Abstract

Let f be an arbitrary Boolean function depending on n variables and let A be a network computing them, i.e., A has n inputs and one output and for an arbitrary Boolean vector a of length n outputs f(a). Assume we have to compute simultaneously the values f(a1), …,f(ar) of f on r arbitrary Boolean vectors a1, …,ar. Then we can do it by r copies of A. But in most cases it can be done more efficiently (with a smaller complexity) by one network with nr inputs and r outputs (as already shown in Uhlig (1974)). In this paper we present a new and simple proof of this fact based on a new construction method. Furthermore, we show that the depth of our network is “almost” minimal.

Introduction

Let us consider (combinatorial) networks. Precise definitions are given in [Lu58, Lu65, Sa76, We87]. We assume that a complete set G of gates is given, i.e., every Boolean function can be computed (realized) by a network consisting of gates of G. For example, the set consisting of 2-input AND, 2-input OR and the NOT function is complete. A cost C(Gi) (a positive number) is associated with each of the gates Gi ∈ G. The complexity C(A) of a network A is the sum of the costs of its gates. The complexity C(f) of a Boolean function f is defined by C(f) = min C(A) where A ranges over all networks computing f.

By Bn we denote the set of Boolean functions {0, l}n → {0, 1}.

INTRODUCTION

Before turning to some selected issues in type theory, type-shifting and the semantics of noun phrases, which form the main topic of this chapter, I will briefly mention some other, very exciting topics of current research in formal semantics that are relevant to the theme of this book, but which do not receive the attention they deserve in what follows.

One is the centrality of context dependence, context change, and a more dynamic view of the basic semantic values. This view of meanings, in terms of functions from contexts to contexts, is important both for the way it helps us make better sense of such things as temporal anaphora, and also for the way it solves some of the crucial foundational questions about presuppositions and the principles governing presupposition projection. The work of Irene Heim [86] is central in this area.

Another crucial area that will not be covered is the recently emerging emphasis on the algebraic structure of certain model-theoretic domains, such as entities, events or eventualities. As well as the work of Godehard Link [137] and Jan-Tore Lønning [140] in this area, referred to by Fenstad [54], I would also mention the attempt by Emmon Bach [6] to generalize the issue into questions about natural language metaphysics. An example of such a question is whether languages that grammaticize various semantic distinctions (such as mass versus count nouns) differently have different ontologies, or whether Link's use of Boolean algebras, which may be either atomic or not necessarily atomic, provides a framework in which to differentiate those structures that are shared from additional ones that only some languages express through grammar.

INTRODUCTION

In the linguistic study of syntax, various formats have been developed for measuring the combinatorial power of natural languages. In particular, there is Chomsky's well-known hierarchy of grammars which can be employed to calibrate syntactic complexity. No similar tradition exists in linguistic semantics, however. Existing powerful formalisms for stating truth conditions, such as Montague Semantics, are usually presented as monoliths like Set Theory or Intensional Type Theory, without any obvious way of expressive fine-tuning. The purpose of this chapter is to show how semantics has its fine-structure too, when viewed from a proper logical angle.

The framework used to demonstrate this claim is that of Categorial Grammar (see Buszkowski et al. [24] or Oehrle et al. [173] for details). In this linguistic paradigm, syntactic sensitivity resides in the landscape of logical calculi of implication, which manipulate functional types as conditional propositions. The landscape starts from a classical Ajdukiewicz system with modus ponens only and then ascends to intuitionistic conditional logic, which allows also subproofs with conditionalisation. A well-known intermediate system is the Lambek Calculus (Moortgat [156], van Benthem [232]), which takes special care in handling occurrences of propositions or types. These calculi represent various ‘engines’ for driving categorial parsing, which can be studied as to their formal properties by employing the tools of logical proof theory.

RELATING THE DISCIPLINES

My role in the original workshop which gave rise to this book was to sum up what had been said during the week, trying to relate the different contributions one to another. In preparing for what threatened to be an arduous task, it occurred to me that a fundamental assumption underlying the organization of such a workshop needed to be taken out and examined overtly: calling a workshop ‘Computational Linguistics and Formal Semantics’ assumes that these two areas of academic endeavour have something to say to each other; may, even, be inextricably related.

