Intrinsic theories and computational complexity

Abstract

We introduce a new proof theoretic approach to computational complexity. With each free algebra $\mathbb{A}$we associate a first order “intrinsic theory for $\mathbb{A}$”, IT ($\mathbb{A}$), with no initial functions other than the constructors of $\mathbb{A}$, and no axioms for them other than the generative and inductive axioms, which delineate $\mathbb{A}$. The case most relevant to traditional proof theory is $\mathbb{A}$=ℕ (the unary natural numbers), and the case most relevant to computer science is $\mathbb{A}$=$\mathbb{W}$={0,1}. An algorithm is provable if it provably maps inputs in $\mathbb{A}$to values in $\mathbb{A}$, and a function is provable if it has a provable algorithm. We show that the provable functions of IT(N) are exactly the provably recursive functions of Peano Arithmetic.

We further show that function provability is equivalent to computational complexity for the following pairs theory/complexity-class: (1) A ramified variant RT($\mathbb{A}$) of IT ($\mathbb{A}$) and elementary functions. (2) N($\mathbb{W}$) with quantifier-free induction and poly-time; (3) RT(N) with quantifier-free induction and linear space (on register machines).

Intrinsic theories combine lean axiomatics with expressive flexibility, since they permit explicit (uncoded) reference to arbitrary computable functions. Thus, the characterizations above provide user-friendly formalisms for feasible mathematics, in which non-feasible algorithms can be mentioned freely. Moreover, natural deduction calculi for these formalisms correspond directly, via formula-as-type homomorphisms, to applicative programs.