10,133 research outputs found
Mutation of Directed Graphs -- Corresponding Regular Expressions and Complexity of Their Generation
Directed graphs (DG), interpreted as state transition diagrams, are
traditionally used to represent finite-state automata (FSA). In the context of
formal languages, both FSA and regular expressions (RE) are equivalent in that
they accept and generate, respectively, type-3 (regular) languages. Based on
our previous work, this paper analyzes effects of graph manipulations on
corresponding RE. In this present, starting stage we assume that the DG under
consideration contains no cycles. Graph manipulation is performed by deleting
or inserting of nodes or arcs. Combined and/or multiple application of these
basic operators enable a great variety of transformations of DG (and
corresponding RE) that can be seen as mutants of the original DG (and
corresponding RE). DG are popular for modeling complex systems; however they
easily become intractable if the system under consideration is complex and/or
large. In such situations, we propose to switch to corresponding RE in order to
benefit from their compact format for modeling and algebraic operations for
analysis. The results of the study are of great potential interest to mutation
testing
Boundedness in languages of infinite words
We define a new class of languages of -words, strictly extending
-regular languages.
One way to present this new class is by a type of regular expressions. The
new expressions are an extension of -regular expressions where two new
variants of the Kleene star are added: and . These new
exponents are used to say that parts of the input word have bounded size, and
that parts of the input can have arbitrarily large sizes, respectively. For
instance, the expression represents the language of infinite
words over the letters where there is a common bound on the number of
consecutive letters . The expression represents a similar
language, but this time the distance between consecutive 's is required to
tend toward the infinite.
We develop a theory for these languages, with a focus on decidability and
closure. We define an equivalent automaton model, extending B\"uchi automata.
The main technical result is a complementation lemma that works for languages
where only one type of exponent---either or ---is used.
We use the closure and decidability results to obtain partial decidability
results for the logic MSOLB, a logic obtained by extending monadic second-order
logic with new quantifiers that speak about the size of sets
From Finite Automata to Regular Expressions and Back--A Summary on Descriptional Complexity
The equivalence of finite automata and regular expressions dates back to the
seminal paper of Kleene on events in nerve nets and finite automata from 1956.
In the present paper we tour a fragment of the literature and summarize results
on upper and lower bounds on the conversion of finite automata to regular
expressions and vice versa. We also briefly recall the known bounds for the
removal of spontaneous transitions (epsilon-transitions) on non-epsilon-free
nondeterministic devices. Moreover, we report on recent results on the average
case descriptional complexity bounds for the conversion of regular expressions
to finite automata and brand new developments on the state elimination
algorithm that converts finite automata to regular expressions.Comment: In Proceedings AFL 2014, arXiv:1405.527
Digraph Complexity Measures and Applications in Formal Language Theory
We investigate structural complexity measures on digraphs, in particular the
cycle rank. This concept is intimately related to a classical topic in formal
language theory, namely the star height of regular languages. We explore this
connection, and obtain several new algorithmic insights regarding both cycle
rank and star height. Among other results, we show that computing the cycle
rank is NP-complete, even for sparse digraphs of maximum outdegree 2.
Notwithstanding, we provide both a polynomial-time approximation algorithm and
an exponential-time exact algorithm for this problem. The former algorithm
yields an O((log n)^(3/2))- approximation in polynomial time, whereas the
latter yields the optimum solution, and runs in time and space O*(1.9129^n) on
digraphs of maximum outdegree at most two. Regarding the star height problem,
we identify a subclass of the regular languages for which we can precisely
determine the computational complexity of the star height problem. Namely, the
star height problem for bideterministic languages is NP-complete, and this
holds already for binary alphabets. Then we translate the algorithmic results
concerning cycle rank to the bideterministic star height problem, thus giving a
polynomial-time approximation as well as a reasonably fast exact exponential
algorithm for bideterministic star height.Comment: 19 pages, 1 figur
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