Superficially, of course, this seems likely to be true: an investigation of language and its use could be seen as the core interest of both disciplines. But, on looking a little more closely, the intimate connection tends to evaporate. For most computational linguists, the task is to define and implement an adequate treatment of (some subset of) some natural language within the framework of a computer application. Even when they are primarily interested in demonstrating that a particular linguistic theory or a particular paradigm of computation is superior to its rivals, the argument will frequently be couched in application-oriented terms: such and such a theory is superior because it leads to clearer linguistic descriptions which are easier to modify and to debug, for example, or such and such a computational paradigm is superior because it allows linguistic programmers to compartmentalize their knowledge of a language, describing sub-parts independently without having to worry about all the possible intricate complexities of interactions between them.

INTRODUCTION

This paper is a tutorial on property-theoretic semantics: our aim is to provide an accessible account of property theory and its application to the formal semantics of natural language. We develop a simple version of property theory and provide the semantics for a fragment of English in the theory. We shall say more about the particular form of the theory in the next section but to begin with we outline the main reasons why we believe property-theoretic semantics to be worthy of attention.

INTENSIONALITY

The main motivation for the development of a theory of propositions and properties stems from the desire to develop an adequate account of intensionality in natural language. In this section we briefly review some possible approaches to intensionality in order to motivate the approach we shall eventually adopt.

INTENSIONALITY VIA POSSIBLE WORLDS

Traditional approaches to intensionality employ the notion of possible world. Propositions are taken to be sets of possible worlds and properties to be functions from individuals to propositions. The modal and doxastic operators are then unpacked as functions from propositions to propositions. For example, the modal operator of necessity is analysed as that function which maps a proposition P to that proposition which consists of all those worlds whose accessible worlds are elements of P. Different choices of the relation of accessibility between worlds yield different modal and doxastic notions. Kripke [127] introduced the possible world analysis of necessity and possibility while Hintikka [88] extended the analysis to deal with belief and knowledge.

The workshop that inspired this book was the high point of a collaborative research project entitled ‘Incorporation of Semantics into Computational Linguistics’, supported by the EEC through the COST13 programme. The workshop was also the first official academic event to be organized by IDSIA, a newly inaugurated institute of the Dalle Molle Foundation situated in Lugano, Switzerland devoted to artificial intelligence research.

A few words about the project may help to place the contents of the book in its proper context. The underlying theme was that although human language is studied from the standpoint of many different disciplines, there is rather less constructive interaction between the disciplines than might be hoped for, and that for such interaction to take place, a common framework of some kind must be established. Computational linguistics (CL) and artificial intelligence (AI) were singled out as particular instances of such disciplines: each has focused on rather different aspects of the relationship between language and computation.

Thus, what we now call CL grew out of attempts to apply the concepts of computer science to generative linguistics (e.g. Zwicky and his colleagues [255], Petrick [181]). Given these historical origins, CL unwittingly inherited some of the goals for linguistic theory that we normally associate with Chomskyan linguistics. One such goal, for example, is to finitely describe linguistic competence, that is, the knowledge that underlies the human capacity for language by means of a set of universal principles that characterize the class of possible languages, and a set of descriptions for particular languages.

INTRODUCTION

This paper is written from the point of view of one who works in artificial intelligence (AI): the attempt to reproduce interesting and distinctive aspects of human behaviour with a computer, which, in my own case, means an interest in human language use.

There may seem little of immediate relevance to cognition or epistemology in that activity. And yet it hardly needs demonstration that AI, as an aspiration and in practice, has always been of interest to philosophers, even to those who may not accept the view that AI is, essentially, the pursuit of metaphysical goals by non-traditional means.

As to cognition in particular, it is also a commonplace nowadays, and at the basis of cognitive science, that the structures underlying AI programs are a guide to psychologists in their empirical investigations of cognition. That does not mean that AI researchers are in the business of cognition, nor that there is any direct inference from how a machine does a task, say translating a sentence from English to Chinese, to how a human does it. It is, however, suggestive, and may be the best intellectual model we currently have of how the task is done. So far, so well known and much discussed in the various literatures that make up cognitive science.

