There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality.

Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

I and i

We propose the class of visibly pushdown languages as embeddings of context-free languages that is rich enough to model program analysis questions and yet is tractable and robust like the class of regular languages. In our definition, the input symbol determines when the pushdown automaton can push or pop, and thus the stack depth at every position. We show that the resulting class Vpl of languages is closed under union, intersection, complementation, renaming, concatenation, and Kleene-*, and problems such as inclusion that are undecidable for context-free languages are Exptime-complete for visibly pushdown automata. Our framework explains, unifies, and generalizes many of the decision procedures in the program analysis literature, and allows algorithmic verification of recursive programs with respect to many context-free properties including access control properties via stack inspection and correctness of procedures with respect to pre and post conditions. We demonstrate that the class Vpl is robust by giving two alternative characterizations: a logical characterization using the monadic second order (MSO) theory over words augmented with a binary matching predicate, and a correspondence to regular tree languages. We also consider visibly pushdown languages of infinite words and show that the closure properties, MSO-characterization and the characterization in terms of regular trees carry over. The main difference with respect to the case of finite words turns out to be determinizability: nondeterministic Buchi visibly pushdown automata are strictly more expressive than deterministic Muller visibly pushdown automata.

/pdf/visibly-pushdown-languages-4wbafpk2oq.pdf

Visibly pushdown languages

On the basis of regular tree grammars, we present a formal framework for XML schema languages. This framework helps to describe, compare, and implement such schema languages in a rigorous manner. Our main results are as follows: (1) a simple framework to study three classes of tree languages (local, single-type, and regular); (2) classification and comparison of schema languages (DTD, W3C XML Schema, and RELAX NG) based on these classes; (3) efficient document validation algorithms for these classes; and (4) other grammatical concepts and advanced validation algorithms relevant to an XML model (e.g., binarization, derivative-based validation).

/pdf/taxonomy-of-xml-schema-languages-using-formal-language-4rlpk660zg.pdf

Taxonomy of XML schema languages using formal language theory

We propose the model of nested words for representation of data with both a linear ordering and a hierarchically nested matching of items. Examples of data with such dual linear-hierarchical structure include executions of structured programs, annotated linguistic data, and HTML/XML documents. Nested words generalize both words and ordered trees, and allow both word and tree operations. We define nested word automata—finite-state acceptors for nested words, and show that the resulting class of regular languages of nested words has all the appealing theoretical properties that the classical regular word languages enjoys: deterministic nested word automata are as expressive as their nondeterministic counterparts; the class is closed under union, intersection, complementation, concatenation, Kleene-a, prefixes, and language homomorphisms; membership, emptiness, language inclusion, and language equivalence are all decidable; and definability in monadic second order logic corresponds exactly to finite-state recognizability. We also consider regular languages of infinite nested words and show that the closure properties, MSO-characterization, and decidability of decision problems carry over.The linear encodings of nested words give the class of visibly pushdown languages of words, and this class lies between balanced languages and deterministic context-free languages. We argue that for algorithmic verification of structured programs, instead of viewing the program as a context-free language over words, one should view it as a regular language of nested words (or equivalently, a visibly pushdown language), and this would allow model checking of many properties (such as stack inspection, pre-post conditions) that are not expressible in existing specification logics.We also study the relationship between ordered trees and nested words, and the corresponding automata: while the analysis complexity of nested word automata is the same as that of classical tree automata, they combine both bottom-up and top-down traversals, and enjoy expressiveness and succinctness benefits over tree automata.

/pdf/adding-nesting-structure-to-words-14v1ybheac.pdf

Adding nesting structure to words

The handbook of language contact

Mixed languages

Balanced grammars are a generalization of parenthesis grammars in two directions. First, several kind of parentheses are allowed. Next, the set of right-hand sides of productions may be an infinite regular language. XML-grammars are a special kind of balanced grammars. This paper studies balanced grammars and their languages. It is shown that there exists a unique minimal balanced grammar equivalent to a given one. Next, balanced languages are characterized through a property of their syntactic congruence. Finally, we show how this characterization is related to previous work of McNaughton and Knuth on parenthesis languages.

Balanced grammars and their languages

This chapter is concerned with the design and analysis of algorithms for minimizing finite automata. Getting a minimal automaton is a fundamental issue in the use and implementation of finite automata tools in frameworks like text processing, image analysis, linguistic computer science, and many other applications. There are two main families of minimization algorithms. The first by a sequence of refinements of a partition of the set of states, the second by a sequence of fusions or merges of states. Hopcroft's and Moore's algorithms belong to the first family, the linear-time minimization of acyclic automata of Revuz belongs to the second family. 
One of our studies is upon the comparison of the nature of Moore's and Hopcroft's algorithms. This gives some new insight in both algorithms. As we shall see, these algorithms are quite different both in behavior and in complexity. In particular, we show that it is not possible to simulate the computations of one of the algorithm by the other. We describe the minimization algorithm by fusion for so-called local automata. A special case of minimization is the construction o minimal automata for finite sets. We consider briefly this case, and in particular describe incremental algorithms. Finally, we consider the case of updating a minimal automaton when a word is added or removed from the set it recognizes.

Minimization of Automata

There is an increasing interest in the shuffle product on formal languages, mainly because it is a standard tool for modeling process algebras. It still remains a mysterious operation on regular languages. Antonio Restivo proposed as a challenge to characterize the smallest class of languages containing the singletons and closed under Boolean operations, product and shuffle. This problem is still widely open, but we present some partial results on it. We also study some other smaller classes, including the smallest class containing the languages composed of a single word of length 2 which is closed under Boolean operations and shuffle by a letter (resp. shuffle by a letter and by the star of a letter). The proof techniques have both an algebraic and a combinatorial flavor.

http://www-igm.univ-mlv.fr/~berstel/Articles/2010ExpressivPowerShuffleProduct.pdf

The expressive power of the shuffle product

We consider Sturmian trees as a natural generalization of Sturmian words. A Sturmian tree is a tree having n+1 distinct subtrees of height n for each n. As for the case of words, Sturmian trees are irrational trees of minimal complexity.

We prove that a tree is Sturmian if and only if the minimal automaton associated to its language is slow, that is if the Moore minimization algorithm splits exactly one equivalence class at each step. We give various examples of Sturmian trees, and we introduce two parameters on Sturmian trees, called the degree and the rank. We show that there is no Sturmian tree of finite degree at least 2 and having finite rank. We characterize the family of Sturmian trees of degree 1 and having finite rank by means of a structural property of their minimal automata.

Luc Boasson

Papers

Mixed languages

Balanced grammars and their languages

Minimization of Automata

The expressive power of the shuffle product

Sturmian Trees