68 research outputs found
Quotient Complexity Of Closed Languages
The final publication is available at Springer via http://dx.doi.org/10.1007/s00224-013-9515-7A language L is prefix-closed if, whenever a word w is in L, then every prefix of w is also in L. We define suffix-, factor-, and subword-closed languages in an analogous way, where by factor we mean contiguous subsequence, and by subword we mean scattered subsequence. We study the state complexity (which we prefer to call quotient complexity) of operations on prefix-, suffix-, factor-, and subword-closed languages. We find tight upper bounds on the complexity of the subword-closure of arbitrary languages, and on the complexity of boolean operations, concatenation, star, and reversal in each of the four classes of closed languages. We show that repeated applications of positive closure and complement to a closed language result in at most four distinct languages, while Kleene closure and complement give at most eight.Natural Sciences and Engineering Research Council of Canada [OGP0000871]VEGA grant [2/0183/11][APVV-0035-10
Boundary-Border Extensions of the Kuratowski Monoid
The Kuratowski monoid is generated under operator composition by
closure and complement in a nonempty topological space. It satisfies
. The Gaida-Eremenko (or GE) monoid
extends by adding the boundary operator. It satisfies
. We show that when the GE monoid
is determined by . When if the interior of the
boundary of every subset is clopen, then . This defines a new
type of topological space we call . Otherwise
. When applied to an arbitrary subset the GE monoid collapses
in one of possible ways. We investigate how these collapses and
interdepend, settling two questions raised by Gardner and
Jackson. Computer experimentation played a key role in our research.Comment: 48 pages, 9 figure
Regular tree languages in low levels of the Wadge Hierarchy
In this article we provide effective characterisations of regular languages
of infinite trees that belong to the low levels of the Wadge hierarchy. More
precisely we prove decidability for each of the finite levels of the hierarchy;
for the class of the Boolean combinations of open sets (i.e.
the union of the first levels); and for the Borel class
(i.e. for the union of the first levels)
Abstract GSOS Rules and a Modular Treatment of Recursive Definitions
Terminal coalgebras for a functor serve as semantic domains for state-based
systems of various types. For example, behaviors of CCS processes, streams,
infinite trees, formal languages and non-well-founded sets form terminal
coalgebras. We present a uniform account of the semantics of recursive
definitions in terminal coalgebras by combining two ideas: (1) abstract GSOS
rules l specify additional algebraic operations on a terminal coalgebra; (2)
terminal coalgebras are also initial completely iterative algebras (cias). We
also show that an abstract GSOS rule leads to new extended cia structures on
the terminal coalgebra. Then we formalize recursive function definitions
involving given operations specified by l as recursive program schemes for l,
and we prove that unique solutions exist in the extended cias. From our results
it follows that the solutions of recursive (function) definitions in terminal
coalgebras may be used in subsequent recursive definitions which still have
unique solutions. We call this principle modularity. We illustrate our results
by the five concrete terminal coalgebras mentioned above, e.\,g., a finite
stream circuit defines a unique stream function
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