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Monographs in Theoretical Computer Science An EATCS Series Editors: W. Brauer G. Rozenberg A. Salomaa Advisory Board: G. Ausiello M. Broy S. Even 1. Hartmanis N. Jones T. Leighton M. Nivat C. Papadimitriou D. Scott Springer Berlin Heidelberg New York Barcelona Budapest Hong Kong London Milan Paris Tokyo Areski N ait Abdallah The Logic of Partial Information With 160 Figures Springer Author Prof. Dr. Areski Nait Abdallah University of Western Ontario Middlesex College, Department of Computer Science London, Ontario, Canada N6A 5B7 Series Editors Prof. Dr. Wilfried Brauer Fakultat fUr Informatik, Technische Universitat Munchen Arcisstrasse 21, D-80333 Miinchen, Germany Prof. Dr. Grzegorz Rozenberg Institute of Applied Mathematics and Computer Science University of Leiden, Niels-Bohr-Weg 1, P.O. Box 9512 NL-2300 RA Leiden, The Netherlands Prof. Dr. Arto Salomaa Turku Centre for Computer Studies Data City, 4th Floor FIN-20520 Turku, Finland Cataloging-in-Publication Data applied for Die Deutsche Bibliothek - CIP-Einheitsaufnahme Nait Abdallah, Areski: The logic of partial information / Areski Nait Abdallah. - Berlin: Heidelberg; New York: London: Paris; Tokyo; Hong Kong; Barcelona; Budapest: Springer. 1995 (Monographs in theoretical computer science -an EATCS series) CR Subject Classification (1991): 1.2.3-4, F.4.1, F.3.1-2, D.l.6, D.3.1 ISBN-13: 978-3-642-78162-9 e-ISBN: 978-3-642-78160-5 DOl: 10.1007/978-3-642-78160-5 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer-Verlag. Violations are liable for prosecution under the German Copyright Law. © Springer-Verlag Berlin Heidelberg 1995 Softcover reprint of the hardcover I st edition 1995 The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Cover design: MetaDesign, Berlin Typesetting: Data conversion by K. Mattes, Heidelberg SPIN 10099661 45/3140 -5 432 I 0 -Printed on acid-free paper To Wnissa, Yamina, Mu~end w Atili and Malika Man, as we realize if we reflect for a moment, never perceives anything com pletely. He can see, hear, touch and taste; but how far he sees, how well he hears, what his touch tells him, and what he tastes depend upon the number and quality of his senses. These limit his perception of the world around him. . . . Every experience contains an indefinite number of unknown factors, not to speak of the fact that every concrete object is always unknown in certain respects, because we cannot know the ultimate nature of matter itself. (Carl G. Jung) Preface One must be able to say at all times - in stead of points, straight lines, and planes - tables, chairs and beer mugs. (David Hilbert) One service mathematics has rendered the human race. It has put common sense back where it belongs, on the topmost shelf next to the dusty canister labelled "discarded nonsense. " (Eric T. Bell) This book discusses reasoning with partial information. We investigate the proof theory, the model theory and some applications of reasoning with par tial information. We have as a goal a general theory for combining, in a principled way, logic formulae expressing partial information, and a logical tool for choosing among them for application and implementation purposes. We also would like to have a model theory for reasoning with partial infor mation that is a simple generalization of the usual Tarskian semantics for classical logic. We show the need to go beyond the view of logic as a geometry of static truths, and to see logic, both at the proof-theoretic and at the model-theoretic level, as a dynamics of processes. We see the dynamics of logic processes bear with classical logic, the same relation as the one existing between classical mechanics and Euclidean geometry. This need implies that algebraic logic, which plays here the role of geometry, has to be generalized to a notion of analytic logic, in order to provide the necessary dynamic tools that are needed: Galileo's mechanics would have never come into being, had not the notion of natural number been generalized, and the notion of continuum been clarified. The approach developed here is based on the novel notions of partial in formation ion, partial model and regular model of a logic system. A novel view of reasoning about actions, inspired from theoretical physics, is also pre sented, and its applications investigated. This view leads to a geometrization of practical reasoning with partial information. VIII Preface A considerable amount of research has been, and continues to be done into the problem of representing and reasoning about "default" knowledge (e.g. if you don't know better, assume that a bird flies.) All previous solu tions have been based around some version of "non-monotonic logic." The notion of "non-monotonic logic" is a radical departure from classical logic. The resulting approaches are complex; the whole area of commonsense as studied in (theoretical) artificial intelligence is sometimes rather confusing. Few mathematical properties have been proved in "non-monotonic logics;" this feature may be due to the fact that they address objects that, intrinsi cally, have few properties. In his overview of pure mathematics, Dieudonne [13J classifies mathematical problems into six groups: (i) still-born problems, (ii) problems without posterity, (iii) problems generating a method, (iv) problems that are organized around a fruitful and lively general theory, (v) wilting theories, (vi) theories going through a process of thinning down ("Verdiinnung"). The fact that few theorems have been proved in "non-monotonic logics" leads to the understandable fear that they may not be in groups (iii) or (iv) of Dieudonne's classification. A new concept should generate more than the elaboration of its own the ory. As an example, the A-calculus was intended by Church [8J as a system of logic where a certain A-operation, representing the abstraction of a func tion from its unspecified variable, played a central role. It was a surprise to discover its equivalence with Turing machines, the link with Scott topology, denotational semantics [96J, and more recently the isomorphism between nor malisation in A-calculus and normalisation in proof theory. In comparison, is non-monotonicity anything more than just a surface effect? We propose here a new approach, that stays "monotonic" whilst still pre senting a logic able to handle default reasoning, and all the problem exam ples in the literature. The cost is a change to the semantics of first-order logic, albeit a natural one: the notion of partial models. This notion is not new in the field of computer science, since it is the basic notion of stan dard programming language theory. The benefit is that the usual "conceptual book-keeping" problems met in several "non-monotonic logics" are eliminated straightaway, and a reasonable computational status for reasoning with "de fault" knowledge is restored. ("Non-monotonic first-order logics" are not even semi-decidable.) This allows us then to see new links between reasoning with partial information and neighbouring areas such as algebraic logic, program ming language theory, theoretical physics, scientific epistemology, et cetera. The results obtained go along the same lines as Hintikka's claim [34J that "The methods best suited to increase conceptual clarity are here, as in many other areas of logic, the semantical ones (in the sense of the term in which Preface IX it has been applied to Carnap's and Tarski's studies.