The Craig interpolation property for a logic states that if , then there exists a formula in the language of such that and , and every propositional variable appearing in appears in and . This property has many useful applications: it can be used to prove Beth definability ; in computer-aided verification it can be used to split a large problem involving into smaller problems involving and ; and in knowledge representation (uniform) interpolation can be used to conceal or forget irrelevant or confidential information in ontology querying . Therefore, demonstrating that a logic possesses the Craig interpolation property is of practical value.
Interpolation can be proved semantically or syntactically. In the semantic method, is the set of valid formulae, thereby requiring a semantics for . In the syntactic method, often known as Maehara’s method , is the set of theorems, thereby requiring a proof-calculus. The syntactic approach constructs the interpolant by induction on the (usually cut-free) derivation of , and usually also provides derivations witnessing and .
Over the past forty years, Gentzen’s original sequent calculus has been extended in many different ways to handle a plethora of logics. The four main extensions are hypersequent calculi , display calculi , nested sequent calculi [1, 10, 20], and labelled calculi . Various interpolation results have been found using these calculi but the only general methodology that we know of is the recent work of Kuznets  with Lellman . Although they use extended sequent calculi, binary-relational Kripke semantical arguments are crucial for their methodology, and extending their method to other semantics is left as further work. They also construct the interpolants using a language containing (interpreted) meta-level connectives which are external to the logic at hand, and do not handle logics containing converse modalities such as tense logic. Finally, their method does not yield derivations witnessing and .
We give a general, purely syntactic, methodology for proving Craig interpolation using nested sequent calculi for a variety of propositional, non-classical logics including normal tense logics, their extensions with path axioms, and bi-intuitionistic logic. Our methodology does not utilise semantics, does not embed one logic into another, and does not utilise logical connectives which are external to the underlying logical language.
The first novelty of our approach is a generalisation of the notion of interpolant from formulas to sets of sequents. The second is a notion of orthogonality which gives rise to a notion of duality via cut: if two interpolants are orthogonal, then the empty sequent is derivable from the sequents in the interpolants using only the cut and the contraction rules. This duality via cut allows us to relate our more general notion of interpolants (as sets of sequents) to the usual notion of interpolants (as formulas). Moreover, given a derivation of , our orthogonality condition not only allows us to construct the interpolant , but also the derivations witnessing and . This fact shows that our approach possesses a distinct complexity-theoretic advantage over the semantic approach: to verify that is indeed the interpolant of , one need only check the derivations of and , which is a PTIME process. In the semantic approach, to verify that and are indeed valid (and that is in fact an interpolant of ) one must construct proofs of the implications, which is generally much harder (e.g., finding a proof of a validity in one of the tense logics presented in Sec. 3 is PSPACE complete).
Related work. Interpolation has been heavily investigated in the description logic community, where it is used to hide or forget information . In this setting, the logic ALC is a syntactic variant of the multimodal normal modal logic Kn while its extension with inverse roles, ALCI, is a variant of the multimodal normal tense logic Ktn. Cate et al  utilise a complexity-optimal tableau algorithm to prove interpolation for ALC via Maehara’s method. They then embed ALCI into ALC and extend their interpolation result to ALCI.
By contrast, our methodology is direct: we obtain interpolation for the normal tense logic Kt, and can then extract interpolation for the normal modal logic K by simply observing that our nested sequent calculus obeys the separation property: if the end-sequent contains no occurrences of the black (converse) modalities, then neither does the interpolant.
As mentioned earlier, the work of Kuznets et al. [6, 13, 14] on interpolation for modal logics in nested sequent calculi is closest to ours. Our construction of interpolants for tense logics shares some similarity with theirs. One crucial difference is that our interpolants are justified purely through syntactic and proof-theoretic means, whereas their interpolants are justified via semantic arguments. Another important difference is that our method extends to the bi-modal case and also (bi-)intuitionistic case, and it is straightforward to adapt our work to the multi-modal case, e.g., using nested sequent calculi as in . Kowalski and Ono  showed interpolation for bi-intuitionistic logic using a sequent calculus with analytic cut. In contrast, our proof is based on a cut-free nested sequent calculus .
