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A Constructive Proof of Rice's Theorem and the Halting Problem via Hilbert's Tenth Problem
首次提交 2026-04-12 04:45
版本更新 2026-10-29 09:03
雷达观测 2026-10-30 11:25
摘要
Rice's theorem states that no non-trivial semantic property of programs is decidable. Classical proofs proceed by reduction from the halting problem, invoking the law of excluded middle (LEM) twice: once through diagonalization, and once through a case split on whether the always-diverging program bot satisfies the property in question. We present a proof that is constructive relative to the undecidability of Hilbert's Tenth Problem (MRDP): valid in intuitionistic logic, requiring neither diagonalization nor self-reference, and adding no classical reasoning beyond the MRDP assumption itself. The key idea is a two-witness construction. Given a non-trivial property P, we attach to each Diophantine polynomial D a pair of programs S^0_D, S^1_D that behave like the negative and positive witnesses for P when D is solvable, and both diverge identically when it is not. A hypothetical decider for P would therefore decide Diophantine solvability via the difference delta_D = DecideP(S^1_D) - DecideP(S^0_D) -- contradicting the MRDP theorem. The argument is structured as two separate implications, never asserting a disjunction about solvability, and never examining P(bot). The undecidability of the halting problem follows as an immediate corollary: a single application of Rice's theorem to the Terminates property. A formalization in the Rocq proof assistant confirms both results within a step-indexed model of computation, with the undecidability of Hilbert's Tenth Problem as the sole external axiom. Both Rice_Theorem and Halting_Problem are closed under the global context.
作者备注：46 pages, Rocq (Coq 8.18+) formalization included. Source and C witness: https://github.com/endrazine/rice-constructive
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v2 · 2026-10-29 09:03 · 最新观测版本
A Constructive Proof of Rice's Theorem and the Halting Problem via Hilbert's Tenth Problem
Rice's theorem states that no non-trivial semantic property of programs is decidable. Classical proofs proceed by reduction from the halting problem, invoking the law of excluded middle (LEM) twice: once through diagonalization, and once through a case split on whether the always-diverging program bot satisfies the property in question. We present a proof that is constructive relative to the undecidability of Hilbert's Tenth Problem (MRDP): valid in intuitionistic logic, requiring neither diagonalization nor self-reference, and adding no classical reasoning beyond the MRDP assumption itself. The key idea is a two-witness construction. Given a non-trivial property P, we attach to each Diophantine polynomial D a pair of programs S^0_D, S^1_D that behave like the negative and positive witnesses for P when D is solvable, and both diverge identically when it is not. A hypothetical decider for P would therefore decide Diophantine solvability via the difference delta_D = DecideP(S^1_D) - DecideP(S^0_D) -- contradicting the MRDP theorem. The argument is structured as two separate implications, never asserting a disjunction about solvability, and never examining P(bot). The undecidability of the halting problem follows as an immediate corollary: a single application of Rice's theorem to the Terminates property. A formalization in the Rocq proof assistant confirms both results within a step-indexed model of computation, with the undecidability of Hilbert's Tenth Problem as the sole external axiom. Both Rice_Theorem and Halting_Problem are closed under the global context.
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