The simulation hypothesis has recently excited renewed interest, especially in the physics and philosophy communities. However, the hypothesis specifically concerns {computers} that simulate physical universes, which means that to properly investigate it we need to couple computer science theory with physics. Here I do this by exploiting the physical Church-Turing thesis. This allows me to introduce a preliminary investigation of some of the computer science theoretic aspects of the simulation hypothesis. In particular, building on (...) Kleene's second recursion theorem, I prove that it is mathematically possible for us to be in a simulation that is being run on a computer \textit{by us}. In such a case, there would be two identical instances of us; the question of which of those is ``really us'' is meaningless. I also show how Rice's theorem provides some interesting impossibility results concerning simulation and self-simulation; briefly describe the philosophical implications of fully homomorphic encryption for (self-)simulation; briefly investigate the graphical structure of universes simulating universes simulating universes, among other issues. I end by describing some of the possible avenues for future research that this preliminary investigation reveals. (shrink)

The fundamental ideas concerning computation and recursion naturally find their place at the interface between logic and theoretical computer science. The contributions in this book, by leaders in the field, provide a picture of current ideas and methods in the ongoing investigations into the pure mathematical foundations of computability theory. The topics range over computable functions, enumerable sets, degree structures, complexity, subrecursiveness, domains and inductive inference. A number of the articles contain introductory and background material which it (...) is hoped will make this volume an invaluable resource for specialists and non-specialists alike. (shrink)

Hay and, then, Johnson extended the classic Rice and Rice-Shapiro Theorems for computably enumerable sets, to analogs for all the higher levels in the finite Ershov Hierarchy. The present paper extends their work to analogs in the transfinite Ershov Hierarchy. Some of the transfinite cases are done for all transfinite notations in Kleene's important system of notations, equation image. Other cases are done for all transfinite notations in a very natural, proper subsystem equation image of equation image, where equation image (...) has at least one notation for each constructive ordinal. In these latter cases it is open as to what happens for the entire set of transfinite notations in equation image. (shrink)

