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Computability and recursion
 BULL. SYMBOLIC LOGIC
, 1996
"... We consider the informal concept of “computability” or “effective calculability” and two of the formalisms commonly used to define it, “(Turing) computability” and “(general) recursiveness.” We consider their origin, exact technical definition, concepts, history, general English meanings, how they b ..."
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Cited by 36 (0 self)
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We consider the informal concept of “computability” or “effective calculability” and two of the formalisms commonly used to define it, “(Turing) computability” and “(general) recursiveness.” We consider their origin, exact technical definition, concepts, history, general English meanings, how they became fixed in their present roles, how they were first and are now used, their impact on nonspecialists, how their use will affect the future content of the subject of computability theory, and its connection to other related areas. After a careful historical and conceptual analysis of computability and recursion we make several recommendations in section §7 about preserving the intensional differences between the concepts of “computability” and “recursion.” Specifically we recommend that: the term “recursive ” should no longer carry the additional meaning of “computable” or “decidable;” functions defined using Turing machines, register machines, or their variants should be called “computable” rather than “recursive;” we should distinguish the intensional difference between Church’s Thesis and Turing’s Thesis, and use the latter particularly in dealing with mechanistic questions; the name of the subject should be “Computability Theory” or simply Computability rather than
Constructing Recursion Operators in Intuitionistic Type Theory
 Journal of Symbolic Computation
, 1984
"... MartinLöf's Intuitionistic Theory of Types is becoming popular for formal reasoning about computer programs. To handle recursion schemes other than primitive recursion, a theory of wellfounded relations is presented. Using primitive recursion over higher types, induction and recursion are for ..."
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Cited by 23 (5 self)
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MartinLöf's Intuitionistic Theory of Types is becoming popular for formal reasoning about computer programs. To handle recursion schemes other than primitive recursion, a theory of wellfounded relations is presented. Using primitive recursion over higher types, induction and recursion are formally derived for a large class of wellfounded relations. Included are < on natural numbers, and relations formed by inverse images, addition, multiplication, and exponentiation of other relations. The constructions are given in full detail to allow their use in theorem provers for Type Theory, such as Nuprl. The theory is compared with work in the field of ordinal recursion over higher types.
Inductive Inference with Procrastination: Back to Definitions
 Fundamenta Informaticae
, 1999
"... In this paper, we reconsider the denition of procrastinating learning machines. In the original denition of Freivalds and Smith [FS93], constructive ordinals are used to bound mindchanges. We investigate possibility of using arbitrary linearly ordered sets to bound mindchanges in similar way. It ..."
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Cited by 9 (2 self)
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In this paper, we reconsider the denition of procrastinating learning machines. In the original denition of Freivalds and Smith [FS93], constructive ordinals are used to bound mindchanges. We investigate possibility of using arbitrary linearly ordered sets to bound mindchanges in similar way. It turns out that using certain ordered sets it is possible to dene inductive inference types dierent from the previously known ones. We investigate properties of the new inductive inference types and compare them to other types. This research was supported by Latvian Science Council Grant No.93.599 and NSF Grant 9421640. Some of the results from this paper were presented earlier [AFS96]. y The third author was supported in part by NSF Grant 9301339. 1 Introduction We study inductive inference using the model developed by Gold [Gol67]. There is a well known hierarchy of larger and larger classes of learnable sets of phenomena based on the number of time a learning machine is all...
Recursion Theoretic Models of Learning: Some Results and Intuitions
 Annals of Mathematics and Artificial Intelligence
, 1995
"... View of Learning To implement a program that somehow "learns" it is neccessary to fix a set of concepts to be learned and develop a representation for the concepts and examples of the concepts. In order to investigate general properties of machine learning it is neccesary to work in as re ..."
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Cited by 5 (2 self)
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View of Learning To implement a program that somehow "learns" it is neccessary to fix a set of concepts to be learned and develop a representation for the concepts and examples of the concepts. In order to investigate general properties of machine learning it is neccesary to work in as representation independent fashion as possible. In this work, we consider machines that learn programs for recursive functions. Several authors have argued that such studies are general enough to include a wide array of learning situations [2,3,22,23,24]. For example, a behavior to be learned can be modeled as a set of stimulus and response pairs. Assuming that any behavior associates only one response to each possible stimulus, behaviors can be viewed as functions from stimuli to responses. Some behaviors, such as anger, are not easily modeled as functions. Our primary interest, however, concerns the learning of fundamental behaviors such as reading (mapping symbols to sounds), recognition (mapping pa...
