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A Little Challenge Problem for Real Number Theorem ... - Lemma 1!

A Little Challenge Problem for Real Number Theorem ... - Lemma 1!

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A Little Challenge Problem for Real Number Theorem Proving 2 ✬ Decidability for theories of real arithmetic ✩ • Truth in the first order language L R of R is decidable (Tarski). • If N(x) ⇔ x ∈ N, then truth in L R + N is undecidable since if Q(x 1 , . . .,x k ) is a quantifier-free formula of natural number arithmetic N |= ∃x 1 . . .x n· Q(x 1 , . ..,x n ) iff R |= ∃x 1 ...x n· N(x 1 ) ∧ . . .N(x n ) ∧ Q(x 1 , . . .,x n ) (ii) and the truth of (i) is undecidable in general (Matiyasevich). • E.g., truth in L R + sin is undecidable since N(x) ⇔ x ∈ N where N(x) := x ≥ 0 ∧ ∃π· 0 < π < 4 ∧ sin π = sin xπ = 0 (i) • What other f make truth in L R + f undecidable? ✫ ✪

A Little Challenge Problem for Real Number Theorem Proving 3 ✬ Encoding sin in a bounded concave γ. I ✩ γ e g = p(s, sin s) ✫ e(x, y) = ( x 1−x , y 1−y ) so γ(x) = q(g(p(x))) where p(x) = x 1−x , q(s) = s 1+s and g(s) = Ks + s 2 + 1 M sin s Question: do any K and M make γ concave? ✪

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