ABSTRACT The individual-community relationship has always been one of the most fundamental topics of social sciences. In sociology, this is known as the micro-macro relationship while in economics it refers to the processes, through which, individual actions lead to macroeconomic phenomena. Based on philosophical discourse and systems theory, many sociologists even use the term "emergence" in their understanding of micro-macro relationship, which refers to collective phenomena that are created by the cooperation of individuals, but cannot be reduced to individual actions. (...) "Emergence" theories attempt to explain the nature of society as a complex system by examining how individuals and their relationships lead to the creation of integrated and macro-social phenomena such as markets, educational systems, cultural beliefs, and shared social practices. As a prelude to activity, every researcher has to answer the question from the methodological point of view, how is it possible to study the behavior of social groups and how can we gain knowledge about the laws related to social groups? Anyone who deals with humanities and social sciences or any reality and phenomenon that affects human beings, inevitably deals with the reality that is emerging. In fact, emergence occurs when one level of reality emerges radically from another level. Examples of emergent levels of reality include how the mind emerges from the body; or the way society emerges from human beings. Therefore, when there is an emerging factor, different scientific disciplines should be used, because it is inevitable to talk about social affairs, psychology and neurobiology, as well as physical and even chemical. -/- INTRODUCTION Since its beginning in the 19th century, sociology has faced the fundamental question of what a social phenomena stands for? At a glance, sociologists deal with the study of social groups, collective behavior, institutions, social structures, social networks, and social dynamics, and ultimately all such phenomena are composed of individuals. Therefore, social phenomena seem to have no independent ontological status, and in this case, sociology is ultimately about reducible individuals. Different approaches about the micro-macro relationship, using the concept of "emergence", argue that although collective phenomena are created by individuals, they cannot be reduced to individual action. The term "emergence" is used to distinguish "emergent results" from " Procedural results". Results are called Procedural that can only be calculated by adding or subtracting the causes that act together. On the contrary, they are emergent outcomes that are qualitatively novel compared to the causes from which they originate. An example of such emergent results is mental characteristics that emerge from neural processes, but are not considered among the characteristics of neural processes of components - which are their origin. Philosophical debates about emergentism and reductionism focus on the mind-brain relationship, but as many philosophers have noted, they can be generalized to apply to any hierarchical set of features. The cognitive revolution reactivated a 19th century controversy between theorists of Identity and dualism. The theorists of Identity believe in reductionist materialism and elimination of metaphysics, according to which the mind is nothing more than the biological, while the dualists believe that the mind and the brain are distinct from each other. Emergence has been considered as a third way between the theories of Identity and dualism. -/- METHODOLOGY Interdisciplinarity Is fundamentally indicated by the dominance of open systems as well as social and emerging systems at different levels. Therefore, as long as we are facing a new emergentism, there will necessarily be different fields and an interdisciplinary method. The commonality of social sciences and cognitive sciences on the issue of emergence provides the possibility of interaction and exchange of concepts between these two for deeper research. Therefore, in this research, apart from the concepts of Subvenience and downward causation, we have used multiple realization and wildly disjunctive, which are used to explain the emerging concept in cognitive sciences, including neurobiology, psychology, and philosophy. For analytical and interdisciplinary approaches, after examining the concepts of subvenience, and wildly disjunctive, Individualistic and collectivist emergentism have been reviewed. And then their adequacy is criticized and evaluated in meeting strong emergence conditions. -/- FINDINGS Philosophers of the mind have been able to define the conditions of irreducibility and downward causation, which are essential for strong emergence, by using the principle of subvenience with the help of multiple realization and wildly disjunctive principles. Therefore, by definition, emergentist individualism suffers from the problem of how to explain downward causality and irreducibility, and emergentist collectivism faces the challenge of reification. The analysis of the challenges facing the two methodological approaches of individualism and collectivism leads us to three unresolved issues about the emerging phenomenon, which are realism, downward causality, and mechanism. Despite the differences in the stances of these two approaches regarding the aforementioned triple problems, individualists and collectivists both agree that some social characteristics are reducible; and some others are irreducible due to their complexity, the determination of which, depends on the case study of the mechanism of emergence of each social characteristic. -/- CONCLUSION An emergence that is consistent with reductionism is called a weak emergence. And strong emergence includes the principle of irreducibility and downward causality. According to strong emergence, the characteristics of social phenomena have downward causal power, which cannot be reduced to the causal forces. The principle of subvenience, together with multiple realization and wildly disjunctive, can provide the conditions of irreducibility and downward causality, which are essential conditions of strong emergence. Based on the philosophical analysis of two methodological approaches, individualist and collectivist, it can be concluded that emergentist individualism suffers from the problem of how to explain downward causality and irreducibility, and reification collectivism faces the challenge of reification. The analysis of the problems faced by these two approaches leads us to three unresolved issues, which are realism, downward causality, and mechanism. individualist emergentists acknowledge the existence of emergent social characteristics, but despite this, they believe that these characteristics are not real but merely analytical constructs that require explanation based on individuals and their mutual actions. But collectivists argue that emergentism yields a categorical ontology and supports social realism. They accept the dependence of social characteristics on individuals, but claim that social entities and structures are ontologically independent. Individualists and collectivists both agree that some social characteristics are reducible; and others are irreducible due to their complexity; and the only way to recognize this is to engage in empirical studies of the mechanisms and temporal processes of "emergence" that create social characteristics. -/- NOVELTY The newness of this compact article is as follows: a) Presenting an interdisciplinary approach using the analysis of philosophy of cognitive sciences, including the philosophy of mind, neurobiology and psychology to formulate the issue of social emergence; b) Criticizing individualist and collectivist methodological approaches to the issue of social emergence and explaining the sufficiency and shortcomings of both in providing weak and strong emerging conditions; c) Explaining and categorizing the challenges facing methodological approaches in explaining the phenomenon of social emergence, in three categories: realism, downward causality, and mechanism. BIBLIOGRAPHY Beckermann, A., Flohr, H., & Kim, J. (eds.) (1992). Emergence or reduction? Essays on the prospects of nonreductive physicalism. Berlin: Walter de Gruyter. Bhaskar, R. (1979). The possibility of Naturalism. New York: Routledge. Bhaskar, R., Danermark, B. & Price, L. (2018). Interdisciplinarity and wellbeing: A critical realist general theory of interdisciplinarity. London and New York: Routledge. Blau, P.M. (1981). Introduction: diverse views of social structure and their common denominator. In Blau, P.M. & Merton, R.K. (eds.), Continuity in Structural Inquiry (pp. 1–23). Beverly Hills: Sage. Blau, P.M. (1987). Microprocess and macrostructure. In K.S. Cook (ed.), Social Exchange Theory (pp. 83–100). Newbury Park, CA: Sage. Brooks, R.A., & Maes, P. (eds). (1994). Artificial life IV; proceedings of the Fourth International Workshop on the Synthesis and Simulation of Living Systems. Cambridge, Mass: MIT Press. Clark, A. (1997). Being there: Putting brain, body, and world together again. Cambridge, Mass.: MIT Press. Coleman, J. S. (1987). Microfoundations and macrosocial behavior. In: J. C. Alexander, B. Giesen, R. Munch, & Neil J. Smelser (Eds.). The Micro-Macro Link (pp. 153–73). Berkeley: University of California. Coleman, J. S. (1990). Foundations of social theory. Cambridge, MA: Harvard University Press. Elster, J. (1985). Making sense of Marx. New York: Cambridge University Press. Fodor, J.A. (1974). Special sciences (Or: The disunity of science as a working hypothesis). Synthese, 28(2), 97–115. Fodor, J.A. (1989). Making mind matter more. Philosophical Topics, 17, 59-79. Giddens, A. (1984). The constitution of society: Outline of the theory of structuration. Berkeley: University of California Press. Hempel, C.G. (1965). Aspects of scientific explanation and other essays in the philosophy of science. New York: The Free Press. Horgan, T. (1993). From supervenience to superdupervenience: meeting the demands of a material world. Mind, 102, 555-586. Humphreys, P. (1997). How properties emerge”. Philosophy of Science, 64, 1-17. Kim, J. (1999). Making sense of emergence. Philosophical Studies, 95, 3–36. Kincaid, H. (1997). Individualism and the unity of science. New York: Rowman & Littlefield. Lewis, D.K. (1986). The plurality of worlds. Oxford: Oxford University Press. Nigel, G., & Conte, R. (eds.) (1995). Artificial societies; The computer simulation of social life. London: UCL Press. O’Connor, T. (1994). Emergent properties. American Philosophical Quarterly, 31, 91-104. Popper, K.R. (1968). Conjectures and refutations: The Growth of Scientific Knowledge. New York: Harper & Row. Sawyer, R. K. (2001). Emergence in sociology: contemporary philosophy of mind and some implications for sociological theory. American Journal of Sociology, 107, 551-585. Sawyer, R. K. (2004). The mechanisms of emergence. Philosophy of the Social Sciences, 34, 260–282. Sawyer, R. K. (2005). Social emergence: societies as complex systems. Cambridge: Cambridge University Press. Sawyer, R.K. (2003). Nonreductive individualism: Part 2, Social causation. Philosophy of the Social Sciences, 33, 203–224. Winch, P. (1993). The idea of a social science and its relation to philosophy‭ (SAMT, Trans.). Tehran, Iran: SAMT. (shrink)

What can computers do in principle? What are their inherent theoretical limitations? These are questions to which computer scientists must address themselves. The theoretical framework which enables such questions to be answered has been developed over the last fifty years from the idea of a computable function: intuitively a function whose values can be calculated in an effective or automatic way. This book is an introduction to computability theory (or recursion theory as it is traditionally known to mathematicians). (...) Dr Cutland begins with a mathematical characterisation of computable functions using a simple idealised computer (a register machine); after some comparison with other characterisations, he develops the mathematical theory, including a full discussion of non-computability and undecidability, and the theory of recursive and recursively enumerable sets. The later chapters provide an introduction to more advanced topics such as Gildel's incompleteness theorem, degrees of unsolvability, the Recursion theorems and the theory of complexity of computation. Computability is thus a branch of mathematics which is of relevance also to computer scientists and philosophers. Mathematics students with no prior knowledge of the subject and computer science students who wish to supplement their practical expertise with some theoretical background will find this book of use and interest. (shrink)

Textbook on Gödel’s incompleteness theorems and computability theory, based on the Open Logic Project. Covers recursive function theory, arithmetization of syntax, the first and second incompleteness theorem, models of arithmetic, second-order logic, and the lambda calculus.

