Four different kinds of grammars that can define crossing dependencies in human language are compared here: (i) context sensitive rewrite grammars with rules that depend on context, (ii) matching grammars with constraints that filter the generative structure of the language, (iii) copying grammars which can copy structures of unbounded size, and (iv) generating grammars in which crossing dependencies are generated from a finite lexical basis. Context sensitive rewrite grammars are syntactically, semantically and computationally unattractive. Generating grammars have a collection of (...) nice properties that ensure they define only “mildly context sensitive” languages, and Joshi has proposed that human languages have those properties too. But for certain distinctive kinds of crossing dependencies in human languages, copying or matching analyses predominate. Some results relevant to the viability of mildly context sensitive analyses and some open questions are reviewed. (shrink)

Head-driven phrase structure grammar (HPSG) is one of the most prominent theories employed in deep parsing of natural language. Many linguistic theories are arguably best formalized in extensions of modal or dynamic logic (Keller, Feature logics, infinitary descriptions and grammar, 1993; Kracht, Linguistics Philos 18:401–458, 1995; Moss and Tiede, In: Blackburn, van Benthem, and Wolther (eds.) Handbook of modal logic, 2006), and HPSG seems to be no exception. Adequate extensions of dynamic logic have not been studied in detail, however; the (...) most important aspect is the reference to sets of substructures. In this paper, an adequate extension is identified, and some important results are established: Satisfiability is highly undecidable, and model checking is shown to be in EXPTIME and PSPACE-hard. A fragment with polynomial time model checking procedures is identified; it is shown to cover considerable fragments of HPSG. (shrink)

We sketch an axiomatic reformalization of Generalized Phrase StructureGrammar (GPSG) – a definition purely within the language ofmathematical logic of the theory GPSG embodies. While this treatment raisesa number of theoretical issues for GPSG, our focus is not thereformalization itself but rather the method we employ. The model-theoreticapproach it exemplifies can be seen as a natural step in the evolution ofconstraint-based theories from their grammar-based antecedents. One goal ofthis paper is to introduce this approach to a broader audience and todemonstrate (...) its application to an existing theory. As such, it joins agrowing literature of similar studies. Prior studies, however, have had anumber of weaknesses – they generally offer little in the way ofconcrete examples of the advantages the approach has to offer, theytypically ignore significant portions of the theories they address, and, byfully abstracting away from the notion of grammar mechanism, they largelyabandon the possibility of establishing meaningful complexity results. Thesecond goal of the paper is to address these issues. Our thrust is to sketchthe reformalization sufficiently to illustrate the way in which it capturesFeature Specification Defaults (FSDs) and the Exhaustive Constant PartialOrdering (ECPO) property. Our definition of FSDs is considerably simplifiedrelative to the original formalization and is free of the procedural flavorthat has led some to assume that FSDs are inherently dynamic. Our treatmentof ECPO uncovers a gap in its definition in the context of partialcategories that has heretofore gone unnoticed. We offer these as ademonstration of the kind of insight that a model-theoretic reinterpretationcan bring to an existing theory. Further, since these are the types ofproperties that prior studies in this genre have failed to address, FSDs andECPO provide a means for us to explore the limitations of these approachesand to offer ways of overcoming them. Finally, the logical framework weemploy has a well defined generative capacity – definability in thisframework characterizes strong context-freeness in a particular sense. Thus,despite being more abstract than its constraint-based predecessors, themodel-theoretic approach, as exemplified here, can offer stronger complexityresults than are typically available in the constraint-based framework. (shrink)

We present a rendering of some common grammatical formalisms in terms of evolving algebras. Though our main concern in this paper is on constraint-based formalisms, we also discuss the more basic case of context-free grammars. Our aim throughout is to highlight the use of evolving algebras as a specification tool to obtain grammar formalisms.

