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Reflective inductive inference of recursive functions

2008
*
Theoretical Computer Science
*

In this paper, we investigate

doi:10.1016/j.tcs.2008.02.022
fatcat:dbu64ujtxbgmlfaiwjzqlrxfku
*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. ...##
###
Inductive Inference on the Base of Fixed Point Theory

1987
*
International Joint Conference on Artificial Intelligence
*

The selected

dblp:conf/ijcai/GergelyS87
fatcat:qe4xp3226zcf5j7hqxpsn4gfsi
*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 ...##
###
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*...

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

1977
*
Studies in Logic and the Foundations of Mathematics
*

Transfinite

doi:10.1016/s0049-237x(08)71124-8
fatcat:ewrump3wpfhbjewfhzxz3j7lwi
*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*. ...##
###
Unfolding Schematic Systems
[chapter]

2017
*
Outstanding Contributions to Logic
*

finitist arithmetic, feasible arithmetic, and theories

doi:10.1007/978-3-319-63334-3_8
fatcat:hn5lmfhpiba6hexg3xitjumh3a
*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. ...##
###
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

1997
*
Theoretical Computer Science
*

logic

doi:10.1016/s0304-3975(97)00012-1
fatcat:f7kh3tldgrfrbmxov5v4uacjdi
*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*. ...##
###
Proof theory of reflection

1994
*
Annals of Pure and Applied Logic
*

A

doi:10.1016/0168-0072(94)90074-4
fatcat:zi36fkkdf5hapod3tynwib2nxm
*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. ...##
###
A Sneak Preview of Proof Theory of Ordinals(Infinity in Philosophy and Mathematics)

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

doi:10.4288/jafpos.20.0_29
fatcat:5jz7rp3a55bd7cigvarcpimhu4
*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*. ...##
###
The witness function method and provably recursive functions of peano arithmetic
[chapter]

1995
*
Studies in Logic and the Foundations of Mathematics
*

This paper presents a new proof

doi:10.1016/s0049-237x(06)80038-8
fatcat:rbzpcmwscbfydg3xq3hcdu4kma
*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 . ...##
###
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

1997
*
SIGPLAN notices
*

Our system handles a much richer language than that presented by Danvy, including all

doi:10.1145/258994.258999
fatcat:7bctzlnb7jas7h6mcmcy7ujweq
*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. ...##
###
Calling context abstraction with shapes

2011
*
SIGPLAN notices
*

Specifically, our approach exploits the

doi:10.1145/1925844.1926406
fatcat:zx7d5eqbkzclropcprnbynlymy
*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. ...##
###
A type-directed, on-line, partial evaluator for a polymorphic language

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

doi:10.1145/258993.258999
dblp:conf/pepm/Sheard97
fatcat:ix3c6zhk7beetmqh5cngaqg5bq
*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. ...
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