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arxiv: 2606.31887 · v1 · pith:ET6MLTN6new · submitted 2026-06-30 · 💻 cs.LO

Uniform Interpolation of Basic Tense Logic

Katsuhiko Sano (Hokkaido University) This is my paper

Pith reviewed 2026-07-01 02:32 UTC · model grok-4.3

classification 💻 cs.LO
keywords uniform interpolationbasic tense logicmodal logiclayered bisimulationconverse modalityKripke semanticsinterpolation theorem
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The pith

Basic tense logic has the uniform interpolation property.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper proves that basic tense logic possesses the uniform interpolation property. Basic tense logic adds a converse modality to basic modal logic, with the accessibility relation defined as the reverse in Kripke models. The proof extends Visser's layered bisimulation argument to cover both directions of the relation. If the extension succeeds, the property holds for this logic introduced by Prior. A reader would care because it supplies the missing case for two-way modal logics.

Core claim

This paper establishes the uniform interpolation theorem for basic tense logic by extending Visser's semantic argument formulated in terms of layered bisimulation to the converse accessibility relation.

What carries the argument

Layered bisimulation construction adapted to the converse accessibility relation in Kripke models.

If this is right

  • Basic tense logic satisfies uniform interpolation.
  • The layered bisimulation method generalizes from one-way modal logic to two-way variants.
  • No extra frame conditions are imposed by the result.
  • The uniform interpolation property is preserved under addition of the converse operator.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same adaptation might apply to other operators in temporal logics.
  • This could support algorithms that compute uniform interpolants for tense logic formulas.
  • Comparisons of interpolation strength become possible across modal logics with and without converse.

Load-bearing premise

Visser's layered bisimulation construction extends directly to the converse accessibility relation without requiring additional semantic conditions or adjustments.

What would settle it

A concrete formula in basic tense logic that lacks any uniform interpolant in the common variables would disprove the theorem.

Figures

Figures reproduced from arXiv: 2606.31887 by Katsuhiko Sano (Hokkaido University).

Figure 1
Figure 1. Figure 1: A model M in the proof of Lemma 7.8, with reflexive loops and transitive compositions omitted. The following result extends Ghilardi and Zawadowski’s [23] theorem on the failure of the uniform interpolation property in S4 to a tense-logical setting. Our proof builds upon the one given in [43, Theorem 16.4], which provides an alternative proof of their theorem. The model used in our proof (see [PITH_FULL_I… view at source ↗
read the original abstract

This paper establishes the uniform interpolation theorem for basic tense logic, which is also known as two-way modal logic or modal logic with converse. First introduced by Arthur Prior, basic tense logic is a syntactic expansion of basic modal logic with a converse modality. Its corresponding accessibility relation is defined as the converse of the standard accessibility relation in a given Kripke model. Although basic tense logic has been widely studied since its introduction, its uniform interpolation property has yet to be fully established. For basic modal logic K, Albert Visser (1996) provided a semantic argument formulated in terms of layered (or bounded) bisimulation, explicitly attributing the uniform interpolation property of K to Silvio Ghilardi. This paper extends Visser's semantic argument to demonstrate that basic tense logic also enjoys the uniform interpolation property.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 2 minor

Summary. The paper claims to establish the uniform interpolation theorem for basic tense logic (K_t) by extending Visser's 1996 layered-bisimulation semantic argument from basic modal logic K, taking the accessibility relation to be bidirectional to accommodate the converse (past) modality introduced by Prior.

Significance. If the extension holds, the result fills a documented gap for tense logic and shows that the layered-bisimulation technique scales symmetrically to two-way models without new semantic conditions or parameters. The argument is presented as a direct adaptation of an existing, non-circular construction.

major comments (1)
  1. [proof of the main theorem (around the extension of the layered bisimulation)] The central claim rests on the assertion that Visser's construction extends directly once accessibility is made bidirectional. The manuscript should include an explicit check (perhaps in the proof of the main theorem) that the bisimulation clauses for the converse modality preserve the layered depth bounds and the interpolation extraction step; without this verification the adaptation step remains unexamined even if the one-directional case is sound.
minor comments (2)
  1. [Introduction / abstract] The abstract and introduction cite Visser (1996) and Ghilardi but do not give a precise pointer to the theorem or section of Visser's paper that is being adapted; adding the reference would aid readers.
  2. [preliminaries] Notation for the two accessibility relations (R and R^{-1}) and the corresponding modalities should be introduced once and used consistently; occasional shifts between 'future'/'past' and 'forward'/'backward' reduce clarity.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the detailed reading and the helpful suggestion regarding the proof presentation. We address the comment below and will revise accordingly.

read point-by-point responses

  1. Referee: [proof of the main theorem (around the extension of the layered bisimulation)] The central claim rests on the assertion that Visser's construction extends directly once accessibility is made bidirectional. The manuscript should include an explicit check (perhaps in the proof of the main theorem) that the bisimulation clauses for the converse modality preserve the layered depth bounds and the interpolation extraction step; without this verification the adaptation step remains unexamined even if the one-directional case is sound.

    Authors: We agree that an explicit verification would make the adaptation clearer. In the revised manuscript we will insert, immediately after the statement of the main theorem, a short paragraph that checks the two required properties: (i) that the additional bisimulation clause for the converse modality respects the layered depth bound (by symmetry with the forward clause), and (ii) that the interpolation extraction step carries over verbatim once the accessibility relation is treated as bidirectional. The argument relies only on the fact that the layered bisimulation relation is defined symmetrically for both directions and that the depth bound is preserved under the converse relation; no new semantic parameters are introduced. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation extends external semantic argument

full rationale

The paper's central result is an extension of Visser's 1996 layered-bisimulation argument (explicitly attributed to Ghilardi) from modal logic K to basic tense logic K_t. The construction is described as carrying over once the accessibility relation is treated as bidirectional, with no equations, parameters, or definitions that reduce the target theorem to the paper's own inputs by construction. No self-citations appear; the cited sources are independent prior work. The argument is therefore self-contained against the external benchmark of Visser's construction and does not exhibit any of the enumerated circularity patterns.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The claim rests on standard Kripke semantics for tense logic and the validity of extending the bisimulation argument; no new free parameters or invented entities are introduced.

axioms (1)
  • domain assumption Kripke models with converse accessibility relation for the tense modalities
    The paper assumes the standard semantic definition where the accessibility for the converse modality is the inverse of the forward relation.

pith-pipeline@v0.9.1-grok · 5652 in / 1072 out tokens · 56558 ms · 2026-07-01T02:32:22.744421+00:00 · methodology

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