Valiant introduces a precise computational model for concept learning. Concept learning is about learning to decide whether a certain data belongs to a certain concept (is this a table? Is there an elephant in the data?). The paper is the birth of PAC-learning (PAC = Probably Approximately Correct). What is a concept? A concept is simply a subset of some domain. A concept comes from a some conc...

Class field theory furnishes an intrinsic description of the abelian extensions of a number field which is in many cases not of an immediate algorithmic nature. We outline the algorithms available for the explicit computation of such extensions.

This paper aims to bring out the significant contribution of the Professor Lotfi A. Zadeh,"the father of Fuzzy Logic" in the field of Soft Computing. The computational Theory of Perceptions (CTP) deals with the automated processing of human knowledge expressed in Natural language. The development of computational system based on the Computational Theory of Perceptions is a challenging...

In the fall of 1997 I gave a series of lectures at Queen’s University on algorithms in invariant theory of finite groups. This article is an expanded version of the material presented there. The main topic is the calculation of the invariant ring of a finite group acting on a polynomial ring by linear transformations of the indeterminates. By “calculation” I mean finding a finite system of gene...

Complexity theory is the part of theoretical computer science that attempts to prove that certain transformations from input to output are impossible to compute using a reasonable amount of resources. Theorem 1 below illustrates the type of ‘‘impossibility’’ proof that can sometimes be obtained (1); it talks about the problem of determining whether a logic formula in a certain formalism (abbrev...

The class NP captures the difficulty of finding certificates. However, in many contexts, one is interested not just in a single certificate, but actually counting the number of certificates. In this manuscript we look at a famous result involving #P, (pronounced “sharp p”), a complexity class that captures this notion. Counting problems arise in diverse fields, often in situations having to do ...