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Formal learning theory is one of several mathematical approaches to the study of intelligent adaptation to the environment. The analysis developed in this book is based on a number theoretical approach to learning and uses the tools of recursive-function theory to understand how learners come to an accurate view of reality. This revised and expanded edition of a successful text provides a comprehensive, self-contained introduction to the concepts and techniques of the theory. Exercises throughout the text provide experience in the use of computational arguments to prove facts about learning. Bradford Books imprint

We develop a static complexity analysis for a higher-order functional language with structural list recursion. The complexity of an expression is a pair consisting of a cost and a potential. The former is defined to be the size of the expression's evaluation derivation in a standard big-step operational semantics. The latter is a measure of the "future" cost of using the value of that expression. A translation function · maps target expressions to complexities. Our main result is the following Soundness Theorem: If t is a term in the target language, then the cost component of t is an upper bound on the cost of evaluating t. The proof of the Soundness Theorem is formalized in Coq, providing certified upper bounds on the cost of any expression in the target language.

Berman and Hartmanis BH77] conjectured that there is a polynomialtime computable isomorphism between any two languages complete for NP with respect to polynomial-time computable many-one (Karp) reductions. Joseph and Young JY85] gave a structural de nition of a class of NP-complete sets|the k-creative sets|and de ned a class of sets (the K k f 's) that are necessarily k-creative. They went on to conjecture that certain of these K k f 's are not isomorphic to the standard NP-complete sets. Clearly, the Berman{Hartmanis and Joseph{Young conjectures cannot both be correct.We introduce a family of strong one-way functions, the scrambling functions. If f is a scrambling function, then K k f is not isomorphic to the standard NP-complete sets, as Joseph and Young conjectured, and the Berman-Hartmanis conjecture fails. Indeed, if scrambling functions exist, then the isomorphism also fails at higher complexity classes such as EXP and NEXP. As evidence for the existence of scrambling functions, we show that much more powerful one-way functions|the annihilating functions|exist relative to a random oracle.

We introduce a typed programming formalism, type-2 inflationary tiered loop programs or ITLP 2 , that characterizes the type-2 basic feasible functionals. ITLP 2 is based on Bellantoni and Cook's [BC92] and Leivant's [Lei95] type-theoretic characterization of polynomial-time and turns out to be closely related to Kapron and Cook's [KC91,KC96] machine-based characterization of the type-2 basic feasible functionals.