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Reflective inductive inference of recursive functions

Gunter Grieser
2008 Theoretical Computer Science  
In this paper, we investigate reflective inductive inference of recursive functions. A reflective IIM is a learning machine that is additionally able to assess its own competence.  ...  Here, we arrive at four different types of reflection: reflection in the limit, optimistic, pessimistic and exact reflection.  ...  Acknowledgements The author wants to thank the anonymous referees for giving many helpful comments, especially for providing Theorem 2 as an extension of the original result.  ... 
doi:10.1016/j.tcs.2008.02.022 fatcat:dbu64ujtxbgmlfaiwjzqlrxfku

Inductive Inference on the Base of Fixed Point Theory

Tamás Gergely, Zoltán Szabó
1987 International Joint Conference on Artificial Intelligence  
The selected recursive function 1s a new one within the standard model of arithmetics.  ...  from (Angluin and Smith 1983)) the majority of Inductive methods defines the regularity in question as e recursive function From the point of view of logic this means that these methods use the standard  ... 
dblp:conf/ijcai/GergelyS87 fatcat:qe4xp3226zcf5j7hqxpsn4gfsi

Page 4412 of Mathematical Reviews Vol. , Issue 2001G [page]

2001 Mathematical Reviews  
functions of Péter (recall that by a well-known result of Parsons, the p.t.c.f.s of [Z; are the primitive recursive ones as well).  ...  Kaye by showing that the provably total computable functions (p.t.c.fs) of IT are exactly the primitive recursive ones, and that (somewhat unexpectedly) the p.t.c.f.s of IZ,;+ IM; are the doubly recursive  ... 

Proof Theory: Some Applications of Cut-Elimination [chapter]

Helmut Schwichtenberg
1977 Studies in Logic and the Foundations of Mathematics  
Transfinite induction and the reflection principle 5.1. We now consider Z without set and function variables.  ...  The proof is by induction on the build-up of < £ () -recursive functionals.  ... 
doi:10.1016/s0049-237x(08)71124-8 fatcat:ewrump3wpfhbjewfhzxz3j7lwi

Unfolding Schematic Systems [chapter]

Thomas Strahm
2017 Outstanding Contributions to Logic  
finitist arithmetic, feasible arithmetic, and theories of inductive definitions. * To Solomon Feferman, with gratitude for his intellectual inspiration and friendship  ...  After a short summary of precursors of the unfolding program, we survey the unfolding procedure and discuss the main results obtained for various schematic systems S, including non-finitist arithmetic,  ...  Moreover, induction must now be given as a rule of inference involving such sequents.  ... 
doi:10.1007/978-3-319-63334-3_8 fatcat:hn5lmfhpiba6hexg3xitjumh3a

Page 5899 of Mathematical Reviews Vol. , Issue 98I [page]

1998 Mathematical Reviews  
Summary: “In a special sense, reflection means to think about one’s own capabilities. This phenomenon is studied in the field of inductive inference of recursive functions.  ...  from some agreed class of recursive functions.  ... 

Foreword

Thomas Zeugmann
1997 Theoretical Computer Science  
logic of/for machine discovery, inductive inference of recursive functions and recursively enumerable languages, learning via queries, learning by artificial and biological neural networks, pattern recognition  ...  On that occasion, it merged with the International Workshop on Analogical and Inductive Inference.  ... 
doi:10.1016/s0304-3975(97)00012-1 fatcat:f7kh3tldgrfrbmxov5v4uacjdi

Proof theory of reflection

Michael Rathjen
1994 Annals of Pure and Applied Logic  
A recursively Mahlo ordinal K is recursively a-Mahlo if for every K-recursive function f : K -K there exists an admissible ordinal p < K closed under f such that p is recursively P-Mahlo for all p < CY  ...  An ordinal K is recursively Mahlo if for every K-recursive function f : K -+ K there exists an admissible ordinal p < K that is closed under f.  ...  Acknowledgements I would like to thank Professors Tim Carlson and Kurt Schiitte I5 for valuable comments on an earlier version of the manuscript.  ... 
doi:10.1016/0168-0072(94)90074-4 fatcat:zi36fkkdf5hapod3tynwib2nxm

