A sup-interpretation is a tool which provides an upper bound on the size of a value computed by some symbol of a program. Supinterpretations have shown their interest to deal with the complexity of first order functional programs. For instance, they allow to characterize all the functions bitwise computable in Alogtime. This paper is an attempt to adapt the framework of sup-interpretations to a fragment of oriented-object programs, including distinct encodings of numbers through the use of constructor symbols, loop and while constructs and non recursive methods with side effects. We give a criterion, called brotherly criterion, which ensures that each brotherly program computes objects whose size is polynomially bounded by the inputs sizes. 1

2 DIAL∀l µ l 4 2.1 The dual typing system............................ 4 2.2 Data structures and notations......................... 5