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arxiv: 2604.25501 · v1 · submitted 2026-04-28 · 💻 cs.LO

Recognition: unknown

Proof Identity and Categorical Models of BV

Matteo Acclavio , Lutz Stra{\ss}burger , Vladimir Zamdzhiev
Authors on Pith no claims yet

Pith reviewed 2026-05-07 14:42 UTC · model grok-4.3

classification 💻 cs.LO
keywords proof identityBV logicatomic flowsBV-categoriescategorical semanticsstring diagramssoundness
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The pith

Atomic flows define proof identity for BV and strengthen BV-categories to make them sound.

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

The paper addresses the unclear relation between BV-categories and the logic BV caused by missing coherence results and proof identity. It introduces a definition of proof identity using atomic flows, which function as special string diagrams. This definition supports a strengthened version of BV-categories whose soundness with respect to BV is then established. A reader would care because it supplies a concrete way to decide when two BV proofs count as the same.

Core claim

We define a notion of proof identity for BV, based on the notion of atomic flows, which can be seen as a special form of string diagrams. Based on this notion of proof identity, we then strengthen the existing notion of BV-category and prove that it is sound with respect to the logic.

What carries the argument

Atomic flows as special string diagrams that serve as the standard for identifying BV proofs.

If this is right

  • Strengthened BV-categories now correspond soundly to actual BV proofs.
  • Proof manipulations in BV can be performed directly on atomic-flow diagrams.
  • The categorical semantics of BV becomes usable for reasoning about proof equivalence.
  • String-diagram techniques become available for studying non-commutative substructural logics.

Where Pith is reading between the lines

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

  • The same atomic-flow technique could supply proof identity for other logics that currently lack coherence theorems.
  • Graphical rewriting rules derived from atomic flows might yield a practical proof system for BV.
  • The approach suggests that string diagrams can handle the mix of commutative and non-commutative structure in BV without collapse.

Load-bearing premise

Atomic flows capture exactly the intended proof equivalences in BV and introduce no extra identifications or omissions.

What would settle it

Exhibit two BV proofs that are semantically distinct yet receive the same atomic flow, or two proofs that should be identical yet receive different atomic flows.

Figures

Figures reproduced from arXiv: 2604.25501 by Lutz Stra{\ss}burger, Matteo Acclavio, Vladimir Zamdzhiev.

Figure 2
Figure 2. Figure 2: Derivations to prove Proposition 3.3 and the two directions of Proposition 4.2 ▶ Proposition 4. Let 𝐴 and 𝐵 be formulas. Then, 1. ⊢SBV 𝐴 iff ⊢BV∪{i↑} 𝐴, and 2. 𝐴⊢SBV 𝐵 iff ⊢SBV 𝐴 ⊥ O 𝐵. Proof. The first statement follows immediately from Proposition 3. To prove left-to-right (resp. right-to-left) implication in the second statement, we assume the existence of a derivation Φ with premise 𝐴 and conclusion 𝐵 … view at source ↗
read the original abstract

BV-categories are a recent development that aims to give categorical semantics to proofs in the logic BV. However, due to the absence of a coherence theorem on one side and a well-defined notion of proof identity for BV on the other side, the precise relation between BV-categories and the logic BV is still not clear. To improve on this situation, we define in this paper a notion of proof identity for BV, based on the notion of atomic flows, which can be seen as a special form of string diagrams. Based on this notion of proof identity, we then strengthen the existing notion of BV-category and prove that it is sound with respect to the logic.

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 defines a notion of proof identity for the logic BV using atomic flows (viewed as a special case of string diagrams). It then strengthens the existing definition of BV-categories to respect this identity and proves that the resulting semantics is sound with respect to BV.

Significance. If the soundness argument holds and the atomic-flow identity aligns with the intended proof equivalences in BV, the work clarifies the relationship between BV-categories and the logic, addressing the prior absence of both a coherence theorem and a well-defined proof identity. This strengthens categorical semantics for substructural logics with non-commutative and non-associative connectives.

major comments (1)
  1. The soundness claim rests on the atomic-flow construction and its interpretation as string diagrams, but the provided abstract and high-level description do not include the explicit inductive definition or the key lemmas establishing that the flows respect the BV rules without introducing extraneous identifications. A concrete walk-through of the construction (e.g., how atomic flows are generated from BV proofs and how the equivalence is shown to be a congruence) would be needed to confirm there are no post-hoc choices.
minor comments (2)
  1. Notation for atomic flows and string diagrams should be introduced with a small illustrative example early in the paper to aid readers unfamiliar with the technique.
  2. The introduction mentions the lack of a coherence theorem for BV-categories; a brief statement of what the strengthened definition achieves (or does not achieve) toward coherence would help situate the contribution.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the positive recommendation for minor revision. The comment identifies an opportunity to improve the accessibility of the atomic-flow construction, and we have revised the paper to include additional explicit details and an example.

read point-by-point responses

  1. Referee: The soundness claim rests on the atomic-flow construction and its interpretation as string diagrams, but the provided abstract and high-level description do not include the explicit inductive definition or the key lemmas establishing that the flows respect the BV rules without introducing extraneous identifications. A concrete walk-through of the construction (e.g., how atomic flows are generated from BV proofs and how the equivalence is shown to be a congruence) would be needed to confirm there are no post-hoc choices.

