Inductive Inference and Reverse Mathematics

The present work investigates inductive inference from the perspective of reverse mathematics. Reverse mathematics is a framework which relates the proof strength of theorems and axioms throughout many areas of mathematics in an interdisciplinary way. The present work looks at basic notions of learnability including Angluin's tell-tale condition and its variants for learning in the limit and for conservative learning. Furthermore, the more general criterion of partial learning is investigated. These notions are studied in the reverse mathematics context for uniformly and weakly represented families of languages. The results are stated in terms of axioms referring to domination and induction strength.

[1]  Thomas Zeugmann,et al.  Characterizations of Monotonic and Dual Monotonic Language Learning , 1995, Inf. Comput..

[2]  E. Mark Gold,et al.  Language Identification in the Limit , 1967, Inf. Control..

[3]  C. Chong,et al.  THE METAMATHEMATICS OF STABLE RAMSEY'S THEOREM FOR PAIRS , 2014 .

[4]  Stephen G. Simpson,et al.  Subsystems of second order arithmetic , 1999, Perspectives in mathematical logic.

[5]  Susumu Hayashi.  Mathematics based on incremental learning - Excluded middle and inductive inference , 2006, Theor. Comput. Sci..

[6]  Frank Stephan,et al.  A Cohesive Set which is not High , 1993, Math. Log. Q..

[7]  Robert H. Sloan,et al.  BOOK REVIEW: "SYSTEMS THAT LEARN: AN INTRODUCTION TO LEARNING THEORY, SECOND EDITION", SANJAY JAIN, DANIEL OSHERSON, JAMES S. ROYER and ARUN SHARMA , 2001 .

[8]  Denis R. Hirschfeldt.  Slicing the Truth - On the Computable and Reverse Mathematics of Combinatorial Principles , 2014, Lecture Notes Series / Institute for Mathematical Sciences / National University of Singapore.

[9]  Dana Angluin,et al.  Inference of Reversible Languages , 1982, JACM.

[10]  Damir D. Dzhafarov,et al.  On the strength of the finite intersection principle , 2013 .

[11]  Dana Angluin,et al.  Inductive Inference of Formal Languages from Positive Data , 1980, Inf. Control..

[12]  Akihiro Yamamoto,et al.  Mind change complexity of inferring unbounded unions of restricted pattern languages from positive data , 2010, Theor. Comput. Sci..

[13]  Manuel Blum,et al.  Inductive inference and unsolvability , 1991, Journal of Symbolic Logic.

[14]  Daniel N. Osherson,et al.  Systems That Learn: An Introduction to Learning Theory for Cognitive and Computer Scientists , 1990 .

[15]  Frank Stephan,et al.  Correction to "A Cohesive Set which is not High" , 1997, Math. Log. Q..

[16]  John Case,et al.  Machine Inductive Inference and Language Identification , 1982, ICALP.

[17]  Akihiro Yamamoto,et al.  Topological properties of concept spaces (full version) , 2010, Inf. Comput..