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Classes of languages generated by the Kleene star of a word
In this paper, we study the lattice and the Boolean algebra, possibly closed under quotient, generated by the languages of the form u*, where u is a word. We provide effective equational characterisations of these classes, i.e. one can decide using our descriptions whether a given regular language belongs or not to each of them
On the Structure and Complexity of Rational Sets of Regular Languages
In a recent thread of papers, we have introduced FQL, a precise specification
language for test coverage, and developed the test case generation engine
FShell for ANSI C. In essence, an FQL test specification amounts to a set of
regular languages, each of which has to be matched by at least one test
execution. To describe such sets of regular languages, the FQL semantics uses
an automata-theoretic concept known as rational sets of regular languages
(RSRLs). RSRLs are automata whose alphabet consists of regular expressions.
Thus, the language accepted by the automaton is a set of regular expressions.
In this paper, we study RSRLs from a theoretic point of view. More
specifically, we analyze RSRL closure properties under common set theoretic
operations, and the complexity of membership checking, i.e., whether a regular
language is an element of a RSRL. For all questions we investigate both the
general case and the case of finite sets of regular languages. Although a few
properties are left as open problems, the paper provides a systematic semantic
foundation for the test specification language FQL
Unified theory for finite Markov chains
We provide a unified framework to compute the stationary distribution of any
finite irreducible Markov chain or equivalently of any irreducible random walk
on a finite semigroup . Our methods use geometric finite semigroup theory
via the Karnofsky-Rhodes and the McCammond expansions of finite semigroups with
specified generators; this does not involve any linear algebra. The original
Tsetlin library is obtained by applying the expansions to , the set of
all subsets of an element set. Our set-up generalizes previous
groundbreaking work involving left-regular bands (or -trivial
bands) by Brown and Diaconis, extensions to -trivial semigroups by
Ayyer, Steinberg, Thi\'ery and the second author, and important recent work by
Chung and Graham. The Karnofsky-Rhodes expansion of the right Cayley graph of
in terms of generators yields again a right Cayley graph. The McCammond
expansion provides normal forms for elements in the expanded . Using our
previous results with Silva based on work by Berstel, Perrin, Reutenauer, we
construct (infinite) semaphore codes on which we can define Markov chains.
These semaphore codes can be lumped using geometric semigroup theory. Using
normal forms and associated Kleene expressions, they yield formulas for the
stationary distribution of the finite Markov chain of the expanded and the
original . Analyzing the normal forms also provides an estimate on the
mixing time.Comment: 29 pages, 12 figures; v2: Section 3.2 added, references added,
revision of introduction, title change; v3: typos fixed and clarifications
adde
Symbolic Algorithms for Language Equivalence and Kleene Algebra with Tests
We first propose algorithms for checking language equivalence of finite
automata over a large alphabet. We use symbolic automata, where the transition
function is compactly represented using a (multi-terminal) binary decision
diagrams (BDD). The key idea consists in computing a bisimulation by exploring
reachable pairs symbolically, so as to avoid redundancies. This idea can be
combined with already existing optimisations, and we show in particular a nice
integration with the disjoint sets forest data-structure from Hopcroft and
Karp's standard algorithm. Then we consider Kleene algebra with tests (KAT), an
algebraic theory that can be used for verification in various domains ranging
from compiler optimisation to network programming analysis. This theory is
decidable by reduction to language equivalence of automata on guarded strings,
a particular kind of automata that have exponentially large alphabets. We
propose several methods allowing to construct symbolic automata out of KAT
expressions, based either on Brzozowski's derivatives or standard automata
constructions. All in all, this results in efficient algorithms for deciding
equivalence of KAT expressions
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