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Situation Theory: a Survey ABSTRACT SITUATION THEORY: A SURVEY Situation semantics was developed as an alternative to possible-worlds semantics. While possible-worlds semantics defines the informational content of sentences in terms of complete descriptions of the way the world is or might be, situation semantics defines the informational content of sentences in terms of partial worlds called situations. Situation theory is a novel informational ontology developed to give situation semantics a rigorous mathematical foundation. In reviewing the literature, we observed that although there are a few texts introducing situation theory in varying levels of detail and systematicity, there has not yet been published, until now, a general survey of the situation-theory literature that might introduce and guide future scholars. We note that a similar, if perhaps less-pressing need exists for a survey of the situation-semantics literature, but regret that we must leave such a survey for others. Jacob Ian Lee December 2011 SITUATION THEORY: A SURVEY by Jacob Ian lee A thesis submitted in partial fulfillment of the requirements for the degree of Master of Science in Computer Science in the College of Science and Mathematics California State University, Fresno December 2011 © 2011 Jacob Ian Lee APPROVED For the Department of Computer Science: We, the undersigned, certify that the thesis of the following student meets the required standards of scholarship, format, and style of the university and the student's graduate degree program for the awarding of the master's degree. Jacob Ian Lee Thesis Author Todd Wilson (Chair) Computer Science Shigeko Seki Computer Science Shih-Hsi Liu Computer Science For the University Graduate Committee: Dean, Division of Graduate Studies AUTHORIZATION FOR REPRODUCTION OF MASTER’S THESIS X I grant permission for the reproduction of this thesis in part or in its entirety without further authorization from me, on the condition that the person or agency requesting reproduction absorbs the cost and provides proper acknowledgment of authorship. Permission to reproduce this thesis in part or in its entirety must be obtained from me. Signature of thesis author: ACKNOWLEDGMENTS To my wife Paulina for her love and patience, to my daughters, Helena and Matylda, for all the times I could not play with them, to my parents, grandparents, and siblings for their unconditional support, to my chair Todd Wilson and committee members, Shih-Hsi Liu and Shigeko Seki, for their guidance, to the staff of the Computer Science department, and especially Gloria Riojas, for the many times they went out of their way to help me get things done, and finally to all of my friends in the infonosphere, from Anthro-L, Dead Voles, and beyond, for their encouragement and criticisms, I give my sincere gratitude. Thank you. TABLE OF CONTENTS Page LIST OF TABLES ................................................................................................viii LIST OF FIGURES.................................................................................................ix INTRODUCTION....................................................................................................1 The Need for a Theory of Semantic Information..............................................1 Received Views of Semantic Information ........................................................2 What is Situation Theory and Situation Semantics?.........................................6 Overview of the Literature................................................................................9 Organization of Thesis ....................................................................................15 SITUATION THEORY..........................................................................................17 Infons and the Space of Issues ........................................................................17 Situations.........................................................................................................22 The Support Relation Between Situations and Infons ....................................23 Parameters and Abstraction ............................................................................25 Complex Infons...............................................................................................46 Non-Wellfounded Infons and Situations.........................................................54 Unsaturated Infons ..........................................................................................61 Infon Identity...................................................................................................65 The World of Situations and Their Parts ........................................................75 Perspectival Infons and Situations ..................................................................91 SITUATION SEMANTICS.................................................................................100 Brief Introduction to Situation Semantics.....................................................100 Meaning of Conditionals...............................................................................103 Generalized Quantifiers in Situation Theory ................................................106 vii Page Further Reading in Situation Semantics........................................................112 INFORMATION FLOW......................................................................................113 Introduction to Information Flow .................................................................113 Classic View of Information Flow in Situation Theory................................114 Early Modifications to Theory of Information Flow ....................................124 Early Channel Theory ...................................................................................149 Intermediate Channel Theory........................................................................164 Later Channel Theory ...................................................................................173 Applications and Additions to Later Channel Theory ..................................194 CONCLUSION ....................................................................................................200 REFERENCES.....................................................................................................201 LIST OF TABLES Page Table 1. Fragment of Classification from Perspective of Holmes, ..................146 Table 2. Fragment of Classification from Perspective of Watson, .................146 Table 3. Fragment of Classification from Perspective of Lestrade, .................147 Table 4. Traffic Light Classification T.................................................................185 Table 5. Traffic Light Channel Core C ................................................................186 Table 6. Traffic Light Infomorphisms..................................................................187 LIST OF FIGURES Page Figure 1. a) Left is a well-founded APG. b) Center is the circular APG picturing . c) Right is the unfolded APG picturing ..........................58 Figure 2. Universal Mapping Property of Sums...................................................177 Figure 3. Minimal Cover......................................................................................179 Figure 4. Canonical Minimal Cover of a Distributed System..............................180 Figure 5. Traffic Light Channel ...........................................................................184 Figure 6. Minimal Cover of Channel ...................................................................192 Figure 7. Diagram obtained from Figure 6 using Universal Mapping Property..193 INTRODUCTION The Need for a Theory of Semantic Information Information and information flows are ubiquitous and polymorphic. They are evident in circumstances as disparate as the seating arrangements of a fono in a Samoan village, the mating signals of Photinus pyralis, the vast heterogeneous computer network known as the internet, and protein bio-synthesis. Talk of information permeates the discourse of numerous academic and professional disciplines. But what is information? A variety of sophisticated approaches to the understanding and measurement of information have been developed. The most successful theories have been quantitative in nature; they attempt to answer the question of how much information is contained in a given message or data set. These theories most famously include mathematical communication theory, commonly called information or coding theory, proposed originally by Claude Shannon (1948), and the algorithmic information theory of Andrey Kolmogorov, Gregory Chaitin and others 1. However, quantitative theories of information cannot tell us what information is contained in a message or data set, or how it can inform us about the world 2. For that, we need a theory of semantic information and a theory of information flow. 1 Roughly speaking, in mathematical communication theory, the amount of information of a received message is inversely proportional to its probability. In algorithmic information theory, the information content of a collection of data is something like the size, in bits, of the smallest program that would outputs that data if executed. Algorithmic information content increases with the randomness of the data. Algorithmic information may be described as a measure of descriptive complexity. For very useful overviews to these and other mathematical theories of information,
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