This much-needed book provides a thorough account of temporal logic, one of the most important areas of logic in computer science today. The book begins with a solid introduction to semantical and axiomatic approaches to temporal logic. It goes on to cover predicate temporal logic, meta-languages, general theories of axiomatization, many dimensional systems, propositional quantifiers, expressive power, Henkin dimension, temporalization of other logics, and decidability results. With its inclusion of cutting-edge results and unifying methodologies, this book is an (...) indispensable reference for both the pure logician and the theoretical computer scientist. (shrink)

Depending on what one means by the main connective of logic, the -if..., then... -, several systems of logic result: classic and modal logics, intuitionistic logic or relevance logic. This book presents the underlying ideas, the syntax and the semantics of these logics. Soundness and completeness are shown constructively and in a uniform way. Attention is paid to the interdisciplinary role of logic: its embedding in the foundations of mathematics and its intimate connection with philosophy, in particular the philosophy (...) of language. Set theory is presented both as a conditio sine qua non for logic and as a interesting exact ontology. The study of infinite sets yields perplexing results. Formalization of informal number theory results in formal number theory; Godel's incompleteness is treated. At appropriate places attention is paid to paradoxes, intuitionism, conditionals, the historical development of logic, to logic programming and automated theorem proving for classical logic.". (shrink)

Mathematical logic is a branch of mathematics that takes axiom systems and mathematical proofs as its objects of study. This book shows how it can also provide a foundation for the development of information science and technology. The first five chapters systematically present the core topics of classical mathematical logic, including the syntax and models of first-order languages, formal inference systems, computability and representability, and Gödel’s theorems. The last five chapters present extensions and developments of classical (...) mathematical logic, particularly the concepts of version sequences of formal theories and their limits, the system of revision calculus, proschemes (formal descriptions of proof methods and strategies) and their properties, and the theory of inductive inference. All of these themes contribute to a formal theory of axiomatization and its application to the process of developing information technology and scientific theories. The book also describes the paradigm of three kinds of language environments for theories and it presents the basic properties required of a meta-language environment. Finally, the book brings these themes together by describing a workflow for scientific research in the information era in which formal methods, interactive software and human invention are all used to their advantage. The second edition of the book includes major revisions on the proof of the completeness theorem of the Gentzen system and new contents on the logic of scientific discovery, R-calculus without cut, and the operational semantics of program debugging. This book represents a valuable reference for graduate and undergraduate students and researchers in mathematics, information science and technology, and other relevant areas of natural sciences. Its first five chapters serve as an undergraduate text in mathematical logic and the last five chapters are addressed to graduate students in relevant disciplines. (shrink)

Attempts to lay a foundation for the sciences based on modern mathematics are questioned. In particular, it is not clear that computer science should be based on set-theoretic mathematics. Set-theoretic mathematics has difficulties with its own foundations, making it reasonable to explore alternative foundations for the sciences. The role of computation within an alternative framework may prove to be of great potential in establishing a direction for the new field of computer science.Whitehead''s theory of (...) reality is re-examined as a foundation for the sciences. His theory does not simply attempt to add formal rigor to the sciences, but instead relies on the methods of the biological and social sciences to construct his world-view. Whitehead''s theory is a rich source of notions that are intended to explain every element of experience. It is a product of Whitehead''s earlier attempt to provide a mathematical foundation for the physical sciences and is still consistent with modern physics. (shrink)

This book constitutes the refereed proceedings of the Third International Symposium on Stochastic Algorithms: Foundations and Applications, SAGA 2005, held in Moscow, Russia in October 2005. The 14 revised full papers presented together with 5 invited papers were carefully reviewed and selected for inclusion in the book. The contributed papers included in this volume cover both theoretical as well as applied aspects of stochastic computations whith a special focus on new algorithmic ideas involving stochastic decisions and the design (...) and evaluation of stochastic algorithms within realistic scenarios. (shrink)

This book presents a set of historical recollections on the work of Martin Davis and his role in advancing our understanding of the connections between logic, computing, and unsolvability. The individual contributions touch on most of the core aspects of Davis’ work and set it in a contemporary context. They analyse, discuss and develop many of the ideas and concepts that Davis put forward, including such issues as contemporary satisfiability solvers, essential unification, quantum computing and generalisations of Hilbert’s tenth problem. (...) The book starts out with a scientific autobiography by Davis, and ends with his responses to comments included in the contributions. In addition, it includes two previously unpublished original historical papers in which Davis and Putnam investigate the decidable and the undecidable side of Logic, as well as a full bibliography of Davis’ work. As a whole, this book shows how Davis’ scientific work lies at the intersection of computability, theoretical computer science, foundations of mathematics, and philosophy, and draws its unifying vision from his deep involvement in Logic. (shrink)

This volume presents the proceedings from the Eleventh Brazilian Logic Conference on Mathematical Logic held by the Brazilian Logic Society in Salvador, Bahia, Brazil. The conference and the volume are dedicated to the memory of professor Mario Tourasse Teixeira, an educator and researcher who contributed to the formation of several generations of Brazilian logicians. Contributions were made from leading Brazilian logicians and their Latin-American and European colleagues. All papers were selected by a careful refereeing processs and were revised and (...) updated by their authors for publication in this volume. There are three sections: Advances in Logic, Advances in Theoretical Computer Science, and Advances in Philosophical Logic. Well-known specialists present original research on several aspects of model theory, proof theory, algebraic logic, category theory, connections between logic and computer science, and topics of philosophical logic of current interest. Topics interweave proof-theoretical, semantical, foundational, and philosophical aspects with algorithmic and algebraic views, offering lively high-level research results. (shrink)

