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; the overall result is sometimes called the Craig–Lyndon theorem.

Example

In propositional logic, let

$\phi =\mathrm{\neg }(P\wedge Q)\to (\mathrm{\neg }R\wedge Q)$

$\psi =(S\to P)\vee (S\to \mathrm{\neg }R)$.

Then $\phi$ tautologically implies $\psi$. This can be verified by writing $\phi$ in conjunctive normal form:

$\phi \equiv (P\vee \mathrm{\neg }R)\wedge Q$.

Thus, if $\phi$ holds, then $P\vee \mathrm{\neg }R$ holds.

$\rho =(P\vee \mathrm{\neg }R)$.

In turn, $P\vee \mathrm{\neg }R$ tautologically implies $\psi$. Because the two propositional variables occurring in $P\vee \mathrm{\neg }R$ occur in both $\phi$ and $\psi$, this means that $P\vee \mathrm{\neg }R$ is an interpolant for the implication $\phi \to \psi$.

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.

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, proofs in modular specifications, modular ontologies.

Further reading

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• 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.