Deterministic Two-way Finite Automaton Research Articles

On the descriptional power of heads, counters, and pebbles

We investigate the descriptional complexity of deterministic two-way k -head finite automata ( k -DHA). It is shown that between non-deterministic pushdown automata and any k -DHA, k ⩾ 2 , there are savings in the size of description which cannot be bounded by any recursive function. The same is true for the other end of the hierarchy. Such non-recursive trade-offs are also shown between any k -DHA, k ⩾ 1 , and DSPACE ( log ) = multi - DHA . We also address the particular case of unary languages. In general, it is possible that non-recursive trade-offs for arbitrary languages reduce to recursive trade-offs for unary languages. Here we present huge lower bounds for the unary trade-offs between non-deterministic finite automata and any k -DHA, k ⩾ 2 . Furthermore, several known simulation results imply the presented trade-offs for other descriptional systems, e.g., deterministic two-way finite automata with k pebbles or with k linearly bounded counters.

Query automata over finite trees

A main task in document transformation and information retrieval is locating subtrees satisfying some pattern. Therefore, unary queries, i.e., queries that map a tree to a set of its nodes, play an important role in the context of structured document databases. The motivation of this work is to understand how the natural and well-studied computation model of tree automata can be used to compute such queries. We define a query automaton (QA) as a deterministic two-way finite automaton over trees that has the ability to select nodes depending on the state and the label at those nodes. We study QAs over ranked as well as over unranked trees. Unranked trees differ from ranked ones in that there is no bound on the number of children of nodes. We characterize the expressiveness of the different formalisms as the unary queries definable in monadic second-order logic (MSO). In contrast to the ranked case, special stay transitions had to be added to QAs over unranked trees to capture MSO. We establish the complexity of the non-emptiness, containment, and equivalence of QAs to be complete for EXPTIME.

Positional simulation of two-way automata: Proof of a conjecture of R. Kannan and generalizations

R. Kannan conjectured that every non-deterministic two-way finite automaton can be positionally simulated by a deterministic two-way finite automaton. The conjecture is proved here by reduction to a similar problem about finite monoids. The method and the result are then generalized to alternating two-way finite automata, certain alternating one-tape linear-time Turing machines, and one-pebble automata.

STRICT LOCAL TESTABILITY OF THE FINITE CONTROL OF TWO-WAY AUTOMATA AND OF REGULAR PICTURE DESCRIPTION LANGUAGES

We prove that every regular language is recognized by a deterministic two-way finite automaton whose control unit is strictly locally testing. Similarly, every picture language which can be described by a regular language can actually be described by a strictly locally testable language; this strengthens a result of Friedhelra Hinz.

Lower bounds on the size of sweeping automata

A sweeping automaton is a deterministic two-way finite automaton which only changes head direction at the ends of the input tape. It is shown that for every n, there is a language which is accepted by an n-state nondeterministic one-way finite automaton, yet which is not accepted by any sweeping automaton with fewer than 2 n states.

On two-way multihead automata

For each positive integer n , let ℒ N ( n ) be the class of sets accepted by a family of automata of type N , each with a read-only input with endmarkers and n two-way input heads. The following result, which is applicable to most types of two-way multihead devices, is proved: If for each positive integer n , there is some integer M n > n such that ℒ N ( n ) is properly contained in ℒ N ( M n ), then ℒ N ( n ) is properly contained in ℒ N ( n + c N ) for each n , where c N =1 or 2, depending on the type of the device. As a consequence, it is shown that deterministic two-way finite automata with n +2 heads are strictly more powerful than deterministic two-way finite automata with n heads for each positive integer n . It is also shown that the class of sets accepted by deterministic (nondeterministic) two-way pushdown automata with n heads is properly included in the class of sets accepted by deterministic (nondeterministic) two-way pushdown automata with n +1 heads.