This is a concise book that introduces students to the basics of logical thinking and important mathematical structures that are critical for a solid understanding of logical formalisms themselves as well as for building the necessary background to tackle other fields that are based on these logical principles. Despite its compact and small size, it includes many solved problems and quite a few end-of-section exercises that will help readers consolidate their understanding of the material. This textbook is essential reading (...) for anyone interested in the logical foundations of Informatics, Computer Science, Data Science, Artificial Intelligence, and other related areas. Written with undergraduate students in these disciplines in mind, this book can very well serve the needs of interested and curious readers who wish to get a grasp of the logical principles upon which these fields are built. This book does not require readers to possess math skills beyond those learned in high school. (shrink)

We present a new manifestation of Gödel’s second incompleteness theorem and discuss its foundational significance, in particular with respect to Hilbert’s program. Specifically, we consider a proper extension of Peano arithmetic ( $\mathbf {PA}$ ) by a mathematically meaningful axiom scheme that consists of $\Sigma ^0_2$ -sentences. These sentences assert that each computably enumerable ( $\Sigma ^0_1$ -definable without parameters) property of finite binary trees has a finite basis. Since this fact entails the existence of polynomial time algorithms, it is (...) relevant for computer science. On a technical level, our axiom scheme is a variant of an independence result due to Harvey Friedman. At the same time, the meta-mathematical properties of our axiom scheme distinguish it from most known independence results: Due to its logical complexity, our axiom scheme does not add computational strength. The only known method to establish its independence relies on Gödel’s second incompleteness theorem. In contrast, Gödel’s theorem is not needed for typical examples of $\Pi ^0_2$ -independence (such as the Paris–Harrington principle), since computational strength provides an extensional invariant on the level of $\Pi ^0_2$ -sentences. (shrink)

In the 21st century, digitalization is a global challenge of mankind. Even for the public, it is obvious that our world is increasingly dominated by powerful algorithms and big data. But, how computable is our world? Some people believe that successful problem solving in science, technology, and economies only depends on fast algorithms and data mining. Chances and risks are often not understood, because the foundations of algorithms and information systems are not studied rigorously. Actually, they are deeply (...) rooted in logics, mathematics, computer science and philosophy. Therefore, this book studies the foundations of mathematics, computer science, and philosophy, in order to guarantee security and reliability of the knowledge by constructive proofs, proof mining and program extraction. We start with the basics of computability theory, proof theory, and information theory. In a second step, we introduce new concepts of information and computing systems, in order to overcome the gap between the digital world of logical programming and the analog world of real computing in mathematics and science. The book also considers consequences for digital and analog physics, computational neuroscience, financial mathematics, and the Internet of Things (IoT). (shrink)

A quasi-order Q induces two natural quasi-orders on $${\mathcal{P}(Q)}$$, but if Q is a well-quasi-order, then these quasi-orders need not necessarily be well-quasi-orders. Nevertheless, Goubault-Larrecq (Proceedings of the 22nd Annual IEEE Symposium 4 on Logic in Computer Science (LICS’07), pp. 453–462, 2007) showed that moving from a well-quasi-order Q to the quasi-orders on $${\mathcal{P}(Q)}$$ preserves well-quasi-orderedness in a topological sense. Specifically, Goubault-Larrecq proved that the upper topologies of the induced quasi-orders on $${\mathcal{P}(Q)}$$ are Noetherian, which means that (...) they contain no infinite strictly descending sequences of closed sets. We analyze various theorems of the form “if Q is a well-quasi-order then a certain topology on (a subset of) $${\mathcal{P}(Q)}$$ is Noetherian” in the style of reverse mathematics, proving that these theorems are equivalent to ACA 0 over RCA 0. To state these theorems in RCA 0 we introduce a new framework for dealing with second-countable topological spaces. (shrink)

Designed to provide a firm foundation in mathematical logic, this work aims to serve as an elementary textbook for both graduate study and for applications of logic, such as logic programming and format specification and verification.

Constructibility and complexity play central roles in recent research in computer science, mathematics and physics. For example, scientists are investigating the complexity of computer programs, constructive proofs in mathematics and the randomness of physical processes. But there are different approaches to the explication of these concepts. This volume presents important research on the state of this discussion, especially as it refers to quantum mechanics. This `foundational debate' in computer science, mathematics and physics was already fully (...) developed in 1930 in the Vienna Circle. A special section is devoted to its real founder Hans Hahn, referring to his contribution to the history and philosophy of science. The documentation section presents articles on the early Philipp Frank and on the Vienna Circle in exile. Reviews cover important recent literature on logical empiricism and related topics. (shrink)

My purpose in this brief paper is to consider the implications of a radically different computer architecure to some fundamental problems in the foundations of Cognitive Science. More exactly, I wish to consider the ramifications of the 'Gödel-Minds-Machines' controversy of the late 1960s on a dynamically changing computer architecture which, I venture to suggest, is going to revolutionize which 'functions' of the human mind can and cannot be modelled by (non-human) computational automata. I will proceed on (...) the presupposition that the reader is familiar with some of the fundamentals of computational theory and mathematical logic. (shrink)

