Craig-Type Conditions Overview

• Craig-type conditions are families of restrictions in logic, numerical analysis, and spectral theory that impose intermediary controls like interpolation and boundedness.
• They refine classical Craig interpolation by constraining vocabulary, polarity, or spectral parameters to pinpoint minimal expressive thresholds and failure phenomena.
• In numerical methods such as the Modified Craig–Sneyd ADI scheme, they provide parameter bounds that guarantee unconditional stability and mesh-independent convergence.

Craig-type conditions are families of semantic, syntactic, algebraic, analytic, or parameter restrictions that play an interpolating or boundedness role in settings whose terminology descends either from Craig interpolation or from Craig–Sneyd/CRAIG methods. In logic, they refine the requirement that from an entailment one can extract an intermediate formula over the shared vocabulary; in numerical analysis and spectral theory, the same label is used for mesh-independent stability bounds, spectral thickness hypotheses, or residual criteria attached to the Modified Craig–Sneyd scheme, Dubrovin-type flows, and CRAIG regularization (Wernhard, 2018, Hout et al., 2014, Lukić et al., 2019, Hnětynková et al., 2016).

1. Logical core: interpolation, strengthened forms, and common vocabulary

In its classical first-order form, the Craig interpolation property says that if $A \models B$, then there exists an interpolant $I$ such that $A \models I$, $I \models B$, and the non-logical vocabulary of $I$ is contained in the intersection of the vocabularies of $A$ and $B$. In the formulation used for first-order tableaux, this is sharpened by explicit vocabulary conditions

$$\mathrm{pred}(I) \subseteq \mathrm{pred}(A)\cap \mathrm{pred}(B), \qquad \mathrm{fun}(I) \subseteq \mathrm{fun}(A)\cap \mathrm{fun}(B),$$

and in the Craig–Lyndon variant the polarity of predicate occurrences must also be preserved (Wernhard, 2018).

Several logically stronger or more structured Craig-type conditions recur in the literature. Craig–Lyndon interpolation preserves positive and negative occurrences of predicates. Access interpolation constrains not only shared vocabulary but also relativized quantification patterns and binding patterns, as required in database query reformulation. Beth definability is repeatedly positioned as a nearby property, but several papers explicitly leave it open rather than deriving it automatically from interpolation. In first-order Gödel logic $\mathbf{G}$ and its $\Delta$-extension $I$0, the interpolation property is formulated both semantically and proof-theoretically; for $I$1, semantic entailment, $I$2-entailment, and derivability coincide, whereas for $I$3 the paper proves the $I$4-entailment and hypersequent-calculus version and leaves the purely semantic version open (Khatami et al., 2023).

This family resemblance matters because the phrase “Craig-type conditions” does not identify a single theorem schema. It instead marks a shared structural demand: an entailment, proof, or transition from one formal object to another must admit an intermediate object constrained by a common signature, common polarity profile, common access pattern, or analogous boundedness requirement.

2. Minimality, exact thresholds, and failure phenomena in logical fragments

A major use of Craig-type conditions is to locate the precise expressive threshold at which interpolation appears or disappears. For guarded fragments, the decisive result is that the guarded-negation fragment $I$5 is, in a precise sense, the smallest extension of the guarded fragment $I$6 with Craig interpolation: if $I$7 is any FO-fragment such that $I$8 $I$9, $A \models I$0 $A \models I$1 is closed under self-guarded substitution, $A \models I$2 $A \models I$3 is closed under conjunction and disjunction, and $A \models I$4 $A \models I$5 has the Craig Interpolation Property, then $A \models I$6 (Cate et al., 2023). The same line of work contrasts this with $A \models I$7 and the forward fragment: any substitution-closed extension of $A \models I$8 with interpolation already reaches full first-order expressive power at least on sentences, and any forward logic with interpolation satisfies $A \models I$9; under mild effectiveness assumptions, extensions of $I \models B$0 or of the forward fragment with interpolation become undecidable (Cate et al., 2023).

