The Challenge of Active Learning Deterministic finite automata (DFAs) are fundamental computational models used across computer science. However, learning an unknown DFA in the passive setting faces complexity issues. As the number of states in the target DFA grows, passively labeled examples alone are insufficient for efficient learning. This motivates the need for active learning…
The Challenge of Learning Automata Finite automata are fundamental computational models with applications in many areas of computer science and engineering. A key question is whether unknown finite automata can be learned, or inferred, from examples of their input-output behavior. This problem of learning an unknown mechanism from examples of its behavior is a central…
Regular expressions are a powerful tool for matching patterns in textual data. However, designing an optimal regular expression to match a specific pattern can be challenging. This article explores the problem of automatically inducing minimal regular expressions from examples, known as regular expression induction. The Challenge of Learning Regular Expressions Manual coding of regular expressions…
Overview of Shor’s Quantum Factoring Algorithm Shor’s algorithm for quantum computing provides an exponential speedup for integer factorization compared to the best known classical algorithms. Developed by mathematician Peter Shor in 1994, this quantum algorithm leverages the unique capabilities of quantum physics to quickly find the prime factors of large integers. The integer factorization problem,…
Evidence for One-Way Functions A one-way function is a mathematical function that is easy to compute on every input, but hard to invert given the image of a random input. More precisely, a function f: X → Y is one-way if: There exists a polynomial-time algorithm to compute f(x) for any x in X For…
Factorization Algorithms & NP-Hardness Integer factorization is the decomposition of a composite number into a product of smaller integers. This problem is fundamental in number theory with implications for cryptography and complexity theory. Factoring large integers is known to be an NP-intermediate problem – as hard as the hardest problems in NP but not known…
Defining the Bounded Halting Problem The bounded halting problem refers to determining if a Turing machine will halt within a set number of computation steps on a given input. Formally, we define the bounded halting problem as follows: Given a Turing machine M, input w, and computation bound t(n), where t is a function of…
Identifying the Problem Algorithm design is fundamentally about creating solutions to computational problems. Thus, the first key step is identifying the specific problem we want the algorithm to solve. This involves clearly defining the inputs and desired outputs. For example, if tasked with sorting a list of numbers, we need to understand properties of the…
Demystifying Computational Complexity Computational complexity theory is the study of the inherent difficulty of computational problems. It seeks to classify problems into complexity classes based on the computational resources required to solve them. Some key concepts in complexity theory include: Time complexity – how the runtime scales with the size of the input Space complexity…
Seeking More Efficient Solutions to Hard Problems As computational tasks grow ever more complex in our increasingly data-driven world, computer scientists urgently seek faster algorithms to tackle critical problems. Exploding dataset sizes strain even the most powerful supercomputers, motivating the quest for efficiency gains. Many tasks lack sufficiently swift solutions, including central challenges in fields…