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It is not the case that The Church-Turing thesis establishes model-invariance for computability, but no analogous thesis has been proven for computational complexity.
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Reasons For
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1.
Polynomial-time equivalence across reasonable models (RAM, Turing machines, circuits) is empirically robust, suggesting implicit complexity-level invariance exists.
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2.
The Church-Turing thesis itself lacks formal proof—it's a conceptual claim. Complexity invariance may simply require different validation methods, not impossibility.
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3.
Encoding overhead (log-factor polynomials) are bounded across standard models, establishing practical model-invariance sufficient for complexity theory's purposes.
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Reasons Against
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Reason against
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1.
All known Turing-complete models (lambda calculus, register machines, etc.) compute identical function classes, validating Church-Turing's universality claim.
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2.
Complexity classes vary across models: polynomial-time on RAM differs from polynomial-time on Turing machines, preventing unified complexity theory.
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3.
P vs NP remains open precisely because no complexity thesis guarantees hardness properties transfer across computational models and encodings.
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