In a fluent, clear, and lively style this translation by two of Maslov's junior colleagues brings the work of the late Soviet scientist S. Yu. Maslov to a wider audience. Maslov was considered by his peers to be a man of genius who was making fundamental contributions in the fields of automatic theorem proving and computational logic. He published little, and those few papers were regarded as notoriously difficult. This book, however, was written for a broad audience of readers and (...) describes elegant examples of applications in such fields as computer science, artificial intelligence, operations research, economic modeling, and biological modeling, among others. The book also brings to light the work by the American mathematician E. L. Post, which inspired Maslov's own work in the development of a general theory and which has been long neglected by mathematicial logicians and systems theorists in the United States. The book's first chapter introduces the Rules of the Game. Part I, Mathematics of Calculi, covers E. L. Post's canonical systems, deductive systems and algorithms, and probabilistic calculi and deductive information. Part II, Horizonal Modeling, takes up a "toy" economy, the calculi of technological possibilities, and the development of rules. Part III, Vertical Modeling, deals with the topics of "to fight and to search" and the consequences of the asymmetry of cognitive mechanisms. Vladimir Lifschitz is affiliated with the Department of Computer Science at Stanford University, and Michael Gelfond with the Department of Electrical Engineering and Computer Science at the University of Texas, El Paso. Theory of Deductive Systems and Its Applicationsis included in the Foundation of Computing Series, edited by Michael Garey. (shrink)

Necessity and sufficiency are the building blocks of all successful explanations. Yet despite their importance, these notions have been conceptually underdeveloped and inconsistently applied in explainable artificial intelligence, a fast-growing research area that is so far lacking in firm theoretical foundations. In this article, an expanded version of a paper originally presented at the 37th Conference on Uncertainty in Artificial Intelligence, we attempt to fill this gap. Building on work in logic, probability, and causality, we establish the central role of (...) necessity and sufficiency in XAI, unifying seemingly disparate methods in a single formal framework. We propose a novel formulation of these concepts, and demonstrate its advantages over leading alternatives. We present a sound and complete algorithm for computing explanatory factors with respect to a given context and set of agentive preferences, allowing users to identify necessary and sufficient conditions for desired outcomes at minimal cost. Experiments on real and simulated data confirm our method’s competitive performance against state of the art XAI tools on a diverse array of tasks. (shrink)

This fourth edition of one of the classic logic textbooks has been thoroughly revised by John Burgess. The aim is to increase the pedagogical value of the book for the core market of students of philosophy and for students of mathematics and computer science as well. This book has become a classic because of its accessibility to students without a mathematical background, and because it covers not simply the staple topics of an intermediate logic course such as Godel's (...) Incompleteness Theorems, but also a large number of optional topics from Turing's theory of computability to Ramsey's theorem. John Burgess has now enhanced the book by adding a selection of problems at the end of each chapter, and by reorganising and rewriting chapters to make them more independent of each other and thus to increase the range of options available to instructors as to what to cover and what to defer. (shrink)

In the paper some applications of Gödel's incompleteness theorems to discussions of problems of computer science are presented. In particular the problem of relations between the mind and machine (arguments by J.J.C. Smart and J.R. Lucas) is discussed. Next Gödel's opinion on this issue is studied. Finally some interpretations of Gödel's incompleteness theorems from the point of view of the information theory are presented.

BackgroundIncreasingly, hospitals and research institutes are developing technical solutions for sharing patient data in a privacy preserving manner. Two of these technical solutions are homomorphic encryption and distributed ledger technology. Homomorphic encryption allows computations to be performed on data without this data ever being decrypted. Therefore, homomorphic encryption represents a potential solution for conducting feasibility studies on cohorts of sensitive patient data stored in distributed locations. Distributed ledger technology provides a permanent record on all (...) transfers and processing of patient data, allowing data custodians to audit access. A significant portion of the current literature has examined how these technologies might comply with data protection and research ethics frameworks. In the Swiss context, these instruments include the Federal Act on Data Protection and the Human Research Act. There are also institutional frameworks that govern the processing of health related and genetic data at different universities and hospitals. Given Switzerland’s geographical proximity to European Union (EU) member states, the General Data Protection Regulation (GDPR) may impose additional obligations.MethodsTo conduct this assessment, we carried out a series of qualitative interviews with key stakeholders at Swiss hospitals and research institutions. These included legal and clinical data management staff, as well as clinical and research ethics experts. These interviews were carried out with two series of vignettes that focused on data discovery using homomorphic encryption and data erasure from a distributed ledger platform.ResultsFor our first set of vignettes, interviewees were prepared to allow data discovery requests if patients had provided general consent or ethics committee approval, depending on the types of data made available. Our interviewees highlighted the importance of protecting against the risk of reidentification given different types of data. For our second set, there was disagreement amongst interviewees on whether they would delete patient data locally, or delete data linked to a ledger with cryptographic hashes. Our interviewees were also willing to delete data locally or on the ledger, subject to local legislation.ConclusionOur findings can help guide the deployment of these technologies, as well as determine ethics and legal requirements for such technologies. (shrink)

Computable model theory, also called effective or recursive model theory, studies algorithmic properties of mathematical structures, their relations, and isomorphisms. These properties can be described syntactically or semantically. One of the major tasks of computable model theory is to obtain, whenever possible, computability-theoretic versions of various classical model-theoretic notions and results. For example, in the 1950's, Fröhlich and Shepherdson realized that the concept of a computable function can make van der Waerden's intuitive notion of an explicit field precise. This led (...) to the notion of a computable structure. In 1960, Rabin proved that every computable field has a computable algebraic closure. However, not every classical result “effectivizes”. Unlike Vaught's theorem that no complete theory has exactly two nonisomorphic countable models, Millar's and Kudaibergenov's result establishes that there is a complete decidable theory that has exactly two nonisomorphic countable models with computable elementary diagrams. In the 1970's, Metakides and Nerode [58], [59] and Remmel [71], [72], [73] used more advanced methods of computability theory to investigate algorithmic properties of fields, vector spaces, and other mathematical structures. (shrink)

