"The field has since expanded to include the study of generalized computability and definability. In these areas, computability theory overlaps with proof theory and effective descriptive set theory."
Recursion theory is the study of the computability of functions and the sets that they generate. It is used in computer science and mathematical logic.
Recursive Functions: Functions that can be defined using a set of rules and conditions, and can be computed in a finite number of steps.
Computability: The study of which functions can be computed using some algorithm or mechanical process.
Turing Machines: Abstract machines that can simulate any algorithm and can be used to define the concept of computability.
Church-Turing Thesis: A conjecture which states that any function that can be computed by an algorithm (in the informal sense) can also be computed by a Turing machine.
Halting Problem: The problem of determining, for a given input and a program, whether the program will halt or run forever.
Reducibility: The concept of reducing one problem to another, such that if the second problem can be solved, then the first problem can also be solved.
Degrees of Unsolvability: A hierarchy of sets that are progressively harder to compute or solve, such that each level contains sets that are harder to solve than the previous level.
Recursion Theorem: A fundamental theorem in recursion theory which states that any computable function can be computed by a program that calls itself.
Incompleteness Theorems: Two theorems by Kurt Gödel that establish the limitations of formal systems, and show that there are true statements that cannot be proved within a formal system.
Arithmetical Hierarchy: A hierarchy of sets that are defined using arithmetic operations (addition, multiplication, etc.), such that each level contains sets that are harder to compute or define than the previous level.
Computability theory: This is the study of the limits of computation, and seeks to determine which problems can and cannot be solved by computers.
Descriptive set theory: This area of recursion theory studies the properties of sets of real numbers, with a particular focus on those which are definable in terms of simpler sets.
Proof theory: This type of recursion theory concerns the structure and properties of logical proofs, and seeks to develop formal systems for representing and manipulating these proofs.
Model theory: This type of recursion theory focuses on the properties of mathematical models and their relationships with logical systems, and determining when a model is valid or not.
Recursive function theory: Also called computable function theory, this is the branch of recursion theory which explores the properties and behavior of computable functions, and provides a framework for their study.
Lambda calculus: This is a formal system designed to study the behavior of functions in a purely mathematical way, similar to that of symbolic logic.
Type theory: This is the study of the properties and relationships of different types of objects within a formal system, and seeks to determine which types are valid or not.
"Computability theory, also known as recursion theory, is a branch of mathematical logic, computer science, and the theory of computation..."
"Basic questions addressed by computability theory include: What does it mean for a function on the natural numbers to be computable?"
"The field has since expanded to include the study of generalized computability and definability. In these areas, computability theory overlaps with proof theory and effective descriptive set theory."
"How can noncomputable functions be classified into a hierarchy based on their level of noncomputability?"
"Although there is considerable overlap in terms of knowledge and methods, mathematical computability theorists study the theory of relative computability, reducibility notions, and degree structures..."
"Those in the computer science field focus on the theory of subrecursive hierarchies, formal methods, and formal languages."
"Computability theory, also known as recursion theory..."
"...computability theory overlaps with proof theory and effective descriptive set theory."
"...originated in the 1930s with the study of computable functions and Turing degrees."
"The field has since expanded to include the study of generalized computability and definability."
"Basic questions addressed by computability theory include: What does it mean for a function on the natural numbers to be computable?"
"...mathematical computability theorists study the theory of relative computability, reducibility notions, and degree structures."
"...a branch of mathematical logic, computer science, and the theory of computation..."
"...computability theory overlaps with proof theory..."
"...computability theory overlaps with... effective descriptive set theory."
"...focus on the theory of subrecursive hierarchies, formal methods, and formal languages."
"...classified into a hierarchy based on their level of noncomputability."
"...a branch of... the theory of computation..."
"Computability theory, also known as recursion theory..."