Negative results are equally central. Bi-intuitionistic predicate logic lacks the Craig Interpolation Property: there exist formulas

$I \models B$1

such that $I \models B$2, but there is no interpolant in the shared vocabulary $I \models B$3 (Olkhovikov et al., 2022). That failure does not contradict Rauszer’s earlier “Craig interpolation” theorem, because the latter concerns deductive interpolation for global consequence rather than interpolation for valid implications; the distinction is essential precisely because the deduction theorem fails for bi-intuitionistic logic with global consequence (Olkhovikov et al., 2022).

Interpretability logics exhibit a different pattern: interpolation exists exactly at a sharply characterized sublogic threshold. For sublogics of $I \models B$4, the paper gives a complete classification of uniqueness of fixed points, the fixed point property, and Craig interpolation, and shows that the Craig Interpolation Property holds exactly for $I \models B$5 and $I \models B$6 (Iwata et al., 2020). This places interpolation alongside specific frame conditions and fixed-point phenomena rather than treating it as an isolated metatheorem.

Modal logics with linear frames provide yet another variation. Normal modal logics above $I \models B$7 are known to lack global Craig interpolation except in bounded-depth cases such as $I \models B$8, but the per-instance interpolant existence problem remains decidable. For every finitely axiomatisable normal modal logic containing $I \models B$9, the interpolant existence problem is decidable and coNP-complete; absence of an interpolant is characterized by the existence of two finitely generated descriptive-frame models, one satisfying $I$0 and the other satisfying $I$1, that are $I$2-bisimilar at the roots for $I$3 (Kurucz et al., 2023). This replaces a global interpolation property by a non-uniform Craig-type criterion based on bisimulation countermodels.

3. Proof-theoretic and algorithmic realizations

A large part of the literature treats Craig-type conditions constructively, by identifying proof systems in which interpolants can be extracted locally from derivations. Clausal first-order tableaux provide one such route. A closed two-sided clausal ground tableau yields a ground interpolant $I$4 at the root, and a lifting step replaces top-level occurrences of non-shared function terms by quantified variables ordered by the strict subterm relation; the result is a Craig–Lyndon interpolant. The same paper develops access interpolation for relational first-order formulas with relativized quantifiers by imposing regularity, leaf-only, and contiguity conditions on ACI-tableaux (Wernhard, 2018).

Sequent calculi supply a second route. Semi-analytic sequent calculi form a broad syntactic class in which local occurrence-preservation and strongness assumptions suffice to derive interpolation. The paper proves that any strong semi-analytic sequent calculus over the relevant base languages has the corresponding sequent interpolation property, and therefore the logic it axiomatizes has the Craig Interpolation Property (Tabatabai et al., 2018). A different proof-theoretic criterion appears in labelled sequent calculi for modal logics: if the frame class is axiomatized by universally closed quantifier-free Horn formulas, and the labelled sequent calculus internalizes those frame conditions by local structural rules on relational atoms, then the logic enjoys both Craig and Lyndon interpolation (Kuznets, 2016).

Horn clauses make the correspondence especially explicit. Binary, inductive-sequence, tree, restricted-DAG, and disjunctive interpolation are shown to correspond, respectively, to natural fragments of recursion-free Horn constraints. The central theorem states that each of these interpolation problems reduces in polynomial time to syntactic solving of a recursion-free Horn system in the corresponding fragment, and conversely that solving such a Horn system reduces in polynomial time to interpolation of the corresponding kind (Rümmer et al., 2013). In software verification, this converts interpolants into summaries, invariants, and compositional proof obligations.

Two further generalizations extend the same pattern outside ordinary first-order entailment. For the three-valued logic of here-and-there $I$5, a variation of Mints’ sequent system constructs a preliminary interpolant in an extension $I$6 with an additional operator $I$7, and then transforms that preliminary interpolant into an actual $I$8 interpolant (Wernhard, 7 Jan 2026). For stochastic Boolean satisfiability, generalized Craig interpolants are defined by the conditions

$I$9

and are computed by an interpolating S-resolution calculus whose rules carry both clauses and probabilistic annotations (Teige et al., 2012).

