Regularity of 2CrevLIN

Prove or disprove that the class 2CrevLIN, consisting of all languages accepted by reversible complete 2-head finite automata (i.e., deterministic and backward deterministic 2-head automata that have exactly one defined transition for every nonempty input configuration), contains only regular languages.

Background

The paper introduces several classes of languages recognized by two-head finite automata with varying restrictions: 2detLIN (deterministic 2-head automata), 2revLIN (reversible deterministic 2-head automata), 2rev1LIN (1-limited reversible 2-head automata), and 2CrevLIN (complete reversible 2-head automata). They establish a strict hierarchy 2CrevLIN ⊂ 2rev1LIN ⊂ 2revLIN ⊂ 2detLIN and show structural constraints on complete reversible automata, including that completeness forces 1-limited behavior and strong connectedness.

They also show that certain nonregular languages cannot be accepted by complete reversible automata (e.g., L_{ba} = {b^n a^n, b^{n+1} a^n | n ≥ 0}), and that 2CrevLIN is closed under complementation and reversal. Motivated by these observations, the authors conjecture that 2CrevLIN may coincide with a subclass of the regular languages, and explicitly pose the conjecture that 2CrevLIN contains only regular languages.