In mathematical logic, Craig's interpolation theorem is a result about the relationship between different logical theories. Roughly stated, the theorem says that if a formula φ implies a formula ψ, and the two have at least one atomic variable symbol in common, then there is a formula ρ, called an interpolant, such that every non-logical symbol in ρ occurs both in φ and ψ, φ implies ρ, and ρ implies ψ. The theorem was first proved for first-order logic by William Craig in 1957. Variants of the theorem hold for other logics, such as propositional logic. A stronger form of Craig's interpolation theorem for first-order logic was proved by Roger Lyndon in 1959;^[1][2] the overall result is sometimes called the Craig–Lyndon theorem.

Example

In propositional logic, let

${\displaystyle \varphi =\lnot (P\land Q)\to (\lnot R\land Q)}$

${\displaystyle \psi =(S\to P)\lor (S\to \lnot R)}$.

Then ${\displaystyle \varphi }$ tautologically implies ${\displaystyle \psi }$. This can be verified by writing ${\displaystyle \varphi }$ in conjunctive normal form:

${\displaystyle \varphi \equiv (P\lor \lnot R)\land Q}$.

Thus, if ${\displaystyle \varphi }$ holds, then ${\displaystyle P\lor \lnot R}$ holds. In turn, ${\displaystyle P\lor \lnot R}$ tautologically implies ${\displaystyle \psi }$. Because the two propositional variables occurring in ${\displaystyle P\lor \lnot R}$ occur in both ${\displaystyle \varphi }$ and ${\displaystyle \psi }$, this means that ${\displaystyle P\lor \lnot R}$ is an interpolant for the implication ${\displaystyle \varphi \to \psi }$.

Lyndon's interpolation theorem

Suppose that S and T are two first-order theories. As notation, let S ∪ T denote the smallest theory including both S and T; the signature of S ∪ T is the smallest one containing the signatures of S and T. Also let S ∩ T be the intersection of the languages of the two theories; the signature of S ∩ T is the intersection of the signatures of the two languages.

Lyndon's theorem says that if S ∪ T is unsatisfiable, then there is an interpolating sentence ρ in the language of S ∩ T that is true in all models of S and false in all models of T. Moreover, ρ has the stronger property that every relation symbol that has a positive occurrence in ρ has a positive occurrence in some formula of S and a negative occurrence in some formula of T, and every relation symbol with a negative occurrence in ρ has a negative occurrence in some formula of S and a positive occurrence in some formula of T.

Proof of Craig's interpolation theorem

We present here a constructive proof of the Craig interpolation theorem for propositional logic.^[3] Formally, the theorem states:

If ⊨φ → ψ then there is a ρ (the interpolant) such that ⊨φ → ρ and ⊨ρ → ψ, where atoms(ρ) ⊆ atoms(φ) ∩ atoms(ψ). Here atoms(φ) is the set of propositional variables occurring in φ, and ⊨ is the semantic entailment relation for propositional logic.

Proof. Assume ⊨φ → ψ. The proof proceeds by induction on the number of propositional variables occurring in φ that do not occur in ψ, denoted |atoms(φ) − atoms(ψ)|.

Base case |atoms(φ) − atoms(ψ)| = 0: Since |atoms(φ) − atoms(ψ)| = 0, we have that atoms(φ) ⊆ atoms(φ) ∩ atoms(ψ). Moreover we have that ⊨φ → φ and ⊨φ → ψ. This suffices to show that φ is a suitable interpolant in this case.

Let’s assume for the inductive step that the result has been shown for all χ where |atoms(χ) − atoms(ψ)| = n. Now assume that |atoms(φ) − atoms(ψ)| = n+1. Pick a q ∈ atoms(φ) but q ∉ atoms(ψ). Now define:

φ' := φ[⊤/q] ∨ φ[⊥/q]

Here φ[⊤/q] is the same as φ with every occurrence of q replaced by ⊤ and φ[⊥/q] similarly replaces q with ⊥. We may observe three things from this definition:

⊨φ' → ψ

(1)

|atoms(φ') − atoms(ψ)| = n

(2)

⊨φ → φ'

(3)

From (1), (2) and the inductive step we have that there is an interpolant ρ such that:

⊨φ' → ρ

(4)

⊨ρ → ψ

(5)

But from (3) and (4) we know that

⊨φ → ρ

(6)

Hence, ρ is a suitable interpolant for φ and ψ.

QED

Since the above proof is constructive, one may extract an algorithm for computing interpolants. Using this algorithm, if n = |atoms(φ') − atoms(ψ)|, then the interpolant ρ has O(exp(n)) more logical connectives than φ (see Big O Notation for details regarding this assertion). Similar constructive proofs may be provided for the basic modal logic K, intuitionistic logic and μ-calculus, with similar complexity measures.

Craig interpolation can be proved by other methods as well. However, these proofs are generally non-constructive:

• model-theoretically, via Robinson's joint consistency theorem: in the presence of compactness, negation and conjunction, Robinson's joint consistency theorem and Craig interpolation are equivalent.
• proof-theoretically, via a sequent calculus. If cut elimination is possible and as a result the subformula property holds, then Craig interpolation is provable via induction over the derivations.
• algebraically, using amalgamation theorems for the variety of algebras representing the logic.
• via translation to other logics enjoying Craig interpolation.

Applications

Craig interpolation has many applications, among them consistency proofs, model checking,^[4] proofs in modular specifications, modular ontologies.

References

Further reading

• John Harrison (2009). Handbook of Practical Logic and Automated Reasoning. Cambridge, New York: Cambridge University Press. ISBN 978-0-521-89957-4.
• Hinman, P. (2005). Fundamentals of Mathematical Logic. A K Peters. ISBN 1-56881-262-0.
• Dov M. Gabbay; Larisa Maksimova (2006). Interpolation and Definability: Modal and Intuitionistic Logics (Oxford Logic Guides). Oxford science publications, Clarendon Press. ISBN 978-0-19-851174-8.
• Eva Hoogland, Definability and Interpolation. Model-theoretic investigations. PhD thesis, Amsterdam 2001.
• W. Craig, Three uses of the Herbrand-Gentzen theorem in relating model theory and proof theory, The Journal of Symbolic Logic 22 (1957), no. 3, 269–285.