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Logic and the Challenge of Computer Science
, 1988
"... Nowadays computer science is surpassing mathematics as the primary field of logic applications, but logic is not tuned properly to the new role. In particular, classical logic is preoccupied mostly with infinite static structures whereas many objects of interest in computer science are dynamic objec ..."
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Nowadays computer science is surpassing mathematics as the primary field of logic applications, but logic is not tuned properly to the new role. In particular, classical logic is preoccupied mostly with infinite static structures whereas many objects of interest in computer science are dynamic objects with bounded resources. This chapter consists of two independent parts. The first part is devoted to finite model theory; it is mostly a survey of logics tailored for computational complexity. The second part is devoted to dynamic structures with bounded resources. In particular, we use dynamic structures with bounded resources to model Pascal.
What Is an Algorithm?
, 2000
"... Machines and Recursive Definitions 2.1 Abstract Machines The bestknown model of mechanical computation is (still) the first, introduced by Turing [18], and after half a century of study, few doubt the truth of the fundamental ChurchTuring Thesis : A function f : N # N on the natural numbers (o ..."
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Cited by 23 (3 self)
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Machines and Recursive Definitions 2.1 Abstract Machines The bestknown model of mechanical computation is (still) the first, introduced by Turing [18], and after half a century of study, few doubt the truth of the fundamental ChurchTuring Thesis : A function f : N # N on the natural numbers (or, more generally, on strings from a finite alphabet) is computable in principle exactly when it can be computed by a Turing Machine. The ChurchTuring Thesis grounds proofs of undecidability and it is essential for the most important applications of logic. On the other hand, it cannot be argued seriously that Turing machines model faithfully all algorithms on the natural numbers. If, for example, we code the input n in binary (rather than unary) notation, then the time needed for the computation of f(n) can sometimes be considerably shortened; and if we let the machine use two tapes rather than one, then (in some cases) we may gain a quadratic speedup of the computation, see [8]. This mea...
Complexity and Real Computation: A Manifesto
 International Journal of Bifurcation and Chaos
, 1995
"... . Finding a natural meeting ground between the highly developed complexity theory of computer science with its historical roots in logic and the discrete mathematics of the integers and the traditional domain of real computation, the more eclectic less foundational field of numerical analysis ..."
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Cited by 11 (0 self)
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. Finding a natural meeting ground between the highly developed complexity theory of computer science with its historical roots in logic and the discrete mathematics of the integers and the traditional domain of real computation, the more eclectic less foundational field of numerical analysis with its rich history and longstanding traditions in the continuous mathematics of analysis presents a compelling challenge. Here we illustrate the issues and pose our perspective toward resolution. This article is essentially the introduction of a book with the same title (to be published by Springer) to appear shortly. Webster: A public declaration of intentions, motives, or views. k Partially supported by NSF grants. y International Computer Science Institute, 1947 Center St., Berkeley, CA 94704, U.S.A., lblum@icsi.berkeley.edu. Partially supported by the LettsVillard Chair at Mills College. z Universitat Pompeu Fabra, Balmes 132, Barcelona 08008, SPAIN, cucker@upf.es. P...
Prospects for mathematical logic in the twentyfirst century
 BULLETIN OF SYMBOLIC LOGIC
, 2002
"... The four authors present their speculations about the future developments of mathematical logic in the twentyfirst century. The areas of recursion theory, proof theory and logic for computer science, model theory, and set theory are discussed independently. ..."
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The four authors present their speculations about the future developments of mathematical logic in the twentyfirst century. The areas of recursion theory, proof theory and logic for computer science, model theory, and set theory are discussed independently.
Arithmetic complexity
 ACM Transactions on Computational Logic
, 2007
"... My purpose in this lecture is to explain how the representation of algorithms by recursive programs can be used in complexity theory, especially in the derivation of lower bounds for worstcase time complexity, which apply to all—or, at least, a very large class of—algorithms. It may be argued that ..."
