Abstract

We present a model of classical linear logic based on the notion of strong stability that was introduced in [BE], a work about sequentiality written jointly with Antonio Bucciarelli.

Introduction

The present article is a new version of an article already published, with the same title, in Mathematical Structures in Computer Science (1993), vol. 3, pp. 365–385. It is identical to this previous version, except for a few minor details.

In the denotational semantics of purely functional languages (such as PCF [P, BCL]), types are interpreted as objects and programs as morphisms in a cartesian closed category (CCC for short). Usually, the objects of this category are at least Scott domains, and the morphisms are at least continuous functions. The goal of denotational semantics is to express, in terms of “abstract” properties of these functions, some interesting computational properties of the language.

One of these abstract properties is “continuity”. It corresponds to the basic fact that any computation that terminates can use only a finite amount of data. The corresponding semantics of PCF is the continuous one, where objects are Scott domains, and morphisms continuous functions.

But the continuous semantics does not capture an important property of computations in PCF, namely “determinism”. Vuillemin and Milner are at the origin of the first (equivalent) definitions of sequentiality, a semantic notion corresponding to determinism. Kahn and Plotkin ([KP]) generalized this notion of sequentiality. More precisely, they defined a category of “concrete domains” (represented by “concrete data structures”) and of sequential functions.

Abstract

Girard's execution formula (given in [Gir88a]) is a decomposition of usual β-reduction (or cut-elimination) in reversible, local and asynchronous elementary moves. It can easily be presented, when applied to a λ-term or a net, as the sum of maximal paths on the λ-term/net that are not cancelled by the algebra L* (as was done in [Dan90, Reg92]).

It is then natural to ask for a characterization of those paths, that would be only of geometric nature. We prove here that they are exactly those paths that have residuals in any reduct of the λ-term/net. Remarkably, the proof puts to use for the first time the interpretation of λ-terms/nets as operators on the Hilbert space.

Presentation

λ-Calculus is simple but not completely convincing as a real machine-language. Real machine instructions have a fixed run-time; a β-reduction step does not. Some implementations do map-reductions into sequences of real elementary steps (as in environment machines for example) but they use a global time to achieve this. The “geometry of interaction” (GOI) is an attempt to find a low-level combinatorial code within which β-reduction could be implemented and such that:

• elementary reduction steps are local;

• parallelism shows up and global time disappears;

• some mathematics dealing with syntax is uncovered.

— Goal and organization of this paper.

A persistent path is a path on a λ-term which survives the action of any reduction (defined in [Reg92]). A regular path is a path which is not cancelled by Girard's algebraic device L* (defined in [Gir88a]).

Abstract

We present natural deduction systems for fragments of intuitionistic linear logic obtained by dropping weakening and contractions also on!-prefixed formulas. The systems are based on a two-dimensional generalization of the notion of sequent, which accounts for a clean formulation of the introduction/elimination rules of the modality. Moreover, the different subsystems are obtained in a modular way, by simple conditions on the elimination rule for!. For the proposed systems we introduce a notion of reduction and we prove a normalization theorem.

Introduction

Proof theory of modalities is a delicate subject. The shape of the rules governing the different modalities in the overpopulated world of modal logics is often an example of what a good rule should not be. In the context of sequent calculus, if we want cut elimination, we are often forced to accept rules which are neither left nor right rules, and which completely destroy the deep symmetries the calculus is based upon. In the context of natural deduction the situation is even worse, since we have to admit deduction trees whose subtrees are not deductions, or, in the best case, elimination rules containing in their premise(s) the eliminated connective. On top of this, any such rule do not characterize (in a universal way, as category theoreticians would say) the modality it “defines”: two different modality with the same rules bear no relation among each other (cf. Section 4.1).

Abstract

The paper expounds geometry of interaction, for the first time in the full case, i.e. for all connectives of linear logic, including additives and constants. The interpretation is done within a C*-algebra which is induced by the rule of resolution of logic programming, and therefore the execution formula can be presented as a simple logic programming loop. Part of the data is public (shared channels) but part of it can be viewed as private dialect (defined up to isomorphism) that cannot be shared during interaction, thus illustrating the theme of communication without understanding. One can prove a nilpotency (i.e. termination) theorem for this semantics, and also its soundness w.r.t. a slight modification of familiar sequent calculus in the case of exponential-free conclusions.

Introduction

Towards a monist duality

Geometry of interaction is a new form of semantics. In order to understand what is achieved, one has to discuss the more traditional forms of semantics.

Classical model theory

The oldest view about logic is that of an external observer : there is a preexisting reality (mathematical, let us say) that we try to understand (e.g. by proving theorems). This form of dualism is backed by the so-called completeness theorem of Gödel (1930), which says that a formula is provable iff it is true in all models (i.e. in all realizations). There is strong heterogeneity in the duality world/observer (or model/proof) proposed by model-theory, since the latter is extremely finite whereas the former is infinite. Hilbert's attempt at reducing the gap between the two actors failed because of the renowned incompleteness theorems, also due to Godel (1931), whose basic meaning is that infinity cannot be eliminated.