Computer simulation modeling is an important part of contemporary scientific practice but has not yet received much attention from philosophers. The present project helps to fill this lacuna in the philosophical literature by addressing three questions that arise in the context of computer simulation of Earth's climate. Computer simulation experimentation commonly is viewed as a suspect methodology, in contrast to the trusted mainstay of material experimentation. Are the results of computer simulation experiments (...) somehow deeply problematic in ways that the results of material experiments are not? I argue against categorical skepticism toward the results of computer simulation experiments by revealing important parallels in the epistemologies of material and computer simulation experimentation. It has often been remarked that simple computer simulation models---but not complex ones---contribute substantially to our understanding of the atmosphere and climate system. Is this view of the relative contributions of simple and complex models tenable? I show that both simple and complex climate models can promote scientific understanding and argue that the apparent contribution of simple models depends upon whether a causal or deductive account of scientific understanding is adopted. When two incompatible scientific theories are under consideration, they typically are viewed as competitors, and we seek evidence that refutes at least one of the theories. In the study of climate change, however, logically incompatible computer simulation models are accepted as complementary resources for investigating future climate. How can we make sense of this use of incompatible models? I show that a collection of incompatible climate models persists in part because of difficulties faced in evaluating and comparing climate models. I then discuss the rationale for using these incompatible models together and argue that this climate model pluralism has both competitive and integrative components. (shrink)

This paper addresses the strength of Ramsey's theorem for pairs ($RT^2_2$) over a weak base theory from the perspective of 'proof mining'. Let $RT^{2-}_2$ denote Ramsey's theorem for pairs where the coloring is given by an explicit term involving only numeric variables. We add this principle to a weak base theory that includes weak König's Lemma and a substantial amount of $\Sigma^0_1$-induction (enough to prove the totality of all primitive recursive functions but not of all primitive recursive functionals). In the (...) resulting theory we show the extractability of primitive recursive programs and uniform bounds from proofs of $\forall\exists$-theorems. There are two components of this work. The first component is a general proof-theoretic result, due to the second author, that establishes conservation results for restricted principles of choice and comprehension over primitive recursive arithmetic PRA as well as a method for the extraction of primitive recursive bounds from proofs based on such principles. The second component is the main novelty of the paper: it is shown that a proof of Ramsey's theorem due to Erdős and Rado can be formalized using these restricted principles. So from the perspective of proof unwinding the computational content of concrete proofs based on $RT^2_2$ the computational complexity will, in most practical cases, not go beyond primitive recursive complexity. This even is the case when the theorem to be proved has function parameters f and the proof uses instances of $RT^2_2$ that are primitive recursive in f. (shrink)

Intuitionistic logic is presented here as part of familiar classical logic which allows mechanical extraction of programs from proofs. to make the material more accessible, basic techniques are presented first for propositional logic; Part II contains extensions to predicate logic. This material provides an introduction and a safe background for reading research literature in logic and computer science as well as advanced monographs. Readers are assumed to be familiar with basic notions of first order logic. One device for (...) making this book short was inventing new proofs of several theorems. The presentation is based on natural deduction. The topics include programming interpretation of intuitionistic logic by simply typed lambda-calculus (Curry-Howard isomorphism), negative translation of classical into intuitionistic logic, normalization of natural deductions, applications to category theory, Kripke models, algebraic and topological semantics, proof-search methods, interpolation theorem. The text developed from materal for several courses taught at Stanford University in 1992-1999. (shrink)

This book provides an invaluable overview of the reach of logic. It provides reference to some of the most important, well-established results in logic, while at the same time offering insight into the latest research issues in the area. It also has a balance of theory and practice, containing essays in the areas of modal logic, intuitionistic logic, logic and language, nonmonotonic logic and logic programming, temporal logic, logic and learning, combination of logics, practical reasoning, logic and artificial intelligence, abduction, (...) theorem proving and goal-directed reasoning. It will be invaluable reading for researchers and graduate students in logic and computer science, and a fabulous source of inspiration for research students in search of a topic for a PhD in logic and theoretical computer science. (shrink)

This article presents a fundamental advance in the theory of mental models as an explanation of reasoning about facts, possibilities, and probabilities. It postulates that the meanings of compound assertions, such as conditionals (if) and disjunctions (or), unlike those in logic, refer to conjunctions of epistemic possibilities that hold in default of information to the contrary. Various factors such as general knowledge can modulate these interpretations. New information can always override sentential inferences; that is, reasoning in daily life is defeasible (...) (or nonmonotonic). The theory is a dual process one: It distinguishes between intuitive inferences (based on system 1) and deliberative inferences (based on system 2). The article describes a computer implementation of the theory, including its two systems of reasoning, and it shows how the program simulates crucial predictions that evidence corroborates. It concludes with a discussion of how the theory contrasts with those based on logic or on probabilities. (shrink)

Alan M. Turing, pioneer of computing and WWII codebreaker, is one of the most important and influential thinkers of the twentieth century. In this volume for the first time his key writings are made available to a broad, non-specialist readership. They make fascinating reading both in their own right and for their historic significance: contemporary computational theory, cognitive science, artificial intelligence, and artificial life all spring from this ground-breaking work, which is also rich in philosophical and logical insight. An (...) introduction by leading Turing expert Jack Copeland provides the background and guides the reader through the selection. About Alan Turing Alan Turing FRS OBE, (1912-1954) studied mathematics at King's College, Cambridge. He was elected a Fellow of King's in March 1935, at the age of only 22. In the same year he invented the abstract computing machines - now known simply as Turing machines - on which all subsequent stored-program digital computers are modelled. During 1936-1938 Turing continued his studies, now at Princeton University. He completed a PhD in mathematical logic, analysing the notion of 'intuition' in mathematics and introducing the idea of oracular computation, now fundamental in mathematical recursion theory. An 'oracle' is an abstract device able to solve mathematical problems too difficult for the universal Turing machine. In the summer of 1938 Turing returned to his Fellowship at King's. When WWII started in 1939 he joined the wartime headquarters of the Government Code and Cypher School (GC&CS) at Bletchley Park, Buckinghamshire. Building on earlier work by Polish cryptanalysts, Turing contributed crucially to the design of electro-mechanical machines ('bombes') used to decipher Enigma, the code by means of which the German armed forces sought to protect their radio communications. Turing's work on the version of Enigma used by the German navy was vital to the battle for supremacy in the North Atlantic. He also contributed to the attack on the cyphers known as 'Fish'. Based on binary teleprinter code, Fish was used during the latter part of the war in preference to morse-based Enigma for the encryption of high-level signals, for example messages from Hitler and other members of the German High Command. It is estimated that the work of GC&CS shortened the war in Europe by at least two years. Turing received the Order of the British Empire for the part he played. In 1945, the war over, Turing was recruited to the National Physical Laboratory (NPL) in London, his brief to design and develop an electronic computer - a concrete form of the universal Turing machine. Turing's report setting out his design for the Automatic Computing Engine (ACE) was the first relatively complete specification of an electronic stored-program general-purpose digital computer. Delays beyond Turing's control resulted in NPL's losing the race to build the world's first working electronic stored-program digital computer - an honour that went to the Royal Society Computing Machine Laboratory at Manchester University, in June 1948. Discouraged by the delays at NPL, Turing took up the Deputy Directorship of the Royal Society Computing Machine Laboratory in that year. Turing was a founding father of modern cognitive science and a leading early exponent of the hypothesis that the human brain is in large part a digital computing machine, theorising that the cortex at birth is an 'unorganised machine' which through 'training' becomes organised 'into a universal machine or something like it'. He also pioneered Artificial Intelligence. Turing spent the rest of his short career at Manchester University, being appointed to a specially created Readership in the Theory of Computing in May 1953. He was elected a Fellow of the Royal Society of London in March 1951 (a high honour). (shrink)

