arXiv subjects

Atanas Ilchev

Publications and source records attributed to Atanas Ilchev.

3 recordsLinked to original sources

arXiv2026

Context-Free Fixed Points and Complete Classification of Orbits in Picard Iteration for Guarded Power Language Operators

We study the language-theoretic structure of fixed points and finite Picard iterates for guarded q-power language operators. For the general operator, we prove that a context-free seeded language deter- mines a unique context-free fixed point and give an effective construction of a context-free grammar generating this fixed point, independently of the initial language.We then consider a marked single-guard special case in which two new symbols separate the recursive contribution from the the contribution of the seed. In this setting, the initial language can be traced and recovered exactly from every finite Picard iterate by means of a regular slice and fixed-word quotients. This yields injectivity of the finite-time maps and an abstract finite-time class-preservation prin- ciple. As a consequence, we obtain a classification of the finite Picard iterates according to the exact position of the initial language in the Chomsky hierarchy. Thus the exact language-theoretic complexity may persist at every finite stage, while all Picard orbits converge to the same context-free fixed point.

cs.FL↗

arXiv2026

Banach Spaces Generated by Finite-Valued Functions: Superreflexive Rigidity, Hankel Operators, and Universality

We investigate Banach spaces generated by uniformly bounded families of functions taking values in a fixed finite set and establish a rigidity principle connecting the cardinality of the generating family with the geometry of its closed linear span. We prove that such a space is superreflexive precisely when the generating family is finite, or equivalently, when the resulting Banach space is finite-dimensional. The main ingredient is a finite-range spreading-model obstruction showing that an infinite family of finite-valued functions cannot generate a superreflexive space under the supremum norm. This principle is applied to Banach spaces generated by the characteristic functions of the left derivatives of formal languages. It yields geometric characterizations of regular languages in terms of finite dimensionality, superreflexivity, and the existence of an equivalent uniformly convex norm. We also obtain a canonical representation of the associated language space as a coordinate-function subspace of a space of continuous functions on a compact shift-orbit closure. The corresponding language Hankel operator is shown to be compact exactly for regular languages. In the nonregular case, we determine its exact distance from both the compact and finite-rank operators and compute all its nontrivial approximation numbers. Finally, we construct a single binary language whose associated Banach space contains an isometric copy of every separable real Banach space. These results reveal a sharp contrast between the geometric rigidity associated with regular languages and the universality that may occur in the nonregular setting.

math.FA↗

arXiv2026

Ultrametric Convergence of Guarded Automata and Applications to Structural Input Validation

We equip language-equivalence classes of deterministic finite automata with a distinguishing-word ultrametric and identify the resulting space isometrically with the regular languages. This space is incomplete, while its metric completion is naturally identified with the complete ultrametric space of all formal languages. Guarded language operators induce contractions on the automaton space, and their Picard iterates converge in the completion to the unique language fixed point, which is represented by a finite automaton exactly when it is regular. Motivated by structural input validation, we use this framework to construct depth-capped deterministic finite automata with certified finite-depth correctness. These automata provide efficient pre-filters for nested input structures, such as parenthesised SQL parameters, while avoiding the backtracking risks of regular-expression engines and the runtime overhead of full context-free parsers. We also outline a practical WAF pipeline combining learned grammar models, finite-state construction, and \(O(1)\)-memory runtime validation.

cs.FL↗