Preface -- 1. Propositional logic : proofs from axioms and inference rules -- 2. First order logic : proofs with quantifiers -- 3. Set theory : proofs by detachment, contraposition, and contradiction -- 4. Mathematical induction : definitions and proofs by induction -- 5. Well-formed sets : proofs by transfinite induction with already well-ordered sets -- 6. The axiom of choice : proofs by transfinite induction -- 7. applications : Nobel-Prize winning applications of sets, functions, and relations -- 8. (...) Solutions to some odd-numbered exercises. (shrink)

This dissertation examines aspects of the interplay between computing and scientific practice. The appropriate foundational framework for such an endeavour is rather real computability than the classical computability theory. This is so because physical sciences, engineering, and applied mathematics mostly employ functions defined in continuous domains. But, contrary to the case of computation over natural numbers, there is no universally accepted framework for real computation; rather, there are two incompatible approaches --computable analysis and BSS model--, both claiming to formalise algorithmic (...) computation and to offer foundations for scientific computing. -/- The dissertation consists of three parts. In the first part, we examine what notion of 'algorithmic computation' underlies each approach and how it is respectively formalised. It is argued that the very existence of the two rival frameworks indicates that 'algorithm' is not one unique concept in mathematics, but it is used in more than one way. We test this hypothesis for consistency with mathematical practice as well as with key foundational works that aim to define the term. As a result, new connections between certain subfields of mathematics and computer science are drawn, and a distinction between 'algorithms' and 'effective procedures' is proposed. -/- In the second part, we focus on the second goal of the two rival approaches to real computation; namely, to provide foundations for scientific computing. We examine both frameworks in detail, what idealisations they employ, and how they relate to floating-point arithmetic systems used in real computers. We explore limitations and advantages of both frameworks, and answer questions about which one is preferable for computational modelling and which one for addressing general computability issues. -/- In the third part, analog computing and its relation to analogue (physical) modelling in science are investigated. Based on some paradigmatic cases of the former, a certain view about the nature of computation is defended, and the indispensable role of representation in it is emphasized and accounted for. We also propose a novel account of the distinction between analog and digital computation and, based on it, we compare analog computational modelling to physical modelling. It is concluded that the two practices, despite their apparent similarities, are orthogonal. (shrink)

This dissertation examines aspects of the interplay between computing and scientific practice. The appropriate foundational framework for such an endeavour is rather real computability than the classical computability theory. This is so because physical sciences, engineering, and applied mathematics mostly employ functions defined in continuous domains. But, contrary to the case of computation over natural numbers, there is no universally accepted framework for real computation; rather, there are two incompatible approaches --computable analysis and BSS model--, both claiming to formalise algorithmic (...) computation and to offer foundations for scientific computing. The dissertation consists of three parts. In the first part, we examine what notion of 'algorithmic computation' underlies each approach and how it is respectively formalised. It is argued that the very existence of the two rival frameworks indicates that 'algorithm' is not one unique concept in mathematics, but it is used in more than one way. We test this hypothesis for consistency with mathematical practice as well as with key foundational works that aim to define the term. As a result, new connections between certain subfields of mathematics and computer science are drawn, and a distinction between 'algorithms' and 'effective procedures' is proposed. In the second part, we focus on the second goal of the two rival approaches to real computation; namely, to provide foundations for scientific computing. We examine both frameworks in detail, what idealisations they employ, and how they relate to floating-point arithmetic systems used in real computers. We explore limitations and advantages of both frameworks, and answer questions about which one is preferable for computational modelling and which one for addressing general computability issues. In the third part, analog computing and its relation to analogue (physical) modelling in science are investigated. Based on some paradigmatic cases of the former, a certain view about the nature of computation is defended, and the indispensable role of representation in it is emphasized and accounted for. We also propose a novel account of the distinction between analog and digital computation and, based on it, we compare analog computational modelling to physical modelling. It is concluded that the two practices, despite their apparent similarities, are orthogonal. (shrink)

Constructibility and complexity play central roles in recent research in computer science, mathematics and physics. For example, scientists are investigating the complexity of computer programs, constructive proofs in mathematics and the randomness of physical processes. But there are different approaches to the explication of these concepts. This volume presents important research on the state of this discussion, especially as it refers to quantum mechanics. This `foundational debate' in computer science, mathematics and physics was already fully (...) developed in 1930 in the Vienna Circle. A special section is devoted to its real founder Hans Hahn, referring to his contribution to the history and philosophy of science. The documentation section presents articles on the early Philipp Frank and on the Vienna Circle in exile. Reviews cover important recent literature on logical empiricism and related topics. (shrink)

Logic has pride of place in mathematics and its 20th century offshoot, computer science. Modern symbolic logic was developed, in part, as a way to provide a formal framework for mathematics: Frege, Peano, Whitehead and Russell, as well as Hilbert developed systems of logic to formalize mathematics. These systems were meant to serve either as themselves foundational, or at least as formal analogs of mathematical reasoning amenable to mathematical study, e.g., in Hilbert’s consistency program. Similar efforts (...) continue, but have been expanded by the development of sophisticated methods to study the properties of such systems using proof and model theory. In parallel with this evolution of logical formalisms as tools for articulating mathematical theories (broadly speaking), much progress has been made in the quest for a mechanization of logical inference and the investigation of its theoretical limits, culminating recently in the development of new foundational frameworks for mathematics with sophisticated computer-assisted proof systems. In addition, logical formalisms developed by logicians in mathematical and philosophical contexts have proved immensely useful in describing theories and systems of interest to computer scientists, and to some degree, vice versa. Three examples of the influence of logic in computer science are automated reasoning, computer verification, and type systems for programming languages. (shrink)

