Computer Science > Theory of Computation > Computable Functions
Description:
The study of computable functions forms a fundamental part of the theory of computation, a core area in computer science concerned with what problems can be solved using algorithms and to what extent. At the heart of this investigation is the concept of a function being “computable,” meaning that there exists a systematic procedure (an algorithm) that can efficiently produce the function’s output from its input within a finite amount of time.
Definition and Formalization
A computable function, also referred to as a recursive function, is a type of mathematical function that can be calculated by a mechanical process. Formally, a function \( f: \mathbb{N} \to \mathbb{N} \) (from the set of natural numbers to itself) is said to be computable if there exists a Turing machine \( T \), which, given any input \( x \in \mathbb{N} \), halts after a finite number of steps and produces \( f(x) \) as output.
This formal concept is best understood through several equivalent representations:
1. Turing Machines: The original computational model proposed by Alan Turing, which comprises an infinite tape and a read-write head that follows a set of instructions.
2. Lambda Calculus: Introduced by Alonzo Church, it’s a formal system for expressing computation by function abstraction and application.
3. Recursive Functions: A class of functions built using initial functions (like constant functions, successor functions, and projection functions) and closed under operations like composition, primitive recursion, and minimization.
Fundamental Concepts
- Decidability: A decision problem is decidable if the indicator function (which returns 1 for ‘yes’ and 0 for ‘no’) is computable.
- Computational Complexity: Beyond mere computability, this field studies the resources (time and space) needed by the Turing machine to compute the function.
- Church-Turing Thesis: A hypothesis positing that any function which would naturally be regarded as computable can be computed by a Turing machine.
Key Theorems and Results
Rice’s Theorem: Every non-trivial property (i.e., applicable to some but not all) of the language recognized by a Turing machine is undecidable. This implies that it is impossible to have a general algorithm that decides whether any given Turing machine has particular non-trivial attributes.
Halting Problem: Demonstrated by Turing, it states there is no general algorithm to determine whether a Turing machine will halt on a given input. This problem is a classic example of an undecidable problem.
Example
Consider the function \( \text{factorial}: \mathbb{N} \to \mathbb{N} \) defined as:
\[
\text{factorial}(n) =
\begin{cases}
1 & \text{if } n = 0, \\
n \times \text{factorial}(n-1) & \text{if } n > 0.
\end{cases}
\]
This is a computable function, as there exists a straightforward recursive algorithm, or equivalently, a Turing machine, which can compute the factorial of any natural number \( n \).
Applications
The study of computable functions has profound implications in various domains such as compiler design, software engineering, artificial intelligence, and the development of efficient algorithms. It also underpins many areas of theoretical computer science, providing essential insights into the limitations and capabilities of automated problem-solving.
In summary, understanding computable functions is essential for grasping the theoretical limits of what computers can do. It serves as a cornerstone for deeper topics in the theory of computation, including complexity theory and the study of formal languages.