User:MrSolis/Playground: Difference between revisions

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-The 5-state busy beaver ([[BB(5)]]) winner is {{TM|1RB1LC_1RC1RB_1RD0LE_1LA1LD_1RZ0LA|halt}}. Discovered by Heiner Marxen and Jürgen Buntrock in 1989<ref>H. Marxen and J. Buntrock. Attacking the Busy Beaver 5. Bulletin of the EATCS, 40, pages 247-251, February 1990. https://turbotm.de/~heiner/BB/mabu90.html</ref>, this machine proved that <math>\operatorname{BB}(5)\ge 47176870\phantom{}</math> and <math>\Sigma(5)\ge 4098</math> at the time.
-== Analysis ==
-Let <math>g(x):=0^\infty\;\textrm{<A}\,1^x\;0^\infty</math>. Then,
-<math display="block">\begin{align}
-g(3x)& \xrightarrow{5x^2+19x+15}&g(5x+6),\\
-g(3x+1)&\xrightarrow{5x^2+25x+27}&g(5x+9),\c\
-g(3x+2)&\xrightarrow{6x+12}&0^\infty\;1\;\textrm{Z>}\;01\;001^{x+1}\;1\;0^\infty.
-\end{array}</math>
-<math display="block">\begin{align}
-  g(x) & \to \frac{5x+18}{3} && \text{if }x \equiv 0 \pmod{3} \\
-  g(x) & \to \frac{5x+22}{3} && \text{if }x \equiv 1 \pmod{3} \\
-  g(x) & \to \text{HALT}     && \text{if }x \equiv 2 \pmod{3}
-\end{align}</math>

Latest revision as of 00:57, 17 February 2025