125 research outputs found
Pebble alternating tree-walking automata and their recognizing power
Pebble tree-walking automata with alternation were first investigated by Milo, Suciu and Vianu (2003), who showed that tree languages recognized by these devices are exactly the regular tree languages. We strengthen this by proving the same result for pebble automata with "strong pebble handling" which means that pebbles can be lifted independently of the position of the reading head and without moving the reading head. Then we make a comparison among some restricted versions of these automata. We will show that the deterministic and non-looping pebble alternating tree-walking automata are strictly less powerful than their nondeterministic counterparts, i.e., they do not recognize all the regular tree languages. Moreover, there is a proper hierarchy of recognizing capacity of deterministic and non-looping n-pebble alternating tree-walking automata with respect to the number of pebbles, i.e., for each n ≥ 0, deterministic and non-looping (n+1)-pebble alternating tree-walking automata are more powerful than their n-pebble counterparts
Weighted tree-walking automata
We define weighted tree-walking automata. We show that the class of tree series recognizable by weighted tree-walking automata over a commutative semiring K is a subclass of the class of regular tree series over K. If K is not a ring, then the inclusion is strict
Automata with Nested Pebbles Capture First-Order Logic with Transitive Closure
String languages recognizable in (deterministic) log-space are characterized
either by two-way (deterministic) multi-head automata, or following Immerman,
by first-order logic with (deterministic) transitive closure. Here we elaborate
this result, and match the number of heads to the arity of the transitive
closure. More precisely, first-order logic with k-ary deterministic transitive
closure has the same power as deterministic automata walking on their input
with k heads, additionally using a finite set of nested pebbles. This result is
valid for strings, ordered trees, and in general for families of graphs having
a fixed automaton that can be used to traverse the nodes of each of the graphs
in the family. Other examples of such families are grids, toruses, and
rectangular mazes. For nondeterministic automata, the logic is restricted to
positive occurrences of transitive closure.
The special case of k=1 for trees, shows that single-head deterministic
tree-walking automata with nested pebbles are characterized by first-order
logic with unary deterministic transitive closure. This refines our earlier
result that placed these automata between first-order and monadic second-order
logic on trees.Comment: Paper for Logical Methods in Computer Science, 27 pages, 1 figur
Weighted Automata and Expressions over Pre-Rational Monoids
The Kleene theorem establishes a fundamental link between automata and expressions over the free monoid. Numerous generalisations of this result exist in the literature; on one hand, lifting this result to a weighted setting has been widely studied. On the other hand, beyond the free monoid, different monoids can be considered: for instance, two-way automata, and even tree-walking automata, can be described by expressions using the free inverse monoid. In the present work, we aim at combining both research directions and consider weighted extensions of automata and expressions over a class of monoids that we call pre-rational, generalising both the free inverse monoid and graded monoids. The presence of idempotent elements in these pre-rational monoids leads in the weighted setting to consider infinite sums. To handle such sums, we will have to restrict ourselves to rationally additive semirings. Our main result is thus a generalisation of the Kleene theorem for pre-rational monoids and rationally additive semirings. As a corollary, we obtain a class of expressions equivalent to weighted two-way automata, as well as one for tree-walking automata
The Tree Width of Separation Logic with Recursive Definitions
Separation Logic is a widely used formalism for describing dynamically
allocated linked data structures, such as lists, trees, etc. The decidability
status of various fragments of the logic constitutes a long standing open
problem. Current results report on techniques to decide satisfiability and
validity of entailments for Separation Logic(s) over lists (possibly with
data). In this paper we establish a more general decidability result. We prove
that any Separation Logic formula using rather general recursively defined
predicates is decidable for satisfiability, and moreover, entailments between
such formulae are decidable for validity. These predicates are general enough
to define (doubly-) linked lists, trees, and structures more general than
trees, such as trees whose leaves are chained in a list. The decidability
proofs are by reduction to decidability of Monadic Second Order Logic on graphs
with bounded tree width.Comment: 30 pages, 2 figure
Advances and applications of automata on words and trees : executive summary
Seminar: 10501 - Advances and Applications of Automata on Words and Trees. The aim of the seminar was to discuss and systematize the recent fast progress in automata theory and to identify important directions for future research. For this, the seminar brought together more than 40 researchers from automata theory and related fields of applications. We had 19 talks of 30 minutes and 5 one-hour lectures leaving ample room for discussions. In the following we describe the topics in more detail
Lower Bounds for Graph-Walking Automata
Graph-walking automata (GWA) traverse graphs by moving between the nodes following the edges, using a finite-state control to decide where to go next. It is known that every GWA can be transformed to a GWA that halts on every input, to a GWA returning to the initial node in order to accept, as well as to a reversible GWA. This paper establishes lower bounds on the state blow-up of these transformations: it is shown that making an n-state GWA traversing k-ary graphs return to the initial node requires at least 2(n-1)(k-3) states in the worst case; the same lower bound holds for the transformation to halting automata. Automata satisfying both properties at once must have at least 4(n-1)(k-3) states. A reversible automaton must have at least 4(n-1)(k-3)-1 states. These bounds are asymptotically tight to the upper bounds proved using the methods from the literature
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