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| '''Isis Squillante'''
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| Discord: @isisoftheeast
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| E-Mail: isissquillante@gmail.com
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| Drafts below.
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| =Quasicycler=
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| A '''quasicycler''' is a Turing machine which is neither a [[cycler]] nor a [[translated cycler]] but cycles through states periodically.
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| ==Formal definition==
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| A ''quasicycler'' or ''quasicyclic Turing machine'' <math>\tau</math> is a Turing machine which has the following two properties:
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| # There exists some natural number <math>p\ge1</math> such that for all sufficiently large <math>N</math>, the current state of <math>\tau</math> after <math>N</math> transitions is identical to the current state of <math>\tau</math> after <math>N+p</math> transitions. <math>p</math> is called the ''period length'' of <math>\tau</math>.
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| # There is no natural number <math>x\ge1</math> such that for all sufficiently large <math>N</math>, the <math>N</math>-th transition is identical to the <math>N+px</math>-th transition.
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| This second property is what distinguishes quasicyclers from translated cyclers; translated cyclers are defined by the negation of the second property of quasicyclers.
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| ==Properties==
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| For any quasicyclic Turing machine with preperiod of length <math>r</math> and period of length <math>p</math>, its [[Rate of tape growth|rate of strict tape growth]] <math>\gamma(x)</math> is bounded above by <math>\frac p2\sqrt{x}+\frac r3</math>.
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