1,132 research outputs found
Aperiodicity, Star-freeness, and First-order Definability of Structured Context-Free Languages
A classic result in formal language theory is the equivalence among
noncounting, or aperiodic, regular languages, and languages defined through
star-free regular expressions, or first-order logic. Together with first-order
completeness of linear temporal logic these results constitute a theoretical
foundation for model-checking algorithms. Extending these results to structured
subclasses of context-free languages, such as tree-languages did not work as
smoothly: for instance W. Thomas showed that there are star-free tree languages
that are counting. We show, instead, that investigating the same properties
within the family of operator precedence languages leads to equivalences that
perfectly match those on regular languages. The study of this old family of
context-free languages has been recently resumed to enhance not only parsing
(the original motivation of its inventor R. Floyd) but also to exploit their
algebraic and logic properties. We have been able to reproduce the classic
results of regular languages for this much larger class by going back to string
languages rather than tree languages. Since operator precedence languages
strictly include other classes of structured languages such as visibly pushdown
languages, the same results given in this paper hold as trivial corollary for
that family too
Entropy of regular timed languages
For timed languages, we define size measures: volume for languages with a fixed finite number of events, and entropy (growth rate) as asymptotic measure for an unbounded number of events. These measures can be used for quantitative comparison of languages, and the entropy can be viewed as information contents of a timed language. For languages accepted by deterministic timed automata, we give exact formulas for volumes. We show that automata with non-vanishing entropy ("thick") have a normal (non-Zeno, discretizable etc.) behavior for typical runs. Next, we characterize the entropy, using methods of functional analysis, as the logarithm of the leading eigenvalue (spectral radius) of a positive integral operator. We devise a couple of methods to compute the entropy: a symbolical one for so-called "1 1 ⁄2-clock" automata, and a numerical one (with a guarantee of convergence)
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