Machine Learning of Higher-Order Programs

A generator program for a computable function (by definition) generates an infinite sequence of programs all but finitely many of which compute that function. Machine learning of generator programs for computable functions is studied. To partially motivate these studies, it is shown that, in some cases, interesting global properties for computable functions can be proved from suitable generator programs which can not be proved from any ordinary programs for them. The power (for variants of various learning criteria from the literature) of learning generator programs is compared with the power of learning ordinary programs. The learning power in these cases is also compared to that of learning limiting programs, i.e., programs allowed finitely many mind changes about their correct outputs.

[1]  Manuel Blum,et al.  Toward a Mathematical Theory of Inductive Inference , 1975, Inf. Control..

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

[3]  Hilary Putnam,et al.  Trial and error predicates and the solution to a problem of Mostowski , 1965, Journal of Symbolic Logic.

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

[5]  Manuel Blum,et al.  A Machine-Independent Theory of the Complexity of Recursive Functions , 1967, JACM.

[6]  Elliott Mendelson,et al.  Introduction to Mathematical Logic , 1979 .

[7]  S. Feferman Arithmetization of metamathematics in a general setting , 1959 .

[8]  Rolf Wiehagen,et al.  Research in the theory of inductive inference by GDR mathematicians - A survey , 1980, Inf. Sci..

[9]  John Case,et al.  Comparison of Identification Criteria for Machine Inductive Inference , 1983, Theor. Comput. Sci..

[10]  N. A. Lynch RELATIVIZATION OF THE THEORY OF COMPUTATION COMPLEXITY , 1972 .

[11]  Keh-Jiann Chen,et al.  Tradeoffs in machine inductive inference , 1981 .

[12]  Paul Young,et al.  An introduction to the general theory of algorithms , 1978 .

[13]  Carl H. Smith,et al.  Inductive Inference: Theory and Methods , 1983, CSUR.

[14]  Jeffrey D. Ullman,et al.  Introduction to Automata Theory, Languages and Computation , 1979 .

[15]  Georg Kreisel,et al.  Mathematical significance of consistency proofs , 1958, Journal of Symbolic Logic.

[16]  D. C. Cooper,et al.  Theory of Recursive Functions and Effective Computability , 1969, The Mathematical Gazette.

[17]  H. E. Rose Subrecursion: Functions and Hierarchies , 1984 .

[18]  R. Soare Recursively enumerable sets and degrees , 1987 .

[19]  Patrick C. Fischer Theory of provable recursive functions , 1965 .

[20]  Mark A. Fulk A study of inductive inference machines , 1986 .

[21]  Georg Kreisel,et al.  On the interpretation of non-finitist proofs—Part I , 1951, Journal of Symbolic Logic.

[22]  N. Shapiro Review: E. Mark Gold, Limiting Recursion; Hilary Putnam, Trial and Error Predicates and the Solution to a Problem of Mostowski , 1971 .

[23]  E. Mark Gold,et al.  Limiting recursion , 1965, Journal of Symbolic Logic.

[24]  Hartley Rogers,et al.  Gödel numberings of partial recursive functions , 1958, Journal of Symbolic Logic.

[25]  Jr. Hartley Rogers Theory of Recursive Functions and Effective Computability , 1969 .

[26]  John Case,et al.  On Learning Limiting Programs , 1992, Int. J. Found. Comput. Sci..

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

[28]  William Craig,et al.  On axiomatizability within a system , 1953, Journal of Symbolic Logic.