A tight upper bound for the path length of AVL trees

AbstractWe prove that the internal path length of an AVL tree of size N is bounded from above by 1.4404N(log2 N-log2log2N)+O(N) and show that this bound is achieved by an infinite family of AVL trees, each tree of which is not of maximal height. These results carry over to the comparison cost of brother trees

Checking Sets, Test Sets, Rich Languages and Commutatively Closed Languages

AbstractThe problem of homomorphism equivalence is to decide for some language L over some finite alphabet Σ and two homomorphisms f and g whether or not f (x) = g(x) for all x in L. It has been conjectured that each L can be represented by some finite subset F such that for all pairs of homomorphisms f and g: f (x) = g(x) for all x in F implies f (x) = g(x) for all x in L. This conjecture is proved for the families of rich and commutatively closed languages. Lower and upper bounds are derived for the sizes of these finite subsets and examples of language families are given for which there are effective constructions of these subsets

Standard Generalized Markup Language: Mathematical and Philosophical Issues

. The Standard Generalized Markup Language (SGML), an ISO standard, has become the accepted method of defining markup conventions for text files. SGML is a metalanguage for defining grammars for textual markup in much the same way that Backus--Naur Form is a metalanguage for defining programming-language grammars. Indeed, HTML, the method of marking up a hypertext documents for the World Wide Web, is an SGML grammar. The underlying assumptions of the SGML initiative are that a logical structure of a document can be identified and that it can be indicated by the insertion of labeled matching brackets (start and end tags). Moreover, it is assumed that the nesting relationships of these tags can be described with an extended context-free grammar (the right-hand sides of productions are regular expressions). In this survey of some of the issues raised by the SGML initiative, I reexamine the underlying assumptions and address some of the theoretical questions that SGML raises...

Data structures, algorithms, and performance

xiv, 594 p.; 24 cm

SGML and XML document grammars and exceptions

The Standard Generalized Markup Language (SGML) and the Extensible Markup Language (XML) allow users to define document type definitions (DTDs), which are essentially extended context-free grammars expressed in a notation that is similar to extended Backus-Naur form. The right-hand side of a production, called a content model, is both an extended and a restricted regular expression. The semantics of content models for SGML DTDs can be modified by exceptions (XML DTDs do not allow exceptions). Inclusion exceptions allow named elements to appear anywhere within the content of a content model, and exclusion exceptions preclude named elements from appearing in the content of a content model. We give precise definitions of the semantics of exceptions, and prove that they do not increase the expressive power of SGML DTDs when we restrict DTDs according to accepted practice. We prove the following results: - Exceptions do not increase the expressive power of extended context-free grammars. - For each DTD with exceptions, we can obtain a structurally equivalent extended context-free grammar - For each DTD with exceptions, we can construct a structurally equivalent DTD when we restrict the DTD to adhere to accepted practice - Exceptions are a powerful shorthand notation-eliminating them may cause exponential growth in the size of an extended context-free grammar or DT