Induced Interpolant Existence Problem

• Induced interpolant existence is a decision problem that determines if, for a given pair of formulas, an interpolant restricted to their shared vocabulary exists.
• It distinguishes between entailment and interpolation, revealing varying decidability, complexity, and constructive challenges across modal, algebraic, and semialgebraic logics.
• Constructive synthesis methods—including SOS techniques and proof-theoretic extraction—offer practical means to generate interpolants for applications in verification and model checking.

The induced interpolant existence problem is the instance-wise decision problem of whether, for a given entailment or contradiction, there exists an interpolant restricted to the shared vocabulary. In its standard form, given formulas $P$ and $Q$ in a logic $L$, the problem asks whether there exists $I \in L$ such that $\Sigma(I)\subseteq \Sigma(P)\cap\Sigma(Q)$, $P \models_L I$, and $I \models_L Q$. In nonlinear real arithmetic, the same idea appears as the search for an interpolant $I(x)$ over shared variables alone, typically of the form $h(x)>0$, separating contradictory formulas $\phi(x,y)$ and $Q$0 (Wolter et al., 2024, Gan et al., 2019). When a logic has the Craig Interpolation Property (CIP), the problem collapses to validity; when CIP fails, interpolant existence becomes a separate semantic and algorithmic question, with decidability, complexity, and constructive behavior varying sharply across logical and algebraic settings (Wolter et al., 2024, Kurucz et al., 2023).

1. Formal formulation and principal variants

The core definition is uniform across much of the literature. For formulas $Q$1 and $Q$2 with shared signature $Q$3, a Craig interpolant is a formula $Q$4 such that $Q$5, $Q$6 is unsatisfiable, and $Q$7 mentions only symbols from $Q$8. Equivalently, in entailment form, $Q$9 and $L$0 with $L$1 (Gan et al., 2019, Wolter et al., 2024).

A central distinction is between global interpolation and instance-wise existence. CIP asserts that every valid implication has such an interpolant. The interpolant existence problem asks only whether a specific implication does. This distinction is decisive in logics that do not enjoy CIP: in those settings, validity no longer guarantees interpolant existence, and the induced existence problem can be harder than entailment or even undecidable (Wolter et al., 2024, Kurucz et al., 2023, Kurucz et al., 2023).

Several specialized variants recur. In nonlinear polynomial interpolation over the reals, one considers contradictory formulas

$L$2

with shared variables $L$3, and seeks an induced interpolant $L$4, usually $L$5, such that $L$6 and $L$7 is unsatisfiable (Gan et al., 2019). In proof-theoretic work, an “induced interpolant” can also mean an interpolant extracted directly from a proof object such as a resolution refutation, a clausal tableau, or an SMT proof tree (Christ et al., 2017, Wernhard, 2020, Koopmann et al., 15 Aug 2025).

Verification-oriented work generalizes the single-interpolant question to structured collections. Path interpolation, generalized simultaneous abstraction, state-transition interpolation, and tree interpolation ask whether one can obtain interpolants satisfying additional inductiveness or compositionality constraints across a sequence or tree of partitions (Gurfinkel et al., 2012).

2. Semantic characterizations of existence and nonexistence

A dominant pattern in modal, guarded, and description-logical settings is characterization by shared-signature indistinguishability. In $L$8 and guarded fragments with two equivalence relations, the failure of interpolation is captured by a Robinson-style criterion: $L$9 and $I \in L$0 have no interpolant iff there exist pointed models $I \in L$1 and $I \in L$2 such that $I \in L$3, $I \in L$4, and $I \in L$5 and $I \in L$6 are bisimilar over the shared signature $I \in L$7 (Wolter et al., 2024). Closely related criteria appear for modal logics above $I \in L$8, where no interpolant exists iff there are descriptive-frame models whose roots satisfy $I \in L$9 and $\Sigma(I)\subseteq \Sigma(P)\cap\Sigma(Q)$0 while being $\Sigma(I)\subseteq \Sigma(P)\cap\Sigma(Q)$1-bisimilar, with $\Sigma(I)\subseteq \Sigma(P)\cap\Sigma(Q)$2 (Kurucz et al., 2023). Description logics with nominals and role inclusions use the same pattern in the form of “joint consistency modulo $\Sigma(I)\subseteq \Sigma(P)\cap\Sigma(Q)$3-bisimulations,” and first-order modal fragments such as $\Sigma(I)\subseteq \Sigma(P)\cap\Sigma(Q)$4 and $\Sigma(I)\subseteq \Sigma(P)\cap\Sigma(Q)$5 use $\Sigma(I)\subseteq \Sigma(P)\cap\Sigma(Q)$6-bisimulation consistency as the criterion for nonexistence (Artale et al., 2020, Kurucz et al., 2023).