When people engage in conversation, they tailor their utterances to their conversational partners, whether these partners are other humans or computational systems. This tailoring, or adaptation to the partner takes place in all facets of human language use, and is based on a mental model or a user model of the conversational partner. Such adaptation has been shown to improve listeners' comprehension, their satisfaction with an interactive system, the efficiency with which they execute conversational tasks, and the likelihood of achieving (...) higher level goals such as changing the listener's beliefs and attitudes. We focus on one aspect of adaptation, namely the tailoring of the content of dialogue system utterances for the higher level processes of persuasion, argumentation and advice-giving. Our hypothesis is that algorithms that adapt content for these processes, according to a user model, will improve the usability, efficiency, and effectiveness of dialogue systems. We describe a multimodal dialogue system and algorithms for adaptive content selection based on multi-attribute decision theory. We demonstrate experimentally the improved efficacy of system responses through the use of user models to both tailor the content of system utterances and to manipulate their conciseness. (shrink)

When people engage in conversation, they tailor their utterances to their conversational partners, whether these partners are other humans or computational systems. This tailoring, or adaptation to the partner takes place in all facets of human language use, and is based on a mental model or a user model of the conversational partner. Such adaptation has been shown to improve listeners' comprehension, their satisfaction with an interactive system, the efficiency with which they execute conversational tasks, and the likelihood of achieving (...) higher level goals such as changing the listener's beliefs and attitudes. We focus on one aspect of adaptation, namely the tailoring of the content of dialogue system utterances for the higher level processes of persuasion, argumentation and advice‐giving. Our hypothesis is that algorithms that adapt content for these processes, according to a user model, will improve the usability, efficiency, and effectiveness of dialogue systems. We describe a multimodal dialogue system and algorithms for adaptive content selection based on multi‐attribute decision theory. We demonstrate experimentally the improved efficacy of system responses through the use of user models to both tailor the content of system utterances and to manipulate their conciseness. (shrink)

It is often claimed that scientists can obtain new knowledge about nature by running computer simulations. How is this possible? I answer this question by arguing that computer simulations are arguments. This view parallels Norton’s argument view about thought experiments. I show that computer simulations can be reconstructed as arguments that fully capture the epistemic power of the simulations. Assuming the extended mind hypothesis, I furthermore argue that running the computer simulation is to execute (...) the reconstructing argument. I discuss some objections and reject the view that computer simulations produce knowledge because they are experiments. I conclude by comparing thought experiments and computer simulations, assuming that both are arguments. (shrink)

Quantum cognition is a new theoretical framework for constructing cognitive models based on the mathematical principles of quantum theory. Due to its increasing empirical success, one wonders what it tells us about the underlying process of cognition. Does it imply that we have quantum minds and there is some sort of quantum structure in the brain? In this paper, I address this important issue by using a new result in the research of quantum foundations. Based on the PBR theorem about (...) the reality of the wave function, I show, given certain assumptions, that the wave function assigned to a cognitive system such as our brain, which is used to calculate probabilities of thoughts/judgment outcomes in quantum cognition, is a real representation of the cognitive state of the system, not a mere representation of incomplete knowledge about the state of the system. This result supports a realist interpretation of quantum cognition, according to which the cognitive state of our brain and its dynamics are not classical but quantum in quantum cognition. In short, quantum cognition implies quantum minds. However, this result does not mean that we have quantum minds and our brain is a quantum computer, since quantum cognition by its standard formulation has not been fully confirmed by experiments. The hope is that more crucial experiments can be done in the near future to determine whether or not quantum cognition is real. (shrink)

We question the widespread assumption that linguistic theory should guide the formulation of mechanistic accounts of human language processing. We develop a pseudo‐linguistic theory for the domain of linguistic stress, based on observation of the learning behavior of a perceptron exposed to a variety of stress patterns. There are significant similarities between our analysis of perception stress learning and metrical phonology, the linguistic theory of human stress. Both approaches attempt to identify salient characteristics of the stress systems under examination without (...) reference to the workings of the underlying processor. Our theory and computer simulations exhibit some strikingly suggestive correspondences with metrical theory. We show, however, that our high‐level pseudo‐linguistic account bears no causal relation to processing in the perception, and provides little insight into the nature of this processing. Because of the persuasive similarities between the nature of our theory and linguistic theorizing, we suggest that linguistic theory may be in much the same position. Contrary to the usual assumption, it may not provide useful guidance in attempts to identify processing mechanisms underlying human language. (shrink)

We argue that Gödel's completeness theorem is equivalent to completability of consistent theories, and Gödel's incompleteness theorem is equivalent to the fact that this completion is not constructive, in the sense that there are some consistent and recursively enumerable theories which cannot be extended to any complete and consistent and recursively enumerable theory. Though any consistent and decidable theory can be extended to a complete and consistent and decidable theory. Thus deduction and consistency are not decidable in logic, and an (...) analogue of Rice's Theorem holds for recursively enumerable theories: all the non-trivial properties of them are undecidable. (shrink)

