. We provide first-order axioms for the theories of finite trees with bounded branching and finite trees with arbitrary (finite) branching. The signature is chosen to express, in a natural way, those properties of trees most relevant to linguistic theories. These axioms provide a foundation for results in linguistics that are based on reasoning formally about such properties. We include some observations on the expressive power of these theories relative to traditional language complexity classes. Key words: Trees, First-Order Theories, Axiomatizations, Natural Language Syntax, EhrenfeuchtFra iss'e Games 1. INTRODUCTION There has been, over the last ten or fifteen years, a growing body of research in generative and computational linguistics that depends to a great extent on reasoning formally about trees. For example, there are a number of grammatical formalisms that have been proposed that manipulate logical descriptions of the trees representing the syntactic structure of strings r...

Contents 1 Language Complexity in Generative Grammar 3 Part I The Descriptive Complexity of Strongly Context-Free Languages 11 2 Introduction to Part I 13 3 Trees as Elementary Structures 15 4 L 2 K;P and SnS 25 5 Definability and Non-Definability in L 2 K;P 35 6 Conclusion of Part I 57 DRAFT 2 / Contents Part II The Generative Capacity of GB Theories 59 7 Introduction to Part II 61 8 The Fundamental Structures of GB Theories 69 9 GB and Non-definability in L 2 K;P 79 10 Formalizing X-Bar Theory 93 11 The Lexicon, Subcategorization, Theta-theory, and Case Theory 111 12 Binding and Control 119 13 Chains 131 14 Reconstruction 157 15 Limitations of the Interpretation 173 16 Conclusion of Part II 179 A Index of Definitions 183 Bibliography DRAFT 1<

A modal logic is developed to deal with finite ordered binary trees as they are used in (computational) linguistics. A modal language is introduced with operators for the `mother of', `first daughter of' and `second daughter of' relations together with their transitive reflexive closures. The relevant class of tree models is defined and three linguistic applications of this language are discussed: context free grammars, command relations, and trees decorated with feature structures. An axiomatic proof system is given for which completeness is shown with respect to the class of finite ordered binary trees. A number of decidability results follow.

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This report gives a formal denition of a decription language for syntax graphs, called the TIGER language. This formal definition complements the informal introduction to the TIGER language which is given in the TIGERSearch user's manual [12]. Syntax graphs are close relatives to...

This paper is formal and quite difficult for readers untrained in modal logic; I have no illusions about this and I apologize in advance if I fail to make things as clear and simple as I should. I do believe, however, that much of the complexity in this paper is unavoidable and anything that is simpler will be so at the cost of precision. Almost everything will be defined here, so that the discussion will on the whole be self-contained. But this is really not to say much when it come to mathematical topics. The reader who is seriously interested should perhaps read an introductory book on modal logic and the lucid survey article [ Bull and Segerberg, 1984 ] to get enough background. I can also recommend [ Blackburn, 1993 ] as an introduction into modal logic in connection with avms. It is impossible to go through all technical proofs in great detail; this would be tantamount to writing a book on this topic. But, I hope, the line of argumentation can be understood even without a proper understanding of the technical points. For the message is of wider importance. If I am right, then modal logic, where it fails, fails necessarily Is there a genuine modal perspective on feature structures? 3 -- and no other framework I know of will not under these circumstances. Secondly, it provides enough technical apparatus to allow to prove significant results. To those who remain unimpressed I can only appeal to their sense of beauty and naturalness. Among the persons who have quite generally helped to shape my views on syntax and logic I wish to thank explicitly those who have contributed to the present paper. These are Mark Ellison and two anonymous referees, who had the questionable pleasure of reading an earlier version of this paper. Moreover, the results on modal feature logic...

In H. Kleine Buning, editor, Computer Science Logic `95 , LNCS 1092, pages 86--105, Springer-Verlag, 1996.

Tree descriptions based on dominance constraints are popular in several areas of computational linguistics including syntax, semantics, and discourse. Tree descriptions in the language of context unification have attracted some interest in unification and rewriting theory. Recently, dominance constraints and context unification have both been used in different underspecified approaches to the semantics of scope, parallelism, and their interaction. This raises the question whether both description languages are related. In this paper, we show for a first time that dominance constraints can be expressed in context unification. We also prove that dominance constraints extended with parallelism constraints are equal in expressive power to context unification.

User-definable strategies for the application of rewrite rules provide a means to construct transformation systems that apply rewrite rules in a controlled way. This paper describes a strategy language and its interpretation. The language is used to control the rewriting of terms using labeled rewrite rules. Rule labels are atomic strategies. Compound strategies are formed by means of sequential composition, nondeterministic choice, left choice, fixed point recursion, and two primitives for expressing term traversal. Several complex strategies such as bottom-up and top-down application and (parallel) innermost and (parallel) outermost reduction can be defined in terms of these primitives. The paper contains two case studies of the application of strategies. 1 Introduction Term rewriting is an ideal technique for program transformation where the transformation of one construct into another is defined by means of rewrite rules. Usually, the rewrite engine contracts redexes according to ...

A natural next step in the evolution of constraint-based grammar formalisms from rewriting formalisms is to abstract fully away from the details of the grammar mechanism--to express syntactic theories purely in terms of the properties of the class of structures they license. By focusing on the structural properties of languages rather than on mechanisms for generating or checking structures that exhibit those properties, this model-theoretic approach can offer simpler and significantly clearer expression of theories and can potentially provide a uniform formalization, allowing disparate theories to be compared on the basis of those properties. We discuss L,p, a monadic second-order logical framework for such an approach to syn- tax that has the distinctive virtue of being superficially expressive--supporting direct statement of most linguistically significant syntactic properties -- but having well-defined strong generative capacity -- languages are definable in L,p iff they are strongly context-free. We draw examples from the realms of GPSG and GB.