Copyright (c) 2009 All rights reserved. http://works.bepress.com/htiede Recent documents in Hans Joerg Tiede en-us Tue, 03 Nov 2009 13:26:10 PST 3600 http://works.bepress.com/htiede/11 http://works.bepress.com/htiede/11 Sun, 13 Sep 2009 18:08:02 PDT Model theoretic syntax is concerned with studying the descriptive complexity of grammar formalisms for natural languages by defining their derivation trees in suitable logical formalisms. The central tool for model theoretic syntax has been monadic second-order logic (MSO). Much of the recent research in this area has been concerned with finding more expressive logics to capture the derivation trees of grammar formalisms that generate non-context-free languages. The motivation behind this search for more expressive logics is to describe formally certain mildly context-sensitive phenomena of natural languages. Several extensions to MSO have been proposed, most of which no longer define the derivation trees of grammar formalisms directly, while others introduce logically odd restrictions. We therefore propose to consider first-order transitive closure logic. In this logic, derivation trees can be defined in a direct way. Our main result is that transitive closure logic, even deterministic transitive closure logic, is more expressive in defining classes of tree languages than MSO. (Deterministic) transitive closure logics are capable of defining non-regular tree languages that are of interest to linguistics. Hans Joerg Tiede Model Theoretic Syntax http://works.bepress.com/htiede/10 http://works.bepress.com/htiede/10 Sun, 13 Sep 2009 15:27:12 PDT Geoffrey K. Pullum Methodology of Linguistics http://works.bepress.com/htiede/9 http://works.bepress.com/htiede/9 Sun, 13 Sep 2009 15:24:46 PDT Hans Joerg Tiede Methodology of Linguistics http://works.bepress.com/htiede/8 http://works.bepress.com/htiede/8 Sat, 16 Feb 2008 14:43:05 PST During the last fifteen years, much of the research of proof theoretical grammars has been focused on their weak generative capacity. This research culminated in Pentus' theorem, which showed that Lambek grammars generate precisely the context-free languages. However, during the same period of time, research on other grammar formalisms has stressed the importance of "strong generative capacity," i.e. the derivation or phrase structure trees that grammars assign to strings. The first topic of this thesis is the strong generative capacity of Lambek grammars. The proof theoretic perspective on grammars allows us to consider different notions of what "structure assigned by a Lambek grammar to a string" is taken to mean. For example, we can take any proof tree that establishes that a grammar generates a certain string or only those that are in some normal form. It can be shown that the formal properties of these notions of structure differ. The main result of this part of the thesis is that, although Lambek grammars generate context-free string languages, their derivation trees are more complex than those of context-free grammars. The latter were characterized by Thatcher as coinciding with the local tree languages, while the derivation trees of Lambek grammars include tree languages which are not regular. Even non-associative Lambek grammars, which recently have become more popular variants of categorial grammar, can be used to generate non-local tree languages. However, their normal form tree languages are always regular. Finally, categorial grammars lacking introduction rules have local derivation trees. Thus, there is a genuine hierarchy of proof theoretical grammars with respect to strong generative capacity. Additionally, we consider the semantic aspect of the proof theoretic approach to language, which is given by the correspondence between proof theory and type theory. Here we are interested in giving an algorithm for counting how many different normal derivations a given string has, corresponding to the degree of ambiguity of an expression. In order to count the number of proofs, we use methods from the theory of type assignment and the coherence theorem for residuated categories to characterize the most general types of normal-form terms. Hans Joerg Tiede Proof Theoretical Syntax Ph.D. Thesis http://works.bepress.com/htiede/7 http://works.bepress.com/htiede/7 Wed, 13 Feb 2008 16:44:49 PST While monadic second-order logic (MSO) has played a prominent role in model theoretic syntax, modal logics have been used in this context since its inception. When comparing propositional dynamic logic (PDL) to MSO over trees, Kracht (1997) noted that there are tree languages that can be defined in MSO that can only be defined in PDL by adding new features whose distribution is predictable. He named such features "inessential features". We show that Kracht's observation can be extended to other modal logics of trees in two ways. First, we demonstrate that for each stronger logic, there exists a tree language that can only be defined in a weaker logic with inessential features. Second, we show that any tree language that can be defined in a stronger logic, but not in some weaker logic, can be defined with inessential features. Additionally, we consider Kracht's definition of inessential features more closely. It turns out that there are features whose distribution can be predicted, but who fail to be inessential in Kracht's sense. We will look at ways to modify his definition. Hans Joerg Tiede Model Theoretic Syntax http://works.bepress.com/htiede/6 http://works.bepress.com/htiede/6 Wed, 13 Feb 2008 16:42:03 PST Lawrence Moss Applied Modal Logic http://works.bepress.com/htiede/5 http://works.bepress.com/htiede/5 Wed, 13 Feb 2008 16:39:41 PST Hans Joerg Tiede Proof Theoretical Syntax http://works.bepress.com/htiede/4 http://works.bepress.com/htiede/4 Wed, 13 Feb 2008 16:36:48 PST In this paper, we continue our investigation of the strong generative capacity of proof theoretical grammars using natural deduction proof trees as the structures that grammars assign to their languages. We review the results that were previously obtained for associative Lambek grammars and extend the methods used there to non-associative Lambek grammars. The main result of this paper is that non-associative Lambek grammars, which generate precisely the context-free string languages, can assign structures to their languages that are not local, thus more complex than the structures context-free grammars can assign. When only proof trees in normal form are considered, the tree languages assigned by non-associative Lambek grammars are always regular, thus their structures are less complex than those of associative Lambek grammars, which have been shown to be able to assign non-regular tree languages to their languages. As AB grammars (also known as categorial grammars) only assign local tree languages to their languages, we arrive at a hierarchy of proof theoretical grammars with respect to their strong generative capacity. Hans Joerg Tiede Proof Theoretical Syntax http://works.bepress.com/htiede/3 http://works.bepress.com/htiede/3 Wed, 13 Feb 2008 16:32:54 PST This paper is concerned with the study of the number of proofs of a sequent in the commutative Lambek calculus. We show that in order to count how many different proofs in \beta \eta -normal form a given sequent \Gamma \vdash \alpha has, it suffices to enumerate all the \Delta \vdash \beta which are "minimal", such that \Gamma \vdash \alpha is a substitution instance of \Delta \vdash \beta. As a corollary we obtain van Benthem's finiteness theorem for the Lambek calculus, which states that every sequent has finitely many different normal form proofs in the Lambek calculus. Hans Joerg Tiede Proof Theoretical Syntax http://works.bepress.com/htiede/2 http://works.bepress.com/htiede/2 Wed, 13 Feb 2008 16:28:33 PST We investigate the indentifiability in the limit of subclasses of generalized quantifiers definable in Presburger arithmetic. Hans Joerg Tiede Learnability Theory