Computability and Complexity

Computation forms the foundation of computer science and underpins our digital infrastructure. To advance computation, we must formalize mathematical models that capture its essential properties related to computability, complexity, algorithms, and problems. These models provide theoretical frameworks to analyze fundamental questions about what can and cannot be computed, how efficiently problems can be solved, and…

PAC learning, an influential framework in computational learning theory, provides mathematical tools to determine if a class of concepts can be efficiently learned from examples. However, unrestricted classes often defy efficient PAC learning. This article examines how restricting concept classes through Occam’s razor can enable PAC learning, using deterministic finite automata (DFAs) as an illustrative…

Finding Factors Efficiently with Randomness The problem of finding the prime factors of large integers is fundamental in number theory and cryptography. However, factoring large integers is known to be extremely computationally intensive. For an integer n, the best known classical factoring algorithms run in sub-exponential time in the size of n. Quantum algorithms can…