# tianyicui/EOTTOC

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tianyicui committed Nov 20, 2011
1 parent 358043f commit 633f8363e569202a24cadf9ebfcda3f15a378c83
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2. EOTTOC.pdf
 @@ -2926,5 +2926,135 @@ recursive . \end_layout +\begin_layout Section +Numerical Functions +\end_layout + +\begin_layout Standard +Define +\emph on +zero, id +\emph default +, +\emph on +succ, +\emph default +they're primative recursive functions +\end_layout + +\begin_layout Standard +plus( +\begin_inset Formula $m$ +\end_inset + +, +\begin_inset Formula $0$ +\end_inset + +) = +\begin_inset Formula $m$ +\end_inset + +, plus( +\begin_inset Formula $m$ +\end_inset + +, +\begin_inset Formula $n+1$ +\end_inset + +) = succ(plus( +\begin_inset Formula $m$ +\end_inset + +, +\begin_inset Formula $n$ +\end_inset + +)); mult( +\begin_inset Formula $m$ +\end_inset + +, +\begin_inset Formula $0$ +\end_inset + +) = zero( +\begin_inset Formula $m$ +\end_inset + +), mult( +\begin_inset Formula $m$ +\end_inset + +, +\begin_inset Formula $n+1$ +\end_inset + +) = plus( +\begin_inset Formula $m$ +\end_inset + +, mult( +\begin_inset Formula $m$ +\end_inset + +, +\begin_inset Formula $n$ +\end_inset + +)); exp, pred, substraction... +\end_layout + +\begin_layout Standard +Primitive recursive predicate: iszero. + Then greater-than, etc... +\end_layout + +\begin_layout Standard +Then we have function defined by cases: div, rem, +\end_layout + +\begin_layout Standard +The set of primitive recursive functions is +\emph on +enumerable +\emph default +. + We can use diagonalization argument to construct a function which is computable + but not primitive recursive. +\end_layout + +\begin_layout Standard +Call a function +\begin_inset Formula $\mu$ +\end_inset + +-recursive if it can be obtained from the basic functions by the operations + of composition, recursive definition, and +\emph on +minimalization of minimalizable functions +\emph default +, which is in turn +\emph on +recursive +\emph default +. +\end_layout + +\begin_layout Chapter +Undecidability +\end_layout + +\begin_layout Section +The Church-Turing Thesis +\end_layout + +\begin_layout Standard +The Church-Turing thesis: algorithm can be rendered as Turing machine that + is guaranteed to halt on all inputs, and all such maachines will be called + algorithms. +\end_layout + \end_body \end_document
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