Degrees of Free Word-Order and Freely Rewriting Restarting Automata

In natural languages with a high degree of word-order freedom syntactic phenomena like dependencies (subordinations) or valencies do not depend on the word-order (or on the individual positions of the individual words). This means that some permutations of sentences of these languages are in some (important) sense syntactically equivalent. Here we study this phenomenon in a formal way. Various types of j-monotonicity for restarting automata can serve as parameters for the degree of word-order freedom and for the complexity of word-order in sentences (languages). Here we combine two types of parameters on computations of restarting automata: 1. the degree of j-monotonicity, and 2. the number of rewrites per cycle. We study these notions formally in order to obtain an adequate tool for modelling and comparing formal descriptions of (natural) languages with different degrees of word-order freedom and word-order complexity.

The English novel in the time of Shakespeare

The rise of the English novel needs rethinking after it has been confined to the "formal realism" of Defoe, Richardson, and Fielding (Watt, 1957), to "antecedents, forerunners" (Schlauch, 1968; Klein, 1970) or to mere "prose fiction" (McKillop, 1951; Davis, Richetti, 1969; Fish, 1971; Salzman, 1985; Kroll, 1998). My paper updates a book by Jusserand under the same title (1890) by proving that the social and moral history of the long prose genre admits no strict separation of "novel" and "romance", as both concepts are intertwined in most fiction (Cuddon, Preston, 1999; Mayer, 2000). The rise of the novel, seen in its European context, mainly in France and Spain (Kirsch, 1986), and equally in England, was due to the melting of the nobility and high bourgeoisie into a "meritocracy", or to its failure, to become the new bearer of the national culture, around 1600. (Brink, 1998). My paper will concentrate on Euphues (1578), a negative romance, Euphues and His England (1580), a novel of manners, both by Lyly; Arcadia (1590-93) by Sidney, a political roman à clef in the disguise of a Greek pastoral romance; The Unfortunate Traveller (1594) by Nashe, the first English picaresque novel, and on Jack of Newbury (1596-97) by Deloney, the first English bourgeois novel. My analysis of the central values in these novels will prove a transition from the aristocratic cardinal virtues of WISDOM, JUSTICE, COURAGE, and HONOUR to the bourgeois values of CLEVERNESS, FAIR PLAY, INDUSTRY, and VIRGINITY. A similar change took place from the Christian virtues of LOVE, FAITH, HOPE to business values like SERVICE, TRUST, and OPTIMISM. Thus, the legacy of history proves that the main concepts of the novel of manners, of political romance, of picaresque and middle-class fiction were all developed in the time of Shakespeare.

The Degree of Word-Expansion of Lexicalized RRWW-Automata

Restarting automata can be seen as analytical variants of classical automata as well as of regulated rewriting systems. We study a measure for the degree of nondeterminism of (context-free) languages in terms of deterministic restarting automata that are (strongly) lexicalized. This measure is based on the number of auxiliary symbols (categories) used for recognizing a language as the projection of its characteristic language onto its input alphabet. This type of recognition is typical for analysis by reduction, a method used in linguistics for the creation and verification of formal descriptions of natural languages. Our main results establish a hierarchy of classes of context-free languages and two hierarchies of classes of non-context-free languages that are based on the expansion factor of a language.

On Vessiot's Theory of Partial Differential Equations

The object of research presented here is Vessiot's theory of partial differential equations: for a given differential equation one constructs a distribution both tangential to the differential equation and contained within the contact distribution of the jet bundle. Then within it, one seeks n-dimensional subdistributions which are transversal to the base manifold, the integral distributions. These consist of integral elements, and these again shall be adapted so that they make a subdistribution which closes under the Lie-bracket. This then is called a flat Vessiot connection. Solutions to the differential equation may be regarded as integral manifolds of these distributions. In the first part of the thesis, I give a survey of the present state of the formal theory of partial differential equations: one regards differential equations as fibred submanifolds in a suitable jet bundle and considers formal integrability and the stronger notion of involutivity of differential equations for analyzing their solvability. An arbitrary system may (locally) be represented in reduced Cartan normal form. This leads to a natural description of its geometric symbol. The Vessiot distribution now can be split into the direct sum of the symbol and a horizontal complement (which is not unique). The n-dimensional subdistributions which close under the Lie bracket and are transversal to the base manifold are the sought tangential approximations for the solutions of the differential equation. It is now possible to show their existence by analyzing the structure equations. Vessiot's theory is now based on a rigorous foundation. Furthermore, the relation between Vessiot's approach and the crucial notions of the formal theory (like formal integrability and involutivity of differential equations) is clarified. The possible obstructions to involution of a differential equation are deduced explicitly. In the second part of the thesis it is shown that Vessiot's approach for the construction of the wanted distributions step by step succeeds if, and only if, the given system is involutive. Firstly, an existence theorem for integral distributions is proven. Then an existence theorem for flat Vessiot connections is shown. The differential-geometric structure of the basic systems is analyzed and simplified, as compared to those of other approaches, in particular the structure equations which are considered for the proofs of the existence theorems: here, they are a set of linear equations and an involutive system of differential equations. The definition of integral elements given here links Vessiot theory and the dual Cartan-Kähler theory of exterior systems. The analysis of the structure equations not only yields theoretical insight but also produces an algorithm which can be used to derive the coefficients of the vector fields, which span the integral distributions, explicitly. Therefore implementing the algorithm in the computer algebra system MuPAD now is possible.