Since its conception nearly 20 years ago, Logic Programming - the idea of using logic as a programming language - has been developed to the point where it now plays an important role in areas such as database theory, artificial intelligence and software engineering. However, there are still many challenging research issues to be addressed and the UK branch of the Association for Logic Programming was set up to provide a forum where the flourishing research community could discuss important issues (...) of Logic Programming which were often by-passed at the large international conferences. This volume contains the twelve papers which were presented at the ALPUK's 3rd conference which was held in Edinburgh, 10-12 April 1991. The aim of the conference was to give a broad but detailed technical insight into the work currently being done in this field, both in the UK and by researchers as far afield as Canada and Bulgaria. The breadth of interest in this area of Computer Science is reflected in the range of the papers which cover - amongst other areas - massively parallel implementation, constraint logic programming, circuit modelling, algebraic proof of program properties, deductive databases, specialised editors and standardisation. The resulting volume gives a good overview of the current progress being made in the field and will be of interest to researchers and students of any aspects of logic programming, parallel computing or database techniques and management. (shrink)

This textbook, which is completely self-contained and can be read by anyone with a secondary school education, is the result of the author's material prepared over the past 15 years of teaching introductory natural language semantics to graduate and undergraduate students at McGill University. The intended audience comprises undergraduate and graduate students in linguistics as well as those in philosophy, computer science and psychology with an interest in natural language semantics. The aim of the textbook is to teach (...) the fundamentals of natural language semantics, with English as the principal, though not the exclusive, target language. The principal aim of the textbook is to provide the student with all the basics necessary to be able him or her read research in natural language semantics, regardless of the theory assumed by the author of the research. In contrast to other textbooks in the field, which aim to introduce students to a particular theory of natural language semantics, this textbook seeks to give to its reader a clear overview of the kind of natural language phenomena which, at least prima facie, fall within the ambit of an investigation of natural language semantics; to assist the reader in developing a mastery of the mathematical tools used to formulate hypotheses about the natural language phenomena studied; and to impart to the reader an appreciation of the interplay between hypothesis and data. As a result, the textbook is rich in its presentation of the patterns of English, drawing extensively on the best and most contemporary descriptive grammars of English. The textbook also has more than 500 exercises, with answers to half of them. (shrink)

Haag's theorem has been interpreted as establishing that quantum field theory cannot consistently represent interacting fields. Earman and Fraser have clarified how it is possible to give mathematically consistent calculations in scattering theory despite the theorem. However, their analysis does not fully address the worry raised by the result. In particular, I argue that their approach fails to be a complete explanation of why Haag's theorem does not undermine claims about the empirical adequacy of particular quantum field theories. I (...) then show that such empirical adequacy claims are protected from Haag's result by the techniques that are required to obtain theoretical predictions for realistic experimental observables. I conclude by showing how Haag's theorem is illustrative of a general tension between the foundational significance of results that can be obtained in perturbation theory and non-perturbative characterizations of the content of quantum field theory. (shrink)

Haag’s theorem has been interpreted as establishing that quantum field theory cannot consistently represent interacting fields. Earman and Fraser have clarified how it is possible to give mathematically consistent calculations in scattering theory despite the theorem. However, their analysis does not fully address the worry raised by the result. In particular, I argue that their approach fails to be a complete explanation of why Haag’s theorem does not undermine claims about the empirical adequacy of particular quantum field theories. I (...) then show that such empirical adequacy claims are protected from Haag’s result by the techniques that are required to obtain theoretical predictions for realistic experimental observables. I conclude by showing how Haag’s theorem is illustrative of a general tension between the foundational significance of results that can be obtained in perturbation theory and non-perturbative characterizations of the content of quantum field theory. 1 Introduction2 Haag’s Theorem and the Interaction Picture3 Earman and Fraser on the Success of Scattering Theory4 Haag’s Theorem and Empirical Adequacy5 Conclusion. (shrink)

We describe a dual-process theory of how individuals estimate the probabilities of unique events, such as Hillary Clinton becoming U.S. President. It postulates that uncertainty is a guide to improbability. In its computer implementation, an intuitive system 1 simulates evidence in mental models and forms analog non-numerical representations of the magnitude of degrees of belief. This system has minimal computational power and combines evidence using a small repertoire of primitive operations. It resolves the uncertainty of divergent evidence for single (...) events, for conjunctions of events, and for inclusive disjunctions of events, by taking a primitive average of non-numerical probabilities. It computes conditional probabilities in a tractable way, treating the given event as evidence that may be relevant to the probability of the dependent event. A deliberative system 2 maps the resulting representations into numerical probabilities. With access to working memory, it carries out arithmetical operations in combining numerical estimates. Experiments corroborated the theory's predictions. Participants concurred in estimates of real possibilities. They violated the complete joint probability distribution in the predicted ways, when they made estimates about conjunctions: P, P, P, disjunctions: P, P, P, and conditional probabilities P, P, P. They were faster to estimate the probabilities of compound propositions when they had already estimated the probabilities of each of their components. We discuss the implications of these results for theories of probabilistic reasoning. (shrink)

Shapiro and Sober claim that Walsh, Ariew, Lewens, and Matthen give a mistaken, a priori defense of natural selection and drift as epiphenomenal. Contrary to Shapiro and Sober’s claims, we first argue that WALM’s explanatory doctrine does not require a defense of epiphenomenalism. We then defend WALM’s explanatory doctrine by arguing that the explanations provided by the modern genetical theory of natural selection are ‘autonomous-statistical explanations’ analogous to Galton’s explanation of reversion to mediocrity and an explanation of the diffusion ofgases. (...) We then argue that whereas Sober’s theory of forces is an adequate description of Darwin’s theory, WALM’s explanatory doctrine is required to understand how themodern genetical theory of natural selection explains large-scale statistical regularities. 1 Introduction2 Shapiro and Sober’s ‘Epiphenomenalism Do’s and Don’ts’3 WALM’s Explanatory Doctrine4 Galton’s Autonomous-Statistical Explanation5 A Second Example: The Statistical Explanation of the Diffusion of Gases6 Distinguishing Two Theories of Evolution by Natural Selection7 A Possible Objection: Are Statistical Laws Sufficient for Explanation?8 Conclusion. (shrink)

A Guide to Plato's Republic provides an integral interpretation of the Republic which is accessible even to readers approaching Plato's masterwork for the first time. Written at a level understandable to undergraduates, it is ideal for students and other readers who have little or no background in philosophy or political theory. Rice anticipates their inevitable reactions to the Republic and treats them seriously, opening the way to an appreciation of the complexities of the text without oversimplifying it. While many books (...) on the Republic never stray far from explicating Plato's text, this work contrasts Plato's responses to perennial issues in philosophy and political theory with those of several key subsequent thinkers. It uses engaging examples to show the continuing relevance of Plato's arguments and introduces some basic vocabulary of philosophy and political theory, going beyond terse dictionary definitions by illustrating what technical terms mean in the context of Plato's work. The author's interpretative posture is appreciative but respectfully critical of Plato's vision. Stressing the relationship between Plato's politics and metaphysics, Rice argues that Plato's reluctance to accept the reality and consequences of finitude accounts for much of what many readers find objectionable in his politics. Lively, relatively brief, and designed to provoke discussion in the classroom, A Guide to Plato's Republic is ideal for political theory and introduction to philosophy courses as well as other courses that assign the Republic as a primary text. (shrink)

