Did Turing prove the undecidability of the halting problem? (2407.00680v2)

Abstract: We discuss the accuracy of the attribution commonly given to Turing's 1936 paper "On computable numbers..." for the computable undecidability of the halting problem, coming eventually to a nuanced conclusion.

• The paper demonstrates that Turing’s work, though not explicitly about the halting problem, laid the core framework through his proof of the symbol-printing problem.
• It employs a detailed reduction and self-referential argument to connect Turing's methods with modern computability theory techniques.
• The analysis highlights historical attribution practices, emphasizing Turing's central, albeit qualified, role in establishing undecidability.

Did Turing Prove the Undecidability of the Halting Problem?

Joel David Hamkins and Theodor Nenu explore the mathematical attributions surrounding Alan Turing's 1936 seminal paper, "On Computable Numbers, with an Application to the Entscheidungsproblem." The central question of their investigation is whether Turing actually proved the undecidability of the halting problem. This paper presents a thorough examination of the precise contributions of Turing's original work, juxtaposed with subsequent interpretations and attributions.

The Halting Problem and Turing's Work

The authors begin by articulating the halting problem: determining whether a given program halts on a given input. Through contemporary lenses, it is understood to be computably undecidable—no Turing machine can solve the problem for all possible program-input pairs.

Turing's 1936 work introduces pivotal concepts in computability theory, including Turing machines, universal computation, and several undecidable problems. However, Hamkins and Nenu emphasize that Turing did not explicitly address the halting problem in this paper. Specifically, the modern formulation of the halting problem, as well as the common self-referential proofs of its undecidability, are absent.

Undecidability of the Circle-Free Problem

Central to Turing's undecidability results is the circle-free problem, which involves determining whether a given program will produce an infinite binary sequence. The authors describe how Turing uses a diagonalization argument reminiscent of Cantor's diagonal proof to demonstrate the undecidability of this problem. The circle-free problem is critical, as it's strictly harder in the hierarchy of computational complexity than the halting problem, classified as $\Pi^0_2$ complete, compared to the halting problem's $\Sigma^0_1$ complexity.

The Symbol-Printing Problem

Hamkins and Nenu highlight a key undecidability result from Turing's paper: the symbol-printing problem. Turing proved that it's undecidable whether a given Turing machine will ever print a specific symbol. This undecidability is shown through a clever reduction argument, avoiding the straightforward computability reduction paradigm.

Relationship to the Halting Problem

The crux of the argument that Turing essentially proved the undecidability of the halting problem lies in the symbolic equivalence between the symbol-printing problem and the halting problem. The authors argue that by considering models where halting is triggered by printing a designated halt symbol, the problems become practically identical.

Hamkins and Nenu also present a simpler self-referential proof for the symbol-printing problem's undecidability, akin to the standard proof for the halting problem, reinforcing their near-equivalence.

Historical and Cultural Attribution

A notable section discusses the practice of mathematical attribution. Mathematicians often credit results and methods to earlier work if the foundational ideas were laid therein. Examples include attributing the irrationality proof of $\sqrt{2}$ to the Pythagoreans or the Chinese remainder theorem to Sunzi. This cultural practice of generous attribution extends to Turing's work.

Conclusion and Nuanced Attribution

In their nuanced conclusion, Hamkins and Nenu assert that while Turing did not explicitly state or prove the undecidability of the halting problem, he provided the foundational framework and core ideas that directly lead to this conclusion. Given the equivalency between the symbol-printing and the halting problems, attributing the undecidability of the halting problem to Turing is seen as essentially accurate, if slightly qualified.

This analysis implicates both the practical implications and the theoretical underpinnings of Turing's original work in computability theory. Moving forward, the paper suggests that Turing's contributions, even if not explicitly formulated in terms of the halting problem, remain central to the field's development. Future developments in AI or computational theory might continue to uncover new interpretations or applications of his foundational ideas.