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Fish number 4 (F4), is a number defined by Japanese googologist Fish in 2002. It is the smallest of the Fish numbers that is defined using an uncomputable function. s'(1) map is a function which maps functions to functions, as follows. Function \(s'(1)f\) is a busy beaver function for an oracle machine having an oracle which calculates function \(f\). That is, the maximum possible numbers of ones that can be written with an n-state, two-color oracle Turing machine is \(s'(1)f(n)\). For \(n>1\), \(s'(n)\) map is defined similar to the s(n) map, After this, the definition is similar to Fish number 3;

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  • Fish number 4
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  • Fish number 4 (F4), is a number defined by Japanese googologist Fish in 2002. It is the smallest of the Fish numbers that is defined using an uncomputable function. s'(1) map is a function which maps functions to functions, as follows. Function \(s'(1)f\) is a busy beaver function for an oracle machine having an oracle which calculates function \(f\). That is, the maximum possible numbers of ones that can be written with an n-state, two-color oracle Turing machine is \(s'(1)f(n)\). For \(n>1\), \(s'(n)\) map is defined similar to the s(n) map, After this, the definition is similar to Fish number 3;
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  • Fish number 4 (F4), is a number defined by Japanese googologist Fish in 2002. It is the smallest of the Fish numbers that is defined using an uncomputable function. s'(1) map is a function which maps functions to functions, as follows. Function \(s'(1)f\) is a busy beaver function for an oracle machine having an oracle which calculates function \(f\). That is, the maximum possible numbers of ones that can be written with an n-state, two-color oracle Turing machine is \(s'(1)f(n)\). By comparing with the order-n busy beaver function \(\Sigma_n(x)\), let \(f\) be a computable function. Then it's easy to see that (exponents mean interation of the map here): \begin{eqnarray*} s'(1)f & = & \Sigma_1(x)\\ s'(1)^2f & = & \Sigma_2(x)\\ s'(1)^3f & = & \Sigma_3(x)\\ s'(1)^nf & = & \Sigma_n(x)\\ s'(1)^xf & = & \Sigma_x(x)\end{eqnarray*} For \(n>1\), \(s'(n)\) map is defined similar to the s(n) map, \begin{eqnarray*} s'(n)f & = & s'(n-1)^{x}f(x) ( ext{for } n>1) \\ \end{eqnarray*} After this, the definition is similar to Fish number 3; \begin{eqnarray*} ssʹ(1)f & = & sʹ(x)f(x) \\ ssʹ(n)f & = & [ssʹ(n − 1)^{x}]f(x) ( ext{for } n>1) \\ F_4(x) & = & ssʹ(2)^{63}f; f(x) = x + 1 \\ F_4 & = & F_4^{63}(3) \end{eqnarray*}
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