Many books explain what is known about the universe. This book investigates what cannot be known. Rather than exploring the amazing facts that science, mathematics, and reason have revealed to us, this work studies what science, mathematics, and reason tell us cannot be revealed. In The Outer Limits of Reason, Noson Yanofsky considers what cannot be predicted, described, or known, and what will never be understood. He discusses the limitations of computers, physics, logic, and our own thought processes. (...) Yanofsky describes simple tasks that would take computers trillions of centuries to complete and other problems that computers can never solve; perfectly formed English sentences that make no sense; different levels of infinity; the bizarre world of the quantum; the relevance of relativity theory; the causes of chaos theory; math problems that cannot be solved by normal means; and statements that are true but cannot be proven. He explains the limitations of our intuitions about the world -- our ideas about space, time, and motion, and the complex relationship between the knower and the known. Moving from the concrete to the abstract, from problems of everyday language to straightforward philosophical questions to the formalities of physics and mathematics, Yanofsky demonstrates a myriad of unsolvable problems and paradoxes. Exploring the various limitations of our knowledge, he shows that many of these limitations have a similar pattern and that by investigating these patterns, we can better understand the structure and limitations of reason itself. Yanofsky even attempts to look beyond the borders of reason to see what, if anything, is out there. (shrink)

This paper consists primarily of a survey of results of Harvey Friedman about some proof-theoretic aspects of various forms of Kruskal's tree theorem, and in particular the connection with the ordinal Γ0. We also include a fairly extensive treatment of normal functions on the countable ordinals, and we give a glimpse of Verlen hierarchies, some subsystems of second-order logic, slow-growing and fast-growing hierarchies including Girard's result, and Goodstein sequences. The central theme of this paper is a powerful theorem due to (...) Kruskal, the ‘tree theorem’, as well as a ‘finite miniaturization’ of Kruskal's theorem due to Harvey Friedman. These versions of Kruskal's theorem are remarkable from a proof-theoretic point of view because they are not provable in relatively strong logical systems. They are examples of so-called ‘natural independence phenomena’, which are considered by most logicians as more natural than the metamathematical incompleteness results first discovered by Gödel. Kruskal's tree theorem also plays a fundamental role in computer science, because it is one of the main tools for showing that certain orderings on trees are well founded. These orderings play a crucial role in proving the termination of systems of rewrite rules and the correctness of Knuth-Bendix completion procedures. There is also a close connection between a certain infinite countable ordinal called Γo and Kruskal's theorem. Previous definitions of the function involved in this connection are known to be incorrect, in that, the function is not monotonic. We offer a repaired definition of this function, and explore briefly the consequences of its existence. (shrink)

In five works spanning a decade, Philip E. Tetlock's interest in counterfactuals has changed. He began with an optimistic desire to make social science more rigorous by identifying best practices in the absence of non-imagined controls for experimentation. Soon, however, he adopted a more pessimistic analysis of the cognitive and psychological barriers facing experts. This shift was brought on by an awareness that experts are not rational Bayesians who continually update their theories to keep up with new information; but (...) instead are affected by political, cognitive, and psychological heuristics, including hindsight bias, cognitive conservatism, and the fundamental attribution error. But techniques of computational simulation-involving the rigorous production of large numbers of counterfactual worlds-make it possible to mitigate both problems that Tetlock identifies: that history, produced only once, is a lousy teacher; and that humans, with their collection of non-Bayesian heuristics, are lousy pupils. Tetlock was wrong to reject this approach as theoretically promising but rhetorically and practically impractical. (shrink)

In five works spanning a decade, Philip E. Tetlock's interest in counterfactuals has changed. He began with an optimistic desire to make social science more rigorous by identifying best practices in the absence of non-imagined controls for experimentation. Soon, however, he adopted a more pessimistic analysis of the cognitive and psychological barriers facing experts. This shift was brought on by an awareness that experts are not rational Bayesians who continually update their theories to keep up with new information; but (...) instead are affected by political, cognitive, and psychological heuristics, including hindsight bias, cognitive conservatism, and the fundamental attribution error. But techniques of computational simulation-involving the rigorous production of large numbers of counterfactual worlds-make it possible to mitigate both problems that Tetlock identifies: that history, produced only once, is a lousy teacher; and that humans, with their collection of non-Bayesian heuristics, are lousy pupils. Tetlock was wrong to reject this approach as theoretically promising but rhetorically and practically impractical. (shrink)

Predictive processing as a computational motif of the neocortex needs to be elaborated into theories of higher cognitive functions that include simulating future behavioural outcomes. We contribute to the neuroscientific perspective of predictive processing as a foundation for the proposed representational architectures of the mind.

A thought experiment focuses attention on the kinds of commonalities and differences to be found in two small parts of visual cortical areas during responses to stimuli that are either identical in quality, but different in location, or identical in location and different only in the one visible property of colour. Reflection on this thought experiment leads to the view that patterns of neural activation are the best candidates for causes of qualitatively conscious events . This view faces a strong (...) objection, namely, that patterns can be realized in many media, and thus candidates for patterns that cause qualia might be realized in ways that would not plausibly result in consciousness. It is argued that this objection can be overcome if qualia-causing patterns of events must be realized within small spatial and temporal regions. Much more importantly, it is argued that this restriction on region size need not be ad hoc. The key concept needed to establish this important point is ‘natural salience’, i.e., distinction from background noise that does not depend on application of a criterion of selection. It is explained how natural salience could figure in an empirically-based theory that would entail size restrictions for qualia-causing neural activation patterns. The question is then raised as to how the resulting view diverges from Chalmers’ account, which relies on the Principle of Organizational Invariance. A second thought experiment envisages replacement of neurons by computer chips with synaptic interfaces. Reflection on this thought experiment enables us to conceptually, and possibly empirically, separate the two views. An argument for preferring the patterns- as-causes , or PACQ, view is given. Because natural salience does not plausibly produce strictly discontinuous boundaries between pattern and noise, questions naturally arise as to the relation of the PACQ view to panpsychism and to ‘emergence’. The PACQ view is distinguished from panpsychism, and it is explained how the former avoids what Seager calls ‘the combination problem', and is thus preferable to panpsychism. The relation of the PACQ view to ‘emergence’ is explained. The conclusion of the paper is that the PACQ view is a philosophically defensible and potentially scientifically fruitful view that offers qualia realists the best hypothesis concerning the neural causes of qualia. (shrink)