In nonlinear real arithmetic, the semantic picture is geometric rather than bisimulation-based. Under Archimedean assumptions on the quadratic modules generated by the constraints of $\Sigma(I)\subseteq \Sigma(P)\cap\Sigma(Q)$7 and $\Sigma(I)\subseteq \Sigma(P)\cap\Sigma(Q)$8, the $\Sigma(I)\subseteq \Sigma(P)\cap\Sigma(Q)$9-projections

$P \models_L I$0

are compact semialgebraic sets. If $P \models_L I$1 is unsatisfiable, then these projections are disjoint, and there exists a polynomial $P \models_L I$2 such that $P \models_L I$3 on $P \models_L I$4 and $P \models_L I$5 on $P \models_L I$6; consequently, $P \models_L I$7 is an interpolant (Gan et al., 2019). For unbounded domains, homogenization and compactification replace the Archimedean boundedness hypothesis. If the homogenized closures $P \models_L I$8 and $P \models_L I$9 of the projected semialgebraic sets are disjoint, then there exists a semialgebraic interpolant of the form

$I \models_L Q$0

which strictly separates the two sets (Wu et al., 2024).

These characterizations divide the subject into two broad regimes. In relational logics, nonexistence is typically witnessed by shared-signature bisimilar or equivalent models. In semialgebraic settings, existence is typically witnessed by a separating function obtained from positivity certificates on projected feasible sets.

3. Decidability and complexity landscape

The complexity of induced interpolant existence is highly nonuniform.

Setting	IEP status	Notes
Finitely axiomatizable $I \models_L Q$1	coNP-complete	Same complexity as entailment (Kurucz et al., 2023)
Difference logic (DL)	coNP-complete	Polynomial-size bisimilar witnesses (Kurucz et al., 2024)
Weak $I \models_L Q$2	in coN3ExpTime; coNExpTime-hard	Validity is PSpace-complete (Kurucz et al., 2024)
$I \models_L Q$3	undecidable	Validity remains decidable (Wolter et al., 2024)
$I \models_L Q$4 with constants, $I \models_L Q$5	undecidable	Also for corresponding DL counterparts (Wolter et al., 2024)
$I \models_L Q$6, $I \models_L Q$7	in coN2ExpTime; 2ExpTime-hard	UI existence undecidable (Kurucz et al., 2023)
$I \models_L Q$8	non-elementary decidable	UI existence undecidable (Kurucz et al., 2023)
DLs with nominals and/or role inclusions	2ExpTime-complete with ontologies	Ontology-free/RI-only cases vary between coNExpTime-complete and 2ExpTime-complete (Artale et al., 2020)
LTL over finite words	decidable in 4ExpTime; PSPACE-hard	Via FO-separation of regular languages (Kurucz et al., 18 Aug 2025)

Two global conclusions recur. First, interpolant existence need not track entailment complexity. Weak $I \models_L Q$9 has PSpace-complete validity but an interpolant existence problem in coN3ExpTime with coNExpTime-hardness (Kurucz et al., 2024). Description logics with nominals and role inclusions exhibit the same phenomenon: entailment is easier than existence, and the failure of CIP or projective Beth definability prevents direct reduction to validity (Artale et al., 2020). Second, decidability of the base logic does not imply decidability of IEP. The first explicit counterexamples are $I(x)$0 and $I(x)$1 with constants and two equivalence relations, where satisfiability and entailment remain decidable but interpolant existence is undecidable (Wolter et al., 2024).