This book sets out the foundations, methodology, and practice of a formal framework for the description of language. The approach embraces the trends of lexicalism and compositional semantics in computational linguistics, and theoretical linguistics more broadly, by developing categorial grammar into a powerful and extendable logic of signs. Taking Montague Grammar as its point of departure, the book explains how integration of methods from philosophy (logical semantics), computer science (type theory), linguistics (categorial grammar) and meta-mathematics (mathematical logic ) provides a (...) categorial foundation with coverage including intensionality, quantification, featural polymorphism, domains and constraints. For the first time, the book systematises categorial thinking into a unified program which is at once both logically secured, and a practical tool for pure lexical grammar development with type-theoretic semantics. It should be of interest to all those active in computational linguistics and formal grammar and is suitable for use at advanced undergraduate, postgraduate, and research levels. (shrink)

We define an order independent version of default unification on typed feature structures. The operation is one where default information in a feature structure typed with a more specific type, will override default information in a feature structure typed with a more general type, where specificity is defined by the subtyping relation in the type hierarchy. The operation is also able to handle feature structures where reentrancies are default. We provide a formal semantics, prove order independence and demonstrate the utility (...) of this version of default unification in several linguistic applications. First, we show how it can be used to define multiple orthogonal default inheritance in the lexicon in a fully declarative fashion. Secondly, we show how default lexical specifications (introduced via default lexical inheritance) can be made to usefully persist beyond the lexicon and interact with syntagmatic rules. Finally, we outline how persistent default unification might underpin default feature propagation principles and a more restrictive and constraint-based approach to lexical rules. (shrink)

This paper investigates whether the fundamental linguistic insights and intuitions of Lexical Functional Grammar, which is usually presented as a constraint-based linguistic theory, can be reformulated in a resource sensitive framework using a substructural modal logic. In the approach investigated here, LFG's f-descriptions are replaced with expressions from a multi-modal propositional logic. In effect, the feature structure unification basis of LFG's f-structures is replaced with a very different resource based mechanism. It turns out that some linguistic analyses that required non-monotonic (...) devices in LFG analysis of Icelandic) can be straightforwardly expressed in the framework presented here. Moreover, a Curry– Howard correspondence between proofs in this logic and -terms provides a semantic interpretation as a by-product of the process of showing syntactic well-formedness. (shrink)

Unification grammars are known to be Turing-equivalent; given a grammar G and a word w, it is undecidable whether w L(G). In order to ensure decidability, several constraints on grammars, commonly known as off-line parsability (OLP), were suggested, such that the recognition problem is decidable for grammars which satisfy OLP. An open question is whether it is decidable if a given grammar satisfies OLP. In this paper we investigate various definitions of OLP and discuss their interrelations, proving that some of (...) the OLP variants are indeed undecidable. We then present a novel, decidable OLP constraint which is more liberal than the existing decidable ones. (shrink)

This article presents a detailed formal approach to concepts and concept combination. Sense generation is a competence‐level theory that attempts to respect constraints from the various cognitive sciences, and postulates “quasi‐classical” conceptual structures where attributes receive only one value (but are defeasible and so do not represent necessary and sufficient conditions on category membership) and where classification is binary (but explicitly context‐sensitive). It is also argued that any general theory of concepts must account for “privative” combinations (e.g., stone lion, fake (...) gun, apparent friend) as extreme test‐cases of representational and classificatory flexibility. The approach presented therefore provides a treatment of these combinations. The approach differentiates between the “lexical concept” (the stable information represented in a mental lexicon) which acts as a base from which the various “senses” (flexible contents associated with words and phrases in context, and used in classification) are “generated.” Generation allows nonmonotonicity, so that in different circumstances, different attributes may be defeated or modified. Classification is treated as relative to the perspective adopted, so that a classification acceptable from one perspective may be unacceptable from another, without contradiction. The result is a view that assumes bottom‐up priority in concept combination, where the range of senses generated by bottom‐up rules of combination is tempered by pragmatic‐communicative constraints on classification. An account of the representational and classification behavior of privative combinations is outlined, and the article concludes with a discussion of some of the implications of the approach. (shrink)

Unification grammars are widely accepted as an expressive means for describing the structure of natural languages. In general, the recognition problem is undecidable for unification grammars. Even with restricted variants of the formalism, off-line parsable grammars, the problem is computationally hard. We present two natural constraints on unification grammars which limit their expressivity and allow for efficient processing. We first show that non-reentrant unification grammars generate exactly the class of context-free languages. We then relax the constraint and show that one-reentrant (...) unification grammars generate exactly the class of mildly context-sensitive languages. We thus relate the commonly used and linguistically motivated formalism of unification grammars to more restricted, computationally tractable classes of languages. (shrink)

Feature logics are the logical basis for so-called unification grammars studied in computational linguistics. We investigate the expressivity of feature terms with negation and the functional uncertainty construct needed for the description of long-distance dependencies and obtain the following results: satisfiability of feature terms is undecidable, sort equations can be internalized, consistency of sort equations is decidable if there is at least one atom, and consistency of sort equations is undecidable if there is no atom.