depends on the value of . As a tag system For the Collatz function in the form Hailstone sequences can be computed by the 2-tag system with production rules , , . In this system, the positive integer is represented by a string of copies of , and iteration of the tag operation halts on any word of length less than 2. (Adapted from De Mol.) The Collatz conjecture equivalently states that this tag system, with an arbitrary finite string of as the initial word, eventually halts (see Tag system for a worked example). Extensions to larger domains Iterating on all integers An extension to the Collatz conjecture is to include all integers, not just positive integers. Leaving aside the cycle 0 → 0 which cannot be entered from outside, there are a total of four known cycles, which all nonzero integers seem to eventually fall into under iteration of . These cycles are listed here, starting with the well-known cycle for positive : Odd values are listed in large bold. Each cycle is listed with its member of least absolute value (which is always odd) first. The generalized Collatz conjecture is the assertion that every integer, under iteration by , eventually falls into one of the four cycles above or the cycle 0 → 0. (The 0 → 0 cycle is only included for the sake of completeness.) Iterating on rationals with odd denominators The Collatz map can be extended to (positive or negative) rational numbers which have odd denominators when written in lowest terms. The number is taken to be 'odd' or 'even' according to whether its numerator is odd or even. Then the formula for the map is exactly the same as when the domain is the integers: an 'even' such rational is divided by 2; an 'odd' such rational is multiplied by 3 and then 1 is added. A closely related fact is that the Collatz map extends to the ring of 2-adic integers, which contains the ring of rationals with odd denominators as a subring. When using the "shortcut" definition of the Collatz map, it is known that any periodic parity sequence is generated by exactly one rational. Conversely, it is conjectured that every rational with an odd denominator has an eventually cyclic parity sequence (Periodicity Conjecture). If a parity cycle has length and includes odd numbers exactly times at indices , then the unique