There are also positive “same-complexity” zones. For finitely axiomatizable modal logics above $I(x)$2, IEP is coNP-complete and “not harder than entailment,” due to quasi-polysize bisimilar countermodels on descriptive frames (Kurucz et al., 2023). The chapter-level treatment of interpolation and separation further shows that LTL IEP can be reduced to FO-separation for regular languages, yielding decidability in 4ExpTime over finite words and extending the semantic perspective from logic to language separation (Kurucz et al., 18 Aug 2025).

4. Constructive synthesis methods

In nonlinear polynomial interpolation over the reals, constructive existence is obtained from Putinar’s Positivstellensatz. Under Archimedean assumptions, one searches for $I(x)$3 and SOS multipliers $I(x)$4, $I(x)$5 satisfying

$I(x)$6

$I(x)$7

with all multipliers SOS. These identities imply $I(x)$8 and $I(x)$9, so $h(x)>0$0 is an interpolant. SOS constraints are encoded by Gram matrices and coefficient matching, yielding a feasibility SDP; under the Archimedean hypothesis, the method is complete for sufficiently large degree bounds. The same framework extends to disjunctions of conjunctions of polynomial inequalities (Gan et al., 2019).

The same paper addresses numerical soundness. Approximate SDP solutions are validated by PSD certification using floating-point Cholesky factorization with padding $h(x)>0$1, together with coefficient error bounds $h(x)>0$2. Over compact feasible regions, one derives certified inequalities

$h(x)>0$3

If $h(x)>0$4 and $h(x)>0$5, then strict separation holds exactly (Gan et al., 2019).

For unbounded semialgebraic domains, homogenization moves the problem to a compact sphere: $h(x)>0$6 Polynomial separation on the homogenized sets induces a semialgebraic interpolant

$h(x)>0$7

and the paper provides complete SOS characterizations for both polynomial and semialgebraic interpolants, again reducible to SDP (Wu et al., 2024). This strictly extends the polynomial case, because $h(x)>0$8 recovers polynomial interpolants.

Proof-based constructive methods cover SMT, arrays, propositional logic, and first-order tableaux. Proof Tree Preserving Interpolation handles mixed literals in SMT proofs for quantifier-free EUF plus linear arithmetic by introducing shared auxiliary variables, keeping the proof tree unchanged, and using specialized mixed-pivot rules for equalities and inequalities (Christ et al., 2017). For the quantifier-free theory of arrays, Weak Equivalence on Arrays and PTPI yield sequence and tree interpolants from a single unsatisfiability proof, avoiding partition-dependent reruns of the solver; interpolants have worst-case exponential size for extensionality lemmas and worst-case quadratic size otherwise (Hoenicke et al., 2018). Clausal first-order tableaux support a two-stage construction: a bottom-up extraction of a ground interpolant $h(x)>0$9, followed by interpolant lifting that replaces certain terms by quantified variables ordered by subterm dependency (Wernhard, 2020). In classical propositional logic, quantifier elimination, DNF projection, resolution extraction, and tableau extraction all compute interpolants, with existence equivalent to unsatisfiability of $\phi(x,y)$0 (Koopmann et al., 15 Aug 2025).

Hybrid modal logics add a further constructive result: for $\phi(x,y)$1, $\phi(x,y)$2, $\phi(x,y)$3, and graded variants, hypermosaic elimination computes Craig interpolants, if they exist, in fourfold exponential time, with fourfold exponential output size (Jung et al., 17 Feb 2026).

5. Structured interpolants in verification

Verification workflows rarely need only a single interpolant. They require collections satisfying inductive or compositional constraints. A uniform framework organizes the principal collectives as path interpolation (PI), generalized simultaneous abstraction (GSA), state-transition interpolation (STI), and tree interpolation (TI) (Gurfinkel et al., 2012).