Tylman has recently pointed out some striking conceptual and methodological analogies between philosophy and computer science. In this paper, I focus on one of Tylman’s most convincing cases, viz. the similarity between Plato’s theory of Ideas and the object-oriented programming paradigm, and analyze it in some more detail. In particular, I argue that the platonic doctrine of the Porphyrian tree corresponds to the fact that most object-oriented programming languages do not support multiple inheritance. This analysis further reinforces Tylman’s (...) point regarding the conceptual continuity between classical metaphysical theorizing and contemporary computer science. (shrink)

In computational complexity theory, decision problems are divided into complexity classes based on the amount of computational resources it takes for algorithms to solve them. In theoretical computer science, it is commonly accepted that only functions for solving problems in the complexity class P, solvable by a deterministic Turing machine in polynomial time, are considered to be tractable. In cognitive science and philosophy, this tractability result has been used to argue that only functions in P can feasibly (...) work as computational models of human cognitive capacities. One interesting area of computational complexity theory is descriptive complexity, which connects the expressive strength of systems of logic with the computational complexity classes. In descriptive complexity theory, it is established that only first-order (classical) systems are connected to P, or one of its subclasses. Consequently, second-order systems of logic are considered to be computationally intractable, and may therefore seem to be unfit to model human cognitive capacities. This would be problematic when we think of the role of logic as the foundations of mathematics. In order to express many important mathematical concepts and systematically prove theorems involving them, we need to have a system of logic stronger than classical first-order logic. But if such a system is considered to be intractable, it means that the logical foundation of mathematics can be prohibitively complex for human cognition. In this paper I will argue, however, that this problem is the result of an unjustified direct use of computational complexity classes in cognitive modelling. Placing my account in the recent literature on the topic, I argue that the problem can be solved by considering computational complexity for humanly relevant problem solving algorithms and input sizes. (shrink)

This work locates the historical and conceptual foundations of cognitive science in the "commonsense" psychology of the philosopher Thomas Reid. I begin with Reid's attack on his rationalist and empiricist competitors of the 17th and 18th centuries. I then present his positive theory as a sophisticated faculty psychology appealing to innateness of mental structure. Reidian psychological faculties are equally trustworthy, causally independent mental powers, and I argue that they share nine distinct properties. This distinguishes Reidian 'intentionalism' from idealist (...) 'representationalism,' which derive cognitive content either from the inherited structure of faculties of from the occurrent structure of sensory activity. ;Next, I turn to consciousness and reflection for a contemporary Reidian response to traditional phenomenology. Unlike reflection, faculties of reason and remembrance are not causally mediated by consciousness. My interpretation of Reid is that 'Humean causation' of individual faculty structure accounts only for 'natural intentionality,' while 'efficient causation' of faculty interrelations accounts for cognitive 'personal intentionality.' ;I then proceed by adopting a form of computational description for reconstructing this view as a computational theory of mind. I contrast functional analyses in explanations of some capacities with computational and componential analyses in explanations of other, intentional capacities, in which some processes must be taken to semantically encode and govern the roles of others. This step reconstructs the Reidian notion of intentional operations as requiring an explanation of component faculties and their representation-governed interactions. I argue that properties of faculties delimiting these interactions under Reid's theory parallel those in Fodor's essay on the "modularity of mind," although the reasons given for individual criteria are often very different. Fodor also proposes a trichotomous mental structure, but I find that a third level of "central systems" is a myth engendered by causal information theory. Such an analysis cannot capture generalizations over the internal representation of semantic roles that determines the character of faculty relations. This requirement for any computational account of cognition is precisely the motivation for the reconstructed Reidian theory. Thus, it comports more favorably with the explanatory program constitutive of a computational cognitive science. (shrink)

This book is designed to make accessible to nonspecialists the still evolving concepts of quantum mechanics and the terminology in which these are expressed. The opening chapters summarize elementary concepts of twentieth century quantum mechanics and describe the mathematical methods employed in the field, with clear explanation of, for example, Hilbert space, complex variables, complex vector spaces and Dirac notation, and the Heisenberg uncertainty principle. After detailed discussion of the Schrödinger equation, subsequent chapters focus on isotropic vectors, used to (...) construct spinors, and on conceptual problems associated with measurement, superposition, and decoherence in quantum systems. Here, due attention is paid to Bell's inequality and the possible existence of hidden variables. Finally, progression toward quantum computation is examined in detail: if quantum computers can be made practicable, enormous enhancements in computing power, artificial intelligence, and secure communication will result. This book will be of interest to a wide readership seeking to understand modern quantum mechanics and its potential applications. (shrink)

