[Paper Review] On provability logics with linearly ordered modalities

Lev D. Beklemishev, David Fernández–Duque+1 more|arXiv (Cornell University)|Oct 17, 2012

Logic, Reasoning, and Knowledge15 references4 citations

TL;DR

This paper introduces the provability logic ${\sf GLP}_{\Lambda}$ for any linearly ordered set $\Lambda$, generalizing Japaridze's logic ${\sf GLP}_{\omega}$. It provides a finitary reduction of ${\sf GLP}_{\Lambda}$ to ${\sf GLP}_{\omega}$, yielding a constructive proof of irreflexivity of the ordering $<_{0}$ on modal words and establishing decidability and a restricted axiomatization for the variable-free fragment of ${\sf GLP}_{\Lambda}$.

ABSTRACT

We introduce the logics GLP(Λ), a generalization of Japaridze's polymodal provability logic GLP(ω) where Λis any linearly ordered set representing a hierarchy of provability operators of increasing strength. We shall provide a reduction of these logics to GLP(ω) yielding among other things a finitary proof of the normal form theorem for the variable-free fragment of GLP(Λ) and the decidability of GLP(Λ) for recursive orderings Λ. Further, we give a restricted axiomatization of the variable-free fragment of GLP(Λ).

Motivation & Objective

• To generalize Japaridze's polymodal provability logic ${\sf GLP}_{\omega}$ to arbitrary linearly ordered index sets $\Lambda$, enabling analysis of stronger proof-theoretic systems.
• To provide a finitary, purely modal proof of irreflexivity of the ordering $<_{0}$ on modal words in ${\sf GLP}_{\Lambda}$, avoiding reliance on soundness assumptions or complex semantics.
• To establish the decidability of ${\sf GLP}_{\Lambda}$ for recursive orderings $\Lambda$ using a reduction to ${\sf GLP}_{\omega}$.
• To give a more restricted and streamlined axiomatization of the variable-free fragment of ${\sf GLP}_{\Lambda}$ than previously known.
• To extend the framework to non-well-ordered linear orderings $\Lambda$, broadening applicability to recursive progressions of theories.

Proposed method

• A reduction technique is developed to translate formulas in ${\sf GLP}_{\Lambda}$ into equivalent formulas in ${\sf GLP}_{\omega}$, preserving provability and enabling finitary reasoning.
• The proof of irreflexivity of $<_{0}$ is established via this reduction, leveraging known finitary proofs in ${\sf GLP}_{\omega}$.
• A positive-style normal form theorem is formulated for variable-free formulas in ${\sf GLP}_{\Lambda}$, avoiding reliance on irreflexivity in certain derivations.
• The logic ${\sf GLP}_{\Lambda}$ is shown to be conservative over its restrictions to subsets of modalities, using the reduction method.
• The proof strategy avoids topological or Kripke models, instead using syntactic manipulations and modal distribution laws.
• The restricted axiomatization is derived by refining the normal form construction and eliminating redundant axioms from prior systems.

Experimental results

Research questions

• RQ1 Can the irreflexivity of the ordering $<_{0}$ on modal words in ${\sf GLP}_{\Lambda}$ be proven using finitary, purely modal methods rather than semantic or arithmetical assumptions?
• RQ2 Is the variable-free fragment of ${\sf GLP}_{\Lambda}$ decidable for recursive orderings $\Lambda$, and can this be shown via reduction to ${\sf GLP}_{\omega}$?
• RQ3 Can a more restricted and streamlined axiomatization be given for the variable-free fragment of ${\sf GLP}_{\Lambda}$ than previously known?
• RQ4 How can the framework of ${\sf GLP}_{\Lambda}$ be extended to non-well-ordered linear orderings $\Lambda$ while preserving key proof-theoretic properties?
• RQ5 To what extent can the normal form theorem for ${\sf GLP}_{\Lambda}$ be re-derived in a positive, constructive way that avoids assumptions like irreflexivity?

Key findings

• The ordering $<_{0}$ on modal words in ${\sf GLP}_{\Lambda}$ is irreflexive, and this is proven via a finitary reduction to ${\sf GLP}_{\omega}$, avoiding reliance on soundness or semantics.
• The variable-free fragment of ${\sf GLP}_{\Lambda}$ is decidable for recursive orderings $\Lambda$, as a consequence of the reduction to ${\sf GLP}_{\omega}$.
• The logic ${\sf GLP}_{\Lambda}$ is conservative over any of its restrictions to subsets of modalities, meaning no new theorems are provable in the full logic beyond those derivable from its fragments.
• A restricted axiomatization of the variable-free fragment of ${\sf GLP}_{\Lambda}$ is provided, improving upon earlier systems by eliminating redundant axioms.
• The normal form theorem for variable-free formulas in ${\sf GLP}_{\Lambda}$ is re-derived in a positive way, which simplifies derivations and removes unnecessary assumptions.
• The results are established in a general setting where $\Lambda$ is any linearly ordered set, not necessarily well-ordered, broadening the scope of the framework for proof-theoretic applications.