Idefix-Closed Languages Overview

• Idefix-closed languages are three regular language families (infix-closed, prefix-closed, and suffix-closed) defined by taking infixes, prefixes, or suffixes of accepted words.
• The structural analysis reveals proper, asymmetric containments among these families, distinguishing their behavior within subregular hierarchies.
• They are applied as control mechanisms in external and internal contextual grammars, enhancing the precision of language generation and automata complexity studies.

Idefix-closed languages are the three regular-language families of infix-closed, prefix-closed, and suffix-closed languages, denoted $INF$, $PRE$, and $SUF$. In the terminology of "Idefix-Closed Languages and Their Application in Contextual Grammars" (Ködding et al., 21 Jul 2025), these families are grouped under a single name because they share the same structural pattern: closure under taking infixes, prefixes, or suffixes of words already in the language. The grouping is not merely terminological. It places three classical closure properties into a common subregular hierarchy and uses them as control families for external and internal contextual grammars, where they generate distinct language families and refine earlier hierarchy results.

1. Definition and formal family structure

Let $L$ be a language over an alphabet $V$. The paper introducing the term defines the idefix-closed families as follows (Ködding et al., 21 Jul 2025).

Family	Notation	Defining condition
Infix-closed	$INF$	$xyz\in L \implies y\in L$, equivalently $$L = Inf(L) := \{\,y \mid xyz\in L \text{ for some } x,z\in V^*\,\}$$
Prefix-closed	$PRE$	$xy\in L \implies x\in L$, equivalently $PRE$0
Suffix-closed	$PRE$1	$PRE$2, equivalently $PRE$3

These are closure conditions on factors of accepted words. The paper also notes that if $PRE$4 is regular, then $PRE$5 and $PRE$6 are regular, and that the idefix-closed families are all subfamilies of $PRE$7. In the contextual-grammar applications developed there, the relevant instances are regular selection languages.

The collective label “idefix-closed” is therefore not a new closure operator on its own; it is a uniform designation for three established classes. Its main value is organizational: it allows comparison of infix-, prefix-, and suffix-based closure within one hierarchy and within one grammar-theoretic framework.

2. Relations among $PRE$8, $PRE$9, and $SUF$0

The internal structure of the idefix-closed families is asymmetric. The principal containments proved in the hierarchy are

$SUF$1

and the paper shows that all displayed inclusions are proper (Ködding et al., 21 Jul 2025).

Properness is established by explicit witnesses. The language

$SUF$2

witnesses $SUF$3. The language

$SUF$4

lies in $SUF$5, while

$SUF$6

lies in $SUF$7. Hence

$SUF$8

This asymmetry is one of the central structural facts about idefix-closed languages. Infix-closure sits below both prefix-closure and suffix-closure, but prefix-closure and suffix-closure do not collapse to one another. A plausible implication is that any grammar or automaton construction sensitive to the direction of context addition can distinguish $SUF$9-controlled and $L$0-controlled behavior even when both remain within the same “idefix-closed” umbrella.

3. Position in the subregular landscape

The 2025 hierarchy places idefix-closed languages between simple regular families and the larger power-separating family. In particular,

$L$1

and these inclusions are proper (Ködding et al., 21 Jul 2025). One witness used for $L$2 is

$L$3

which is presented in the paper as an element of $L$4, and therefore also outside $L$5 while still lying in the larger hierarchy above it.

The comparison with other subregular families is mostly by incomparability rather than containment. The paper proves that every family in the following groups is incomparable to both $L$6 and $L$7: $L$8

$L$9

$V$0

$V$1

$V$2

Representative witnesses are given explicitly. For example,

$V$3

is used to separate $V$4 from $V$5, while

$V$6

separates $V$7 from $V$8. The language

$V$9

lies in $INF$0, and

$INF$1

lies in $INF$2.

These comparisons show that idefix-closed languages do not simply replicate familiar “low-complexity” regular subclasses. They form a separate axis in the subregular lattice, with strong closure behavior but without the algebraic simplifications that characterize classes such as definite, non-counting, circular, or star languages.

4. Contextual grammars as the main application setting

The main application in (Ködding et al., 21 Jul 2025) is to contextual grammars with selection languages. A contextual grammar with selection in a family $INF$3 is a triple

$INF$4

where $INF$5 is an alphabet, $INF$6 is a finite set of selection pairs $INF$7, $INF$8 is the selection language, $INF$9 is a finite set of contexts, and $xyz\in L \implies y\in L$0 is the axiom set.

The paper considers two derivation modes. In an external contextual grammar, a word $xyz\in L \implies y\in L$1 derives $xyz\in L \implies y\in L$2 if $xyz\in L \implies y\in L$3 and

$xyz\in L \implies y\in L$4

for some $xyz\in L \implies y\in L$5. In an internal contextual grammar, a word $xyz\in L \implies y\in L$6 derives $xyz\in L \implies y\in L$7 if $xyz\in L \implies y\in L$8, $xyz\in L \implies y\in L$9, and

$$L = Inf(L) := \{\,y \mid xyz\in L \text{ for some } x,z\in V^*\,\}$$0

A key structural principle is monotonicity: $$L = Inf(L) := \{\,y \mid xyz\in L \text{ for some } x,z\in V^*\,\}$$1 and the paper proves this for both external and internal contextual grammars (Ködding et al., 21 Jul 2025). This principle lifts containments between base language families directly to containments between the corresponding generated-language families, making the idefix-closed hierarchy usable as a control hierarchy for grammar formalisms.

