Actual arithmetic and feasibility
Résumé
This paper presents a methodology for reasoning about the computational complexity of functional programs. We introduce a first order arithmetic \StrictTa which is a syntactic restriction of Peano arithmetic. We establish that the set of functions which are provably total in \StrictTa, is exactly the set of polynomial time functions.The cut-elimination process is polynomial time computable. Compared to others feasible arithmetics, \StrictTa is conceptually simpler. The main feature of \StrictTa concerns the treatment of the quantification. The range of quantifiers is restricted to the set of {\em actual terms} which is the set of constructor terms with variables. The inductive formulas are restricted to conjunctions of atomic formulas.