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Step By Recursive Step: Church's Analysis Of Effective Calculability
 BULLETIN OF SYMBOLIC LOGIC
, 1997
"... Alonzo Church's mathematical work on computability and undecidability is wellknown indeed, and we seem to have an excellent understanding of the context in which it arose. The approach Church took to the underlying conceptual issues, by contrast, is less well understood. Why, for example, wa ..."
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Alonzo Church's mathematical work on computability and undecidability is wellknown indeed, and we seem to have an excellent understanding of the context in which it arose. The approach Church took to the underlying conceptual issues, by contrast, is less well understood. Why, for example, was "Church's Thesis" put forward publicly only in April 1935, when it had been formulated already in February/March 1934? Why did Church choose to formulate it then in terms of G odel's general recursiveness, not his own #definability as he had done in 1934? A number of letters were exchanged between Church and Paul Bernays during the period from December 1934 to August 1937; they throw light on critical developments in Princeton during that period and reveal novel aspects of Church's distinctive contribution to the analysis of the informal notion of e#ective calculability. In particular, they allow me to give informed, though still tentative answers to the questions I raised; the char...
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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. 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...
On the Search for Tractable Ways of Reasoning about Programs
, 2001
"... This paper traces the important steps in the history up to around 1990 of research on reasoning about programs. The main focus is on sequential imperative programs but some comments are made on concurrency. Initially, researchers focussed on ways of verifying that a program satifies its specific ..."
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This paper traces the important steps in the history up to around 1990 of research on reasoning about programs. The main focus is on sequential imperative programs but some comments are made on concurrency. Initially, researchers focussed on ways of verifying that a program satifies its specification (or that two programs were equivalent). Over time it has become clear that post facto verification is only practical for small programs and attention turned to verification methods which support the development of programs; for larger programs it is necesary to exploit a notion of composability.
Bounded Immunity and BttReductions
 MLQ Math. Log. Q
, 1999
"... We define and study a new notion called kimmunity that lies between immunity and hyperimmunity in strength. Our interest in kimmunity is justified by the result that # # does not ktt reduce to a kimmune set, which improves a previous result by Kobzev [7, 13]. We apply the result to show that ..."
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We define and study a new notion called kimmunity that lies between immunity and hyperimmunity in strength. Our interest in kimmunity is justified by the result that # # does not ktt reduce to a kimmune set, which improves a previous result by Kobzev [7, 13]. We apply the result to show that # # does not bttreduce to MIN, the set of minimal programs. Other applications include the set of Kolmogorov random strings, and retraceable and regressive sets. We also give a new characterization of e#ectively simple sets and show that simple sets are not bttcuppable. Keywords: Computability, Recursion Theory, bounded reducibilities, minimal programs, immunity, kimmune, regressive, retraceable, e#ectively simple, cuppable. 1 Introduction There seems to be a large gap between immunity and hyperimmunity (himmunity) that is waiting to be filled. What happens, one wonders if the disjoint strong arrays that try to witness that a set is not himmune are subjected to additional conditions...
How much can analog and hybrid systems be proved (super)Turing
 Applied Mathematics and Computation
, 2006
"... Church thesis and its variants say roughly that all reasonable models of computation do not have more power than Turing Machines. In a contrapositive way, they say that any model with superTuring power must have something unreasonable. Our aim is to discuss how much theoretical computer science can ..."
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Church thesis and its variants say roughly that all reasonable models of computation do not have more power than Turing Machines. In a contrapositive way, they say that any model with superTuring power must have something unreasonable. Our aim is to discuss how much theoretical computer science can quantify this, by considering several classes of continuous time dynamical systems, and by studying how much they can be proved Turing or superTuring. 1
Interactive smallstep algorithms I: Axiomatization,
, 2006
"... In earlier work, the Abstract State Machine Thesis — that arbitrary algorithms are behaviorally equivalent to abstract state machines — was established for several classes of algorithms, including ordinary, interactive, smallstep algorithms. This was accomplished on the basis of axiomatizations o ..."
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In earlier work, the Abstract State Machine Thesis — that arbitrary algorithms are behaviorally equivalent to abstract state machines — was established for several classes of algorithms, including ordinary, interactive, smallstep algorithms. This was accomplished on the basis of axiomatizations of these classes of algorithms. Here we extend the axiomatization and, in a companion paper, the proof, to cover interactive smallstep algorithms that are not necessarily ordinary. This means that the algorithms (1) can complete a step without necessarily waiting for replies to all queries from that step and (2) can use not only the environment’s replies but also the order in which the replies were received.
The Church–Turing thesis: consensus and opposition
 Logical Approaches to Computational Barriers: Second Conference on Computability in Europe, CiE 2006
, 2006
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Alan Turing and the Mathematical Objection
 Minds and Machines 13(1
, 2003
"... Abstract. This paper concerns Alan Turing’s ideas about machines, mathematical methods of proof, and intelligence. By the late 1930s, Kurt Gödel and other logicians, including Turing himself, had shown that no finite set of rules could be used to generate all true mathematical statements. Yet accord ..."
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Abstract. This paper concerns Alan Turing’s ideas about machines, mathematical methods of proof, and intelligence. By the late 1930s, Kurt Gödel and other logicians, including Turing himself, had shown that no finite set of rules could be used to generate all true mathematical statements. Yet according to Turing, there was no upper bound to the number of mathematical truths provable by intelligent human beings, for they could invent new rules and methods of proof. So, the output of a human mathematician, for Turing, was not a computable sequence (i.e., one that could be generated by a Turing machine). Since computers only contained a finite number of instructions (or programs), one might argue, they could not reproduce human intelligence. Turing called this the “mathematical objection ” to his view that machines can think. Logicomathematical reasons, stemming from his own work, helped to convince Turing that it should be possible to reproduce human intelligence, and eventually compete with it, by developing the appropriate kind of digital computer. He felt it should be possible to program a computer so that it could learn or discover new rules, overcoming the limitations imposed by the incompleteness and undecidability results in the same way that human mathematicians presumably do. Key words: artificial intelligence, ChurchTuring thesis, computability, effective procedure, incompleteness, machine, mathematical objection, ordinal logics, Turing, undecidability The ‘skin of an onion ’ analogy is also helpful. In considering the functions of the mind or the brain we find certain operations which we can express in purely mechanical terms. This we say does not correspond to the real mind: it is a sort of skin which we must strip off if we are to find the real mind. But then in what remains, we find a further skin to be stripped off, and so on. Proceeding in this way, do we ever come to the ‘real ’ mind, or do we eventually come to the skin which has nothing in it? In the latter case, the whole mind is mechanical (Turing, 1950, p. 454–455). 1.