Information seeking, like learning and problem solving, demands general cognitive facility and special knowledge and skills and is influenced by attitudes and preferences. General cognitive facility – what is commonly called intelligence – includes our abilities to remember, make inferences, and monitor our intellectual activity. Special knowledge and skills of three types also interact to determine information-seeking performance: knowledge and skills related to the problem domain, knowledge and skills specific to the search system and setting, and knowledge and skills related to information seeking itself. Attitudes such as motivation, confidence, tenacity, tolerance for ambiguity and uncertainty, curiosity, and preferences for social interaction and media influence when and how we apply information-seeking knowledge and skills. Taken together, these types of knowledge, skills, and attitudes compose our personal information infrastructures. Information problems are always embedded in a context that determines which facets of our personal information infrastructure are brought to bear in a specific situation. Personal information infrastructures develop as we gain knowledge of the information-seeking factors and skills in managing the information-seeking process. Information professionals apply their general cognitive abilities to building knowledge and skills concerning sources and systems that contain information, techniques for mapping users' needs to tasks, and strategies for seeking and representing information. Knowing what knowledge and skills are useful in manual environments and today's electronic environments will lead to better designs for future information systems and to better training for professionals and end users alike.

For a quarter century we have been inundated by prognostications about the information society and changes in the world's economies and cultures. With apologies to Samuel Clemens, I believe that reports of the demise of society and culture have been greatly exaggerated. I do believe, however, that information technologies (computers and communication networks) are bringing about qualitative changes in how we learn and work. In particular, our abilities and capabilities to seek and use information are strongly influenced by these environments. This book aims to explicate some of these changes so that information workers can better prepare for the ongoing changes ahead and system designers can better understand the needs and perspectives of information seekers.

As a teacher, I have always been troubled by students' confusion about memory and learning. Memory is necessary but not sufficient for learning and understanding, and this confusion reflects larger distinctions among information, knowledge, and wisdom. It is this concern that has led me to consider information seeking as a broader process rather than the more limited notion of information retrieval. The book presents a framework for understanding information seeking and applies the framework to an analysis of search strategies and how they have been affected by electronic technology. Based on 10 years of user studies, this book describes how formal, analytical search strategies have been made more powerful by technology and argues for systems that also support intuitive browsing strategies.

As more information becomes available in electronic form, more systems are developed to support electronic information seeking, and more people gain experience using such systems, our overall expectations change and evolve about the value of information and the roles it plays in our lives. In the previous chapters we considered how technical developments have led to complex and rapidly changing electronic environments. These developments include

• Hardware advances in storage, processing, display, and networking.

• Integration of application software such as text management, database management, communications, and hypermedia.

• Retrieval algorithms and techniques such as inverted indexes, vector representations, and clustering.

• Human–computer interface developments such as user-centered design, direct manipulation, and graphical user interfaces.

These electronic environments have influenced information-seeking by amplifying what is possible in manual environments and requiring new information-seeking strategies. In this chapter, we summarize how electronic environments have already changed information seeking and examine some of the constraining conditions that moderate these continuing changes.

Effects of electronic environments

To examine how information seeking is affected by electronic environments, we should distinguish between the physical and intellectual consequences of information in electronic form. This distinction must be qualified as one of convenience, because physical and intellectual changes are interrelated. Physical changes include greater volumes of information, remote access (allows users to transcend space), transfer speed (allows users to minimize time requirements), multiple formats and flexible management of those formats, behavioral actions of users, and capital investments.

Information seeking involves a number of personal and environmental factors and processes. In this chapter we identify these factors and processes and see how they work together to define and constrain information seeking. Before reading further, stop and consider the many information-seeking activities you perform each day. Suppose you have a well-defined information need such as finding a phone number for a business in a foreign city. What do you need to know to begin? What things do you already know about telephones, businesses, and information seeking that will help in your search? What sources could help? How can you determine whether they are available? How do you use them? What are the costs in time or money? How will you know when you have found the correct number? What kinds of questions can you imagine for a more openended but better-focused information problem such as understanding the implications of the European Common Market's trade agreement with Japan on what investments to make for a child's college trust fund? How would your strategies differ for a fuzzy problem like gaining information to improve one's knowledge of a domain of interest? Clearly, we encounter many varieties of information problems and apply varied information-seeking strategies to solve these problems. To understand this variety, it is useful to have a framework that explicates factors and processes common to information seeking in general.

