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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...
Under consideration for publication in Math. Struct. in Comp. Science Reversible Combinatory Logic
, 2006
"... The λcalculus is destructive: its main computational mechanism – beta reduction – destroys the redex and makes it thus impossible to replay the computational steps. Combinatory logic is a variant of the λcalculus which maintains irreversibility. Recently, reversible computational models have been ..."
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The λcalculus is destructive: its main computational mechanism – beta reduction – destroys the redex and makes it thus impossible to replay the computational steps. Combinatory logic is a variant of the λcalculus which maintains irreversibility. Recently, reversible computational models have been studied mainly in the context of quantum computation, as (without measurements) quantum physics is inherently reversible. However, reversibility also changes fundamentally the semantical framework in which classical computation has to be investigated. We describe an implementation of classical combinatory logic into a reversible calculus for which we present an algebraic model based on a generalisation of the notion of group. 1.
Quantitative Computation by Hilbert Machines
 UMC'98  Unconventional Models of Computation
, 1998
"... . In order to extend classical models of computing with symbols we introduce a model for quantitative computation which is based on infinitedimensional topological linear structures. In particular machines which operate on data taken from Hilbert spaces will be looked at. These Hilbert machines ( ..."
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. In order to extend classical models of computing with symbols we introduce a model for quantitative computation which is based on infinitedimensional topological linear structures. In particular machines which operate on data taken from Hilbert spaces will be looked at. These Hilbert machines (and other topological linear machines) allow the adequate treatment of concepts like infiniteness and similarity as they are based on a combination of a simple algebraic structure together with a topological one. We first discuss the differences between qualitative data representation using symbols and quantitative descriptions. Hilbert machines are then introduced on a concrete as well as on a purely abstract  C algebraic  level. Furthermore, the relation to other computational models will be investigated. It will also be shown that various novel computational models like Quantum Computation, Girard's Geometry of Interaction, or Neural Networks are actually instances of Hilbert mach...
Under consideration for publication in Math. Struct. in Comp. Science Reversible Combinatory Logic
, 2006
"... The λcalculus is destructive: its main computational mechanism – beta reduction – destroys the redex and makes it thus impossible to replay the computational steps. Combinatory logic is a variant of the λcalculus which maintains irreversibility. Recently, reversible computational models have been ..."
Abstract
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The λcalculus is destructive: its main computational mechanism – beta reduction – destroys the redex and makes it thus impossible to replay the computational steps. Combinatory logic is a variant of the λcalculus which maintains irreversibility. Recently, reversible computational models have been studied mainly in the context of quantum computation, as (without measurements) quantum physics is inherently reversible. However, reversibility also changes fundamentally the semantical framework in which classical computation has to be investigated. We describe an implementation of classical combinatory logic into a reversible calculus for which we present an algebraic model based on a generalisation of the notion of group. 1.
Quantitative Computation by Hilbert Machines (Extended Abstract)
"... In order to extend classical models of computing with symbols we introduce a model for quantitative computation which is based on infinitedimensional topological linear structures. In particular machines which operate on data taken from Hilbert spaces will be looked at. These Hilbert machines ..."
Abstract
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In order to extend classical models of computing with symbols we introduce a model for quantitative computation which is based on infinitedimensional topological linear structures. In particular machines which operate on data taken from Hilbert spaces will be looked at. These Hilbert machines (and other topological linear machines) allow the adequate treatment of concepts like infiniteness and similarity as they are based on a combination of a simple algebraic structure together with a topological one. We first discuss the differences between qualitative data representation using symbols and quantitative descriptions. Hilbert machines are then introduced on a concrete as well as on a purely abstract  C algebraic  level. Furthermore, the relation to other computational models will be investigated. It will also be shown that various novel computational m...