We examine two very different approaches to formalising real computation, commonly referred to as “Computable Analysis” and “the BSS approach”. The main models of computation underlying these approaches—bit computation and BSS, respectively—have also been put forward as appropriate foundations for scientific computing. The two frameworks offer useful computability and complexity results about problems whose underlying domain is an uncountable space. Since typically the problems dealt with in physical sciences, applied mathematics, economics, and engineering are also defined in uncountable domains, (...) it is fitting that we choose between these two approaches a foundational framework for scientific computing. However, the models are incompatible as to their results. What is more, the BSS model is highly idealised and unrealistic; yet, it is the de facto implicit model in various areas of computational mathematics, with virtually no problems for the everyday practice. This paper serves three purposes. First, we attempt to delineate what the goal of developing foundations for scientific computing exactly is. We distinguish between two very different interpretations of that goal, and on the separate basis of each one, we put forward answers about the appropriateness of each framework. Second, we provide an account of the fruitfulness and wide use of BSS, despite its unrealistic assumptions. Third, according to one of our proposed interpretations of the scope of foundations, the target domain of both models is a certain mathematical structure. In a clear sense, then, we are using idealised models to study a purely mathematical structure. The third purpose is to point out and explain this intriguing phenomenon and attempt to make connections with the typical case of idealised models of empirical domains. (shrink)

This book illustrates the program of Logical-Informational Dynamics. Rational agents exploit the information available in the world in delicate ways, adopt a wide range of epistemic attitudes, and in that process, constantly change the world itself. Logical-Informational Dynamics is about logical systems putting such activities at center stage, focusing on the events by which we acquire information and change attitudes. Its contributions show many current logics of information and change at work, often in multi-agent settings where social behavior is essential, (...) and often stressing Johan van Benthem's pioneering work in establishing this program. However, this is not a Festschrift, but a rich tapestry for a field with a wealth of strands of its own. The reader will see the state of the art in such topics as information update, belief change, preference, learning over time, and strategic interaction in games. Moreover, no tight boundary has been enforced, and some chapters add more general mathematical or philosophical foundations or links to current trends in computer science. (shrink)

The present volume stems from a book-proposal made about two years ago to College Publications, London. The main idea was that of illustrating the interplay between the contemporary work in logic and the mainstream mathematics. The division of the volume in two sections - topics in 'logic' vs topics in 'computing' - is more or less conventional. Some contributions are focussed on historical and technical details meant to put in perspective the impact of the work of some outstanding mathematicians and (...) philosophers on the contemporary research in logic and computing science. Some other papers, also with a historical flavour, were supposed to evidentiate punctual methods of research and specific concepts or topics, as, e.g., decidability, computability, randomness, and computational or descriptive complexity. In general, the papers were intended as specific surveys of results. Other volumes - to be issued subsequently in the same series - will hopefully delineate aspects of the contemporary logic landscape that have not been illustrated here. The intended audience of the book includes graduate students in mathematical logic, foundations of matematics, and computing science, as well as philosophers, mathematicians, and, possibly, other scientists interested in the recent research on logic and computing. (shrink)

Foundational questions in logic, mathematics, computer science and physics are constant sources of epistemological debate in contemporary philosophy. To what extent is the transfinite part of mathematics completely trustworthy? Why is there a general 'malaise' concerning the logical approach to the foundations of mathematics? What is the role of symmetry in physics? Is it possible to build a coherent worldview compatible with a macroobjectivistic position and based on the quantum picture of the world? What account can be (...) given of opinion change in the light of new evidence? These are some of the questions discussed in this volume, which collects 14 lectures on the foundations of science given at the School of Philosophy of Science, Trieste, October 1989. The volume will be of particular interest to any student or scholar engaged in interdisciplinary research into the foundations of science in the context of contemporary debates. (shrink)

This is a short, modern, and motivated introduction to mathematical logic for upper undergraduate and beginning graduate students in mathematics and computer science. Any mathematician who is interested in getting acquainted with logic and would like to learn Gödel’s incompleteness theorems should find this book particularly useful. The treatment is thoroughly mathematical and prepares students to branch out in several areas of mathematics related to foundations and computability, such as logic, axiomatic set theory, model theory, (...) recursion theory, and computability. In this new edition, many small and large changes have been made throughout the text. The main purpose of this new edition is to provide a healthy first introduction to model theory, which is a very important branch of logic. Topics in the new chapter include ultraproduct of models, elimination of quantifiers, types, applications of types to model theory, and applications to algebra, number theory and geometry. Some proofs, such as the proof of the very important completeness theorem, have been completely rewritten in a more clear and concise manner. The new edition also introduces new topics, such as the notion of elementary class of structures, elementary diagrams, partial elementary maps, homogeneous structures, definability, and many more. (shrink)

