6.2 Kleene's second recursion theorem

Kleene's second recursion theorem is a key concept in computability theory. It shows that for any partial recursive function, there's a program that behaves like the result of applying that function to itself. This enables the creation of self-referential programs.

The theorem has wide-ranging applications in computer science and logic. It proves the existence of quines, helps establish recursively inseparable sets, and plays a role in characterizing recursively enumerable sets. Understanding this theorem is crucial for grasping the limits of computation.

Kleene's second recursion theorem

• Fundamental result in computability theory establishes the existence of fixed points for certain types of functions
• Serves as a powerful tool for constructing self-referential programs and proving properties of recursive functions
• Plays a crucial role in understanding the limitations and capabilities of computation

Statement of the theorem

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• For any partial recursive function $f$, there exists a number $n$ such that $\varphi_n = \varphi_{f(n)}$
• In other words, there is a program $n$ that, when executed, behaves exactly like the program obtained by applying $f$ to $n$
• The theorem guarantees the existence of a fixed point for any computable function

Proof of the theorem

• Relies on the construction of a specific partial recursive function $g$ that takes an index $i$ and returns an index $g(i)$
• The function $g$ is defined using the s-m-n theorem and the universal function $\varphi$
• By applying the fixed-point theorem to $g$, we obtain an index $n$ such that $\varphi_n = \varphi_{g(n)}$, which satisfies the desired property

Fixed point theorem

• States that every total computable function has a fixed point
• A fixed point of a function $f$ is a value $x$ such that $f(x) = x$
• The second recursion theorem can be seen as a generalization of the fixed point theorem to partial recursive functions

Applications of Kleene's second recursion theorem

• Demonstrates the existence of self-referential programs and sets
• Enables the construction of complex recursive structures and proofs in computability theory
• Finds applications in various areas of theoretical computer science and logic

Existence of quines

• A quine is a program that produces its own source code as output
• The second recursion theorem guarantees the existence of quines for any programming language that can express partial recursive functions
• Constructing a quine involves defining a function $f$ that takes a program $p$ and returns a program that outputs the source code of $p$

Recursively inseparable sets

• Two sets $A$ and $B$ are recursively inseparable if there is no recursive set $C$ such that $A \subseteq C$ and $B \subseteq \overline{C}$
• The second recursion theorem can be used to prove the existence of recursively inseparable sets
• Examples include the set of valid statements in first-order logic and their negations

Recursively enumerable sets

• A set is recursively enumerable (r.e.) if it is the domain of a partial recursive function
• The second recursion theorem plays a role in characterizing the properties of r.e. sets
• It can be used to prove that the complement of an r.e. set is not necessarily r.e.

Relationship to other recursion theorems

• Kleene's second recursion theorem is part of a family of recursion theorems in computability theory
• It is closely related to other important results, such as Kleene's first recursion theorem and the fixed-point theorem

Kleene's first recursion theorem

• States that for any recursive function $f$, there exists a number $n$ such that $\varphi_n(x) = f(n, x)$ for all $x$
• The first recursion theorem deals with the existence of programs that can access their own index
• It is a special case of the second recursion theorem where the function $f$ is a projection function

Recursion theorem vs fixed-point theorem

• The fixed-point theorem is a special case of the second recursion theorem where the function $f$ is total and computable
• While the fixed-point theorem guarantees the existence of a fixed point for total computable functions, the second recursion theorem extends this to partial recursive functions
• The second recursion theorem can be seen as a generalization and strengthening of the fixed-point theorem

Significance in computability theory

• Kleene's second recursion theorem is a fundamental result in computability theory
• It has far-reaching implications for understanding the nature of computation and the limits of what can be computed

Role in defining partial recursive functions

• The second recursion theorem is used in the definition and characterization of partial recursive functions
• It allows for the construction of self-referential programs and the encoding of complex recursive structures
• The theorem is essential for proving properties and relationships between different classes of computable functions

Connection to universal functions

• The proof of the second recursion theorem relies on the existence of a universal function $\varphi$
• A universal function is a function that can simulate the behavior of any other computable function when given its index
• The second recursion theorem demonstrates the power and flexibility of universal functions in computability theory

Limitations and extensions

• While Kleene's second recursion theorem is a powerful result, it has certain limitations and can be extended in various ways
• Understanding these limitations and extensions helps to gain a deeper understanding of the theorem and its implications

Restrictions on acceptable functions

• The second recursion theorem applies specifically to partial recursive functions
• It does not hold for all functions or for functions that are not computable
• The theorem relies on the existence of a computable enumeration of partial recursive functions

Generalized second recursion theorem

• The second recursion theorem can be generalized to handle more complex recursive structures
• One such generalization is the recursion theorem with parameters, which allows for the construction of fixed points with additional inputs
• The generalized second recursion theorem extends the applicability of the theorem to a wider range of recursive constructions