On some Open Problems for Finite Automata with Translucent Input Letters

Abstract: Finite automata with translucent input letters are a recent model of discontinuous input processing. Basically, classical finite automata are equipped with a translucency function that defines, depending on the state, the set of translucent input symbols. While processing the input, translucent symbols are skipped and only visible symbols are read and processed. It is distinguished between deterministic and nondeterministic models and, in addition, between returning and non-returning models. In the former case, the automaton restarts from the left end of the input after having consumed some visible symbol, whereas in the latter case the automaton restarts from the left end of the input when the right endmarker symbol is seen. Returning finite automata with translucent letters have been introduced by Nagy and Otto and its non-returning variant has been introduced by Mraz and Otto. Many results concerning the computational capacity, relations between deterministic and nondeterministic models, and relations between returning and non-returning models are known. Moreover, some results on closure properties and decidability questions have been obtained as well. However, some questions have still been open since many years. In this paper, we will give answers to some of these open questions. In particular, we show the non-closure under concatenation, Kleene star, reversal, and inverse homomorphism for the non-returning deterministic as well as nondeterministic model. We also obtain non-closure under inverse homomorphism for the returning deterministic and nondeterministic model. Finally, we investigate the emptiness problem for non-returning finite automata with translucent input letters and show the decidability of the problem in case of deterministic as well as nondeterministic automata.

• The paper demonstrates non-closure under concatenation, Kleene star, reversal, inverse homomorphism, and complementation for FA-TL models.
• It employs explicit constructions and state-counting arguments to show how translucent input disrupts classical regular language properties.
• The decidability of emptiness for non-returning FA-TL is established via reduction to regular language emptiness using iterated finite-state transducers.

Open Problems and Closure Properties of Finite Automata with Translucent Input Letters

Introduction and Motivation

Finite automata with translucent input letters (FA-TL) extend classical automata models by incorporating a discontinuous input reading paradigm controlled by a translucency function, which specifies, statewise, input symbols to be considered invisible. As such, FA-TL skip over designated symbols, only processing visible ones, leading to distinctive computational behaviors in both deterministic (DFA-TL) and nondeterministic (NFA-TL) forms, as well as returning and non-returning variants. This model, originally proposed by Nagy and Otto, has been investigated for its computational capabilities, closure properties, and decidability of pivotal problems. Despite significant progress, key closure properties and decision problems remained unresolved; this work settles several of those open questions, specifically regarding non-closure under various language operations and the emptiness problem for non-returning variants.

Technical Background

In FA-TL, the translucency function $\tau : Q \to 2^\Sigma$ designates, for each state $q \in Q$, the set of translucent (invisible) input symbols. Computation advances by reading the next visible (non-translucent) symbol, potentially skipping over translucent ones. Non-returning automata continue scanning forward until hitting the $\lhd$ endmarker, whereas returning automata reset to the input's left end after processing a visible symbol.

Formally, a non-returning nondeterministic FA-TL (nrNFAwtl) is described by $$M = \langle Q, \Sigma, q_0, \lhd, \tau, \delta \rangle$$, with transitions dependent upon visible symbols in the current state. Deterministic versions (nrDFAwtl, DFAwtl) restrict the transition relation accordingly.

Main Results

Non-Closure under Fundamental Language Operations

The authors address several unresolved closure properties for both deterministic and nondeterministic, returning and non-returning FA-TL models:

• Non-Closure Under Concatenation and Kleene Star: The families $L(\mathrm{nrDFAwtl})$ and $L(\mathrm{nrNFAwtl})$ are demonstrated to be non-closed under concatenation and the Kleene star operation. An explicit witness language construction, leveraging state-counting and looping arguments, shows that concatenation and iterative composition of languages accepted by such automata can yield languages unrecognizable by FA-TL models of the same class.
• Non-Closure Under Reversal: Both $L(\mathrm{nrDFAwtl})$ and $L(\mathrm{nrNFAwtl})$ fail to be closed under word reversal. The paper provides a construction of an nrDFAwtl for a language $L$ and demonstrates that $L^R$ (the reversal) is not accepted by any nrNFAwtl, using a two-phase verification approach to encode the non-closure.
• Non-Closure Under Inverse Homomorphism: The work proves that none of the four families—DFAwtl, nrDFAwtl, NFAwtl, nrNFAwtl—are closed under inverse non-erasing homomorphisms. This is shown constructively via homomorphic inverse images that yield non-regular languages lacking letter-equivalent regular sublanguages, in direct contrast to known properties of FA-TL language families.
• Non-Closure Under Complement: Specifically for $q \in Q$0, non-closure under complementation is established via De Morgan's laws and the previously shown non-closure under intersection.

Decidability of the Emptiness Problem

A comprehensive treatment is provided for the emptiness problem for non-returning FA-TL, previously open for non-returning DFAwtl/NFAwtl. The approach utilizes iterated finite-state transducers (IFST), aligning the computation of FA-TL with bounded-sweep IFSTs, for which the accepted language is always regular when sweeps are bounded. The authors establish that any accepted word must be recognized within a number of sweeps bounded by the automaton's state space, enabling an effective reduction to a regular language emptiness problem. Thus, emptiness for both non-returning deterministic and nondeterministic FA-TL is decidable.

Discussion and Implications

The results presented substantially delimit the algebraic robustness of FA-TL language families, in stark contrast to classical regular languages. The systematic demonstration of non-closure for basic regular operations—concatenation, star, reversal, inverse homomorphism, and complement—indicates that translucent input skipping fundamentally disrupts the compositional properties central to regular languages.

From a theoretical perspective, these findings refine the landscape of automata with non-standard reading, clarifying the expressive boundaries and operation invariants, and highlighting sensitivities to input manipulation and symbol transformation. Practically, these non-closure results caution against directly deploying FA-TL for language specification in contexts where standard closure properties (useful in parsing, specification, or code analysis) are required.

The reduction of the emptiness problem to regular language emptiness, via simulation with iterated finite-state transducers, is notable. This not only establishes a technical foundation for decision procedures in this model but may offer a template for handling other decidability questions in automata with unconventional input processing. Furthermore, the proof techniques involving careful state- and sweep-tracking could inspire analogous strategies for variants and extensions, such as FA-TL with two-way motion or devices equipped with auxiliary storage.

Future Directions

The research highlights several avenues for further investigation. The closure of $q \in Q$1 under reversal remains unresolved. Potential areas of exploration include:

• The interplay of translucency with other non-classical extensions (e.g., pushdown or queue automata).
• Quantitative complexity measures (state and jump complexity) and their algorithmic implications.
• Effective minimization and equivalence testing algorithms within the FA-TL paradigm.
• Extending the theory to infinite words and other generalized automata settings.

Conclusion

This work resolves longstanding open questions regarding closure and decidability properties for finite automata with translucent input letters. The established non-closure results sharply differentiate the FA-TL model's algebraic behavior from classical automata, while the decidability of emptiness for non-returning variants enables further algorithmic and practical exploration. These insights contribute to a deeper understanding of discontinuous input processing mechanisms and their computational consequences within automata theory.

Reference: "On some Open Problems for Finite Automata with Translucent Input Letters" (2606.26683).