Combining physics, mathematics and computer science, quantum computing and its sister discipline of quantum information have developed in the past few decades from visionary ideas to two of the most fascinating areas of quantum theory. General interest and excitement in quantum computing was initially triggered by Peter Shor (1994) who showed how a quantum algorithm could exponentially “speed-up” classical computation and factor large numbers into primes far more efficiently than any (known) classical algorithm. Shor’s algorithm was soon followed (...) by several other algorithms that aimed to solve combinatorial and algebraic problems, and in the years since theoretical study of quantum systems serving as computational devices has achieved tremendous progress. Common belief has it that the implementation of Shor’s algorithm on a large scale quantum computer would have devastating consequences for current cryptography protocols which rely on the premise that all known classical worst-case algorithms for factoring take time exponential in the length of their input (see, e.g., Preskill 2005). Consequently, experimentalists around the world are engaged in attempts to tackle the technological difficulties that prevent the realisation of a large scale quantum computer. But regardless whether these technological problems can be overcome (Unruh 1995; Ekert and Jozsa 1996; Haroche and Raimond 1996), it is noteworthy that no proof exists yet for the general superiority of quantum computers over their classical counterparts. -/- The philosophical interest in quantum computing is manifold. From a social-historical perspective, quantum computing is a domain where experimentalists find themselves ahead of their fellow theorists. Indeed, quantum mysteries such as entanglement and nonlocality were historically considered a philosophical quibble, until physicists discovered that these mysteries might be harnessed to devise new efficient algorithms. But while the technology for harnessing the power of 50–100 qubits (the basic unit of information in the quantum computer) is now within reach (Preskill 2018), only a handful of quantum algorithms exist, and the question of whether these can truly outperform any conceivable classical alternative is still open. From a more philosophical perspective, advances in quantum computing may yield foundational benefits. For example, it may turn out that the technological capabilities that allow us to isolate quantum systems by shielding them from the effects of decoherence for a period of time long enough to manipulate them will also allow us to make progress in some fundamental problems in the foundations of quantum theory itself. Indeed, the development and the implementation of efficient quantum algorithms may help us understand better the border between classical and quantum physics (Cuffaro 2017, 2018a; cf. Pitowsky 1994, 100), and perhaps even illuminate fundamental concepts such as measurement and causality. Finally, the idea that abstract mathematical concepts such as computability and complexity may not only be translated into physics, but also re-written by physics bears directly on the autonomous character of computer science and the status of its theoretical entities—the so-called “computational kinds”. As such it is also relevant to the long-standing philosophical debate on the relationship between mathematics and the physical world. (shrink)

Colburn (computer science, U. of Minnesota-Duluth) has a doctorate in philosophy and an advanced degree in computer science; he's worked as a philosophy professor, a computer programmer, and a research scientist in artificial intelligence. Here he discusses the philosophical foundations of artificial intelligence; the new encounter of science and philosophy (logic, models of the mind and of reasoning, epistemology); and the philosophy of computer science (touching on math, abstraction, software, and ontology).

Computational methods such as computer simulations, Monte Carlo methods, and agent-based modeling have become the dominant techniques in many areas of science. Extending Ourselves contains the first systematic philosophical account of these new methods, and how they require a different approach to scientific method. Paul Humphreys draws a parallel between the ways in which such computational methods have enhanced our abilities to mathematically model the world, and the more familiar ways in which scientific instruments have expanded our access (...) to the empirical world. This expansion forms the basis for a new kind of empiricism, better suited to the needs of science than the older anthropocentric forms of empiricism. Human abilities are no longer the ultimate standard of epistemological correctness.Humphreys also includes arguments for the primacy of properties rather than objects, the need to consider technological constraints when appraising scientific methods, and a detailed account of how the path from computational template to scientific application is constructed. This last feature allows us to hold a form of selective realism in which anti-realist arguments based on formal reconstructions of theories can be avoided. One important consequence of the rise of computational methods is that the traditional organization of the sciences is being replaced by an organization founded on computational templates.Extending Ourselves will be of interest to philosophers of science, epistemologists, and to anyone interested in the role played by computers in modern science. (shrink)

Climatology is a paradigmatic complex systems science. Understanding the global climate involves tackling problems in physics, chemistry, economics, and many other disciplines. I argue that complex systems like the global climate are characterized by certain dynamical features that explain how those systems change over time. A complex system's dynamics are shaped by the interaction of many different components operating at many different temporal and spatial scales. Examining the multidisciplinary and holistic methods of climatology can help us better understand the (...) nature of complex systems in general. -/- Questions surrounding climate science can be divided into three rough categories: foundational, methodological, and evaluative questions. "How do we know that we can trust science?" is a paradigmatic foundational question (and a surprisingly difficult one to answer). Because the global climate is so complex, questions like "what makes a system complex?" also fall into this category. There are a number of existing definitions of `complexity,' and while all of them capture some aspects of what makes intuitively complex systems distinctive, none is entirely satisfactory. Most existing accounts of complexity have been developed to work with information-theoretic objects (signals, for instance) rather than the physical and social systems studied by scientists. -/- Dynamical complexity, a concept articulated in detail in the first third of the dissertation, is designed to bridge the gap between the mathematics of contemporary complexity theory (in particular the formalism of "effective complexity" developed by Gell-Mann and Lloyd [2003]) and a more general account of the structure of science generally. Dynamical complexity provides a physical interpretation of the formal tools of mathematical complexity theory, and thus can be used as a framework for thinking about general problems in the philosophy of science, including theories, explanation, and lawhood. -/- Methodological questions include questions about how climate science constructs its models, on what basis we trust those models, and how we might improve those models. In order to answer questions about climate modeling, it's important to understand what climate models look like and how they are constructed. Climate model families are significantly more diverse than are the model families of most other sciences (even sciences that study other complex systems). Existing climate models range from basic models that can be solved on paper to staggeringly complicated models that can only be analyzed using the most advanced supercomputers in the world. I introduce some of the central concepts in climatology by demonstrating how one of the most basic climate models might be constructed. I begin with the assumption that the Earth is a simple featureless blackbody which receives energy from the sun and releases it into space, and show how to model that assumption formally. I then gradually add other factors (e.g. albedo and the greenhouse effect) to the model, and show how each addition brings the model's prediction closer to agreement with observation. After constructing this basic model, I describe the so-called "complexity hierarchy" of the rest of climate models, and argue that the sense of "complexity" used in the climate modeling community is related to dynamical complexity. -/- With a clear understanding of the basics of climate modeling in hand, I then argue that foundational issues discussed early in the dissertation suggest that computation plays an irrevocably central role in climate modeling. "Science by simulation" is essential given the complexity of the global climate, but features of the climate system--the presence of non-linearities, feedback loops, and chaotic dynamics--put principled limits on the effectiveness of computational models. This tension is at the root of the staggering pluralism of the climate model hierarchy, and suggests that such pluralism is here to stay, rather than an artifact of our ignorance. Rather than attempting to converge on a single "best fit" climate model, we ought to embrace the diversity of climate models, and view each as a specialized tool designed to predict and explain a rather narrow range of phenomena. Understanding the climate system as a whole requires examining a number of different models, and correlating their outputs. This is the most significant methodological challenge of climatology. -/- Climatology's role contemporary political discourse raises an unusually high number of evaluative questions for a physical science. The two leading approaches to crafting policy surrounding climate change center on mitigation (i.e. stopping the changes from occurring) and adaptation (making post hoc changes to ameliorate the harm caused by those changes). Crafting an effective socio-political response to the threat of anthropogenic climate change, however, requires us to integrate multiple perspectives and values: the proper response will be just as diverse and pluralistic as the climate models themselves, and will incorporate aspects of both approaches. I conclude by offering some concrete recommendations about how to integrate this value pluralism into our socio-political decision making framework. (shrink)

