In this paper, we introduce 1-way and 2-way quantum finite state automata (1qfa's and 2qfa's), which are the quantum analogues of deterministic, nondeterministic and probabilistic 1-way and 2-way finite state automata. We prove the following facts regarding 2qfa's. 1. For any ffl ? 0, there is a 2qfa M which recognizes the non-regular language L = fa m b m j m 1g with (one-sided) error bounded by ffl, and which halts in linear time. Specifically, M accepts any string in L with probability 1 and rejects any string not in L with probability at least 1 \Gamma ffl. 2. For every regular language L, there is a reversible (and hence quantum) 2-way finite state automaton which recognizes L and which runs in linear time. In fact, it is possible to define 2qfa's which recognize the non-context-free language fa m b m c m jm 1g, based on the same technique used for 1. Consequently, the class of languages recognized by linear time, bounded error 2qfa's properly includes the regular l...

Motivated by formal models recently proposed in the context of XML, we study automata and logics on strings over infinite alphabets. These are conservative extensions of classical automata and logics defining the regular languages on finite alphabets. Specifically, we consider register and pebble automata, and extensions of first-order logic and monadic second-order logic. For each type of automaton we consider one-way and two-way variants, as well as deterministic, nondeterministic, and alternating control. We investigate the expressiveness and complexity of the automata, their connection to the logics, as well as standard decision problems. Some of our results answer open questions of Kaminski and Francez on register automata.

this paper is to investigate the power of tree-walking automata with pebbles. Obviously, the unrestricted use of pebbles leads to a class of tree languages much larger than the regular tree languages, in fact to all tree languages in NSPACE(logn). Thus, we restrict the automaton to the recursive use of pebbles, in the sense that the life times of pebbles, i.e., the times between dropping a pebble and lifting it again, are properly nested. A similar, but stronger, nesting requirement is studied in [13] for 2-way automata on strings. We prove in Section 5 that our restriction indeed guarantees that all tree languages recognized by the tree-walking pebble automaton are regular, but we conjecture that the automaton is not powerful enough to recognize all regular tree languages. In Section 6 we generalize the notion of pebble to that of a \set-pebble", in such a way that the tree-walking set-pebble automaton recognizes exactly the regular tree languages.

We present a novel, yet simple, technique for the specification of context in structured documents that we call caterpillar expressions. Although we are primarily applying this technique in the specification of context-dependent style sheets for HTML, SGML and XML documents, it can also be used for query specification for structured documents, as we shall demonstrate, and for the specification of computer program transformations. From a conceptual point of view, structured documents are trees, and one of the oldest and best-established techniques to process trees and, hence, structured documents are tree automata. We present a number of theoretical results that allow us to compare the expressive power of caterpillar expressions and caterpillar automata, their companions, to the expressive power of tree automata. In particular, we demonstrate that each caterpillar expression describes a regular tree language that is, hence, recognizable by a tree automaton. Finally, we empl...

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. We want to understand how the natural and well-studied computation model of tree automata can be used to compute such queries. We dene a query automaton (QA) as a deterministic two-way nite 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 dier from ranked ones in that there is no bound on the number of children of nodes. We characterize the expressiveness of the dierent formalisms as the unary queries denable in monadic second-order logic (MSO). Surprisingly, in contrast to the ranked case, special stay transitions had to be added to QAs over unranked trees to capture MSO. We es...

: Two-way and multi-pebble automata are considered (the latter appropriately restricted to accept only regular languages), and enriched with additional features, such as nondeterminism and concurrency. We investigate the succinctness of such machines, and the extent to which this succinctness carries over to make the reasoning problem in propositional dynamic logic (PDL) more difficult. The two main results establish that each additional pebble provides inherent exponential power on both fronts. 1 Introduction 1.1 Background This paper continues our work in [H], [DH], [HRV], seeking exponential (or higher) discrepancies in the succinctness of finite automata when augmented with various additional mechanisms. It is well-known, for example, that NFAs are exponentially more succinct than DFAs, in the following upper and lower bound senses (see [RS], [MF]): (i) Any NFA can be simulated by a DFA with at most an exponential growth in size; (ii) There is a family of regular sets, L n , for ...

String transductions that are definable in monadic second-order (mso) logic (without the use of parameters) are exactly those realized by deterministic two-way finite state transducers. Nondeterministic mso definable string transductions (i.e., those definable with the use of parameters) corespond to compositions of two nondeterministic two-way finite state transducers that have the finite visit property. Both families of mso definable string transductions are characterized in terms of Hennie machines, i.e., two-way finite state transducers with the finite visit property that are allowed to rewrite their input tape. Introduction In language theory, it is always a pleasant surprise when two formalisms, introduced with different motivations, turn out to be equally powerful, as this indicates that the underlying concept is a natural one. Additionally, this means that notions and tools from one formalism can be made use of within the other, leading to a better understanding of th...

We describe a new elementary reduction of two-way automata to one-way automata. The reduction is based on the subset construction rather than on crossing sequence analysis.

Over the last 30 years or so many results have appeared on the descriptional complexity of machines with limited resources. Since these results have appeared in a variety of different contexts, o rgo4 here is to pro vide a survey o these results. Particular emphasis is put o limiting reso rces (e.g., no ndeterminism, ambiguity,lo o ahead, etc.) fo vario s types o finite state machines, pushdo wn auto mata, parsers and cellular auto mata ando n the e#ect it haso n their descriptio nal co mplexity. We also address the questio no f how descriptional complexity might help in the future to solve practical issues, such as software reliability.

A "trip" is a triple (g; u; v) where g is, in general, a graph and u and v are nodes of that graph. The trip is from u to v on the graph g. For the special case that g is a tree (or even a string) we investigate ways of specifying and implementing sets of trips. The main result is that a regular set of trips, specified as a regular tree language, can be implemented by a tree-walking automaton that uses marbles and one pebble.