Flexible Arithmetic for Huge Numbers with Recursive Series of Operations

1. 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

2. Finland and 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

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

4. Ackerman’s Recursive Function • Problems: • huge numbers: 999 is so huge that it would require a 1800km piece of paper (with 0.5cm 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

5. 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

6. 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

7. 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

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

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

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

11. 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

12. 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

13. 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

14. 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