The IOth International Congress of Logic, Methodology and Philosophy of Science, which took place in Florence in August 1995, offered a vivid and comprehensive picture of the present state of research in all directions of Logic and Philosophy of Science. The final program counted 51 invited lectures and around 700 contributed papers, distributed in 15 sections. Following the tradition of previous LMPS-meetings, some authors, whose papers aroused particular interest, were invited to submit their works for publication in a (...) collection of selected contributed papers. Due to the large number of interesting contributions, it was decided to split the collection into two distinct volumes: one covering the areas of Logic, Foundations of Mathematics and Computer Science, the other focusing on the general Philosophy of Science and the Foundations of Physics. As a leading choice criterion for the present volume, we tried to combine papers containing relevant technical results in pure and applied logic with papers devoted to conceptual analyses, deeply rooted in advanced present-day research. After all, we believe this is part of the genuine spirit underlying the whole enterprise of LMPS studies. (shrink)

This edited collection bridges the foundations and practice of constructive mathematics and focuses on the contrast between the theoretical developments, which have been most useful for computer science (ie: constructive set and type theories), and more specific efforts on constructive analysis, algebra and topology. Aimed at academic logician, mathematicians, philosophers and computer scientists with contributions from leading researchers, it is up to date, highly topical and broad in scope.

This volume presents the proceedings of the workshop CSL '91 (Computer Science Logic) held at the University of Berne, Switzerland, October 7-11, 1991. This was the fifth in a series of annual workshops on computer sciencelogic (the first four are recorded in LNCS volumes 329, 385, 440, and 533). The volume contains 33 invited and selected papers on a variety of logical topics in computer science, including abstract datatypes, bounded theories, complexity results, cut elimination, denotational (...) semantics, infinitary queries, Kleene algebra with recursion, minimal proofs, normal forms in infinite-valued logic, ordinal processes, persistent Petri nets, plausibility logic, program synthesis systems, quantifier hierarchies, semantics of modularization, stable logic, term rewriting systems, termination of logic programs, transitive closure logic, variants of resolution, and many others. (shrink)

This essay explores how mid-twentieth-century mathematicians at New York University envisioned their discipline, cultural identities and social roles, and how these self-constructed identities materialized in the planning of their new academic building, Warren Weaver Hall. These mathematicians considered their research to be a ‘living part of the stream of science’, requiring a mathematics research library which they equated to a scientific laboratory and a complex of computing rooms which served as an interdisciplinary research centre. Identifying as ‘scientists’, they understood (...) their societal value to be that of researchers, outputting mathematics research valuable to the natural sciences, the emerging field of computer science and the United States government and military, as well as educators. When the building opened in 1965, it was touted by the university administration as an ‘example of excellence’; it later, in 1970, became the site of heated negotiations when university student and faculty protestors staged a sit-in rebuking the Atomic Energy Commission's Computing Center housed on the second floor. A close study of the correspondence between the mathematicians and their peers in the university's administration, private foundations, government agencies and an architectural firm not only illuminates the day-to-day work practices of this eminent group of mathematicians, but also sheds light on their own self-constructed academic and social identities within their contemporaneous Cold War culture. (shrink)

