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

3. AAECC’99 Honolulu November 14-19, 1999 Flexible Arithmetic for Huge Numbers with Recursive Series of Operations by Terziyan, Tsymbal, & Puuronen 3 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

4. 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: 9 9 9 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.

5. 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:

6. 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

7. 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

8. 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

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

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

11. 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: Intervals coded with each such pair are: For example:

12. 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

13. 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?

14. 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