INTRODUCTION

Large practical computational-linguistic applications, such as machine translation systems, require a large number of knowledge and processing modules to be put together in a single architecture and control environment. Comprehensive practical systems must have knowledge about speech situations, goal-directed communicative actions, rules of semantic and pragmatic inference over symbolic representations of discourse meanings and knowledge of syntactic and phonological/graphological properties of particular languages. Heuristic methods, extensive descriptive work on building world models, lexicons and grammars as well as a sound computational architecture are crucial to the success of this paradigm. In this paper we discuss some paradigmatic issues in building computer programs that understand and generate natural language. We then illustrate some of the foundations of our approach to practical computational linguistics by describing a language for representing text meaning and an approach to developing an ontological model of an intelligent agent. This approach has been tested in the dionysus project at Carnegie Mellon University which involved designing and implementing a natural language understanding and generation system.

SEMANTICS AND APPLICATIONS

Natural language processing projects at the Center for Machine Translation of Carnegie Mellon University are geared toward designing and building large computational applications involving most crucial strata of language phenomena (syntactic, semantic, pragmatic, rhetorical, etc.) as well as major types of processing (morphological and syntactic parsing, semantic interpretation, text planning and generation, etc.). Our central application is machine translation which is in a sense the paradigmatic application of computational linguistics.

INTRODUCTION

This paper stands in marked contrast to many of the others in this volume in that it is intended to be entirely tutorial in nature, presupposing little on the part of the reader but a user's knowledge of English, a modicum of good will, and the desire to learn something about the notion of unification as it has come to be used in theoretical and computational linguistics. Most of the ideas I shall be talking about were first introduced to the study of ordinary language by computational linguists and their adoption by a notable, if by no means representative, subset of theoretical linguists represents an important milestone in our field, for it is the first occasion on which the computationalists have had an important impact on linguistic theory. Before going further, there may be some value in pursuing briefly why this rapprochement between these two branches has taken so long to come about. After all, the flow of information in the other direction – from theoretician to computationalist – has continued steadily from the start.

PRODUCTIVITY

The aspects of ordinary language that have proved most fascinating to its students all have something to do with its productivity, that is, with the fact that there appears to be no limit to the different utterances that can be made and understood in any of the languages of the world. Certainly, speakers can make and understand utterances that they have never made or understood before, and it is presumably largely in this fact that the immense power and flexibility of human language resides.

INTRODUCTION

The integration of syntactic and semantic processing has prompted a number of different architectures for natural language systems, such as rule-by-rule interpretation [221], semantic grammars [22] and cascaded ATNs [253]. The relationship between syntax and semantics has also been of central concern in theoretical linguistics, particularly following Richard Montague's work. Also, with the recent rapprochement between theoretical and computational linguistics, Montague's interpretation scheme has made its way into natural language systems. Variations on Montague's interpretation scheme have been adopted and implemented in several syntactic theories with a significant following in computational linguistic circles. The first steps in this direction were taken by Hobbs and Rosenschein [89]. A parser for LFG was augmented with a Montagovian semantics by Halvorsen [76]. GPSG has been similarly augmented by Gawron et al. [65], and Schubert and Pelletier [206] followed with a compositional interpretation scheme using a first order logic rather than Montague's computationally intractable higher-order intensional logic.

In parallel with these developments in the syntax/semantics interface, unification-based mechanisms for linguistic description had significant impact both on syntactic theory and syntactic description. But the Montagovian view of compositionality in semantics and the architectures for configuring the syntax/semantics interface were slower to achieve a similar revaluation in view of unification and the new possibilities for composition which it brought.

INTRODUCTION

This chapter concerns experiments with Situation Schemata as a linguistic representation language, in the first instance for Machine Translation purposes but not exclusively so. The work reported is primarily concerned with the adaptation of the Situation Schemata language presented by Fenstad et al. [56] (but see also Fenstad et al. in this volume) to an implementation within grammars having interestingly broad linguistic coverage. We shall also consider the computational tools required for such an implementation and our underlying motivation. The chapter is therefore not solely concerned with the form of the representation language and its relation to any eventual interpretation language, but also with practical and methodological issues associated with the implementation of current theories in Computational Linguistics. A key theme is the interplay between linguistic information, representation and appropriate mechanisms for abstraction.