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My purpose in this lecture is to explain how the representation of algorithms by recursive programs can be used in complexity theory, especially in the derivation of lower bounds for worstcase time complexity, which apply to all—or, at least, a very large class of—algorithms. It may be argued that recursive programs are not a new computational paradigm, since their manifestation as HerbrandGödelKleene systems was present at the very beginning of the modern theory of computability, in 1934. But they have been dissed as tools for complexity analysis, and part of my mission here is to rehabilitate them. I will draw my examples primarily from van den Dries ’ [1] and the joint work in [3, 2], incidentally providing some publicity for the fine results in those papers. Some of these results are stated in Section 3; before that, I will set the stage in Sections 1 and 2, and in the last Section 4 of this abstract I will outline very briefly some conclusions about recursion and complexity which I believe that they support. 1 Partial Algebras
Algorithms vs. Machines
"... Yiannis Moschovakis argues that some algorithms, and in particular the mergesort algorithm, cannot be adequately described in terms of machines acting on states. We show how to describe the mergesort algorithm, on its natural level of abstraction, in terms of distributed abstract state machines. ..."
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Yiannis Moschovakis argues that some algorithms, and in particular the mergesort algorithm, cannot be adequately described in terms of machines acting on states. We show how to describe the mergesort algorithm, on its natural level of abstraction, in terms of distributed abstract state machines.
1 What Is an Algorithm?
"... When algorithms are defined rigorously in Computer Science literature machines, mathematical models of computers, sometimes idealized by allowing access to “unbounded memory”. 1 My aims here are to argue ..."
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When algorithms are defined rigorously in Computer Science literature machines, mathematical models of computers, sometimes idealized by allowing access to “unbounded memory”. 1 My aims here are to argue
Elementary algorithms and their implementations
"... In the sequence of articles [3, 5, 4, 6, 7], Moschovakis has proposed a mathematical modeling of the notion of algorithm—a settheoretic “definition ” of algorithms, much like the “definition ” of real numbers as Dedekind cuts on the rationals or that of random variables as measurable functions on a ..."
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In the sequence of articles [3, 5, 4, 6, 7], Moschovakis has proposed a mathematical modeling of the notion of algorithm—a settheoretic “definition ” of algorithms, much like the “definition ” of real numbers as Dedekind cuts on the rationals or that of random variables as measurable functions on a probability space. The aim is to provide a traditional foundation for the theory of algorithms, a development of it within axiomatic set theory in the same way as analysis and probability theory are rigorously developed within set theory on the basis of the set theoretic modeling of their basic notions. A characteristic feature of this approach is the adoption of a very abstract notion of algorithm which takes recursion as a primitive operation, and is so wide as to admit “nonimplementable ” algorithms: implementations are special, restricted algorithms (which include the familiar models of computation, e.g., Turing and random access machines), and an algorithm is implementable if it is reducible to an implementation. Our main aim here is to investigate the important relation between an
RECURSION IN COGNITION: A COMPUTATIONAL INVESTIGATION INTO THE REPRESENTATION AND PROCESSING OF LANGUAGE
"... ADVERTIMENT. L'accés als continguts d'aquesta tesi doctoral i la seva utilització ha de respectar els drets de la persona autora. Pot ser utilitzada per a consulta o estudi personal, així com en activitats o materials d'investigació i docència en els termes establerts a l'art. 32 ..."
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ADVERTIMENT. L'accés als continguts d'aquesta tesi doctoral i la seva utilització ha de respectar els drets de la persona autora. Pot ser utilitzada per a consulta o estudi personal, així com en activitats o materials d'investigació i docència en els termes establerts a l'art. 32 del Text Refós de la Llei de Propietat Intel·lectual (RDL 1/1996). Per altres utilitzacions es requereix l'autorització prèvia i expressa de la persona autora. En qualsevol cas, en la utilització dels seus continguts caldrà indicar de forma clara el nom i cognoms de la persona autora i el títol de la tesi doctoral. No s'autoritza la seva reproducció o altres formes d'explotació efectuades amb finalitats de lucre ni la seva comunicació pública des d'un lloc aliè al servei TDX. Tampoc s'autoritza la presentació del seu contingut en una finestra o marc aliè a TDX (framing). Aquesta reserva de drets afecta tant als continguts de la tesi com als seus resums i índexs. ADVERTENCIA. El acceso a los contenidos de esta tesis doctoral y su utilización debe respetar los derechos de la persona autora. Puede ser utilizada para consulta o estudio personal, así como en actividades o materiales de investigación y docencia en los términos establecidos en el art. 32 del Texto Refundido de la Ley de Propiedad Intelectual (RDL 1/1996). Para otros usos se requiere la autorización previa y expresa de la persona autora. En cualquier caso, en la utilización de sus contenidos se deberá indicar de forma clara el nombre y apellidos de la persona autora y el título de la tesis doctoral. No se