 
Summary: A logic for nonincreasing PTIME computation
Ulrich Berger, Martin Hofmann, Helmut Schwichtenberg
1 Introduction
We wish to solve the following equation:
Heyting Arithmetic
G¨odel's T = ???
Calculus from M.H.'s LICS 1999 paper
In other words we seek a logic whose \Pi 0
2 proofs can be realised by terms in M.H.'s
system and thus in particular have nonsizeincreasing polytime Skolem Functions.
Let us call M.H.'s system C (in honour of Caseiro) for further reference.
As a first approximation we aim for a firstorder logic which does not have all
of C in its term language but merely the constructor symbols. This will allow us
to stick to plain firstorder logic. Higherorder constructions are accommodated
indirectly through the use of complex formulas in inductions.
2 Definition of the logic
We start from a usual first order signature.
It should contain in particular constructor symbols for the desired datatypes
with their usual arities. For example, to accommodate unary natural numbers,
booleans, and lists we would include constants 0; nil; tt; ff, unary function symbol S,
