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Coding addition Sasha Rubin Cornell REU 2009 Arithmetic on N Addition is space-efficient. eg. Base 10 coding of N . carry propagation procedure for +. Arithmetic on N Addition is space-efficient. eg. Base 10 coding of N . carry

  1. Coding addition Sasha Rubin Cornell REU 2009

  2. Arithmetic on N Addition is space-efficient. eg. Base 10 coding of N . ’carry propagation’ procedure for +.

  3. Arithmetic on N Addition is space-efficient. eg. Base 10 coding of N . ’carry propagation’ procedure for +. But the usual multiplication procedure is not space-efficient. More is true. There is no space-efficient coding of ( N , × ).

  4. Space-efficient presentations of + Basic systems: base 10, base 2, . . .

  5. Space-efficient presentations of + Basic systems: base 10, base 2, . . . Non-standard systems: Fibonacci, − 2, . . .

  6. Space-efficient presentations of + Basic systems: base 10, base 2, . . . Non-standard systems: Fibonacci, − 2, . . . Observation. In all these presentations the codes of the numbers in natural order 0 , 1 , 2 , 3 , · · · are ordered in reverse length-lexicographic order. Eg. 0 , 1 , 10 , 11 , 100 , 101 , 110 , · · ·

  7. Space-efficient presentations of + Basic systems: base 10, base 2, . . . Non-standard systems: Fibonacci, − 2, . . . Observation. In all these presentations the codes of the numbers in natural order 0 , 1 , 2 , 3 , · · · are ordered in reverse length-lexicographic order. Eg. 0 , 1 , 10 , 11 , 100 , 101 , 110 , · · · Conjecture. In every space efficient presentation of + the codes are, essentially, in reverse length-lexicographic ordering

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