The scientific study of consciousness, also referred to as consciousness science, is a young scientific field devoted to understanding how conscious experiences and the brain relate. It comprises a host of theories, experiments, and analyses that aim to investigate the problem of consciousness empirically, theoretically, and conceptually. This thesis addresses some of the questions that arise in these investigations from a formal and mathematical perspective. These questions concern theories of consciousness, experimental paradigms, methodology, and artificial consciousness. -/- Regarding (...) theories of consciousness, the thesis contributes to the understanding of the mathematical structure that some of the formal theories in the field propose. The work presented here targets the theory of consciousness known as Integrated Information Theory (IIT) and the neuroscientific theory known as Predictive Processing Theory or Free Energy Principle in its Active Inference form (AI-PP). The thesis provides axiomatic definitions of the mathematical structures that constitute these theories, and uses these definitions to address some of the open questions surrounding the theories. For AI-PP, this includes a rigorous derivation of the formula for Active Inference via Free Energy minimisation and a proof of compositionality of Free Energy. For IIT, this includes resolutions of some of the criticisms of IIT's formal scope and applications, but also the identification of new issues that concern the formalism and its derivation. When possible, the definitions are provided in the mathematical framework of category theory. -/- Regarding experiments, the thesis addresses the main paradigm for testing and falsifying theories of consciousness currently applied in the field. This paradigm consists of comparing the conscious experience that a theory predicts with the conscious experience that is inferred from behavioural data or report by use of measures of consciousness. The thesis provides a formal model of this paradigm and shows that under a certain condition—if inference and prediction are independent—, any minimally informative theory of consciousness can always be falsified. This is deeply problematic since the field’s reliance on report or behaviour to infer conscious experiences, in conjunction with the general structure of most contemporary theories of consciousness, implies such independence. This observation provides the exact formal underpinning of the well-known unfolding argument. The thesis analyses the origin of the problem and identifies precisely which changes are required to avoid this problem in future research. The thesis furthermore shows that the problem of falsifying theories of consciousness, and of empirical comparisons of theories of consciousness more generally, follows from a pervasive closure paradigm in consciousness science, that consists of taking a neuroscientific account of the brain as input to a theory of consciousness, so as to explain what consciousness is, without allowing for modifications or adaptations of the neuroscientific account that would accommodate consciousness as part of the brain's functioning. As is shown in the thesis, this paradigm has implications that point to a fundamental need of revision. -/- Regarding methodological and conceptual questions, the thesis contributes to the foundations of structural research in consciousness science. Structural research aims to use mathematical structures or mathematical spaces, instead of verbal descriptions or simple categorisations, to represent conscious experiences scientifically, for example when building theories of consciousness, or when exploring new empirical avenues to measure consciousness. Despite considerable advances in this realm, there was, prior to this thesis, no explicit definition of what a mathematical structure of conscious experience should be; that is, how the attribution of mathematical structure to conscious experiences should be systematically understood. Perhaps the most important contribution of this thesis to the field is to propose such a definition. The definition, a structural concept, extends existing approaches wherever available, and provides a basis for developing a common formal language to study consciousness, bridging developments as far apart as psychophysics and phenomenology. In addition, and independently of this proposal, the thesis offers a critical analysis of which metaphysical premises need to be presumed in structural research, whether the use of particular formal tools (such as structure-preserving mappings or homomorphisms) is justified, and how structural theories of consciousness could otherwise be built in the first place. An attempt to expand the results from consciousness to more general problems in philosophy of science is made in the context of the well-known Newman problem. -/- Regarding the question of artificial consciousness—can AI feel?—, the thesis contributes two results that take the form of no-go theorems, as known from physics. The first no-go theorem shows that if consciousness is relevant for the temporal evolution of a system's states—if it is dynamically relevant—, then contemporary AI systems cannot be conscious. That is because AI systems run on CPUs, GPUs, TPUs or other processors which have been designed and verified to adhere to computational dynamics that systematically preclude or suppress deviations. The second no-go theorem is situated in the context of computational functionalism, a view which posits that consciousness is a computation. The theorem shows that if computational functionalism holds true, consciousness cannot be a Turing computation. Rather, it must be a novel type of computation that has recently been proposed by Geoffrey Hinton, called mortal computation. -/- This thesis is part of a global effort to pioneer a mathematical perspective in consciousness science, now called Mathematical Consciousness Science. The hope behind the research carried out in this PhD is to illustrate the power and usefulness of mathematical approaches in different areas of consciousness science, and in doing so, to lay the foundations for future mathematical work that complements and supports empirical and theoretical work in the further development of this exciting field. (shrink)

Open peer commentary on the article “Info-computational Constructivism and Cognition” by Gordana Dodig-Crnkovic. Upshot: Dodig-Crnkovic’s “info-computational constructivism” (IC), as an essential part of a constructivist approach, needs integration with the logical, mathematical and physical evidence coming from quantum field theory (QFT) as the fundamental physics of the emergence of “complex systems” in all realms of natural sciences.

Epistemic logic has grown from its philosophical beginnings to find diverse applications in computer science, and as a means of reasoning about the knowledge and belief of agents. This book provides a broad introduction to the subject, along with many exercises and their solutions. The authors begin by presenting the necessary apparatus from mathematics and logic, including Kripke semantics and the well-known modal logics K, T, S4 and S5. Then they turn to applications in the context of distributed (...) systems and artificial intelligence. These include the notions of common knowledge, distributed knowledge, explicit and implicit belief, the interplays between knowledge and time, and knowledge and action, as well as a graded (or numerical) variant of the epistemic operators. The authors also discuss extensively the problem of logical omniscience. They cover Halpern & Moses' theory of honest formulas, and they make a digression into the realm of nonmonotonic reasoning and preferential entailment. They discuss Moore's autoepistemic logic, together with Levesque's related logic of "all I know". Furthermore, they show how one can base default and counterfactual reasoning on epistemic logic. Graduate students in philosophy or in computer science, especially those with an interest in AI, will find this book useful. (shrink)

This paper describes the author’s experience of infusing an introductory database course with privacy content, and the on-going project entitled Integrating Ethics Into the Database Curriculum, that evolved from that experience. The project, which has received funding from the National Science Foundation, involves the creation of a set of privacy modules that can be implemented systematically by database educators throughout the database design thread of an undergraduate course.