[Paper Review] The $\Sigma$_1 Provability Logic of HA

Mohammad Ardeshir, Mojtaba Mojtahedi|arXiv (Cornell University)|Sep 19, 2014

Logic, Reasoning, and Knowledge13 references3 citations

TL;DR

This paper introduces and proves the soundness and completeness of a modal logic, i.eHσ, for Σ₁-interpretations in Heyting Arithmetic (HA), establishing it as the Σ₁-provability logic of HA. The logic is decidable, and as a consequence, HA + □⊥ is shown to have the de Jongh property, resolving a long-standing open problem in intuitionistic provability logic.

ABSTRACT

For the Heyting Arithmetic HA, HA* is defined as the theory $\\{A\\mid {\\sf HA}\\vdash A^{\\Box}\\}$, where $A^{\\Box}$ is called the box translation of $A$. We characterize the $\\Sigma_1$-provability logic of HA* as a modal theory ${\\sf iH}_\\sigma^*$.

Motivation & Objective

• To axiomatize the Σ₁-provability logic of Heyting Arithmetic (HA), which had remained an open problem since the 1970s.
• To define a modal theory iHσ that captures arithmetical Σ₁-interpretations in HA.
• To prove that iHσ is both sound and complete with respect to Σ₁-interpretations in HA.
• To establish the decidability of iHσ.
• To show that HA + □⊥ satisfies the de Jongh property as a corollary.

Proposed method

• The authors define a new modal logic iHσ based on TNNIL-formulae and the Solovay function, which maps modal formulas to arithmetical sentences.
• They use Kripke models of HA and the notion of q-realizability to analyze the behavior of modal formulas under arithmetical interpretations.
• The Solovay function is constructed to decide the truth of boxed formulas in HA, enabling the translation of modal validity into arithmetical provability.
• The proof relies on a hierarchy of Kripke models and the use of the extended Leivant’s principle to handle Σ₁-formulas.
• They apply the TNNIL and TNNIL⁻ algorithms to characterize the class of formulas closed under certain modal reductions, ensuring completeness.
• The decidability of iHσ is established via a bounded search over finite Kripke models and the use of the box translation to reduce modal validity to arithmetical provability.

Experimental results

Research questions

• RQ1 What is the correct modal logic that captures Σ₁-provability in Heyting Arithmetic?
• RQ2 Is there a decidable modal logic that is sound and complete for Σ₁-interpretations in HA?
• RQ3 Does HA + □⊥ satisfy the de Jongh property, i.e., does it validate the same propositional logic as its modal companion?
• RQ4 Can the provability logic of HA be fully axiomatized using a modal theory that respects the intuitionistic nature of HA?
• RQ5 How can Kripke models and realizability be combined to analyze Σ₁-formulas in HA?

Key findings

• The modal logic iHσ is sound and complete for Σ₁-interpretations in HA, meaning that a modal formula A is provable in iHσ if and only if its arithmetical interpretation A* is provable in HA.
• The logic iHσ is decidable, as the validity of any formula in iHσ can be determined by a finite search over Kripke models and the Solovay function.
• The paper proves that HA + □⊥ has the de Jongh property, meaning that the propositional logic of HA + □⊥ coincides with the logic of its modal companion.
• The extended Leivant’s principle is used to show that Σ₁-formulas in HA are preserved under certain interpretations, supporting the completeness result.
• The TNNIL⁻-algorithm is used to characterize the class of formulas closed under the modal reduction rules, which is essential for proving completeness.
• The Solovay function provides a uniform way to map modal formulas to arithmetical sentences such that provability in iHσ corresponds exactly to provability in HA.