Two-Way Automata Making Choices Only at the Endmarkers

The question of the state-size cost for simulation of two-way nondeterministic automata (2NFAs) by two-way deterministic automata (2DFAs) was raised in 1978 and, despite many attempts, it is still open. Subsequently, the problem was attacked by restricting the power of 2DFAs (e.g., using a restricted input head movement) to the degree for which it was already possible to derive some exponential gaps between the weaker model and the standard 2NFAs. Here we use an opposite approach, increasing the power of 2DFAs to the degree for which it is still possible to obtain a subexponential conversion from the stronger model to the standard 2DFAs. In particular, it turns out that subexponential conversion is possible for two-way automata that make nondeterministic choices only when the input head scans one of the input tape endmarkers. However, there is no restriction on the input head movement. This implies that an exponential gap between 2NFAs and 2DFAs can be obtained only for unrestricted 2NFAs using capabilities beyond the proposed new model. As an additional bonus, conversion into a machine for the complement of the original language is polynomial in this model. The same holds for making such machines self-verifying, halting, or unambiguous. Finally, any superpolynomial lower bound for the simulation of such machines by standard 2DFAs would imply L<>NL. In the same way, the alternating version of these machines is related to L =? NL =? P, the classical computational complexity problems.

💡 Research Summary

The paper tackles the long‑standing open problem of quantifying the state‑size blow‑up when simulating two‑way nondeterministic finite automata (2NFAs) by two‑way deterministic finite automata (2DFAs). While many works have obtained exponential gaps by restricting the power of the deterministic model (e.g., sweeping, oblivious, or sub‑linear reversal machines), these restrictions produce machines that are weaker than standard 2DFAs. In contrast, the authors adopt the opposite strategy: they strengthen the deterministic side just enough to still allow a sub‑exponential conversion from a stronger nondeterministic model to ordinary 2DFAs.

The central contribution is the definition of two‑way outer‑nondeterministic finite automata (2ONFAs). A 2ONFA is a standard 2NFA in which nondeterministic choices are permitted only when the input head scans one of the two endmarkers (⊢ or ⊣). Apart from this restriction, the head may move arbitrarily left or right on the interior of the tape, and the input alphabet is unrestricted. The authors also introduce the alternating counterpart, two‑way outer‑alternating finite automata (2OAFAs), where universal and existential choices are likewise confined to the endmarkers.

A key technical tool is a deterministic “segment‑reachability” procedure. Given a 2ONFA, two states p and q, and an input w, the procedure decides whether there exists a computation segment that starts at the left endmarker in state p, traverses the interior without visiting any endmarker, and returns to the left endmarker in state q. This subroutine runs in polynomial time with respect to the number of states and is used repeatedly to (i) eliminate infinite loops, (ii) construct self‑verifying automata, and (iii) simulate the nondeterministic behavior deterministically.

Using this tool, the authors obtain several simulation results:

1. Self‑Verifying Simulation – Every n‑state 2ONFA can be transformed into a two‑way self‑verifying automaton (2SVFA) with O(n⁸) states. Consequently, complementing a 2ONFA language requires only polynomially many states, a stark contrast with the exponential blow‑up known for sweeping 2NFAs.

2. Sub‑Exponential Deterministic Simulation – An n‑state 2ONFA can be simulated by a standard 2DFA with O(n log n) states. The construction stores, for each pair of states, a compact description of reachable segments, allowing the deterministic machine to mimic the nondeterministic choices without enumerating all computation paths.

3. Conditional Polynomial Simulation – Assuming L = NL, the same n‑state 2ONFA admits a polynomial‑size 2DFA simulation. Hence, any super‑polynomial lower bound for 2ONFA‑to‑2DFA simulation would separate L from NL.

4. Unambiguous Simulation – Every 2ONFA can be converted into an equivalent unambiguous 2ONFA with only a polynomial increase in states.

5. Alternating Variant – If L = P, then any n‑state 2OAFA can be simulated by a polynomial‑size 2DFA; if NL = P, the same holds for a simulation by a 2NFA. Therefore, a super‑polynomial lower bound for 2OAFAs would imply L = NL = P.

All these results extend earlier constructions for unary 2NFAs to the general alphabet case, but without relying on a normal form that forces head reversals to the endmarkers. Instead, the authors adapt techniques originally developed for deterministic machines (Sipser’s sweeping constructions, later refinements) to the setting where nondeterminism is confined to the tape boundaries.

The paper concludes that limiting nondeterministic choices to the endmarkers does not drastically reduce expressive power: the resulting 2ONFAs retain essentially the same state‑complexity as unrestricted 2NFAs, yet they admit efficient deterministic simulations and complementations. Moreover, the simulation bounds create a direct bridge between automata‑theoretic state complexity and classical complexity class separations (L vs. NL, NL vs. P). The authors suggest that further tightening of lower bounds for 2ONFA simulations could yield breakthroughs on these long‑standing open problems.