A Parameterized Halting Problem

Chen, Yijia; Flum, Jörg

doi:10.1007/978-3-642-30891-8_17

Cited by 2 publications

(1 citation statement)

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“…Theories: Theories are assumed to be PA or an extension of PA-any arithmetically sound theory that can encode and (if true) prove p↓. 7 The standard definition of PA is modified so binary strings are encoded in arithmetic sentences as natural numbers, in binary not unary, adding a leading "1" to avoid losing leading zeros. PA under a standard approach writes numbers in unary, for instance, "SSS0", the third successor of 0, encodes the number 3.…”

Section: Preliminariesmentioning

confidence: 99%

Isomorphisms Between Impossible and Hard Tasks

Monroe¹

2023

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If no efficient proof shows that an unprovable arithmetic sentence 'x is Kolmogorov random' ('x∈R') lacks a length t proof, an isomorphism associates for each x impossible and hard tasks: ruling out any proof and length t proofs respectively. This resembles Pudlák's feasible incompleteness. This possible isomorphism implies widely-believed complexity theoretic conjectures hold-in effect, translating theorems from noncomputability about proof speedup and average-case hardness directly to complexity.Formally, we conjecture: sentence "Peano arithmetic (PA) lacks any length t proof of 'x∈R'" lacks t O(1) length proofs in any consistent extension T of PA if and only if T cannot prove 'x∈R'. If so, tautologies encoding the sentence lack t O(1) length proofs in any proof system P for x∈R sufficiently long (relative to the description of a program enumerating theorems of a theory T proving 'P is sound'). R's density implies: TAUT / ∈AvgP, Feige's hypothesis holds, and, a new conjecture, P 's nonoptimality has dense witnesses. If the isomorphism holds for any Π 0 sentence, PH does not collapse, because the arithmetic hierarchy does not collapse. alent since P A 'x∈R' implies T 'x∈R' by the assumption that T extends PA.

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Section: Preliminariesmentioning

confidence: 99%