Counting arithmetic formulas

An arithmetic formula is an expression involving only the constant 1, and the binary operations of addition and multiplication, with multiplication by 1 not allowed. We obtain an asymptotic formula for the number of arithmetic formulas evaluating to n as n goes to infinity, solving a conjecture of E.K.?Gnang and D.?Zeilberger. We give also an asymptotic formula for the number of arithmetic formulas evaluating to n and using exactly k multiplications. Finally we analyze three specific encodings for producing arithmetic formulas. For almost all integers n , we compare the lengths of the arithmetic formulas for n that each encoding produces with the length of the shortest formula for n (which we estimate from below). We briefly discuss the time-space tradeoff offered by each.

[1]  Edinah K. Gnang.  Integer formula encoding SageTeX package , 2014, ArXiv.

[2]  W. D. Melo,et al.  The cost of computing integers , 1996 .

[3]  R. Stanley.  Enumerative Combinatorics: Volume 1 , 2011 .

[4]  Qi Cheng.  On the ultimate complexity of factorials , 2004, Theor. Comput. Sci..

[5]  On the number of arithmetic formulas , 2015 .

[6]  R. Remmert,et al.  Theory of Complex Functions , 1990 .

[7]  Herbert S. Wilf,et al.  Generating functionology , 1990 .

[8]  G. Andrews.  ENUMERATIVE COMBINATORICS, VOLUME 2 (Cambridge Studies in Advanced Mathematics 62) By R ICHARD P. S TANLEY : 581 pp., £45.00 (US$69.95), ISBN 0 521 56069 1 (Cambridge University Press, 1999). , 2000 .

[9]  Heng Huat Chan.  Analytic Number Theory for Undergraduates , 2009, Monographs in Number Theory.

[10]  Peter Bürgisser.  On Defining Integers And Proving Arithmetic Circuit Lower Bounds , 2009, computational complexity.

[11]  Carlos Gustavo Moreira.  On asymptotic estimates for arithmetic cost functions , 1997 .

[12]  Zeroless arithmetic: representing integers ONLY using ONE , 2013, 1303.0885.

[13]  Edinah K. Gnang,et al.  Some integer formula-encodings and related algorithms , 2013, Adv. Appl. Math..

[14]  R. L. Goodstein,et al.  On the restricted ordinal theorem , 1944, Journal of Symbolic Logic.

[15]  Pascal Koiran.  Valiant’s model and the cost of computing integers , 2004, computational complexity.

[16]  N. J. A. Sloane,et al.  The On-Line Encyclopedia of Integer Sequences , 2003, Electron. J. Comb..