 
Summary: An arithmetic for nonsizeincreasing polynomialtime computation
Klaus Aehlig \Lambda
Abstract
An arithmetical system is presented in such a way that
from every proof a realizing term can be extracted that is
definable in a certain affine linear typed variant of G¨odel's
T and therefore is a nonsizeincreasing polynomial time
computable function with time complexity O(X n ) if the
proof contains at most n nested uses of induction.
1 Introduction
In [5] Hofmann presented a restriction of G¨odel's T allow
ing iteration in all types with the property that all de
finable functions are nonsizeincreasing polynomialtime
computable. G¨odel's T [4] is related to Heyting Arithmetic
via realizability interpretation. If, for example, 8x ø 9 \Lambda y ae A
with A quantifier free can be proved, then there exists a
term of type ø ! ae in G¨odel's T calculating the y for a
given x. For more details on realizability interpretations
see [8, IIIx4].
We suggest a restriction of Heyting Arithmetic with the
