15,922 research outputs found
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
Graphs Encoded by Regular Expressions
In the conversion of finite automata to regular expressions, an exponential blowup in size can generally not be avoided. This is due to graph-structural properties of automata which cannot be directly encoded by regular expressions and cause the blowup combinatorially. In order to identify these structures, we generalize the class of arc-series-parallel digraphs to the acyclic case. The resulting digraphs are shown to be reversibly encoded by linear-sized regular expressions. We further derive a characterization of our new class by a finite set of forbidden minors and argue that these minors constitute the primitives causing the blowup in the conversion from automata to expressions
Probabilistic Logic, Probabilistic Regular Expressions, and Constraint Temporal Logic
The classic theorems of BĂĽchi and Kleene state the expressive equivalence of finite automata to monadic second order logic and regular expressions, respectively. These fundamental results enjoy applications in nearly every field of theoretical computer science. Around the same time as BĂĽchi and Kleene, Rabin investigated probabilistic finite automata. This equally well established model has applications ranging from natural language processing to probabilistic model checking.
Here, we give probabilistic extensions BĂĽchi\\\''s theorem and Kleene\\\''s theorem to the probabilistic setting. We obtain a probabilistic MSO logic by adding an expected second order quantifier. In the scope of this quantifier, membership is determined by a Bernoulli process. This approach turns out to be universal and is applicable for finite and infinite words as well as for finite trees. In order to prove the expressive equivalence of this probabilistic MSO logic to probabilistic automata, we show a Nivat-theorem, which decomposes a recognisable function into a regular language, homomorphisms, and a probability measure.
For regular expressions, we build upon existing work to obtain probabilistic regular expressions on finite and infinite words. We show the expressive equivalence between these expressions and probabilistic Muller-automata. To handle Muller-acceptance conditions, we give a new construction from probabilistic regular expressions to Muller-automata. Concerning finite trees, we define probabilistic regular tree expressions using a new iteration operator, called infinity-iteration. Again, we show that these expressions are expressively equivalent to probabilistic tree automata.
On a second track of our research we investigate Constraint LTL over multidimensional data words with data values from the infinite tree. Such LTL formulas are evaluated over infinite words, where every position possesses several data values from the infinite tree. Within Constraint LTL on can compare these values from different positions. We show that the model checking problem for this logic is PSPACE-complete via investigating the emptiness problem of Constraint BĂĽchi automata
Automata Tutor v3
Computer science class enrollments have rapidly risen in the past decade.
With current class sizes, standard approaches to grading and providing
personalized feedback are no longer possible and new techniques become both
feasible and necessary. In this paper, we present the third version of Automata
Tutor, a tool for helping teachers and students in large courses on automata
and formal languages. The second version of Automata Tutor supported automatic
grading and feedback for finite-automata constructions and has already been
used by thousands of users in dozens of countries. This new version of Automata
Tutor supports automated grading and feedback generation for a greatly extended
variety of new problems, including problems that ask students to create regular
expressions, context-free grammars, pushdown automata and Turing machines
corresponding to a given description, and problems about converting between
equivalent models - e.g., from regular expressions to nondeterministic finite
automata. Moreover, for several problems, this new version also enables
teachers and students to automatically generate new problem instances. We also
present the results of a survey run on a class of 950 students, which shows
very positive results about the usability and usefulness of the tool
Logical and Algebraic Characterizations of Rational Transductions
Rational word languages can be defined by several equivalent means: finite
state automata, rational expressions, finite congruences, or monadic
second-order (MSO) logic. The robust subclass of aperiodic languages is defined
by: counter-free automata, star-free expressions, aperiodic (finite)
congruences, or first-order (FO) logic. In particular, their algebraic
characterization by aperiodic congruences allows to decide whether a regular
language is aperiodic.
We lift this decidability result to rational transductions, i.e.,
word-to-word functions defined by finite state transducers. In this context,
logical and algebraic characterizations have also been proposed. Our main
result is that one can decide if a rational transduction (given as a
transducer) is in a given decidable congruence class. We also establish a
transfer result from logic-algebra equivalences over languages to equivalences
over transductions. As a consequence, it is decidable if a rational
transduction is first-order definable, and we show that this problem is
PSPACE-complete
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
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