[Paper Review] Constructive Provability Logic

Robert J. Simmons, Bernardo Toninho|arXiv (Cornell University)|May 29, 2012

Logic, programming, and type systems19 references3 citations

TL;DR

This paper introduces constructive provability logic (CPL and CPL*), an intuitionistic modal logic that validates Gödel-Löb's provability rule through logical reflection over provability. It provides equivalent natural deduction and sequent calculus formulations, establishes proof-theoretic justification for stratified negation in logic programming, and formalizes all results in Agda.

ABSTRACT

We present constructive provability logic, an intuitionstic modal logic that validates the Löb rule of Gödel and Löb's provability logic by permitting logical reflection over provability. Two distinct variants of this logic, CPL and CPL*, are presented in natural deduction and sequent calculus forms which are then shown to be equivalent. In addition, we discuss the use of constructive provability logic to justify stratified negation in logic programming within an intuitionstic and structural proof theory.

Motivation & Objective

• To develop a constructive, intuitionistic modal logic that validates the Löb rule and Gödel-Löb's provability logic within a proof-theoretic framework.
• To provide a structural proof theory for stratified negation in logic programming, addressing its non-compositional nature in classical semantics.
• To establish a formal connection between constructive provability logic and logic programming, particularly for forward-chaining and locally stratified negation.
• To present two equivalent formulations—CPL (tethered) and CPL* (de-tethered)—in both natural deduction and sequent calculus forms.
• To lay the groundwork for extending constructive provability logic to first-order logic and higher-order logic programming languages like λProlog and Twelf.

Proposed method

• Formalizing constructive provability logic using a judgmental methodology with explicit worlds and reflection over provability.
• Introducing two variants: CPL (tethered) and CPL* (de-tethered), each with natural deduction and sequent calculus presentations.
• Proving equivalence between natural deduction and sequent calculus formulations for both CPL and CPL* at the level of provability.
• Using focused proof search (uniform proofs) to justify stratified negation in logic programming, avoiding classical truth-value assignments.
• Applying the logic to model bottom-up logic programming with stratified negation, where negation is based on absence of proof in a saturated database.
• Formalizing all theorems and derivations in the Agda proof assistant to ensure correctness and mechanization.

Experimental results

Research questions

• RQ1 How can a constructive, intuitionistic modal logic be designed to validate the Löb rule and capture provability in a proof-theoretic way?
• RQ2 What are the structural and proof-theoretic properties of constructive provability logic, and how do its two variants (CPL and CPL*) differ?
• RQ3 Can constructive provability logic provide a principled, structural justification for stratified negation in logic programming, avoiding classical truth-value semantics?
• RQ4 How can the logic be extended to support first-order quantification and higher-order logic programming?
• RQ5 What is the relationship between constructive provability logic and classical provability logic (e.g., GL), and how do their proof-theoretic presentations compare?

Key findings

• CPL and CPL* are two equivalent formulations of constructive provability logic in both natural deduction and sequent calculus forms, with the latter being de-tethered and more flexible.
• The logic validates the Löb rule and supports logical reflection over provability, enabling a constructive interpretation of provability in intuitionistic settings.
• The paper provides the first focused sequent calculus for a constructive provability logic, which is novel in the literature.
• The logic offers a proof-theoretic foundation for stratified negation in logic programming, avoiding the non-compositional limitations of classical stratification.
• All results are formally verified in Agda, demonstrating the feasibility of mechanized reasoning in this logic.
• The work opens pathways for integrating constructive provability logic into advanced logic programming systems, including those supporting distributed and higher-order computation.