7,344 research outputs found
Quantum automata, braid group and link polynomials
The spin--network quantum simulator model, which essentially encodes the
(quantum deformed) SU(2) Racah--Wigner tensor algebra, is particularly suitable
to address problems arising in low dimensional topology and group theory. In
this combinatorial framework we implement families of finite--states and
discrete--time quantum automata capable of accepting the language generated by
the braid group, and whose transition amplitudes are colored Jones polynomials.
The automaton calculation of the polynomial of (the plat closure of) a link L
on 2N strands at any fixed root of unity is shown to be bounded from above by a
linear function of the number of crossings of the link, on the one hand, and
polynomially bounded in terms of the braid index 2N, on the other. The growth
rate of the time complexity function in terms of the integer k appearing in the
root of unity q can be estimated to be (polynomially) bounded by resorting to
the field theoretical background given by the Chern-Simons theory.Comment: Latex, 36 pages, 11 figure
Complexity of Two-Dimensional Patterns
In dynamical systems such as cellular automata and iterated maps, it is often
useful to look at a language or set of symbol sequences produced by the system.
There are well-established classification schemes, such as the Chomsky
hierarchy, with which we can measure the complexity of these sets of sequences,
and thus the complexity of the systems which produce them.
In this paper, we look at the first few levels of a hierarchy of complexity
for two-or-more-dimensional patterns. We show that several definitions of
``regular language'' or ``local rule'' that are equivalent in d=1 lead to
distinct classes in d >= 2. We explore the closure properties and computational
complexity of these classes, including undecidability and L-, NL- and
NP-completeness results.
We apply these classes to cellular automata, in particular to their sets of
fixed and periodic points, finite-time images, and limit sets. We show that it
is undecidable whether a CA in d >= 2 has a periodic point of a given period,
and that certain ``local lattice languages'' are not finite-time images or
limit sets of any CA. We also show that the entropy of a d-dimensional CA's
finite-time image cannot decrease faster than t^{-d} unless it maps every
initial condition to a single homogeneous state.Comment: To appear in J. Stat. Phy
Sampling-Based Temporal Logic Path Planning
In this paper, we propose a sampling-based motion planning algorithm that
finds an infinite path satisfying a Linear Temporal Logic (LTL) formula over a
set of properties satisfied by some regions in a given environment. The
algorithm has three main features. First, it is incremental, in the sense that
the procedure for finding a satisfying path at each iteration scales only with
the number of new samples generated at that iteration. Second, the underlying
graph is sparse, which guarantees the low complexity of the overall method.
Third, it is probabilistically complete. Examples illustrating the usefulness
and the performance of the method are included.Comment: 8 pages, 4 figures; extended version of the paper presented at IROS
201
MSO definable string transductions and two-way finite state transducers
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) correspond 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.Comment: 63 pages, LaTeX2e. Extended abstract presented at 26-th ICALP, 199
Context-free pairs of groups I: Context-free pairs and graphs
Let be a finitely generated group, a finite set of generators and
a subgroup of . We call the pair context-free if the set of all
words over that reduce in to an element of is a context-free
language. When is trivial, itself is called context-free; context-free
groups have been classified more than 20 years ago in celebrated work of Muller
and Schupp as the virtually free groups.
Here, we derive some basic properties of such group pairs. Context-freeness
is independent of the choice of the generating set. It is preserved under
finite index modifications of and finite index enlargements of . If
is virtually free and is finitely generated then is context-free. A
basic tool is the following: is context-free if and only if the
Schreier graph of with respect to is a context-free graph
On Pebble Automata for Data Languages with Decidable Emptiness Problem
In this paper we study a subclass of pebble automata (PA) for data languages
for which the emptiness problem is decidable. Namely, we introduce the
so-called top view weak PA. Roughly speaking, top view weak PA are weak PA
where the equality test is performed only between the data values seen by the
two most recently placed pebbles. The emptiness problem for this model is
decidable. We also show that it is robust: alternating, nondeterministic and
deterministic top view weak PA have the same recognition power. Moreover, this
model is strong enough to accept all data languages expressible in Linear
Temporal Logic with the future-time operators, augmented with one register
freeze quantifier.Comment: An extended abstract of this work has been published in the
proceedings of the 34th International Symposium on Mathematical Foundations
of Computer Science (MFCS) 2009}, Springer, Lecture Notes in Computer Science
5734, pages 712-72
Synthesising Graphical Theories
In recent years, diagrammatic languages have been shown to be a powerful and
expressive tool for reasoning about physical, logical, and semantic processes
represented as morphisms in a monoidal category. In particular, categorical
quantum mechanics, or "Quantum Picturalism", aims to turn concrete features of
quantum theory into abstract structural properties, expressed in the form of
diagrammatic identities. One way we search for these properties is to start
with a concrete model (e.g. a set of linear maps or finite relations) and start
composing generators into diagrams and looking for graphical identities.
Naively, we could automate this procedure by enumerating all diagrams up to a
given size and check for equalities, but this is intractable in practice
because it produces far too many equations. Luckily, many of these identities
are not primitive, but rather derivable from simpler ones. In 2010, Johansson,
Dixon, and Bundy developed a technique called conjecture synthesis for
automatically generating conjectured term equations to feed into an inductive
theorem prover. In this extended abstract, we adapt this technique to
diagrammatic theories, expressed as graph rewrite systems, and demonstrate its
application by synthesising a graphical theory for studying entangled quantum
states.Comment: 10 pages, 22 figures. Shortened and one theorem adde
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