The rest of the chapter is divided into three further sections. Section 7.2 suggests that the problem of translation can shed light on a range of representational issues on the syntax/semantics border; it contains a brief introduction to Situation Semantics and the original version of Situation Schemata. Section 7.3 outlines the design of a computational toolkit for experimenting with unification-based descriptions, concentrating on details of the grammar formalism and its abstraction mechanisms. Section 7.4 considers the form of Situation Schemata that has been implemented within a grammar of German and some related representational issues that arise out of this work.

INTRODUCTION

Computational semantics lies at the intersection of three disciplines: linguistics, logic and computer science. A natural language system should provide a coherent framework for relating linguistic form and semantic content, and the relationship must be algorithmic.

There have been several important pairwise interactions, as detailed below, between linguistics and computer science, linguistics and logic, and logic and computer science. The point of computational semantics is the insistence on relating all three simultaneously. This is necessary from a cognitive as well as a knowledge engineering point of view.

LINGUISTICS AND COMPUTER SCIENCE

N. Chomsky's Syntactic Structures [34] signalled a renewal of theoretical linguistics, which for some of its theoretical tools drew upon automata and formal language theory. A link was soon established with the emerging computer science, leading to a vigorous field of computational linguistics, focusing on questions of linguistic form, i.e. syntax and morphology. This has proved to be of lasting value for the study of both natural and programming languages.

LINGUISTICS AND LOGIC

Language is more than linguistic form. R. Montague, in a series of papers starting from 1967 (see Thomason [219]), showed how to use the insights from logical semantics to ‘tame’ the meaning component of a natural language. Montague's tool was the model theory of higher order intensional logic. And he convincingly demonstrated how the use of this model theory could explain a wide range of linguistic phenomena.

Introduction

In this chapter a number of existence proofs and theoretical discussions are presented. These are related to the earlier chapters, but were not presented there in order not to distract too much from the main line of those chapters. Sections 9.2 and 9.3 are related to Chapter 1. Sections 9.4 and 9.5 are related to Chapters 2 and 3, respectively. Finally Sections 9.6 and 9.7 are related to Chapter 5.

Undefinedness revisited

In this section we explain precisely how the truth and falsity of COLD-K assertions with respect to the partial many-sorted algebras is established. In particular the issue of undefinedness deserves a careful treatment. In this section we focus on the terms and assertions as presented in in Chapter 1 (see Tables 1.1 and 1.2).

Recall that a partial many-sorted Σ-algebra M is a system of carrier sets SM (one for each sort name S in Σ), partial functions fM (one for each function name f in Σ), and relations rM (one for each relation name r in Σ). The functions fM must be compatible with their typing in the following sense: if f : S1#…# Sm→ V1 #…# Vn is in £ we have that fM is a partial function from to. Similarly the predicates must be compatible with their typing, i.e. if r : S1×…× Sm is in Σ we have that rM is a relation on.

General

In this appendix a concrete syntax for COLD-K is defined. It is concerned with the full language, including the constructs presented in Chapter 10 and 11. The notions of term, expression and statement are integrated into a single syntactical category called <expression>. We give an (extended) BNF grammar defining a set of strings of ASCII characters which are used as concrete representations of the COLD-K constructs.

For user convenience, it is allowed to omit redundant type information. In the applied occurrence of a name the associated type information is generally omitted (otherwise the use of names would become very clumsy). Though in many situations the missing type information can be reconstructed from the context, there are situations where ambiguities may occur. We leave it to the parser to report such ambiguities; there is a special syntactic operator (the cast) to disambiguate the type of an expression.

In the concrete syntax defined here prefix, infix and postfix notations are used for the built-in operators of the language. For the user-defined operators (predicates, functions, procedures) only a prefix notation is provided. The main reason for not introducing infix, postfix or mixfix notations for the latter is simplicity. The possibility to define special notations for user-defined operators is typical for user-oriented versions of COLD, which can be defined as a kind of macro-extension of COLD-K.

Concrete syntax

We define the concrete syntax of COLD-K by means of a context free grammar and priority and associativity rules for the built-in operators. Below we shall define the lexical elements (tokens).