# Craig interpolation

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 ST denote the smallest theory including both S and T; the signature of ST is the smallest one containing the signatures of S and T. Also let ST be the intersection of the languages of the two theories; the signature of ST is the intersection of the signatures of the two languages.

Lyndon's theorem says that if ST is unsatisfiable, then there is an interpolating sentence ρ in the language of ST 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 qatoms(φ) but qatoms(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:

## Applications

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

## References

1. ^ Lyndon, Roger (1959), "An interpolation theorem in the predicate calculus", Pacific Journal of Mathematics, 9: 129–142, doi:10.2140/pjm.1959.9.129.
2. ^ Troelstra, Anne Sjerp; Schwichtenberg, Helmut (2000), Basic Proof Theory, Cambridge tracts in theoretical computer science, vol. 43 (2nd ed.), Cambridge University Press, p. 141, ISBN 978-0-521-77911-1.
3. ^ Harrison pgs. 426–427
4. ^ Vizel, Y.; Weissenbacher, G.; Malik, S. (2015). "Boolean Satisfiability Solvers and Their Applications in Model Checking". Proceedings of the IEEE. 103 (11): 2021–2035. doi:10.1109/JPROC.2015.2455034. S2CID 10190144.