Our world continues to become increasingly complex, interconnected, and dynamic: There are more people and institutions; they engage in more relationships and exchange; and the rates of change continue to grow, largely because of developments in technology and the importance of information to human and technical development. We live in an information society in which more people must manage more information, which in turn requires more technological support, which both demands and creates more information. Electronic technology and information are mutually reinforcing phenomena, and one of the key aspects of living in the information society is the growing level of interactions we have with this complex and increasingly electronic environment. The general consequences of the information society are threefold: larger volumes of information, new forms and aggregations of information, and new tools for working with information.

First, we find ourselves dealing with more information in all aspects of our lives. More of us are “knowledge workers,” generating, managing, and communicating information to produce and provide goods and services for an increasingly global economy. In addition to the often-noted trend toward more people managing more information in the workplace, people must go beyond the workplace to learn new skills and acquire new knowledge to do their jobs.

Evolution proceeds in many waves, some brief by human temporal sensibilities and some lasting for centuries. Some changes in information seeking take place before technological investments are fully amortized (e.g., the latest CPU or software upgrade brings with it access to new information resources), and some take place over careers as strategies and patterns of use learned in school evolve based on new sources and tools. This final chapter examines one long-term change that computing technology brings to cognition in general and to information seeking in particular, considers how different domains interact to influence the evolution of information seeking, and concludes with some ideas about what types of systems we should strive to build.

Amplification and augmentation

Applying computational power to information problems has been a research and design goal from the first days of computing. The dreams of language translation and cybernetic assistants have given way to dreams of artificial realities and intelligent agents, but our fascination with the manipulation of symbolic data and with interactivity remains a driving force behind much of the research in artificial intelligence and engineering. One way to consider how computation may be applied to information problems is to examine how it may be applied to amplify and augment intellect.

Throughout our lives we develop knowledge, skills, and attitudes that allow us to seek and use information. This chapter introduces the notion of personal information infrastructure, which will be used to describe this complex of knowledge, skills, and attitudes. It also introduces the notion of interactivity, a key characteristic of computer technology that allows information seekers to use electronic environments in ways that emulate interactions with human sources of information. The chapter also provides an overview of the technological developments that underlie information seeking in electronic environments.

Personal information infrastructures

The primary activities of scientists, physicians, businesspersons, and other professionals are gathering information from the world, mentally integrating that information with their own knowledge – thus creating new knowledge – and acting on this new knowledge to accomplish their goals. Most often, this knowledge and the consequences of using it are articulated to the external world as information. All humans develop mental structures and skills for conducting such activities according to their individual abilities, experiences, and physical resources. An individual person's collection of abilities, experience, and resources to gather, use, and communicate information are referred to as a personal information infrastructure.

Many specific system features have been shown to invite and support browsing as an information-seeking strategy. We are beginning to acquire a set of techniques that define what is possible in designing such features. Determining what is optimal for different users, tasks, and settings requires systematically testing techniques across the range of information-seeking factors. Because browsing requires users to coordinate physical and mental activities, systems that support browsing must solve both technical and conceptual problems. Technical challenges such as the computational power needed to manipulate huge vector spaces on-the-fly and display problems such as resolution limitations, refresh and scroll rates, window sizes, and juxtapositions are difficult enough in isolation but must be coordinated with other technical problems such as mechanisms for selection and control of information and conceptual problems such as what the best representations of meaning are for specific information items and what should be displayed at what time, in what form, and at what level of granularity. Programs of research are needed that address the technical problems of designing interaction styles for browsing, that determine the physiological and psychological boundaries of browsing activities, and that test various representations for browsable information. These are technical, user, and organizational areas, respectively. Although different researchers and groups typically specialize in one of these problem areas, ultimately the support for browsing will depend on integrating results from all three.

The storage and retrieval of scientific texts were early applications of computers, and by the early 1960s, schemes for automatic indexing and abstracting had emerged (e.g., Doyle, 1965; Luhn, 1957, 1958; O'Connor, 1964; Tasman, 1957). As online systems emerged in the 1960s and 1970s, more databases and new search features were created to give professional intermediaries more power in searching for information. Searching in online systems was complex, and so intermediaries created systematic strategies for eliciting users' needs; selecting terms, synonyms, and morphological variants appropriate to the need and the system; using Boolean operators to formulate precise queries; restricting those queries to specific database fields; forming intermediate sets of results; manipulating those sets; and selecting appropriate display formats. The strategies and tactics that professional intermediaries use are meant to maximize retrieval effectiveness while minimizing online costs. These strategies are goal oriented and systematic and are termed analytical strategies. In this chapter, we describe several analytical strategies to illustrate how electronic environments have changed information seeking by allowing searchers to systematically manipulate large sets of potentially relevant documents. These strategies in turn influenced subsequent designs of online systems. Next we look at studies of novice users working with various online systems, showing how difficult analytical strategies are to learn and apply, and the need for electronic systems that support informal information-seeking strategies for end users.