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arxiv: 2605.03628 · v1 · submitted 2026-05-05 · 💻 cs.LO

Recognition: unknown

Induction rules for Transition Algebra

Go Hashimoto

Pith reviewed 2026-05-07 13:27 UTC · model grok-4.3

classification 💻 cs.LO
keywords Transition AlgebraSequent calculusInduction ruleCompactnessCraig interpolationRewriting systemsProof systemsCompleteness
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The pith

Restricting the Star rule to induction creates a compact sequent proof system for Transition Algebra that is complete under a new semantics and yields a model-theoretic proof of Craig interpolation.

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

Transition Algebra is an infinite logic for reasoning about rewriting systems. Earlier natural deduction systems were complete only for countable signatures yet failed to be compact, limiting their use. The paper replaces natural deduction with a sequent calculus in which the Star rule is restricted to an induction principle. A matching semantics is then defined so that the restricted system becomes complete. This complete system is finally applied to derive Craig interpolation in model-theoretic terms.

Core claim

By restricting the (Star) rule to induction and adopting a sequent proof system instead of natural deduction, we obtain a compact proof system for Transition Algebra. We define an appropriate semantics that renders this system complete. Using this complete system, we provide a model-theoretic proof of the Craig interpolation property.

What carries the argument

The induction-restricted (Star) rule inside the sequent calculus for Transition Algebra, together with the semantics that establishes its completeness.

If this is right

  • The resulting proof system is compact, allowing finite derivations for properties that previously required infinite proofs.
  • Completeness holds with respect to the introduced semantics for all signatures.
  • Craig interpolation is established by a direct model-theoretic argument rather than syntactic methods.
  • The system becomes usable for practical analysis of rewriting systems where compactness matters.

Where Pith is reading between the lines

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

  • The same restriction technique could be tested on other non-compact infinite logics to recover compactness without losing core reasoning power.
  • Model-theoretic interpolation proofs may support new algorithms for checking properties of infinite transition systems.
  • Rewriting-system tools might now incorporate sequent-based provers built on this induction rule.
  • The completeness semantics could be compared directly with existing coalgebraic or categorical semantics for transitions to check agreement on finite models.

Load-bearing premise

Restricting the Star rule to induction preserves enough of the original expressive power to still capture rewriting systems while permitting a semantics under which the sequent system is complete.

What would settle it

A concrete Transition Algebra formula that holds in the new semantics yet is unprovable in the restricted sequent calculus, or a rewriting system in which Craig interpolation fails despite the new proof system.

read the original abstract

Transition Algebra (TA) is a type of infinite logic introduced to discuss rewriting systems. The natural deductive proof systems already introduced in TA satisfy completeness for countable signatures. However, it lacks compactness, making it unsuitable for practical applications. This time, we will create a compact proof system by restricting the (Star) rule to induction. We will also use a sequent proof system instead of natural deduction. Furthermore, we will introduce a semantics that makes this system complete, and its application (a model-theoretic proof of Craig interpolation).

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

2 major / 1 minor

Summary. Transition Algebra (TA) is an infinite logic for rewriting systems whose prior natural-deduction systems are complete for countable signatures but lack compactness. The manuscript proposes a compact sequent calculus obtained by restricting the (Star) rule to induction, introduces a new semantics under which this system is claimed to be complete, and applies the result to a model-theoretic proof of Craig interpolation.

Significance. If the restricted induction rule preserves the essential expressive power of TA while delivering compactness and completeness, the work would supply a practically usable proof system for rewriting logics and a new model-theoretic route to interpolation. The shift to sequent calculus could also improve proof-search properties.

major comments (2)
  1. Abstract: the claim that restricting the (Star) rule to induction yields a compact system that remains complete under the new semantics is asserted without any rule definitions, semantic clauses, or proof sketch; the manuscript therefore supplies no verification that the restriction preserves the theorems of the original system.
  2. Abstract: the model-theoretic proof of Craig interpolation is announced but no interpolation theorem statement, semantic construction, or derivation is supplied, so the load-bearing application cannot be assessed.
minor comments (1)
  1. The abstract is written entirely in future tense ('we will create', 'we will introduce'); for a journal submission this should be revised to present-tense description of completed results.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful review and the recommendation of major revision. We agree that the abstract is too brief and does not supply the supporting definitions or proof outlines needed to verify the central claims. The revised manuscript will incorporate the sequent calculus rules, semantic clauses, completeness argument, interpolation theorem statement, and model-theoretic construction. We address each major comment below.

read point-by-point responses

  1. Referee: Abstract: the claim that restricting the (Star) rule to induction yields a compact system that remains complete under the new semantics is asserted without any rule definitions, semantic clauses, or proof sketch; the manuscript therefore supplies no verification that the restriction preserves the theorems of the original system.