Outline of the paper. In Sec. 2 we give a brief overview of a typical interpolation proof using the traditional sequent calculus, and highlight some issues of extending it to nested sequent calculi, which motivates the generalisation of the interpolation theorem we adopt in this paper. In Sec. 3 we show how the generalised notion of interpolants can be used to prove the Craig interpolation theorem for classical tense logic and its extensions with path axioms , covering all logics in the modal cube and more. We then show how our approach can be extended to bi-intuitionistic logic in Sec. 4. In Sec. 5 we conclude and discuss future work.
2 Overview of our approach
We analyze a typical syntactic interpolation proof for Gentzen sequents, highlight the issues of extending it to nested sequents, and motivate our syntactic approach for interpolation.
Consider, for example, a two-sided sequent calculus for classical logic such as G3c . Interpolation holds when we can prove that for all , if is provable in G3c, then so are both and , for some containing only propositional variables common to both and .
The inductive construction of can be encoded via inference rules over more expressive sequents that specify the splitting of the contexts and the interpolant constructed thus far. In G3c, we write to denote the sequent with its context split into and , and with the interpolant. Inference rules for this extended sequent are similar to the usual ones, with variations encoding the different ways the contexts may be split. For example, the initial rule has the following four variants corresponding to the four splittings of where can occur (with four different interpolants!):
Branching rules, such as the right-introduction rule for , split into two variants, depending on whether the principal formula is in the first or the second partition of the context:
Observe that the interpolants of the conclusion sequents are composed from the interpolants of the premises, but with the main connectives dual to one another: a disjunction in the rule and a conjunction in the rule. These observations also apply for the other rules of G3c, with a slight subtlety for the implication-left rule: see . Interpolation for G3c can then be proved by a straightforward induction on the height of proofs.
Below we discuss some issues with extending this approach to proving interpolation for modal/tense logics and bi-intuitionistic logic using nested sequent calculi, and how these issues lead to the generalisation of the intermediate lemmas we need to prove (which amounts to an interpolation theorem for sequents, rather than formulae).
Classical modal and tense logics
A nested sequent  can be seen as a tree of traditional Gentzen-style sequents. For classical modal logics, single-sided sequents suffice, so a nested sequent in this case can be seen as a nested multiset: i.e. a multiset whose elements can be formulae or multisets. Following the notation in , a sequent nested inside another sequent is prefixed with a , which is the structural proxy for the modal operator. For example, the nested sequent below first left, with two sub-sequents and , represents the formula shown second left:
Nested sequent calculi for modal logics [1, 9, 10] typically contain the propagation rule for diamond shown third left above which “propagates” the into the scope of , when read upwards. Propagation rules complicate the adaptation of the interpolation proof from traditional Gentzen sequent calculi. In particular, it is not sufficient to partition a context into two disjoint multisets. That is, suppose a nested sequent is provable, and we would like to construct an interpolant such that and are provable, where is the negation normal form of . Suppose the proof of ends with a propagation rule, e.g., when and as shown above far right. In this case, by induction, we can construct an interpolant such that the splittings and of the premiss are provable, but it is in general not obvious how to construct the desired interpolant for the conclusion from . For this example, should be , should be , which does not mention at all.
The above issue with propagation rules suggests that we need to strengthen the induction hypothesis to construct interpolants, i.e., by considering splitting the sequent context at every sub-sequent in the nested sequent. For example, the nested sequent above should be split into and when applying the induction hypothesis. Then, is indeed an interpolant: both and are provable. Nevertheless, employing a formula interpolant is not enough to push through the inductive argument in general. Consider, for example, the nested sequent , which is provable with an identity rule, and its partition and . There is no formula such that both and are provable. One solution to this problem is to generalise the interpolation statement to consider a nested sequent as an interpolant: If a nested sequent is provable, then for every ‘partitioning’ of into and (where the partitioning applies to every sub-sequent in a nested sequent; the precise definition will be given in subsequent sections), there exists (the interpolant), and such that
The propositional variables occuring in are in both and ,
splits into and , and splits into and , where denotes the nested sequent with all formula occurrences replaced with their negations, and
Both and are provable.