This paper initiates the reverse mathematics of social choice theory, studying Arrow's impossibility theorem and related results including Fishburn's possibility theorem and the Kirman–Sondermann theorem within the framework of reverse mathematics. We formalise fundamental notions of social choice theory in second-order arithmetic, yielding a definition of countable society which is tractable in RCA0. We then show that the Kirman–Sondermann analysis of social welfare functions can be carried out in RCA0. This approach yields a proof of Arrow's theorem in RCA0, and (...) thus in PRA, since Arrow's theorem can be formalised as a Π01 sentence. Finally we show that Fishburn's possibility theorem for countable societies is equivalent to ACA0 over RCA0. (shrink)

With the advent of powerful parallel computers, efforts have commenced to simulate complete mammalian brains. However, so far none of these efforts has produced outcomes close to explaining even the behavioral complexities of animals. In this article, we suggest four challenges that ground this shortcoming. First, we discuss the connection between hypothesis testing and simulations. Typically, efforts to simulate complete mammalian brains lack a clear hypothesis. Second, we treat complications related to a lack of parameter constraints for large-scale simulations. To (...) demonstrate the severity of this issue, we review work on two small-scale neural systems, the crustacean stomatogastric ganglion and the Caenorhabditis elegans nervous system. Both of these small nervous systems are very thoroughly, but not completely understood, mainly due to issues with variable and plastic parameters. Third, we discuss the hierarchical structure of neural systems as a principled obstacle to whole-brain simulations. Different organizational levels imply qualitative differences not only in structure, but in choice and appropriateness of investigative technique and perspective. The challenge of reconciling different levels also undergirds the challenge of simulating and hypothesis testing, as modeling a system is not the same thing as simulating it. Fourth, we point out that animal brains are information processing systems tailored very specifically for the ecological niches the respective animals live in. (shrink)

The four authors present their speculations about the future developments of mathematical logic in the twenty-first century. The areas of recursion theory, proof theory and logic for computer science, model theory, and set theory are discussed independently.

Set theory, it has been contended, developed from its beginnings through a progression ofmathematicalmoves, despite being intertwined with pronounced metaphysical attitudes and exaggerated foundational claims that have been held on its behalf. In this paper, the seminal results of set theory are woven together in terms of a unifying mathematical motif, one whose transmutations serve to illuminate the historical development of the subject. The motif is foreshadowed in Cantor's diagonal proof, and emerges in the interstices of the inclusion vs. membership (...) distinction, a distinction only clarified at the turn of this century, remarkable though this may seem. Russell runs with this distinction, but is quickly caught on the horns of his well-known paradox, an early expression of our motif. The motif becomes fully manifest through the study of functionsof the power set of a set into the set in the fundamental work of Zermelo on set theory. His first proof in 1904 of his Well-Ordering Theoremis a central articulation containing much of what would become familiar in the subsequent development of set theory. Afterwards, the motif is cast by Kuratowski as a fixed point theorem, one subsequently abstracted to partial orders by Bourbaki in connection with Zorn's Lemma. Migrating beyond set theory, that generalization becomes cited as the strongest of fixed point theorems useful in computer science.Section 1 describes the emergence of our guiding motif as a line of development from Cantor's diagonal proof to Russell's Paradox, fueled by the clarification of the inclusion vs. membership distinction. Section 2 engages the motif as fully participating in Zermelo's work on the Well-Ordering Theorem and as newly informing on Cantor's basic result that there is no bijection. Then Section 3 describes in connection with Zorn's Lemma the transformation of the motif into an abstract fixed point theorem, one accorded significance in computer science. (shrink)

Catherine Elgin has recently argued that a nonfactive conception of understanding is required to accommodate the epistemic successes of science that make essential use of idealizations and models. In this paper, I argue that the fact that our best scientific models and theories are pervasively inaccurate representations can be made compatible with a more nuanced form of scientific realism that I call Understanding Realism. According to this view, science aims at (and often achieves) factive scientific understanding of natural (...) phenomena. I contend that this factive scientific understanding is provided by grasping a set of true modal information about the phenomenon of interest. Furthermore, contrary to Elgin’s view, I argue that the facticity of this kind of scientific understanding can be separated from the inaccuracy of the models and theories used to produce it. (shrink)

An investigation is presented in which a computer simulation model (DIAGNOSER) is used to develop and test predictions for behavior of subjects in a task of medical diagnosis. The first experiment employed a process‐tracing methodology in order to compare hypothesis generation and evaluation behavior of DIAGNOSER with individuals at different levels of expertise (students, trainees, experts). A second experiment performed with only DIAGNOSER identified conditions under which errors in reasoning in the first experiment could be related to interpretation (...) of specific data items. Predictions derived from DIAGNOSER's performance were tested in a third experiment with a new sample of subjects. Data from the three experiments indicated that (1) form of diagnostic reasoning was similar for all subjects trained in medicine and for the simulation model, (2) substance of diagnostic reasoning employed by the simulation model was parable with that of the more expert subjects, and (3) errors in subjects' reasoning were attributable to deficiencies in disease knowledge and the interpretation of specific patient data cues predicted by the simulation model. (shrink)

There is a very simple way in which the safe/normal variable discipline of Bellantoni–Cook recursion [S. Bellantoni, S. Cook, A new recursion theoretic characterization of the polytime functions, Computational Complexity 2 97–110] can be imposed on arithmetical theories like PA: quantify over safes and induct on normals. This weakens the theory severely, so that the provably recursive functions become more realistically computable . Earlier results of D. Leivant [Intrinsic theories and computational complexity, in: D. Leivant , Logic and (...) Computational Complexity, Lecture Notes in Computer Science, vol. 960, Springer-Verlag, 1995, pp. 177–194] are re-worked and extended in this new context, giving proof-theoretic characterizations of complexity classes between Grzegorczyk’s E2 and E3. (shrink)