    Authors: The referee is correct that the submitted version asserts the compactness and completeness results in the abstract without exhibiting the restricted (Star) rule, the new semantic clauses, or any completeness argument. In the revision we will add the formal definition of the induction-restricted sequent calculus, the clauses of the new semantics, and a proof sketch establishing that every sequent derivable in the original system remains derivable (or is semantically valid) under the restricted rule. This will explicitly verify preservation of the original theorems while securing compactness. revision: yes

  2. Referee: Abstract: the model-theoretic proof of Craig interpolation is announced but no interpolation theorem statement, semantic construction, or derivation is supplied, so the load-bearing application cannot be assessed.

    Authors: We acknowledge that the current manuscript announces the interpolation application without stating the theorem, describing the semantic construction, or outlining the derivation. The revised version will contain the precise statement of Craig interpolation for Transition Algebra under the new semantics, the model-theoretic construction used to prove it, and a sketch of the derivation that relies on the completeness result. This will make the application fully assessable. revision: yes

Circularity Check

0 steps flagged

No significant circularity in the derivation chain

full rationale

The paper's abstract outlines a plan to restrict the (Star) rule to induction for compactness, adopt sequent calculus, and introduce an independent semantics to establish completeness with an application to Craig interpolation. No equations, rule definitions, or self-citations are supplied that would reduce the completeness result or the semantics to a definitional tautology, a fitted parameter renamed as prediction, or a self-referential chain. The derivation is therefore self-contained against external benchmarks, with the new semantics serving as independent support rather than a constructed equivalence to the inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract provides no explicit free parameters, axioms, or invented entities; the contribution centers on rule restrictions and semantics definitions.

pith-pipeline@v0.9.0 · 5360 in / 948 out tokens · 47006 ms · 2026-05-07T13:27:05.851549+00:00 · methodology

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Reference graph

Works this paper leans on

9 extracted references · 3 canonical work pages

  1. [1]

    51st International Colloquium on Automata, Languages, and Programming,

    Go Hashimoto and Daniel G. 51st International Colloquium on Automata, Languages, and Programming,. 2024 , doi =

    2024

  2. [2]

    Logical foundations of

    R. Logical foundations of. Theoretical Computer Science , volume =. 2002 , pages =

    2002

  3. [3]

    2007 , timestamp =

    All About Maude -- A High-Performance Logical Framework, How to Specify, Program and Verify Systems in Rewriting Logic , series =. 2007 , timestamp =. doi:10.1007/978-3-540-71999-1 , _bib2doi_selected =

    work page doi:10.1007/978-3-540-71999-1 2007

  4. [4]

    50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025) , pages =

    Hashimoto, Go and G. 50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025) , pages =. 2025 , volume =

    2025

  5. [5]

    2026 , eprint=

    Forcing and Interpolation in First-Order Hybrid Logic with rigid symbols , author=. 2026 , eprint=

    2026

  6. [6]

    The Journal of Symbolic Logic , volume =

    Andrzej Tarlecki , title =. The Journal of Symbolic Logic , volume =. 2024 , doi =

    2024

  7. [7]

    Institution-Independent Model Theory , series =

    R. Institution-Independent Model Theory , series =. 2025 , _bib2doi_finished =

    2025

  8. [8]

    Journal of Logical and Algebraic Methods in Programming , volume =

    The. Journal of Logical and Algebraic Methods in Programming , volume =. 2023 , issn =. doi:10.1016/j.jlamp.2023.100887 , author =

    work page doi:10.1016/j.jlamp.2023.100887 2023

  9. [9]

    Cut-elimination for a logic with definitions and induction , journal =

    Raymond McDowell and Dale Miller , keywords =. Cut-elimination for a logic with definitions and induction , journal =. 2000 , issn =. doi:https://doi.org/10.1016/S0304-3975(99)00171-1 , url =

    work page doi:10.1016/s0304-3975(99)00171-1 2000