65,030 research outputs found
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
Partially Ordered Two-way B\"uchi Automata
We introduce partially ordered two-way B\"uchi automata and characterize
their expressive power in terms of fragments of first-order logic FO[<].
Partially ordered two-way B\"uchi automata are B\"uchi automata which can
change the direction in which the input is processed with the constraint that
whenever a state is left, it is never re-entered again. Nondeterministic
partially ordered two-way B\"uchi automata coincide with the first-order
fragment Sigma2. Our main contribution is that deterministic partially ordered
two-way B\"uchi automata are expressively complete for the first-order fragment
Delta2. As an intermediate step, we show that deterministic partially ordered
two-way B\"uchi automata are effectively closed under Boolean operations.
A small model property yields coNP-completeness of the emptiness problem and
the inclusion problem for deterministic partially ordered two-way B\"uchi
automata.Comment: The results of this paper were presented at CIAA 2010; University of
Stuttgart, Computer Scienc
On equivalence, languages equivalence and minimization of multi-letter and multi-letter measure-many quantum automata
We first show that given a -letter quantum finite automata
and a -letter quantum finite automata over
the same input alphabet , they are equivalent if and only if they are
-equivalent where , , are the
numbers of state in respectively, and . By
applying a method, due to the author, used to deal with the equivalence problem
of {\it measure many one-way quantum finite automata}, we also show that a
-letter measure many quantum finite automaton and a
-letter measure many quantum finite automaton are
equivalent if and only if they are -equivalent
where , , are the numbers of state in respectively,
and .
Next, we study the language equivalence problem of those two kinds of quantum
finite automata. We show that for -letter quantum finite automata, the
non-strict cut-point language equivalence problem is undecidable, i.e., it is
undecidable whether
where
and are -letter quantum finite automata.
Further, we show that both strict and non-strict cut-point language equivalence
problem for -letter measure many quantum finite automata are undecidable.
The direct consequences of the above outcomes are summarized in the paper.
Finally, we comment on existing proofs about the minimization problem of one
way quantum finite automata not only because we have been showing great
interest in this kind of problem, which is very important in classical automata
theory, but also due to that the problem itself, personally, is a challenge.
This problem actually remains open.Comment: 30 pages, conclusion section correcte
Finite automata with advice tapes
We define a model of advised computation by finite automata where the advice
is provided on a separate tape. We consider several variants of the model where
the advice is deterministic or randomized, the input tape head is allowed
real-time, one-way, or two-way access, and the automaton is classical or
quantum. We prove several separation results among these variants, demonstrate
an infinite hierarchy of language classes recognized by automata with
increasing advice lengths, and establish the relationships between this and the
previously studied ways of providing advice to finite automata.Comment: Corrected typo
- …