A Sneak Preview of Proof Theory of Ordinals(Infinity in Philosophy and Mathematics)

Toshiyasu ARAI
2012 Annals of the Japan Association for Philosophy of Science  
For another approach to the ordinal anal}.rses via ep$ilon substitution < wFK i T e M,Jo[K.] for soine recursive well ordering Kdv of type a} (Wo[{] denotes a III-sentence saying tliat K is a well ordering  ...  Although some papers [7][8][9][10][11] have been already appeared, a large part of the ser"ies of papers under the project, 'proof theory for theories of ordinals; [2-5] has not yet published.  ...  maps n-recursive functions to ny-recuTsive functions.  ... 
doi:10.4288/jafpos.20.0_29 fatcat:5jz7rp3a55bd7cigvarcpimhu4

The witness function method and provably recursive functions of peano arithmetic [chapter]

Samuel R. Buss
1995 Studies in Logic and the Foundations of Mathematics  
This paper presents a new proof of the characterization of the provably recursive functions of the fragments IΣ n of Peano arithmetic.  ...  This includes the definition of α-primitive recursive (in Σ k ) functions and normal forms for such functions.  ...  In fact, it is quite simple -an IΣ n -induction inference is handled by primitive recursion in Σ n−1 .  ... 
doi:10.1016/s0049-237x(06)80038-8 fatcat:rbzpcmwscbfydg3xq3hcdu4kma

Page 477 of Mathematical Reviews Vol. , Issue 82b [page]

1982 Mathematical Reviews  
the use of recursion in abstract complexity theory; inductive inference.  ...  By an applica- tion of the inductive definability theorem new faithful extension and reflection properties are obtained.  ... 

Page 3176 of Mathematical Reviews Vol. , Issue 2003e [page]

2003 Mathematical Reviews  
The deflationist is urged to adopt a reflection principle like “for any £)-sentence y, from the formal statement that y is provable in PA, infer y” as a new rule of inference.  ...  generic functions by recursion on this type.”  ... 

A type-directed, on-line, partial evaluator for a polymorphic language

Tim Sheard
1997 SIGPLAN notices  
Our system handles a much richer language than that presented by Danvy, including all of the features functional programmers have come to expect, such as polymorphism x5, inductive datatypes x10, and recursion  ...  The laziness of our implementation also causes the code produced to be more abstract and compact.  ...  functions in particular as outlined above to drive partial evaluation of recursive functions over inductive t ypes.  ... 
doi:10.1145/258994.258999 fatcat:7bctzlnb7jas7h6mcmcy7ujweq

Calling context abstraction with shapes

Xavier Rival, Bor-Yuh Evan Chang
2011 SIGPLAN notices  
Specifically, our approach exploits the inductive structure of a calling context by treating it directly as a stack of activation records.  ...  While possible, computing adequate summaries is difficult, particularly in the presence of recursive procedures.  ...  That is, while we automatically derive a programspecific inductive definition to summarize call stacks, we do not try to infer inductive definitions for recursive heap structures.  ... 
doi:10.1145/1925844.1926406 fatcat:zx7d5eqbkzclropcprnbynlymy

A type-directed, on-line, partial evaluator for a polymorphic language

Tim Sheard
1997 Proceedings of the 1997 ACM SIGPLAN symposium on Partial evaluation and semantics-based program manipulation - PEPM '97  
Our system handles a much richer language than that presented by Danvy, including all of the features functional programmers have come to expect, such as polymorphism x5, inductive datatypes x10, and recursion  ...  The laziness of our implementation also causes the code produced to be more abstract and compact.  ...  functions in particular as outlined above to drive partial evaluation of recursive functions over inductive t ypes.  ... 
doi:10.1145/258993.258999 dblp:conf/pepm/Sheard97 fatcat:ix3c6zhk7beetmqh5cngaqg5bq
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