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.
In propositional logic, let
Thus, if holds, then holds. In turn, tautologically implies . Because the two propositional variables occurring in occur in both and , this means that is an interpolant for the implication .
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
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
|atoms(φ') − atoms(ψ)| = n
⊨φ → φ'
⊨φ' → ρ
⊨ρ → ψ
⊨φ → ρ
Hence, ρ is a suitable interpolant for φ and ψ.
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.
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