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arxiv: 2606.26682 · v1 · pith:GSZGBWFT · submitted 2026-06-25 · cs.FL

Idefix-Free Languages and Their Application in External Contextual Grammars

Marvin K\"odding (PH Heidelberg) , Bianca Truthe (Universit\"at Giessen) This is my paper

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classification cs.FL
keywords contextual grammarsidefix-free languagessubregular languagesexternal contextual grammarslanguage hierarchiesprefix-free languagessuffix-free languagesinfix-free languages
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The pith

External contextual grammars that select derivations with idefix-free languages generate new families that extend known hierarchies of subregular languages.

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

The paper studies infix-free, prefix-free, and suffix-free languages, grouped as idefix-free languages, and uses them as selection mechanisms inside external contextual grammars. It places the resulting language families beside previously studied subregular classes such as finite, monoidal, nilpotent, definite, and star languages. The central move is to locate these new families inside the existing hierarchy diagrams for external contextual grammars and thereby add fresh entries. A reader cares because the comparison shows how a single restriction on the selector changes the generative capacity in a way that is neither trivial nor already covered by earlier restrictions. The work therefore refines the map of how limited regular selectors control the power of contextual grammars.

Core claim

By investigating infix-, prefix-, and suffix-free languages as selection languages for external contextual grammars and comparing the families they generate to the families obtained from finite, monoidal, nilpotent, combinational, definite, ordered, non-counting, power-separating, commutative, circular, union-free, star, and comet languages, the paper obtains new language families that can be inserted into the existing hierarchies, thereby extending those hierarchies.

What carries the argument

idefix-free selection languages inside external contextual grammars, which restrict the context-selection step while still producing families that sit at new positions relative to other subregular selectors.

If this is right

  • The idefix-free families sit strictly between some of the listed subregular families and the full regular languages in the hierarchy.
  • New inclusion or incomparability relations appear among the families generated by external contextual grammars.
  • The generative power of external contextual grammars is shown to be sensitive to the precise free-language restriction placed on the selector.
  • The hierarchies become finer, with additional layers between the previously known levels.

Where Pith is reading between the lines

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

  • The same idefix-free selectors could be tested inside internal contextual grammars to see whether the same extensions appear.
  • Decidability or complexity questions for membership in these new families might be settled by reduction to the corresponding questions for the selector languages.
  • The comparison technique used here could be applied to other named subregular classes not yet examined as selectors.

Load-bearing premise

The idefix-free families must be distinct from the already-listed subregular families and must occupy non-trivial new positions inside the hierarchy of external contextual grammars.

What would settle it

An explicit proof or counter-example showing that every language generated by an external contextual grammar with an idefix-free selector is already generated by one of the previously studied subregular selectors.

Figures

Figures reproduced from arXiv: 2606.26682 by Bianca Truthe (Universit\"at Giessen), Marvin K\"odding (PH Heidelberg).

Figure 1
Figure 1. Figure 1: Resulting hierarchy of subregular language families. [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Resulting hierarchy of language families by external contextual grammars with special selec [PITH_FULL_IMAGE:figures/full_fig_p010_2.png] view at source ↗
read the original abstract

In this paper, we continue the research on the power of contextual grammars with selection languages from subfamilies of the family of regular languages. We investigate infix-, prefix-, and suffix-free languages (referred to as idefix-free languages) and compare such language families to some other subregular families of languages (finite, monoidal, nilpotent, combinational, (symmetric) definite, ordered, non-counting, power-separating, commutative, circular, union-free, star, and comet languages). Further, we compare the families of the hierarchies obtained for external contextual grammars with the language families defined by these new types for the selection. In this way, we extend the existing hierarchies by new language families.

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 / 0 minor

Summary. The manuscript continues research on contextual grammars using subregular selection languages by defining and studying infix-, prefix-, and suffix-free languages (collectively called idefix-free languages). It compares the resulting families to fourteen other subregular families (finite, monoidal, nilpotent, combinational, (symmetric) definite, ordered, non-counting, power-separating, commutative, circular, union-free, star, and comet languages) and then inserts the new families into the hierarchies of languages generated by external contextual grammars, claiming to extend those hierarchies.

Significance. If the comparisons establish that at least one idefix-free family is strictly incomparable to or properly contained in the previously studied subregular classes, thereby inserting new nodes into the external contextual grammar hierarchy, the result would add concrete new families to the known classification. The paper states that comparisons are performed, which, if backed by explicit separating examples, would constitute a non-trivial extension.

major comments (1)
  1. Abstract: the claim that the idefix-free families extend the existing hierarchies by producing new positions requires demonstration that the three new families are distinct from the fourteen listed subregular families in a non-trivial way (proper inclusion or incomparability). The abstract asserts that comparisons are performed, yet the load-bearing step is the provision of witness languages or counter-examples separating the new families; without these, the claimed extension could collapse onto already-known positions in the hierarchy.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the detailed report and the opportunity to clarify the manuscript. We address the major comment point by point below.

read point-by-point responses

  1. Referee: [—] Abstract: the claim that the idefix-free families extend the existing hierarchies by producing new positions requires demonstration that the three new families are distinct from the fourteen listed subregular families in a non-trivial way (proper inclusion or incomparability). The abstract asserts that comparisons are performed, yet the load-bearing step is the provision of witness languages or counter-examples separating the new families; without these, the claimed extension could collapse onto already-known positions in the hierarchy.

    Authors: We agree that explicit separating examples are essential to substantiate the claimed extensions. Sections 3 and 4 of the manuscript contain the required comparisons: for each idefix-free family we provide concrete witness languages demonstrating both proper inclusions and incomparabilities with the fourteen listed subregular families (finite, monoidal, nilpotent, etc.). These witnesses are then used in Section 5 to insert the new families at distinct positions within the external contextual grammar hierarchies. The abstract's reference to performed comparisons is therefore supported by these explicit constructions. Should the referee consider the presentation of any particular witness insufficiently prominent, we can add a dedicated summary table in a revision. revision: no

Circularity Check

0 steps flagged

No circularity; standard family comparisons without self-referential reductions

full rationale

The paper extends hierarchies of external contextual grammars by comparing infix-, prefix-, and suffix-free (idefix-free) selection languages against fourteen listed subregular families. All steps rely on standard definitions from prior literature and explicit inclusion/incomparability arguments; no equations, fitted parameters renamed as predictions, or load-bearing self-citations appear in the provided text. The central claims rest on external definitions and (presumably) witness languages or proofs that are independent of the present work, rendering the derivation self-contained.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Abstract-only review limits visibility into assumptions; no free parameters or invented entities mentioned. Relies on standard definitions of regular language subfamilies and contextual grammar operations from prior work.

axioms (1)
  • standard math Standard definitions and closure properties of regular language subfamilies hold as background.
    Invoked implicitly when comparing families; common in formal language theory.

pith-pipeline@v0.9.1-grok · 5664 in / 1033 out tokens · 21601 ms · 2026-06-26T02:03:42.719003+00:00 · methodology

discussion (0)

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Reference graph

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