Two-way finite automaton

A two-way deterministic finite automaton (2DFA) is an abstract machine, a generalized version of the deterministic finite automaton (DFA) which can revisit characters already processed. As in a DFA, there are a finite number of states with transitions between them based on the current character, but each transition is also labelled with a value indicating whether the machine will move its position in the input to the left, right, or stay at the same position. Equivalently, 2DFAs can be seen as read-only Turing machines with no work tape, only a read-only input tape.

2DFAs were introduced in a seminal 1959 paper by Rabin and Scott,[1] who proved them to have equivalent power to one-way DFAs. That is, any formal language which can be recognized by a 2DFA can be recognized by a DFA which only examines and consumes each character in order. Since DFAs are obviously a special case of 2DFAs, this implies that both kinds of machines recognize precisely the class of regular languages. However, the equivalent DFA for a 2DFA may require exponentially many states, making 2DFAs a much more practical representation for algorithms for some common problems.

2DFAs are also equivalent to read-only Turing machines that use only a constant amount of space on their work tape, since any constant amount of information can be incorporated into the finite control state via a product construction (a state for each combination of work tape state and control state).

Formally, a two-way deterministic finite automaton can be described by the following 8-tuple: ${\displaystyle M=(Q,\Sigma ,L,R,\delta ,s,t,r)}$ where

• ${\displaystyle Q}$ is the finite, non-empty set of states
• ${\displaystyle \Sigma }$ is the finite, non-empty set of input symbols
• ${\displaystyle L}$ is the left endmarker
• ${\displaystyle R}$ is the right endmarker
• ${\displaystyle \delta :Q\times (\Sigma \cup \{L,R\})\rightarrow Q\times \{\mathrm {left,right} \}}$
• ${\displaystyle s}$ is the start state
• ${\displaystyle t}$ is the end state
• ${\displaystyle r}$ is the reject state

In addition, the following two conditions must also be satisfied:

• For all ${\displaystyle q\in Q}$

${\displaystyle \delta (q,L)=(q^{\prime },\mathrm {right} )}$ for some ${\displaystyle q^{\prime }\in Q}$

${\displaystyle \delta (q,R)=(q^{\prime },\mathrm {left} )}$ for some ${\displaystyle q^{\prime }\in Q}$

It says that there must be some transition possible when the pointer reaches either end of the input word.

• For all symbols ${\displaystyle \sigma \in \Sigma \cup \{L\}}$

${\displaystyle \delta (t,\sigma )=(t,R)}$

${\displaystyle \delta (r,\sigma )=(r,R)}$

${\displaystyle \delta (t,R)=(t,L)}$

${\displaystyle \delta (r,R)=(r,L)}$

It says that once the automaton reaches the accept or reject state, it stays in there forever and the pointer goes to the right most symbol and cycles there infinitely.[2]

A two-way nondeterministic finite automaton (2NFA) may have multiple transitions defined in the same configuration. Its transition function is

• ${\displaystyle \delta :Q\times (\Sigma \cup \{L,R\})\rightarrow 2^{Q\times \{\mathrm {left,right} \}}}$.

Like a standard one-way NFA, a 2NFA accepts a string if at least one of the possible computations is accepting. Like the 2DFAs, the 2NFAs also accept only regular languages.

A two-way alternating finite automaton (2AFA) is a two-way extension of an alternating finite automaton (AFA). Its state set is

• ${\displaystyle Q=Q_{\exists }\cup Q_{\forall }}$ where ${\displaystyle Q_{\exists }\cap Q_{\forall }=\emptyset }$.

States in ${\displaystyle Q_{\exists }}$ and ${\displaystyle Q_{\forall }}$ are called existential resp. universal. In an existential state a 2AFA nondeterministically chooses the next state like an NFA, and accepts if at least one of the resulting computations accepts. In a universal state 2AFA moves to all next states, and accepts if all the resulting computations accept.

Two-way and one-way finite automata, deterministic and nondeterministic and alternating, accept the same class of regular languages. However, transforming an automaton of one type to an equivalent automaton of another type incurs a blow-up in the number of states. Christos Kapoutsis[3] determined that transforming an ${\displaystyle n}$-state 2DFA to an equivalent DFA requires ${\displaystyle n(n^{n}-(n-1)^{n})}$ states in the worst case. If an ${\displaystyle n}$-state 2DFA or a 2NFA is transformed to an NFA, the worst-case number of states required is ${\displaystyle {\binom {2n}{n+1}}=O\left({\frac {4^{n}}{\sqrt {n}}}\right)}$. Ladner, Lipton and Stockmeyer.[4] proved that an ${\displaystyle n}$-state 2AFA can be converted to a DFA with ${\displaystyle 2^{n2^{n}}}$ states. The 2AFA to NFA conversion requires ${\displaystyle 2^{\Theta (n\log n)}}$ states in the worst case, see Geffert and Okhotin.[5]

It is an open problem whether every 2NFA can be converted to a 2DFA with only a polynomial increase in the number of states. The problem was raised by Sakoda and Sipser,[6] who compared it to the P vs. NP problem in the computational complexity theory. Berman and Lingas[7] discovered a formal relation between this problem and the L vs. NL open problem, see Kapoutsis[8] for a precise relation.

Sweeping automata are 2DFAs of a special kind that process the input string by making alternating left-to-right and right-to-left sweeps, turning only at the endmarkers. Sipser[9] constructed a sequence of languages, each accepted by an n-state NFA, yet which is not accepted by any sweeping automata with fewer than ${\displaystyle 2^{n}}$ states.

The concept of 2DFAs was in 1997 generalized to quantum computing by John Watrous's "On the Power of 2-Way Quantum Finite State Automata", in which he demonstrates that these machines can recognize nonregular languages and so are more powerful than DFAs. [10]

A pushdown automaton that is allowed to move either way on its input tape is called two-way pushdown automaton (2PDA);[11] it has been studied by Hartmanis, Lewis, and Stearns (1965).[12] Aho, Hopcroft, Ullman (1968)[13] and Cook (1971)[14] characterized the class of languages recognizable by deterministic (2DPDA) and non-deterministic (2NPDA) two-way pushdown automata; Gray, Harrison, and Ibarra (1967) investigated the closure properties of these languages.[15]