For example, the nested sequent , with partitions and , has the interpolant (hence ), and
One remaining issue is that, since we now use a nested sequent as an interpolant, the composition of interpolants needs to be adjusted as well. Recall that in the construction of interpolants for G3c above, in the case involving the right-introduction for , we constructed either or as the interpolant for the conclusion. If and are nested sequents, the expression or would not be well-formed. To solve this remaining issue, we generalise the interpolant further to be a set of nested sequents.
Fitting and Kuznets  similarly generalise the notion of interpolants, but instead of generalising interpolants to a set of (nested) sequents, they introduce ‘meta’ connectives for conjunction and disjunction, applicable only to interpolants, and justified semantically. Our notion of interpolants requires no new logical operators or semantical notions.
Propositional bi-intuionistic logic
Bi-intuitionistic logic is obtained from intuitionistic logic by adding a subtraction (or exclusion) connective that is dual to implication. Its introduction rules are the mirror images of those for implication; in the traditional sequent calculus, these take the form:
However, as shown in , the cut rule cannot be entirely eliminated in a sequent calculus featuring these rules, although they can be restricted to analytic cuts . In , Postniece et al. show how bi-intuitionistic logic can be formalised in a nested sequent calculus. Although interpolation holds for intuitionistic logic, it does not generalise straightforwardly to bi-intuitionistic logic, and only very recently has interpolation for bi-intuitionistic logic been shown . The proof for the interpolation theorem for intuitionistic logic is very similar to the proof of the same theorem for classical logic; one simply needs to restrict the partitioning of the sequent to the form where is empty and contains at most one formula occurrence. Since the (nested) sequent calculus for bi-intuitionistic logic uses multiple-conclusion (nested) sequents, the proof for intuitionistic logic cannot be adapted to the bi-intutionistic case. The problem already shows up in the very simple case involving the identity rule: suppose we have a proof of the initial sequent and we want to partition the sequent as . It is not possible to find an interpolant such that and (otherwise, one would be able to prove the excluded middle , which is not valid in bi-intuitionistic logic, using the cut formula ). In general, the inductive construction of the interpolant for may involve finding an interpolant for the problematic partition of the form , where is non-empty. This case does not arise in the interpolation proof for intuitionistic logic in , due to the restriction to single-conclusion sequents.
We show that the above issue with bi-intutionistic logic can be solved using the same approach as in modal logic: simply extend the interpolant to a set of nested sequents. In particular, for , the generalised interpolation statement only requires finding an interpolating sequent and its ‘dual’ (see below) such that both and are provable, which is achieved by letting , , and
Interpolating sequents and orthogonality
In a simplified form (e.g., sequent calculus), the generalised interpolation result we show can be roughly summarised as follows: given a provable sequent , there exist two sets of sequents and such that
For every sequent , the sequent is provable,
For every sequent , the sequent is provable,
The propositional variables in and occur in both and , and
The sequents in and are orthogonal to each other, that is, the empty sequent is derivable from all sequents in using only the cut rule and possibly structural rules (contraction and/or weakening).
The set is taken to be the (sequent) interpolant.
Last, the orthogonality condition, can be seen as a generalisation of duality. To see how this is the case, consider a degenerate case where is a classical sequent (e.g., in G3c). We show how one can convert a formula interpolant in the usual definition (i.e., formula s.t. and are provable) to a sequent interpolant satisfying the four conditions above, and vice-versa. For the forward direction, simply let and . It is easy to see that is orthogonal to For the converse direction, suppose we have a sequent interpolant and its orthogonal . We illustrate how one can construct a formula interpolant . To simplify the discussion, let us assume that
and that the following sequents are provable:
Let Then it is easy to see that is provable given (1), and is provable given (2) - (5). The formal statement and the proof of the generalised interpolation theorem will be discussed in detail in the next two sections.
