17,183 research outputs found
Towards Nominal Formal Languages
We introduce formal languages over infinite alphabets where words may contain
binders. We define the notions of nominal language, nominal monoid, and nominal
regular expressions. Moreover, we extend history-dependent automata
(HD-automata) by adding stack, and study the recognisability of nominal
languages
Factorization in Formal Languages
We consider several novel aspects of unique factorization in formal
languages. We reprove the familiar fact that the set uf(L) of words having
unique factorization into elements of L is regular if L is regular, and from
this deduce an quadratic upper and lower bound on the length of the shortest
word not in uf(L). We observe that uf(L) need not be context-free if L is
context-free.
Next, we consider variations on unique factorization. We define a notion of
"semi-unique" factorization, where every factorization has the same number of
terms, and show that, if L is regular or even finite, the set of words having
such a factorization need not be context-free. Finally, we consider additional
variations, such as unique factorization "up to permutation" and "up to
subset"
Formal Languages in Dynamical Systems
We treat here the interrelation between formal languages and those dynamical
systems that can be described by cellular automata (CA). There is a well-known
injective map which identifies any CA-invariant subshift with a central formal
language. However, in the special case of a symbolic dynamics, i.e. where the
CA is just the shift map, one gets a stronger result: the identification map
can be extended to a functor between the categories of symbolic dynamics and
formal languages. This functor additionally maps topological conjugacies
between subshifts to empty-string-limited generalized sequential machines
between languages. If the periodic points form a dense set, a case which arises
in a commonly used notion of chaotic dynamics, then an even more natural map to
assign a formal language to a subshift is offered. This map extends to a
functor, too. The Chomsky hierarchy measuring the complexity of formal
languages can be transferred via either of these functors from formal languages
to symbolic dynamics and proves to be a conjugacy invariant there. In this way
it acquires a dynamical meaning. After reviewing some results of the complexity
of CA-invariant subshifts, special attention is given to a new kind of
invariant subshift: the trapped set, which originates from the theory of
chaotic scattering and for which one can study complexity transitions.Comment: 23 pages, LaTe
Dynamic Complexity of Formal Languages
The paper investigates the power of the dynamic complexity classes DynFO,
DynQF and DynPROP over string languages. The latter two classes contain
problems that can be maintained using quantifier-free first-order updates, with
and without auxiliary functions, respectively. It is shown that the languages
maintainable in DynPROP exactly are the regular languages, even when allowing
arbitrary precomputation. This enables lower bounds for DynPROP and separates
DynPROP from DynQF and DynFO. Further, it is shown that any context-free
language can be maintained in DynFO and a number of specific context-free
languages, for example all Dyck-languages, are maintainable in DynQF.
Furthermore, the dynamic complexity of regular tree languages is investigated
and some results concerning arbitrary structures are obtained: there exist
first-order definable properties which are not maintainable in DynPROP. On the
other hand any existential first-order property can be maintained in DynQF when
allowing precomputation.Comment: Contains the material presenten at STACS 2009, extendes with proofs
and examples which were omitted due lack of spac
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