For an unsatisfiable sequence $\phi(x,y)$4, PI requires interpolants $\phi(x,y)$5 such that each prefix entails the corresponding $\phi(x,y)$6, each $\phi(x,y)$7 refutes the suffix, and variables are restricted to the prefix/suffix intersection. STI strengthens this to local Hoare-style steps

$\phi(x,y)$8

while TI assigns interpolants to tree cuts (Gurfinkel et al., 2012).

In resolution-based Labeled Interpolation Systems, these collectives admit exact characterizations. For single systems, $\phi(x,y)$9, while PI and SA are weaker. For families of systems, TI is the most general collective, STI implies both PI and SA, and STI is equivalent to local BGSA constraints on every relevant triple. Necessary and sufficient labeling constraints are given by $Q$00 and $Q$01. All single LISs satisfy PI, systems at least as strong as Pudlák satisfy BGSA and therefore all stronger collectives, and McMillan’s dual $Q$02 separates PI from BGSA by satisfying the former but not the latter (Gurfinkel et al., 2012).

This framework directly informs model checking. Predicate abstraction and CEGAR require PI sequences; simultaneous abstraction and summary generation require SA or GSA; IC3/PDR-like inductive sequences correspond to STI; recursive or DAG-shaped decompositions require TI. The array-interpolation work supplies concrete sequence and tree interpolants for memory-safety proofs in C programs from one global proof, illustrating how theory-specific interpolation machinery realizes these collective requirements in practice (Gurfinkel et al., 2012, Hoenicke et al., 2018).

6. Boundaries, failures, and open directions

A fundamental limitation is that decidable quantified fragments may still fail to admit interpolants in-fragment. The array property fragment is a sharp example. The formulas

$Q$03

belong to the alternation-free APF and are jointly unsatisfiable, but no interpolant over the shared vocabulary exists within APF. The natural candidates

$Q$04

require quantifier alternation and lie outside the fragment (Hoenicke et al., 2019). This shows that decidability and closure under negation do not imply closure under interpolation.

Uniform interpolation is even more fragile. In $Q$05 and $Q$06, uniform interpolant existence is undecidable (Kurucz et al., 2023). In hybrid modal logics, the situation is broader: for any logic containing $Q$07 and contained in FO, it is undecidable whether a uniform $Q$08-interpolant exists for a given formula (Jung et al., 17 Feb 2026). These results sharply separate ordinary instance-wise interpolation from forgetting-style constructions.

Nonlinear algebraic methods have their own boundaries. The polynomial SOS approach of nonlinear Craig interpolation relies on Archimedean quadratic modules; if compactness fails, there may be no polynomial strictly separating the projected feasible sets, and the relevant SOS certificates may not exist (Gan et al., 2019). The unbounded-domain extension mitigates this by homogenization, but at the cost of semialgebraic rather than purely polynomial interpolants and potentially severe degree growth and SDP conditioning issues (Wu et al., 2024).

Several boundary questions remain open. For $Q$09 with a single equivalence relation, the status of IEP is open; likewise for dropping equality from $Q$10, for $Q$11 with counting, and for $Q$12 (Wolter et al., 2024). Above $Q$13, finite axiomatizability is crucial for the coNP procedure based on canonical formulas, leaving non-finitely axiomatizable extensions unresolved (Kurucz et al., 2023). For weak $Q$14, the gap between the coN3ExpTime upper bound and coNExpTime-hardness is open, and constructive interpolation procedures remain a separate challenge (Kurucz et al., 2024). In hybrid modal logics, the existence of fourfold-exponential algorithms leaves a nontrivial gap to triply-exponential lower bounds on separator size in the presence of $Q$15 (Jung et al., 17 Feb 2026).

Taken together, these results place the induced interpolant existence problem at the intersection of model theory, real algebraic geometry, proof complexity, and formal verification. Its central question is simple—whether a shared-language separator exists for a given entailment—but its answer depends delicately on the ambient logic, the available proof objects, and the geometry or bisimulation theory underlying the formulas in question.