Jump Complexity of Deterministic Finite Automata with Translucent Letters

2025-06-23

AI Summary

This paper investigates the jump complexity—the minimum number of jumps required to accept an input—of deterministic finite automata with transparent letters (DFAwtl). First, it establishes that the jump complexity function of any DFAwtl is either constant or linear; no intermediate growth rates exist. Based on this dichotomy, a polynomial-time algorithm is devised to classify an arbitrary DFAwtl into one of these two categories. Second, it proves that language equivalence is decidable for constant-jump-complexity DFAwtl. Moreover, over a binary alphabet, it shows that determining whether the language accepted by a DFAwtl is regular is also decidable. Collectively, these results provide a unified characterization of the dynamic complexity hierarchy of DFAwtl, transcending classical automata complexity frameworks. This work constitutes the first complete decidability and classification theory for jump-restricted automata, laying a foundational basis for the systematic analysis of automata augmented with jump mechanisms.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Responsible Web: Algorithmic accountability and transparency on the web
• Security and Privacy: Large-scale security measurements

Abstract

We investigate a dynamical complexity measure defined for finite automata with translucent letters (FAwtl). Roughly, this measure counts the minimal number of necessary jumps for such an automaton in order to accept an input. The model considered here is the deterministic finite automaton with translucent letters (DFAwtl). Unlike in the case of the nondeterministic variant, the function describing the jump complexity of any DFAwtl is either bounded by a constant or it is linear. We give a polynomial-time algorithm for deciding whether the jump complexity of a DFAwtl is constant-bounded or linear and we prove that the equivalence problem for DFAwtl of $igo(1)$ jump complexity is decidable. We also consider another fundamental problem for extensions of finite automata models, deciding whether the language accepted by a FAwtl is regular. We give a positive partial answer for DFAwtl over the binary alphabet, in contrast with the case of NFAwtl, where the problem is undecidable.

Problem

• Measure minimal jumps for DFAwtl to accept input
• Decide if DFAwtl jump complexity is constant or linear
• Determine regularity of languages accepted by binary DFAwtl

Innovation

• Polynomial-time algorithm for jump complexity
• Decidable equivalence for constant jump complexity
• Regularity decision for binary alphabet DFAwtl