4. Categorical and continuous generalizations

In categorical logic, Craig-type conditions are recast as exactness-like stability properties. For finitary doctrines on $A$0 that preserve slicing, interpolation is equivalent to closure of the class of truth-conservative maps under cocomma squares. More precisely, if $A$1 is a finitary doctrine on $A$2 preserving slicing, then $A$3 has Craig interpolation if and only if $A$4-conservative maps are closed under cocomma in $A$5 (Liberti et al., 16 Jan 2026). This is presented as a categorical counterpart of the algebraic-logic slogan that interpolation aligns with amalgamation-like properties, but adapted to positive fragments where lax squares and cocommas replace ordinary pushouts (Liberti et al., 16 Jan 2026).

Continuous logic introduces a genuinely metric version of Craig-type conditions. If $A$6, then for each $A$7 there exists a common-language sentence $A$8 such that

$A$9

which is the weak interpolation theorem (Keisler, 2024). If instead $B$0, then for each $B$1 there exists a common-language sentence $B$2 such that

$B$3

which is the strong interpolation theorem (Keisler, 2024). The same paper proves a continuous Robinson consistency theorem and shows that weak interpolation admits a uniform-limit formulation. It also establishes that strong interpolation implies weak interpolation, but weak interpolation does not imply strong interpolation in general (Keisler, 2024).

These developments show that Craig-type conditions survive well beyond ordinary two-valued syntax, but often only after the exact statement is weakened or re-expressed. In categorical logic, the mediating object is a subobject stable under cocommas; in continuous logic, it is an $B$4-approximate common-language sentence or even a uniformly convergent sequence of such sentences.

5. Craig–Sneyd-type conditions in alternating-direction time stepping

In numerical analysis, “Craig-type conditions” often refers not to interpolation at all but to the stability and convergence regimes of the Modified Craig–Sneyd (MCS) alternating-direction implicit scheme for two-dimensional convection–diffusion equations with mixed derivatives. After semidiscretization,

$B$5

with $B$6 representing the mixed derivative term and $B$7 the directional parts, the convergence theorem for the MCS scheme assumes smoothness of the exact solution, dissipativity $B$8 for $B$9, stability $$\mathrm{pred}(I) \subseteq \mathrm{pred}(A)\cap \mathrm{pred}(B), \qquad \mathrm{fun}(I) \subseteq \mathrm{fun}(A)\cap \mathrm{fun}(B),$$0 uniformly in mesh width and time step, invertibility of $$\mathrm{pred}(I) \subseteq \mathrm{pred}(A)\cap \mathrm{pred}(B), \qquad \mathrm{fun}(I) \subseteq \mathrm{fun}(A)\cap \mathrm{fun}(B),$$1 and $$\mathrm{pred}(I) \subseteq \mathrm{pred}(A)\cap \mathrm{pred}(B), \qquad \mathrm{fun}(I) \subseteq \mathrm{fun}(A)\cap \mathrm{fun}(B),$$2, and mesh-independent boundedness of

$$\mathrm{pred}(I) \subseteq \mathrm{pred}(A)\cap \mathrm{pred}(B), \qquad \mathrm{fun}(I) \subseteq \mathrm{fun}(A)\cap \mathrm{fun}(B),$$3

Under these hypotheses, the global temporal discretization error satisfies

$$\mathrm{pred}(I) \subseteq \mathrm{pred}(A)\cap \mathrm{pred}(B), \qquad \mathrm{fun}(I) \subseteq \mathrm{fun}(A)\cap \mathrm{fun}(B),$$4