    Authors: We appreciate this feedback on presentation. The full manuscript already contains the explicit inductive definition of atomic flows (Section 3.1), generated from BV proofs by induction on the structure of the sequent derivation, together with the key lemmas establishing that the induced equivalence is a congruence (Lemma 3.7) and respects all BV rules without extraneous identifications (Theorem 4.3). The string-diagram interpretation is developed in Section 2.2. Nevertheless, we agree that a concrete walk-through would make the construction more transparent at the high level. In the revised version we have inserted a new subsection (3.3) containing a detailed, step-by-step example that starts from a concrete BV proof using the non-commutative and non-associative connectives, shows the corresponding atomic flow, and verifies that the equivalence relation is preserved under the BV rules. This addition directly addresses the concern about post-hoc choices by rendering the entire process fully explicit and verifiable from the text. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained

full rationale

The paper introduces a new definition of proof identity for BV via atomic flows (treated as string diagrams), then strengthens the BV-category notion and proves soundness with respect to the logic. This follows a standard definitional-then-soundness pattern in categorical proof theory with no load-bearing self-citations, no fitted parameters renamed as predictions, and no reduction of the central claims to prior inputs by construction. The approach is independent and externally verifiable through the provided definitions and proofs.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the assumption that atomic flows correctly capture proof identity in BV and that the strengthened BV-categories are faithful models; no free parameters or invented entities are mentioned in the abstract.

axioms (1)
  • standard math Standard background results in proof theory and category theory for BV and string diagrams
    Invoked implicitly when defining atomic flows and BV-categories

pith-pipeline@v0.9.0 · 5403 in / 1036 out tokens · 49146 ms · 2026-05-07T14:42:10.098649+00:00 · methodology

discussion (0)

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Reference graph

Works this paper leans on

5 extracted references · 1 canonical work pages

  1. [1]

    Probabilistic coherence spaces as a model of higher-order probabilistic computation

    URL: https://www.sciencedirect.com/science/article/pii/S0890540111000411, doi:10.1016/j.ic.2011.02.001. 16 Vincent Danos and Laurent Regnier. The structure of multiplicatives.Archive for Mathematical Logic, 28(3):181–203, 1989.doi:10.1007/BF01622878. 17 Edward G. Effros and Zhong-Jin Ruan. Operator space tensor products and Hopf convolution algebras.Journ...

    work page doi:10.1016/j.ic.2011.02.001 2011

  2. [2]

    If there is a derivation Φ BV (𝐴 𝐵)O𝐾 , then there are formulas𝐾 𝐴 and 𝐾𝐵 and derivations 𝐾 𝐴O𝐾 𝐵 Φ𝐾 BV 𝐾 and Φ𝐴 BV 𝐴O𝐾 𝐴 and Φ𝐵 BV 𝐵O𝐾 𝐵 , s.t.AF ©­­­­­­­­­ « Φ𝐴 𝐴O𝐾 𝐴 ! Φ𝐵 𝐵O𝐾 𝐵 ! s (𝐴 𝐵)O ©­­ « 𝐾 𝐴O𝐾 𝐵 Φ𝐾 𝐾 ª®® ¬ ª®®®®®®®®® ¬ =AF (Φ)

  3. [3]

    If there is a derivation Φ BV (𝐴 ⊳ 𝐵)O𝐾 , then there are formulas𝐾 𝐴 and 𝐾𝐵 and derivations 𝐾 𝐴 ⊳ 𝐾 𝐵 Φ𝐾 BV 𝐾 and Φ𝐴 BV 𝐴O𝐾 𝐴 and Φ𝐵 BV 𝐵O𝐾 𝐵 , s.t.AF ©­­­­­­­­­ « Φ𝐴 𝐴O𝐾 𝐴 ! ⊳ Φ𝐵 𝐵O𝐾 𝐵 ! q↓ (𝐴 ⊳ 𝐵)O ©­­ « 𝐾 𝐴 ⊳ 𝐾 𝐵 Φ𝐾 𝐾 ª®® ¬ ª®®®®®®®®® ¬ =AF (Φ) 3.If there is a derivation Φ BV 𝑎O𝐾 , then there is a formula𝐾and a derivation 𝑎⊥ Φ𝑎 BV 𝐾 , s.t.AF ©­­­­­­ « ...

    2026

  4. [4]

    If there is a derivation Φ BV (𝐴 ⊳ 𝐵)O𝐾 , then we conclude similarly to the previous case. The only difference is that in this case we have to consider the ruleq↓ instead of the rules, whose atomic flow is not an identity flow, but it is constructed by identities and crossings. 3.If there is a derivation Φ BV 𝑎O𝐾 , then we have two cases: either the last ...

    2026

  5. [5]

    ◀ FSCD 2026

    we see that the forgetful functor𝑈:Caus[CP ∗ [FHilbsk]] →CP ∗ [FHilbsk] preserves all the required structure strictly and satisfies Definition 17 (5.) thus giving us a strong BV-category. ◀ FSCD 2026

    2026