} It is not very helpful merely to put one's intuition into the form of a deductive system, as happens in the syntactical method. They are rarely sharpened in the process. They are usually much sharpened, however, if we inquire into the conditions of truth for the different kinds of sentences that we are dealing with; which is essentially what the semantical method amounts to." Preliminary reports on this research were published in [72], where the notion of partial information ion was first introduced, and [74, 75, 76]. Oth erwise, the material presented here is entirely new. The author has written a program which demonstrates default reasoning on a variety of classically "dif ficult" cases. He will make it available to readers who would like to witness these concepts in action. The reader is expected to have some background in propositional and first order logic. Some knowledge of the elementary semantics of programming languages, although not a requisite, is also useful in helping to put things into perspective. This book consists of an introductory part (Chap. 2), and three main parts: propositional partial information ionic logic (Chaps. 3, and 5 through 13), first-order ionic logic (Chaps. 14 through 17), and applications to partially specified logic systems (Chap. 9) and to the frame problem and reasoning about actions (Chaps. 20 through 22). The applications of partial information logic are made via the use of two fundamental principles: the principle of the statics of logic systems (for partially specified logic systems), and the least action principle (for reasoning about actions.) Programming in ionic logic is discussed in Chap. 18, and the notions of syntactic and semantic path are presented in Chap. 19. A conceptual overview of reasoning with partial information, with some brief descriptions of the applications studied here, is given in Chap. 4. It may be used by the reader as a "geographical map" of the topics presented. Chapter 1 gives a general introduction to the problem of reasoning with partial information, and discusses the motivations of our approach. Chapter 2 shows that many of the novel features of reasoning with partial information have nothing to do with commonsense or non-monotonicity, but have their roots in partial propositional logic, i.e. propositional logic with partial interpretations. This is reminiscent of Kronecker's [41] famous "God created natural numbers. Everything else is the invention of man." Chapter 3 discusses the syntax of propositional partial information ions, and Chaps. 5 and 6 define semantic domains for this new language, and how formulae are to be interpreted in these domains. Chapter 7 discusses the algebra of the logic, which turns out to be a gen eralized Boolean algebra, and the structure of the domain of interpretations seen from the point of view of the language. This structure will be used later in Chaps. 20 and 21 while reasoning about actions, where this domain is seen as a phase space (in the physical sense) in which the physical systems con- X Preface sidered evolve. The fundamental tool here is the Galois connection between interpretations and some suitably ordered classes of interpretations. Chapter 8 presents a first tool for reasoning in the new logic: tableaux. Chapter 9 discusses their application to practical problems ranging from the notion of weak implication in commonsense reasoning to the Heisenberg un certainty principle in quantum mechanics, and the derivation of presupposi tions in natural language. Chapters 10 through 12 discuss the issue ofaxiomatizing partial informa tion ionic logic using a formal deductive system with axioms and proof rules. A syntactic distinction is drawn between ionic logics that respect a separa tion between hard, proven knowledge and soft, conjectural knowledge (called here Lakatosian), and those that do not respect such a separation (called here non-Lakatosian). Chapter 13 shows how the notion of "non-monotonic logic" can be recon structed from the language of partial information ions. The reconstruction is based on a syntactic notion of acceptable justification, and the similar ities and differences with the semantic approach are discussed. Links with "non-monotonic" approaches such as Reiter's default logic, Lukaszewicz' re vised default logic, et cetera, are investigated (see also Chap. 18.) Some links with related work are discussed throughout: truth maintenance systems it la Doyle in Sect. 9.3 (Chap. 9), Skolemization problems in Reiter's logic in Sect. 18.3.1 (Chap. 18), Poole's logical framework for default reasoning in Sect. 18.3.2 (Chap. 18), Sandewall's approach to multiple defeasible inheri tance in Sect. 19.2 (Chap. 19), Hanks and McDermott's and Morris's solutions to the Yale Shooting Problem in Reiter's logic in Sect. 20.4.1 (Chap. 20). Chapters 14 through 17 generalize the material presented so far from propositional to first-order logic. Of particular interest is the treatment of equality, and the introduction of the new notions of actual object and fic tional object. This generalization allows reasoning about fictions, as well as about soft, conjectural objects. Several default reasoning problems are also discussed and solved (Chap. 16). Chapter 18 discusses the issue of logic programming in the style of Prolog in partial information ionic logic. Chapter 19 introduces the notion of evolution path in the phase space asso ciated with reasoning with partial information, and applies it to the problem of multiple defeasible inheritance. Chapters 20 through 22 build upon this notion, and systematize it into a least action principle which is postulated to be a fundamental principle of reasoning about actions in the framework of partial information, very much like Fermat's principle in geometrical optics or Galileo's least action principle in mechanics. This principle yields, in a uniform manner, the solution of a wide range of problems: Yale shooting problem, Robot problem, projection problem, et cetera. These are discussed in Chap. 20 and 21.