5. External and internal hierarchy results

For external contextual grammars, the idefix-closed families insert new layers into the generated-language hierarchy. By monotonicity and the base containments,

$$L = Inf(L) := \{\,y \mid xyz\in L \text{ for some } x,z\in V^*\,\}$$2

hold in the external setting (Ködding et al., 21 Jul 2025). The paper further states that external contextual grammars with idefix-closed selection languages form distinct families with different generative power: $$L = Inf(L) := \{\,y \mid xyz\in L \text{ for some } x,z\in V^*\,\}$$3 is strictly larger than $$L = Inf(L) := \{\,y \mid xyz\in L \text{ for some } x,z\in V^*\,\}$$4, $$L = Inf(L) := \{\,y \mid xyz\in L \text{ for some } x,z\in V^*\,\}$$5 and $$L = Inf(L) := \{\,y \mid xyz\in L \text{ for some } x,z\in V^*\,\}$$6 are incomparable, both are strictly below $$L = Inf(L) := \{\,y \mid xyz\in L \text{ for some } x,z\in V^*\,\}$$7, and many standard generated-language families such as those arising from $$L = Inf(L) := \{\,y \mid xyz\in L \text{ for some } x,z\in V^*\,\}$$8, $$L = Inf(L) := \{\,y \mid xyz\in L \text{ for some } x,z\in V^*\,\}$$9, $PRE$0, $PRE$1, $PRE$2, and $PRE$3 remain incomparable with $PRE$4 and $PRE$5.

For internal contextual grammars, the same broad pattern reappears: $PRE$6 again together with a broad incomparability pattern (Ködding et al., 21 Jul 2025). The paper identifies its main new application result here: it solves an open problem concerning internal contextual grammars with suffix-closed selection languages. The question was whether $PRE$7 is incomparable to $PRE$8 or $PRE$9, or whether it is a subset of one of them. The new hierarchy settles this issue by locating $xy\in L \implies x\in L$0 relative to the surrounding subregular control families.

The significance of these grammar-theoretic results is that idefix-closed selection languages do not merely provide alternative presentations of already known grammar classes. They produce new control classes and sharpen the global picture of what contextual grammars can generate under structurally constrained selection.

6. The suffix-closed branch in automata and complexity theory

Among the three idefix-closed branches, suffix-closed languages have an especially developed automata-theoretic complexity profile. In "Complexity of Left-Ideal, Suffix-Closed and Suffix-Free Regular Languages" (Brzozowski et al., 2016), suffix-closed languages are treated as one of the three special subclasses of suffix-convex regular languages, alongside left ideals and suffix-free languages. The paper emphasizes the complement duality: if $xy\in L \implies x\in L$1, then $xy\in L \implies x\in L$2 is a left ideal. Using this duality, it constructs a most-complex witness stream of suffix-closed languages and proves, for $xy\in L \implies x\in L$3, bounds including syntactic semigroup size

$xy\in L \implies x\in L$4

reversal complexity

$xy\in L \implies x\in L$5

exactly

$xy\in L \implies x\in L$6

atoms, star complexity

$xy\in L \implies x\in L$7

with

$xy\in L \implies x\in L$8

restricted product complexity

$xy\in L \implies x\in L$9

unrestricted product complexity

$PRE$00

and restricted and unrestricted Boolean-operation complexities matching the regular-language upper bounds in the forms stated there.

The earlier paper "Syntactic Complexity of Ideal and Closed Languages" (Brzozowski et al., 2010) places suffix-closed languages in the ideal/closed duality for regular languages. It studies closed languages as complements of ideal languages and presents the suffix-closed case as the complement class of left ideals. Its key quantitative conclusion for this branch is that there exist left ideals and suffix-closed languages of syntactic complexity

$PRE$01

and it formulates

$PRE$02

as the conjectured exact worst-case bound for suffix-closed languages, with exact tightness proved for small $PRE$03 and explicit extremal automata reaching the bound.

These results concern only one member of the idefix-closed triad, but they show that at least the suffix-closed branch is rich enough to support a detailed algebraic and state-complexity theory independent of the contextual-grammar motivation.

7. Terminological scope and neighboring notions

The term idefix-closed should not be conflated with idefix-free. In "Idefix-Free Languages and Their Application in External Contextual Grammars" (Ködding et al., 25 Jun 2026), idefix-free languages are the three families of prefix-free, suffix-free, and infix-free languages. That paper explicitly contrasts them with the closed counterparts $PRE$04, $PRE$05, and $PRE$06, and shows that the free and closed families are largely incomparable. The contrast is particularly sharp in the grammar setting: for external contextual grammars with infix-free selection languages,

$PRE$07

whereas prefix-free and suffix-free selection languages still allow infinite generated languages and remain distinct from the generated-language families arising from prefix-, suffix-, or infix-closed selection.

The word closed also has unrelated meanings elsewhere in formal-language theory. In "Strategical languages of infinite words" (Arfi et al., 2010), a language $PRE$08 is closed in the prefix topology on infinite words, and the paper proves

$PRE$09

Its canonical generating strategy is

$PRE$10

By contrast, "Closures in Formal Languages: Concatenation, Separation, and Algorithms" (0901.3763) uses positive-closed to mean

$PRE$11

that is, closure under nonempty concatenation. These are different notions from idefix-closedness, even though the same adjective appears.

Accordingly, “idefix-closed languages” is best understood as a domain-specific collective name for the three regular families $PRE$12, $PRE$13, and $PRE$14, rather than as a universal notion of closedness across formal-language research. Within that specific domain, the term captures a coherent trio of closure properties, a nontrivial subregular hierarchy, and a productive interface with contextual grammar theory (Ködding et al., 21 Jul 2025).