A note on notation
In what follows, we adopt a representation of nested sequents using restricted labelled sequents where we use the labels and relational atoms to encode the tree structure of a nested sequent. To clarify what we mean, consider the following nested sequent for tense logic :
Graphically, the nested sequent can be represented as a tree (shown below left) with two types of edges and . Alternatively, the nested sequent can be represented as the polytree shown below right with a single type of edge and where the orientation of the edge encodes the two types of structures and of the nested sequent (observe that the -edge from to in the left diagram has been reversed in the right diagram).111A polytree is a directed graph such that its underlying graph—the graph obtained by ignoring the orientation of the edges—is a tree.
|A, B @>[dl]_∘@>[dr]^∘|
A, B @>[dl]@>[dr]
In the latter representation, the structure of the nested sequent can be encoded using a single binary relation: we label each node of the tree corresponding to the nested sequent (as shown above left) with unique labels , , , , encode each edge from a label to a label with a relation , and encode each edge with a relation . The above nested sequent can then be equivalently represented as a labelled sequent where and is a relational symbol:
Inference rules in a nested sequent calculus can be trivially encoded as rules in a restricted labelled calculus seen as a ‘data structure’ rather than a proper labelled sequent calculus.
We stress that our labelled notation to represent nested sequents is just a matter of presentation: the labelled representation is notationally simpler for presenting inference rules and composing nested sequents. For instance, the operation of merging two nested sequents with isomorphic shapes is simply the union of the multiset of labelled formulae.
3 Interpolation for Tense Logics
As usual, we interpret as saying that holds at every point in the immediate future, and as saying that holds at some point in the immediate future. Conversely, the and modalities make reference to the past: says that holds at every point in the immediate past, and says that holds at some point in the immediate past. Last, we take to be the negation of , and use the notation and .
We consider tense formulae in negation normal form (nnf) as this simplifies our calculi while retaining the expressivity of the original language. The language for the tense logics we consider is given via the following BNF grammar:
Since our language excludes an explicit connective for negation, we define it formally below (Def. 3). Using the definition, we may define an implication to be . For a formula , we define the negation recursively on the structure of : if then and if then . The clauses concerning the connectives are as follows: (1) , (2) , (3) , and (4) .
Path axioms are of the form (or, equivalently, ) with . See  for an overview of path axioms.
The tense logics we consider are all extentions of the minimal tense logic (Fig. 1) with path axioms. Thus, is the minimal extension of with all axioms from the finite set of path axioms.
The calculus for , extended with a set of path axioms , is given in Fig. 2. Labelled sequents are defined to be syntactic objects of the form , where is a multiset of relational atoms of the form and is a multiset of labelled formulae of the form , with a tense formula and labels from a countable set .
Note that the side conditions and of the and rules, respectively, depend on the set of path axioms added to . The definition of the relation is founded upon various auxiliary concepts that fall outside the main scope of this paper. We therefore refer the interested reader to App. A where the relation as well as the concepts needed for its definition are explicitly provided. See also [9, 23] for details.
The contraction rules , the weakening rules and are admissible, and all inference rules are invertible in .
Consider the formula , which is a theorem in the logic with . A proof of this formula is provided in Fig. 3.
As stated in Sec. 2, we extend the notion of an interpolant to a set of nested sequents. In our definition of interpolants, we are interested only in duality via cut. In particular, the relational atoms (encoding the tree shape of a nested sequent) are not explicitly represented in the interpolants since they can be recovered from the contexts of the sequents in which the interpolants are used. We therefore define a flat sequent to be a sequent without relational atoms. For classical tense logic, a flat sequent is thus a multiset of labelled formulas.
An interpolant, denoted , is a set of flat sequents.
For example, the set below is an interpolant in our context:
Since our interpolant is no longer a formula, we need to define the dual of an interpolant in order to generalise the statement of the interpolation result to sequents. We have informally explained in Sec. 2 that duality in this case is defined via cut. Intuitively, given an interpolant , its dual is any set of nested sequents such that the empty sequent can be derived from and using cut (possibly with contraction). For example, given , there are several candidates for its dual:
The empty sequent can be derived from , for using cut (and contraction, in the case of ). In principle, any of the dual candidates to can be used, but to make the construction of the interpolants deterministic, our definition below will always choose , as it is relatively straightforward to define as a function of
For an interpolant , the orthogonal is defined as
For example, the orthogonal of is
Let be the interpolant
where does not occur in and define
where an empty disjunction is .
We define an interpolation sequent to be a syntactic object of the form , where is a set of relational atoms, and are multisets of labelled formulae (for ), and is an interpolant. Note that in the interpolation calculus , (see Fig. 4).
The vertical bar in an interpolation sequent marks where the sequent will be partitioned, with the left partition serving as the antecedent and the right partition serving as the consequent in the interpolation statement. For example, the initial interpolation sequent shown below left splits into the two sequents shown below right
where the first member of the split is placed in the left sequent and the second member is placed in the right sequent (note that the relational atoms are inherited by both sequents). We think of the interpolant as being implied by the left sequent, and so, we place it in the left sequent, and we think of the interpolant as implying the right sequent, so we place its negation (viz. ) in the right sequent. Observe that an application of between the two sequents, yields without the interpolant. Performing a in this way syntactically establishes (without evoking the semantics) that the interpolant is indeed an interpolant (so long as the interpolant satisfies certain other properties; cf. Lem. 4 below).
The interpolation calculus (Fig. 4) uses interpolation sequents. More importantly, the calculus succinctly represents our algorithm for constructing interpolants. Most of the rules are straightforward counterparts of the proof system in Fig. 2, except for the orthogonality rule The orthogonality rule is arguably the most novel aspect of our interpolation calculus, as it imposes a strong requirement on our generalised notion of interpolants, that it must respect the underlying duality in the logic. The key to the correctness of this rule is given in the Persistence Lemma below, which shows that double-orthogonal transformation always retains some sequents in the original interpolant.
[Persistence] If , then there exists a such that
Given a formula , we define the set of propositional variables of to be the set . This notation extends straightforwardly to sets of formulae and interpolants.
We write to denote that the sequent is provable in . Similarly, denotes that the sequent is provable in
A logic has the Craig interpolation property iff for every implication in the logic, there is a formula such that (i) and (ii) and are in the logic, where is taken to be the implication connective of the logic.
We now establish that each tense logic possess the Craig interpolation property when the implication connective is taken to be . To achieve this, we begin by showing that an interpolant sequent can be constructed from any cut-free proof. If , then there exists an such that , , and all labels occuring in also occur in or .
Induction on the height of the proof of and by using the rules of . ∎
For all we have and
For all , we have
To prove Craig interpolation, we need to construct formula interpolants. Lem. 4 provides sequent interpolants, so the next step is to show how one can derive a formula interpolant from a sequent interpolant. This is possible if the formulas in an interpolant are all prefixed with the same label. In that case, there is a straightforward interpretation of the interpolant as a formula. More precisely, let , where for all . Then, its formula interpretation is given by Given such an interpolant , we write to denote its formula interpretation. The following lemma is a straightforward consequence of this interpretation. Let be an interpolant with for each . For any multiset of relational atoms and multiset of labelled formulae , if for all , then .
However, the formula-interpolant derived in Lem. 4 gives only one-half of the full picture, as one still needs to show that the orthogonal of a sequent interpolant admits a dual interpretation as a formula. A key to this is the following Duality Lemma that shows that orthogonality behaves like negation. [Duality] Given an interpolant , the empty sequent is derivable from using the rule and the contraction rule.
An interesting consequence of Duality Lemma is that it translates into duality in the above formula interpretation as well, as made precise in the following lemma. Let be an interpolant with for each . For any multiset of relational atoms and multiset of labelled formulae , if for all , then .
Suppose for some By Lem. 4, we have a derivation of the empty sequent from assumptions .
Due to admissibility of weakening (Lem. 3), for each , there is a proof of the sequent . Adding to every leaf sequent in belonging to gives us a derivation :
where and sequents in brackets are provable, and where denotes multiple copies of sequents or rules. By the assumption we know that each is provable, so by adding to each premise sequent in , we get the following proof:
If , then there exists a such that (i) and (ii) and .
Every extension of the (minimal) tense logic with a set of path axioms has the Craig interpolation property.
4 Interpolation for Bi-Intuitionistic Logic
The language for bi-intuitionistic logic is given via the following BNF grammar: