25,049 research outputs found
The omega-inequality problem for concatenation hierarchies of star-free languages
The problem considered in this paper is whether an inequality of omega-terms
is valid in a given level of a concatenation hierarchy of star-free languages.
The main result shows that this problem is decidable for all (integer and half)
levels of the Straubing-Th\'erien hierarchy
Complexity of Left-Ideal, Suffix-Closed and Suffix-Free Regular Languages
A language over an alphabet is suffix-convex if, for any words
, whenever and are in , then so is .
Suffix-convex languages include three special cases: left-ideal, suffix-closed,
and suffix-free languages. We examine complexity properties of these three
special classes of suffix-convex regular languages. In particular, we study the
quotient/state complexity of boolean operations, product (concatenation), star,
and reversal on these languages, as well as the size of their syntactic
semigroups, and the quotient complexity of their atoms.Comment: 20 pages, 11 figures, 1 table. arXiv admin note: text overlap with
arXiv:1605.0669
Higher-Order Operator Precedence Languages
Floyd's Operator Precedence (OP) languages are a deterministic context-free
family having many desirable properties. They are locally and parallely
parsable, and languages having a compatible structure are closed under Boolean
operations, concatenation and star; they properly include the family of Visibly
Pushdown (or Input Driven) languages. OP languages are based on three relations
between any two consecutive terminal symbols, which assign syntax structure to
words. We extend such relations to k-tuples of consecutive terminal symbols, by
using the model of strictly locally testable regular languages of order k at
least 3. The new corresponding class of Higher-order Operator Precedence
languages (HOP) properly includes the OP languages, and it is still included in
the deterministic (also in reverse) context free family. We prove Boolean
closure for each subfamily of structurally compatible HOP languages. In each
subfamily, the top language is called max-language. We show that such languages
are defined by a simple cancellation rule and we prove several properties, in
particular that max-languages make an infinite hierarchy ordered by parameter
k. HOP languages are a candidate for replacing OP languages in the various
applications where they have have been successful though sometimes too
restrictive.Comment: In Proceedings AFL 2017, arXiv:1708.0622
On All Things Star-Free
We investigate the star-free closure, which associates to a class of languages its closure under Boolean operations and marked concatenation. We prove that the star-free closure of any finite class and of any class of groups languages with decidable separation (plus mild additional properties) has decidable separation. We actually show decidability of a stronger property, called covering. This generalizes many results on the subject in a unified framework. A key ingredient is that star-free closure coincides with another closure operator where Kleene stars are also allowed in restricted contexts
Separation for dot-depth two
The dot-depth hierarchy of Brzozowski and Cohen classifies the star-free
languages of finite words. By a theorem of McNaughton and Papert, these are
also the first-order definable languages. The dot-depth rose to prominence
following the work of Thomas, who proved an exact correspondence with the
quantifier alternation hierarchy of first-order logic: each level in the
dot-depth hierarchy consists of all languages that can be defined with a
prescribed number of quantifier blocks. One of the most famous open problems in
automata theory is to settle whether the membership problem is decidable for
each level: is it possible to decide whether an input regular language belongs
to this level?
Despite a significant research effort, membership by itself has only been
solved for low levels. A recent breakthrough was achieved by replacing
membership with a more general problem: separation. Given two input languages,
one has to decide whether there exists a third language in the investigated
level containing the first language and disjoint from the second. The
motivation is that: (1) while more difficult, separation is more rewarding (2)
it provides a more convenient framework (3) all recent membership algorithms
are reductions to separation for lower levels.
We present a separation algorithm for dot-depth two. While this is our most
prominent application, our result is more general. We consider a family of
hierarchies that includes the dot-depth: concatenation hierarchies. They are
built via a generic construction process. One first chooses an initial class,
the basis, which is the lowest level in the hierarchy. Further levels are built
by applying generic operations. Our main theorem states that for any
concatenation hierarchy whose basis is finite, separation is decidable for
level one. In the special case of the dot-depth, this can be lifted to level
two using previously known results
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