemhoff provided a modular method for (intuitionistic) modal logics and intermediate logics with sequent calculi consisting of so-called focused rules, among others establishing the UIP for ${\sf D}$ [14, 15].

There are also algebraic and model-theoretic methods. The UIP for ${\sf GL}$ and ${\sf K}$ is due to Shavrukov [28] and Ghilardi [12] respectively. Interestingly, modal logics ${\sf S4}$ and ${\sf K4}$ do not enjoy the UIP [2, 13] despite enjoying the CIP. Visser provided purely semantic proofs for ${\sf K}$, ${\sf GL}$, and ${\sf IPC}$ based on bounded bisimulation up to atomic propositions [31]. This method was later applied to prove the stronger Lyndon UIP for a wide range of modal logics [17]. The semantic interpretation of uniform interpolation is called bisimulation quantifiers, see [7] for an extended explanation. Bisimulations will also play a role in the current paper.

The proof-theoretic approach has two advantages. First, it enables one to find interpolants constructively rather than merely prove their existence.¹¹1More precisely, the method enables one to find interpolants efficiently rather than by an exhaustive search of all formulas, the search that terminates due to the proven existence of an interpolant. Second, it can turn uniform interpolation into a powerful tool in the study of existence of proof systems. Negative results are obtained in [14, 15] stating that logics without the UIP cannot have certain natural sequent calculi. As a consequence, ${\sf K4}$ and ${\sf S4}$ do not possess such proof systems. Similar negative results are obtained for modal and substructural logics in [29] and [30] using the CIP and UIP. These methods are based on calculi with regular sequents.

The goal of this paper is to extend the same line of research to multisequent formalisms such as hypersequents and nested sequents²²2Nested sequents are also known as tree-hypersequents [25] or deep sequents [5] in the literature.. Such forms of sequent calculi have recently been adapted to prove the CIP of modal logics via nested sequents [10] and hypersequents [18]. A modular proof-theoretic framework encompassing these and also labelled sequents was provided in [19]. The same ideas were extended to multisequent calculi for intermediate logics [21]. The method combines syntactic and semantic reasoning. Generalized Craig interpolants are constructed using the calculus in a purely syntactic manner, but the method’s correctness uses semantic notions from Kripke models of the underlying logic.

This paper extends this method providing proof-theoretic proofs for the UIP based on nested sequents for ${\sf K}$, ${\sf D}$, and ${\sf T}$ and on hypersequents for ${\sf S5}$. The UIP for these logics has been known, but we provide a new method that can hopefully be extended to other logics. Similar to [19], we combine syntactic and semantic reasoning. We use the terminating calculi to define the uniform interpolants and then provide model modifications and use bisimulations to prove the correctness of these interpolants.

Bílková [3] also provided a syntactic method for uniform interpolation for ${\sf K}$ based on nested sequents. She presented proofs based on two nested calculi for ${\sf K}$: one with a standard modality and another that is based on a different modal language with a cover modality $\nabla$. Bílková’s method for nested sequents is closely related to her work based on regular sequents in [2]. The main difference with our method is that we exploit the treelike structure of nested sequents reflecting the treelike models for ${\sf K}$ by incorporating semantic arguments while the algorithm for the interpolant computation remains fully syntactic. We intend our method to form a good basis for generalizing to other logics with multisequent calculi.

The paper is organized as follows. In Sect. 2, we introduce the nested sequent calculi for ${\sf K}$,  ${\sf T}$, and ${\sf D}$, as well as model modifications invariant under bisimulation. In Sect. 3, we present our method to prove uniform interpolation for ${\sf K}$, ${\sf T}$, and ${\sf D}$. In Sect. 4, we show how the method can be adjusted to hypersequents for ${\sf S5}$. Section 5 concludes the paper and maps the immediate next steps.

We define $\overline{\varphi}$ (or $\neg\varphi$) recursively as usual using De Morgan’s laws to push the negation inwards. $\varphi\to\psi\colonequals\overline{\varphi}\lor\psi$.

One way of looking at a nested sequent is to consider a tree of ordinary (one-sided) sequents, i.e., of multisets of formulas. Each structural box in the nested sequent creates a child in the tree. In order to be able to reason about formulas in a particular tree node, we introduce labels. A label is a finite sequence of natural numbers. We denote labels by $\sigma,\tau,\dots$; a label $\sigma\ast n$ (or simply $\sigma n$) denotes the label $\sigma$ extended by the natural number $n$.

The set $\mathcal{L}(\Gamma)$ can be considered as the set of nodes of the corresponding tree of $\Gamma$, with $1$ being the root of this tree. Often, we do not distinguish between a nested sequent $\Gamma$ and its labeled sequent $l(\Gamma)$. For example, we say that $\sigma\in\Gamma$ if $\sigma\in\mathcal{L}(\Gamma)$.

Following [5], we will work with contexts in rules to signify that the rules can be applied in an arbitrary node of the nested sequent. We will work with unary contexts which are nested sequents with exactly one hole, denoted by the symbol $\{\>\}$. Such contexts are denoted by $\Gamma\{\>\}$. The insertion $\Gamma\{\Delta\}$ of a nested sequent $\Delta$ into a context $\Gamma\{\>\}$ is obtained by replacing the occurrence $\{\>\}$ with $\Delta$. The hole $\{\>\}$ can be labeled the same way as formulas. We write $\Gamma\{\>\}_{\sigma}$ to denote the label of the hole.

In this paper we use nested sequent calculi for classical modal logics ${\sf K}$, ${\sf D}$, and ${\sf T}$ from [5]. Recall that ${\sf K}$ consists of all classical tautologies, the ${\sf k}$-axiom $\Box(\varphi\rightarrow\psi)\rightarrow(\Box\varphi\rightarrow\Box\psi)$ and is closed under modus ponens (from $\varphi\rightarrow\psi$ and $\varphi$, infer $\psi$) and necessitation (from $\varphi$, infer $\Box\varphi$). Further, ${\sf D}\colonequals{\sf K}+\Box\varphi\rightarrow\Diamond\varphi$ and ${\sf T}\colonequals{\sf K}+\Box\varphi\rightarrow\varphi$. We now introduce nested sequent calculi and then Kripke semantics for these logics.

The terminating nested sequent calculus NK for the modal logic K consists of all rules in the first two rows in Fig. 1 plus the rule k. This calculus is an extension of the multiset-based version from [5] to the language with Boolean constants $\bot$ and $\top$, necessitating an addition of the rule $\mathsf{id_{\top}}$ for handling these (cf. also the treatment of Boolean constants in [10]). The nested calculus ND (NT) for the modal logic D (T) is obtained by adding to NK the rule ${\sf d}$ (${\sf t}$). As shown in [5], the nested sequent calculi ${\sf NK}$, ${\sf ND}$, and ${\sf NT}$ are sound and complete for modal logics ${\sf K}$, ${\sf D}$, and ${\sf T}$ respectively, i.e., a nested sequent $\Gamma$ is derivable in ${\sf NK}$ (${\sf ND}$, ${\sf NT}$) if and only if its formula interpretation $\iota(\Gamma)$ is a theorem of ${\sf K}$ (${\sf D}$, ${\sf T}$).

The rules from Fig. 1 with embedded contraction are sometimes called Kleene’d rules. Following [5], in order to ensure finite proof search, we only apply a rule when the principal formula in the conclusion is not saturated w.r.t. this rule, i.e., $\varphi\lor\psi$, $\varphi\land\psi$, and $\Box\varphi$ are not ${\sf K}$-saturated, $\Diamond\varphi$ in the rule k is not ${\sf K}$-saturated w.r.t. the label of the bracket containing $\Delta$, $\Diamond\varphi$ in the rule d is not ${\sf D}$-saturated, and $\Diamond\varphi$ in the rule t is not ${\sf T}$-saturated. Since for Kleene’d rules principal formulas are preserved in the premises, the number of applications of each of the rules $\mathsf{k}$, $\mathsf{d}$, and $\mathsf{t}$ is bounded. The way to think of a saturated sequent is that in a bottom-up proof search when we reach a saturated sequent, it does not make sense to apply more rules as these would only lead to duplications.

Intuitively, nested sequents capture the tree structure of Kripke models for modal logics. We define truth for nested sequents in Kripke models and then recall relevant facts about bisimulations and introduce model modifications that we use in the proof of uniform interpolation.

We will use models based on finite intransitive directed trees, usually denoting the root $\rho$. For ${\sf T}$, the accessibility relation $R$ is required to be reflexive, i.e., $\forall w\in WwRw$. For ${\sf D}$, the accessibility relation $R$ must be serial, i.e., $\forall w\in W\exists v\in WwRv$. Note that such seriality implies reflexivity of the leaves of the tree. Finally, we assume $R$ to be irreflexive for ${\sf K}$. From now on we call these models ${\sf T}$-models, ${\sf D}$-models, and ${\sf K}$-models respectively.

Following [19], we now extend the definitions of truth and validity to nested sequents.

The following lemma, which can be easily proved by induction on the structure of $\Gamma$, implies completeness for validity of nested sequents.

We now define bisimulations modulo an atomic proposition $p$, similar to the ones from [7, 31], where uniform interpolation is studied on the basis of bisimulation quantifiers. While those papers focus on purely semantic methods, we embed the semantic tool of bisimulation into our constructive proof-theoretic approach in Sect. 3. Our bisimulations behave largely like standard bisimulations except they do not have to preserve the truth of formulas with occurrences of $p$.

The main property of bisimulations is truth preservation for modal formulas. The following theorem is proved the same way as [4, Theorem 2.20].

We are interested in manipulations of treelike models that preserve bisimulations up to $p$, in particular, in duplicating a part of a model or replacing it with a bisimilar model.

In this section we prove the uniform interpolation theorem for ${\sf K}$, ${\sf T}$, and ${\sf D}$ via their nested sequent calculi ${\sf NK},{\sf NT}$, and ${\sf ND}$ respectively. We define a new notion of uniform interpolation for nested sequents in Def. 3.15 that involves Kripke semantics. We then prove in Lemma 3.13 that this implies the standard definition of uniform interpolation.

For classical-based logics, the existence of left-interpolants ensures the existence of right-interpolants, and vice versa. Assuming $\forall p\varphi$ is defined for each formula $\varphi$, we can define $\exists p\varphi:=\neg\forall p\overline{\varphi}$. Thus, from now on, we focus on $\forall p\varphi$.

We import some of the notation from [19] in order to formulate the uniform interpolation property for nested sequents.

We define the label-erasing function ${\sf form}$ from multiformulas to formulas, as well as multiformula equivalence, and list some of the latter’s easily provable properties.

We now introduce the uniform interpolation property for nested sequents. Here, the uniform interpolants are multiformulas instead of formulas.

The condition on labels in (i) ensures that interpretations of $\Gamma$ are suitable for $A_{p}(\Gamma)$.

Since we use bisimulations up to $p$ to find a model $\mathcal{M}^{\prime}$ in the NUIP(iii) condition, we replace it with a (possibly) stronger condition (iii)^′:

It easily follows from Theorem 2.16 that, like formulas, both nested sequents and multiformulas are invariant under bisimulations: