Maehara-Style Craig Interpolants

• The paper offers a novel, purely syntactic method for constructing Craig interpolants using a step-by-step Maehara-style procedure in nested sequent calculi.
• It demonstrates how modal, tense, and bi-intuitionistic logics are addressed by locally assigning and combining interpolant candidates via structural induction.
• The approach ensures soundness, completeness, and adherence to the shared-vocabulary criterion, highlighting its practical significance in proof theory.

Maehara-style Construction of Craig Interpolants

A Maehara-style construction provides a purely syntactic, rule-by-rule method for extracting Craig interpolants from cut-free proofs in a range of logics beyond classical first-order logic. This methodology operates entirely within the given proof system—in particular, within nested sequent calculi for modal, tense, and bi-intuitionistic logics—by locally assigning interpolant candidates at each step and combining them by structural induction to yield global interpolants expressing the shared vocabulary between two contexts (Lyon et al., 2019).

1. Nested Sequent Calculi for Modal and Bi-Intuitionistic Logics

The foundational systems for Maehara-style interpolation in this setting are nested sequent calculi, which generalize Gentzen-style sequents by encoding modalities, propagation, and relational atoms directly in sequent structure.

• Tense logic Kt and extensions: Formulas are in negation-normal form over atoms $p$ and connectives $\land,\lor$, and four modalities $\Box,\Diamond,\blacksquare,\blackdiamond$ ("always next," "sometime next," "always previous," "sometime previous"). Path axioms induce relational structure through relational atoms $Rxy$ representing transitions between worlds. Sequents are of the form $R \vdash \Gamma$, with $R$ a multiset of $Rxy$ and $\Gamma$ a multiset of labelled formulas $x:A$.
• Bi-intuitionistic logic (BiInt): Formulas comprise $p$, constants $\top,\bot$, Boolean connectives, implication ($\supset$), and exclusion ($-$). The nested sequent system encodes structure with $Rxy$ atoms and uses polarities for formula occurrences, supporting analytic cuts and invertible rules.

All rules, including Boolean, modal, and dual-modal, are height-preserving invertible, and the system admits cut-elimination (Lyon et al., 2019).

2. Syntactic Interpolation Theorem as Sequent Condition

Given a derivable two-sided sequent

$R,\,\Gamma_1,\Gamma_2 \vdash \Delta_1,\Delta_2,$

a Craig-interpolant is encoded as a set $I$ of "flat" sequents with the following properties:

• Left derivability: For all $\Lambda \in I$, $R,\Gamma_1,\Lambda \vdash \Delta_1$ is derivable.
• Right derivability: For all $\Theta \in I^\perp$ (the dual set), $R,\Gamma_2,\Theta \vdash \Delta_2$ holds.
• Shared vocabulary: All atoms in $I \cup I^\perp$ are in the intersection of the left and right context variables.
• Orthogonality: $I$ and $I^\perp$ are orthogonal—cuts on complementary literals suffice to derive the empty sequent.

A standard interpolation scenario arises for $S = \vdash x:\bar A \lor B$, producing a set $I$ whose conjunction-disjunction aggregate

$$C = \bigwedge_{\Lambda \in I} \bigvee_{x:A \in \Lambda} A$$

is the Craig interpolant, satisfying both $\vdash x:\bar A \lor C$ and $\vdash x:\bar C \lor B$ (Lyon et al., 2019).

3. Inductive Maehara Procedure and Definition of Interpolants

The Maehara-style extraction is a bottom-up induction on the cut-free proof of the target sequent, with local interpolant construction assigned to each inference rule:

• Flat sequents: Interpolants are sets $I = \{\Lambda_1,\dots,\Lambda_n\}$ where each $\Lambda_i$ is a flat sequent (one-sided, no relational atoms).
• Duality and orthogonality: The dual $I^\perp$ is generated by negating each formula in constituent $\Lambda_i$, producing all possible choices for cut formulas.

Rule-specific interpolant assignment includes:

• Identity/Boolean rules: Direct context split, interpolants formed by inspecting the side to which each atomic formula is assigned.
• Boolean connectives: Union ($\lor$) preserves existing interpolants; conjunction ($\land$) aggregates interpolant sets from branches.
• Modal propagation: Modal rules induce "boxing" operations, transforming $I$ by substituting modalities as formulas move between worlds.
• Orthogonality: An explicit "orth" step switches contexts and fixes dual interpolants.

Each step ensures that interpolant sets only reference shared vocabulary and maintain local derivability and orthogonality.

Key Lemmas

• Persistence: Any $\Lambda \in (I^\perp)^\perp$ corresponds to some $\Lambda' \in I$ with $\Lambda' \subseteq \Lambda$.
• Local correctness: For $R \vdash \Gamma \mid \Delta \Longrightarrow I$, every $\Lambda \in I$ is left-provable; every $\Theta \in I^\perp$ is right-provable.
• Duality: $I \cup I^\perp$ forms a cut system yielding the empty sequent; when all formulas are identically labelled, this recovers the standard interpolant via conjunction-disjunction (Lyon et al., 2019).

4. Soundness, Completeness, and Variable Condition

The construction is sound: all atoms in constructed interpolants originate from both contexts of the partition. Inductive invariance guarantees preservation of the intersection condition.

Completeness follows from the correctness lemma: From a cut-free proof of $\vdash x:\bar A \lor B$, one extracts $I$ yielding sequents

$$\vdash x:\bar A, x:\bigwedge\bigvee I, \qquad \vdash x:\overline{\bigwedge\bigvee I}, x:B$$

and thus single-formula interpolants $C = \bigwedge\bigvee I$ fulfill both $A \to C$ and $C \to B$, with all variables of $C$ in $\mathrm{Vars}(A) \cap \mathrm{Vars}(B)$ (Lyon et al., 2019).

5. Detailed Example in Tense Logic

For the $Kt$-theorem

$$\vdash\, \bar q \to (\bar p \lor p) \quad (\text{or } \vdash x: q \lor (\bar p \lor p)),$$

the Maehara process proceeds as follows:

1. Atomic occurrence: Deepest identity on $p,\bar p$ yields interpolant $\{w:p\}$.
2. Modality lifting: Two applications of $\Box$ and $\blacksquare$ "box" the interpolant, producing $\{z:\Box p\}$ then $\{y:\blacksquare\Box p\}$.
3. Orthogonality: Flipping sides produces the dual $\{y:\lnot\blacksquare\Box p\}$.
4. Boolean combination: Top-level join yields $I = \{y: \Box p \lor \blacksquare\Box p\}$ and the interpolant $C = \Box p \lor \blacksquare\Box p$.

This interpolant derivation is checked by showing:

$$\vdash q \to (\Box p \lor \blacksquare\Box p), \quad \vdash (\Box p \lor \blacksquare\Box p) \to (\bar p \lor p)$$

and $C$ uses only the common atom $p$, strictly implementing the Craig variable condition (Lyon et al., 2019).

6. Algorithmic Structure of the Construction

The procedure for extracting the Maehara-style interpolant from a nested sequent proof is:

1. Initialize with the root sequent and partition.
2. Traverse bottom-up, assigning to each inference step a local interpolant according to clause-specific rules.
3. Apply modal boxing and Boolean composition where appropriate for each type of inference.
4. At each orthogonality step, update dual interpolant sets.
5. At completion, assemble the global interpolant as the conjunctive-disjunctive formula over the leaf-level interpolant set $I$.
6. Validate soundness/completeness and variable conditions via inductive invariants.

This process is fully constructive and does not employ semantic methods, embeddings, or external interpreted connectives (Lyon et al., 2019).

7. Scope, Extensions, and Novel Features

The Maehara-style construction as realized in nested sequent calculi is applicable to:

• Classical tense logic (including extensions with arbitrary path axioms)
• Bi-intuitionistic logic (with analytic two-sided cuts)
• Logics admitting nested representations not easily formalizable in standard Gentzen systems

A key innovation is the definition of an orthogonality condition structurally enforcing duality between interpolants, extending the original Maehara framework to encompass "converse" modality and propagation phenomena characteristic of tense and bi-intuitionistic logics.

This generalizes the interpolation property to a broader class of modal and intuitionistic systems with cut-free nested calculi, yielding uniform and entirely syntactic Craig interpolants as formulas over the shared propositional atoms of the input formulas (Lyon et al., 2019).