The history and concept of computability
 in Handbook of Computability Theory
, 1999
"... We consider the informal concept of a “computable ” or “effectively calculable” function on natural numbers and two of the formalisms used to define it, computability” and “(general) recursiveness. ” We consider their origin, exact technical definition, concepts, history, how they became fixed in th ..."
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Cited by 5 (1 self)
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We consider the informal concept of a “computable ” or “effectively calculable” function on natural numbers and two of the formalisms used to define it, computability” and “(general) recursiveness. ” We consider their origin, exact technical definition, concepts, history, how they became fixed in their present roles, and how
Turing Oracle Machines, Online Computing, and Three Displacements in Computability Theory
, 2009
"... ..."
Probabilistic and Team PFINtype Learning: General Properties
"... We consider the probability hierarchy for Popperian FINite learning and study the general properties of this hierarchy. We prove that the probability hierarchy is decidable, i.e. there exists an algorithm that receives p1 and p2 and answers whether PFINtype learning with the probability of success ..."
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Cited by 3 (3 self)
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We consider the probability hierarchy for Popperian FINite learning and study the general properties of this hierarchy. We prove that the probability hierarchy is decidable, i.e. there exists an algorithm that receives p1 and p2 and answers whether PFINtype learning with the probability of success p1 is equivalent to PFINtype learning with the probability of success p2. To prove our result, we analyze the topological structure of the probability hierarchy. We prove that it is wellordered in descending ordering and orderequivalent to ordinal ffl0. This shows that the structure of the hierarchy is very complicated. Using similar methods, we also prove that, for PFINtype learning, team learning and probabilistic learning are of the same power.
The Role of True Finiteness in the Admissible Recursively Enumerable Degrees
 Memoirs of the American Mathematical Society
"... Abstract. When attempting to generalize recursion theory to admissible ordinals, it may seem as if all classical priority constructions can be lifted to any admissible ordinal satisfying a sufficiently strong fragment of the replacement scheme. We show, however, that this is not always the case. In ..."
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Cited by 2 (1 self)
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Abstract. When attempting to generalize recursion theory to admissible ordinals, it may seem as if all classical priority constructions can be lifted to any admissible ordinal satisfying a sufficiently strong fragment of the replacement scheme. We show, however, that this is not always the case. In fact, there are some constructions which make an essential use of the notion of finiteness which cannot be replaced by the generalized notion of αfiniteness. As examples we discuss both codings of models of arithmetic into the recursively enumerable degrees, and nondistributive lattice embeddings into these degrees. We show that if an admissible ordinal α is effectively close to ω (where this closeness can be measured by size or by cofinality) then such constructions may be performed in the αr.e. degrees, but otherwise they fail. The results of these constructions can be expressed in the firstorder language of partially ordered sets, and so these results also show that there are natural elementary differences between the structures of αr.e. degrees for various classes of admissible ordinals α. Together with coding work which shows that for some α, the theory of the αr.e. degrees is complicated, we get that for every admissible ordinal
THE ROLE OF TRUE FINITENESS IN THE ADMISSIBLE RECURSIVELY ENUMERABLE DEGREES
, 2004
"... When attempting to generalize recursion theory to admissible ordinals, it may seem as if all classical priority constructions can be lifted to any admissible ordinal satisfying a sufficiently strong fragment of the replacement scheme. We show, however, that this is not always the case. In fact, ther ..."
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When attempting to generalize recursion theory to admissible ordinals, it may seem as if all classical priority constructions can be lifted to any admissible ordinal satisfying a sufficiently strong fragment of the replacement scheme. We show, however, that this is not always the case. In fact, there are some constructions which make an essential use of the notion of finiteness which cannot be replaced by the generalized notion of αfiniteness. As examples we discuss both codings of models of arithmetic into the recursively enumerable degrees, and nondistributive lattice embeddings into these degrees. We show that if an admissible ordinal α is effectively close to ω (where this closeness can be measured by size or by cofinality) then such constructions may be performed in the αr.e. degrees, but otherwise they fail. The results of these constructions can be expressed in the firstorder language of partially ordered sets, and so these results also show that there are natural elementary differences between the structures of αr.e. degrees for various classes of admissible ordinals α. Together with coding work which shows that for some α, the theory of the αr.e. degrees is complicated, we get that for every admissible