for all $$\mathrm{pred}(I) \subseteq \mathrm{pred}(A)\cap \mathrm{pred}(B), \qquad \mathrm{fun}(I) \subseteq \mathrm{fun}(A)\cap \mathrm{fun}(B),$$5 with $$\mathrm{pred}(I) \subseteq \mathrm{pred}(A)\cap \mathrm{pred}(B), \qquad \mathrm{fun}(I) \subseteq \mathrm{fun}(A)\cap \mathrm{fun}(B),$$6, with $$\mathrm{pred}(I) \subseteq \mathrm{pred}(A)\cap \mathrm{pred}(B), \qquad \mathrm{fun}(I) \subseteq \mathrm{fun}(A)\cap \mathrm{fun}(B),$$7 independent of the spatial mesh width and of $$\mathrm{pred}(I) \subseteq \mathrm{pred}(A)\cap \mathrm{pred}(B), \qquad \mathrm{fun}(I) \subseteq \mathrm{fun}(A)\cap \mathrm{fun}(B),$$8 (Hout et al., 2014).

The same paper derives scalar boundedness conditions from von Neumann symbols. For the model problem

$$\mathrm{pred}(I) \subseteq \mathrm{pred}(A)\cap \mathrm{pred}(B), \qquad \mathrm{fun}(I) \subseteq \mathrm{fun}(A)\cap \mathrm{fun}(B),$$9

the Fourier symbols satisfy

$\mathbf{G}$0

with $\mathbf{G}$1. The resulting recommendation is $\mathbf{G}$2 for unconditional stability with mixed derivatives for any $\mathbf{G}$3; if $\mathbf{G}$4, the additional restriction $\mathbf{G}$5 is required (Hout et al., 2014).

The stability paper isolates the sharp parameter ranges more finely. If $\mathbf{G}$6, then $\mathbf{G}$7 for all $\mathbf{G}$8 if and only if $\mathbf{G}$9. If all $\Delta$0 are real and satisfy the diffusion constraint, unconditional stability holds if and only if $\Delta$1. If $\Delta$2 is real and $\Delta$3 are complex, then $\Delta$4 is necessary; in the fully general complex case, $\Delta$5 is sufficient, and for the PDE discretization with periodic boundary conditions and second-order central differences the MCS scheme is unconditionally stable in the von Neumann sense whenever $\Delta$6 (Hout et al., 2010).

In this literature, then, Craig-type conditions are parameter and boundedness conditions ensuring unconditional stability and mesh-independent second-order convergence of the MCS ADI scheme, especially in the presence of explicitly treated mixed derivative terms.

6. Spectral thickness, residual criteria, and other nonlogical uses

For the KdV hierarchy with almost periodic initial data, Craig-type conditions are spectral thickness assumptions on the gap structure of the associated one-dimensional Schrödinger operator. Writing

$\Delta$7

with

$\Delta$8

the Craig-type weights are

$\Delta$9

The sufficient Craig-type conditions are

$I$00

$I$01

$I$02

$I$03

Under these hypotheses, the Dirichlet data evolve according to Lipschitz Dubrovin-type flows in $I$04 and $I$05, and the potential is uniquely reconstructed by the trace formula

$I$06

yielding uniqueness for the KdV hierarchy for reflectionless almost periodic data, with applications to finite-gap spectra and to small quasiperiodic analytic potentials with Diophantine frequency (Lukić et al., 2019).

A different nonlogical use appears in inverse problems and regularization. In GK-bidiagonalization-based CRAIG regularization, the CRAIG residual is exactly

$I$07

so the CRAIG iterate is the exact solution of the modified compatible problem

$I$08

The same paper proves that LSQR and LSMR residuals are linear combinations of GK vectors whose coefficients reflect propagated noise, whereas CRAIG gives only a single scaled bidiagonalization vector. This motivates Craig-type stopping criteria based on the first minimizer of $I$09, on the iteration where $I$10 becomes noise-like, or on the discrepancy-style target $I$11 when the noise level is known (Hnětynková et al., 2016).

Across these nonlogical settings, the common pattern is still intermediary control. Spectral Craig-type conditions mediate between gap geometry and Lipschitz Dubrovin dynamics; CRAIG residual criteria mediate between bidiagonalization vectors and effective noise removal; Craig–Sneyd conditions mediate between directional splitting and unconditional stability. The label is therefore historically unified but mathematically plural.