A sequent calculus with procedure calls

In this paper, we extend the sequent calculus LKF into a calculus LK(T), allowing calls to a decision procedure. We prove cut-elimination of LK(T).

💡 Research Summary

The paper introduces LK(T), an extension of the focused sequent calculus LKF that incorporates calls to an external decision procedure T. The motivation stems from the observation that modern automated reasoning tools often need to combine pure logical inference with domain‑specific theory reasoning (e.g., arithmetic, arrays, bit‑vectors). While LKF already offers a powerful focusing discipline that dramatically reduces proof search space, it is limited to pure propositional or first‑order logic. To bridge this gap, the authors add a new inference rule, called Proc‑Call, which allows a sequent containing a set of “theory atoms” to be handed over to a decision procedure. The procedure returns either a satisfiability answer together with a model (or a witness) or a proof of unsatisfiability. The rule then updates the sequent accordingly: if the atoms are satisfiable, the atoms are removed and the model information may be used to instantiate variables; if they are unsatisfiable, the branch is closed immediately.

The syntax of LK(T) mirrors that of LKF: sequents are of the form ⊢ Γ , Δ, where Γ and Δ are multisets of formulas, but a distinguished subset of formulas is identified as theory atoms. The logical rules (positive/negative focusing, initial, structural) are unchanged, preserving the strong normalization properties of LKF. The crucial addition is the Proc‑Call rule, which is parameterised by a black‑box decision procedure T that satisfies three meta‑properties: determinism (the same input yields the same output), completeness (if the set of atoms is satisfiable, T will answer “sat”), and soundness (the model returned by T indeed satisfies the atoms). Under these assumptions the authors prove soundness (every LK(T) proof corresponds to a valid derivation in the underlying logic) and completeness (any valid entailment can be proved in LK(T)).

The central technical contribution is a cut‑elimination theorem for LK(T). The proof proceeds in two stages. First, cuts that involve only logical rules are eliminated using the standard nested cut‑reduction technique already known for LKF. Second, cuts that involve the Proc‑Call rule require a novel transformation called Procedure Expansion. This transformation extracts the model (or refutation) produced by T, translates it back into logical formulas, and replaces the Proc‑Call occurrence with a purely logical derivation that simulates the effect of the external call. If T reports unsatisfiable, the cut collapses to a contradiction and is removed; if T reports satisfiable, the model is used to instantiate variables, thereby eliminating the need for the external call. By showing that each application of Procedure Expansion strictly reduces a well‑founded measure on proofs, the authors obtain a terminating reduction sequence that yields a cut‑free proof.

Beyond the meta‑theoretical results, the paper discusses implementation considerations. Because focusing is preserved, existing LKF‑based proof search engines can be reused with minimal changes. The Proc‑Call rule can be realised as an API call to modern SMT solvers such as Z3 or CVC4. The model information returned by T can be attached as metadata to proof objects, enabling efficient reconstruction of the logical steps during proof checking. The authors also outline future work, including handling nondeterministic procedures, integrating cost models for procedure calls to guide proof search, and extending the framework to support combinations of multiple theories via a meta‑theory layer.

In summary, the paper provides a rigorous foundation for integrating external decision procedures into a focused sequent calculus. By defining LK(T), proving its soundness, completeness, and cut‑elimination, the authors demonstrate that logical reasoning and theory reasoning can be combined modularly without sacrificing the desirable proof‑theoretic properties of LKF. This work paves the way for more expressive and efficient automated theorem provers that can seamlessly invoke specialized solvers while maintaining a clean, verifiable proof calculus.