Effective Completeness Theorem

• Effective Completeness is a refined theorem that defines a computable transformation from semantically valid formulas to constructible proofs using formal systems like Coq.
• It employs dovetailed enumerations of candidate proofs and models to systematically derive a formal proof for every valid formula.
• The framework integrates classical principles such as Markov’s Principle and Weak König’s Lemma while extending to modal, continuous, and algebraic logics.

The Effective Completeness Theorem is a refined form of the classical completeness theorem that asserts, not only that every valid sentence is syntactically derivable, but that there exists an explicit, computable transformation from semantic validity to a concrete proof object within a given formal system. This transformation is implemented as an algorithm or program, potentially within a proof assistant or type theory, yielding both theoretical and practical consequences. Multiple completed developments—mechanized in Coq, formulated in intuitionistic and classical settings, or extended to continuous and modal logics—demonstrate the depth, technical variety, and foundational implications of effective completeness in mathematical logic (Forster et al., 2020).

1. Formal Statement and Scope in Constructive Type Theory

The effective completeness theorem, as formalized in the Forster–Kirst–Wehr Coq library, is stated for first-order logic over a countable, decidable signature. The central object is a computable function: $f : \mathsf{Form}_L \to \mathsf{Proof}$ such that for every closed formula $\varphi$,

$$\models \varphi \implies \vdash f(\varphi) : \varphi.$$

In Coq, this is realized by a dependent pair:

Definition completeness_effective : { f : formula → proof | ∀ φ, valid φ → yields_proof (f φ) φ }.

Here, valid φ is defined as $$\forall M : \mathrm{Tarski}\text{-model},\; M \vDash \varphi$$; yields_proof d φ means $d$ is a closed proof-term deriving $\varphi$. The emphasis is on providing a witnessing function $f$ which, given a formula valid in all Tarski models, returns a closed formal proof (Forster et al., 2020).

2. Computability-Theoretic Mechanism and Proof Search

The constructive extraction of proofs relies on two mutually dovetailed enumerations:

• Proof-search: enumerate all candidate closed proof terms in the sequent calculus by size, leveraging the enumeration of the proof type as both enumerable and with decidable equality.
• Model-search: enumerate all finite partial Tarski structures; a countermodel search for $\varphi$ proceeds by dovetailing over these.

If $\varphi$ is not valid, the process will eventually uncover a countermodel. If $\varphi$ is valid, the model-search never halts, but the infinite, computable proof-search tree must eventually yield a correct proof-term by a constructive variant of the Henkin–Lindenbaum construction. This realizes a computable map from semantic validity to syntactic derivation (Forster et al., 2020).

3. Dependence on Classical Principles: Markov’s Principle and WKL

In the standard interpretation via Tarski semantics, essential computational steps use either Markov's Principle (MP) or Weak König's Lemma (WKL):

• MP is required to extract a witness from the stability of existential statements stating "no proof exists": $\neg\neg(\exists n.\,P(n)) \to \exists n.\,P(n)$ for computably enumerable predicates (Forster et al., 2020).
• Alternatively, WKL is invoked to extract an infinite branch from a decidable infinite tree, corresponding to the existence of a Henkin construction yielding a genuine proof-term.

In contrast, for algebraic semantics (e.g., complete Heyting/Boolean algebras) or dialogue-game semantics, the completeness proofs produce actual proofs constructively—no use of MP or WKL is necessary; the realiser can be directly constructed, e.g., via normalization by evaluation (Forster et al., 2020).

4. Data-theoretic and Complexity Aspects

Key computability properties supporting the construction include:

• The data types representing formulas and proofs are enumerable with decidable equality.
• For each finite context $\Gamma$ and formula $\psi$, the sequent judgment $\Gamma \vdash \psi$ is enumerable (Fact enumT_form).
• Proof terms can be enumerated as $nth\_proof : \mathbb{N} \to \mathsf{option~proof}$ such that $$\forall d, \text{yields\_proof}\;d\;\varphi \iff \exists n, nth\_proof~n = Some\;d$$.

However, the running time of $f$ is non-elementary in the worst case, matching the nonelementary complexity of full first-order proof search. From the Coq development, $f$ can be extracted into the call-by-value $\lambda$-calculus, becoming an executable program, a fact that underlines its constructive content (Forster et al., 2020).

5. Variants, Applications, and Extensions

The effective completeness methodology generalizes beyond basic first-order logic:

• Algebraic and dialogue semantics: yield constructive completeness theorems (no MP or WKL), with the mapping $f$ given by explicit term extraction.
• Continuous logic and metric structures: generalized the effective completeness theorem further, showing that for a continuous theory $T$ with a computable name, there is a computable presentation of a model, and the Turing degree of the presentation matches the degree of $T$ (Camrud, 2022).
• Modal logics (e.g., S4.3.1): effective completeness is achieved via specialized, linear Kripke frame constructions, producing decidable discrete linear $\omega$-models (Nichols, 2017).
• Propositional calculi and $\omega$-rule: constructive Kalmár-style procedures, and primitive recursive codings for $\omega$-proofs in arithmetic, yield effective completeness in broader settings (Santos, 2023, Frittaion, 2021).

6. Constructive Content and Metamathematical Insights

The constructive internalization of the completeness theorem, i.e., extracting a program from the proof, clarifies the relation between semantic and syntactic notions of validity. The Coq development of Forster–Kirst–Wehr demonstrates that, by tracking the computable enumerations of formulas, proofs, and finite models, explicit completeness procedures are fully mechanizable. The reliance (or avoidance) of classical principles such as MP and WKL is transparent at each step, delineating which variants of effective completeness are available in intuitionistic versus classical settings, and under which semantics (Forster et al., 2020).

The effective completeness theorem thus not only reinforces the connection between model theory and proof theory but also provides a computational foundation for automated reasoning and mechanized mathematics, supporting executable extraction and formal verification of logical entailment.