347,690 research outputs found
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
Decidability of Identity-free Relational Kleene Lattices
National audienceFamilies of binary relations are important interpretations of regular expressions, and the equivalence of two regular expressions with respect to their relational interpretations is decidable: the problem reduces to the equality of the denoted regular languages.Putting together a few results from the literature, we first make explicit a generalisation of this reduction, for regular expressions extended with converse and intersection: instead of considering sets of words (i.e., formal languages), one has to consider sets of directed and labelled graphs.We then focus on identity-free regular expressions with intersection—a setting where the above graphs are acyclic—and we show that the corresponding equational theory is decidable. We achieve this by defining an automaton model, based on Petri Nets, to recognise these sets of acyclic graphs, and by providing an algorithm to compare such automata
Symbolic Solving of Extended Regular Expression Inequalities
This paper presents a new solution to the containment problem for extended
regular expressions that extends basic regular expressions with intersection
and complement operators and consider regular expressions on infinite alphabets
based on potentially infinite character sets. Standard approaches deciding the
containment do not take extended operators or character sets into account. The
algorithm avoids the translation to an expression-equivalent automaton and
provides a purely symbolic term rewriting systems for solving regular
expressions inequalities.
We give a new symbolic decision procedure for the containment problem based
on Brzozowski's regular expression derivatives and Antimirov's rewriting
approach to check containment. We generalize Brzozowski's syntactic derivative
operator to two derivative operators that work with respect to (potentially
infinite) representable character sets.Comment: Technical Repor
On the complexity of minimum inference of regular sets
We prove results concerning the computational tractability of some problems related to determining minimum realizations of finite samples of regular sets by finite automata and regular expressions
On universality of concurrent expressions with synchronization primitives
AbstractConcurrent expressions are a class of extended regular expressions with a shuffle operator (‖) and its closure (). The class of concurrent expressions with synchronization primitives, called synchronized concurrent expressions, is introduced as an extended model of Shaw's flow expressions. This paper discusses some formal properties of synchronized concurrent expressions from a formal language theoretic point of view. It is shown that synchronized concurrent expressions with three signal/wait operations are universal in the sense that they can simulate any semaphore controlled concurrent expressions and they can describe the class of recursively enumerable sets. Some results on semaphore controlled regular expressions are also included to give a taste of more positive results
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
Deciding KAT and Hoare Logic with Derivatives
Kleene algebra with tests (KAT) is an equational system for program
verification, which is the combination of Boolean algebra (BA) and Kleene
algebra (KA), the algebra of regular expressions. In particular, KAT subsumes
the propositional fragment of Hoare logic (PHL) which is a formal system for
the specification and verification of programs, and that is currently the base
of most tools for checking program correctness. Both the equational theory of
KAT and the encoding of PHL in KAT are known to be decidable. In this paper we
present a new decision procedure for the equivalence of two KAT expressions
based on the notion of partial derivatives. We also introduce the notion of
derivative modulo particular sets of equations. With this we extend the
previous procedure for deciding PHL. Some experimental results are also
presented.Comment: In Proceedings GandALF 2012, arXiv:1210.202
- …