An algorithm is a finite set of well-defined instructions used to solve a problem or perform a computation. It is fundamental to computer science and underpins the operation of software and hardware systems, impacting fields from data processing to artificial intelligence.
Computability Theory explores the limits of what problems can be solved by algorithms, examining the capabilities and limitations of computational models. It is foundational in understanding which problems are algorithmically solvable and provides a framework for classifying problems based on their computational complexity.
The Church-Turing Thesis posits that any function that can be effectively computed by a human using a well-defined procedure can also be computed by a Turing machine, serving as a foundational principle for computer science. It bridges the gap between abstract mathematical computation and practical machine-based computation, asserting the limits of what can be algorithmically solved.
Decidability refers to the ability to determine, using an algorithm, whether a statement or problem can be conclusively resolved as either true or false. It is a fundamental concept in computer science and logic, highlighting the limits of algorithmic computation and distinguishing between problems that are solvable and those that are not.
Computability concerns the ability to solve a problem effectively using a finite set of operations or an algorithm. It examines which problems can be solved on a computer in principle, laying the foundation for understanding what can be achieved within computer science and logic.