Flexible Arithmetic for Huge Numbers with Recursive Series

Flexible Arithmetic for Huge Numbers with Recursive Series of Operations Vagan Terziyan*, Alexey Tsymbal**, Seppo Puuronen** *Dept. of CS and ISs, Kharkov State Technical University of Radio-electronics, UKRAINE vagan@kture. cit-ua. net **Department of CS and ISs, University of Jyväskylä, FINLAND {alexey, sepi}@jytko. jyu. fi AAECC’ 99 Honolulu November 1999

Finland Ukraine University of Jyväskylä Finland State Technical University of Radioelectronics Kharkov Ukraine AAECC’ 99 Honolulu November 14 -19, 1999 Flexible Arithmetic for Huge Numbers with Recursive Series of Operations by Terziyan, Tsymbal, & Puuronen 2

Contents u Introduction: Ackerman’s Recursive Function u Infinite Series of Arithmetical Operations u Properties of Recursive Arithmetical Operations u Calculation of Recursive Arithmetical Operations: A Derecursivation Algorithm u Recursive Counters and Coding Huge Numbers u A Relationship Between Recounters and Recursive Arithmetical Operations u Conclusions and Future Work u Contact Info AAECC’ 99 Honolulu November 14 -19, 1999 Flexible Arithmetic for Huge Numbers with Recursive Series of Operations by Terziyan, Tsymbal, & Puuronen 3

Ackerman’s Recursive Function Problems: 9 9 9 is • huge numbers: so huge that it would require a 1800 km piece of paper (with 0. 5 cm on each digit); • recursion of second degree -- labor-consuming algorithm for calculation; • practical applications. AAECC’ 99 Honolulu November 14 -19, 1999 Flexible Arithmetic for Huge Numbers with Recursive Series of Operations by Terziyan, Tsymbal, & Puuronen 4

Infinite Series of Arithmetical Operations Inverse operations: AAECC’ 99 Honolulu November 14 -19, 1999 Flexible Arithmetic for Huge Numbers with Recursive Series of Operations by Terziyan, Tsymbal, & Puuronen 5

Properties of Recursive Arithmetical Operations AAECC’ 99 Honolulu November 14 -19, 1999 Flexible Arithmetic for Huge Numbers with Recursive Series of Operations by Terziyan, Tsymbal, & Puuronen 6

Calculation of Recursive Arithmetical Operations AAECC’ 99 Honolulu November 14 -19, 1999 Flexible Arithmetic for Huge Numbers with Recursive Series of Operations by Terziyan, Tsymbal, & Puuronen 7

Derecursivation Algorithm: Variables AAECC’ 99 Honolulu November 14 -19, 1999 Flexible Arithmetic for Huge Numbers with Recursive Series of Operations by Terziyan, Tsymbal, & Puuronen 8

Derecursivation Algorithm AAECC’ 99 Honolulu November 14 -19, 1999 Flexible Arithmetic for Huge Numbers with Recursive Series of Operations by Terziyan, Tsymbal, & Puuronen 9

Recursive Counters AAECC’ 99 Honolulu November 14 -19, 1999 Flexible Arithmetic for Huge Numbers with Recursive Series of Operations by Terziyan, Tsymbal, & Puuronen 10

Coding Huge Numbers with Recounters We propose to code huge numbers with the following pair: For example: Intervals coded with each such pair are: AAECC’ 99 Honolulu November 14 -19, 1999 Flexible Arithmetic for Huge Numbers with Recursive Series of Operations by Terziyan, Tsymbal, & Puuronen 11

A Relationship Between Recounters and Recursive Arithmetical Operations AAECC’ 99 Honolulu November 14 -19, 1999 Flexible Arithmetic for Huge Numbers with Recursive Series of Operations by Terziyan, Tsymbal, & Puuronen 12

Conclusions and Future Work • A recursive expansion of the set of ordinary arithmetical operations was investigated; • The recursive arithmetical operation was defined, where n is the level of recursion starting with ordinary + (n=1); • Basic properties of recursive operations were investigated, an algorithm for calculation of these operations was considered; • The recursive counters’ apparatus was proposed for representation of huge integers, which are results of recursive operations, in a restricted memory; • Practical applications: random number generation? AAECC’ 99 Honolulu November 14 -19, 1999 Flexible Arithmetic for Huge Numbers with Recursive Series of Operations by Terziyan, Tsymbal, & Puuronen 13

Contact Info Vagan Terziyan*, Alexey Tsymbal**, Seppo Puuronen** *Dept. of CS and ISs, Kharkov State Technical University of Radio-electronics, UKRAINE vagan@kture. cit-ua. net **Department of CS and ISs, University of Jyväskylä, FINLAND {alexey, sepi}@jytko. jyu. fi AAECC’ 99 Honolulu November 14 -19, 1999 Flexible Arithmetic for Huge Numbers with Recursive Series of Operations by Terziyan, Tsymbal, & Puuronen 14