Elementary Linear Logic Revisited for Polynomial Time and an Exponential Time Hierarchy

Abstract

Abstract. Elementary linear logic is a simple variant of linear logic, introduced by Girard and which characterizes in the proofs-as-programs approach the class of elementary functions (computable in time bounded by a tower of exponentials of fixed height). Other systems, like light linear logic have then been defined to capture in a similar way polynomial time functions, but at the price of either a more complicated syntax or of more involved encodings. Such logical systems can then be the basis of type systems to guarantee statically complexity properties on lambdacalculus. Our goal here is to show that despite its simplicity, elementary linear logic can nevertheless be used as a common framework to characterize the different levels of a hierarchy of deterministic time complexity classes, within elementary time. We consider a variant of this logic with type fixpoints and weakening (elementary affine logic with fixpoints). The key ingredients are then the choice of specific types and a finer study of the normalization procedure on proof-nets. We characterize in this way the class P of polynomial time predicates and more generally the hierarchy of classes k-EXP, for k ≥ 0, where k-EXP is the union of DTIME(2 n i k ), for i ≥ 1.

On the expressivity of elementary linear logic: Characterizing Ptime and an exponential time hierarchy

“…The bounds given are polynomially bounded by |R w |, hence by n. As already argued in the proof of Proposition 20 there exists a polynomial Q such that: any reduction step on a proof-net R can be simulated on a Turing machine in time Q (|R|). So together the bound on the size of intermediary proof-nets and the bound on the number of steps for reaching T (4) show that the reduction from T to T (4) can be performed in time polynomial in n. Now, coming back to T (4) , as it is a proof-net of conclusion ! 2 B which does not have any cut at depth inferior or equal to 1, and has only exponential cuts at depth 2, by Lemma 24, applied here at depth 2, T (4) can then be reduced to its normal form in time O (|T (4) | 2 ).…”

“…So together the bound on the size of intermediary proof-nets and the bound on the number of steps for reaching T (4) show that the reduction from T to T (4) can be performed in time polynomial in n. Now, coming back to T (4) , as it is a proof-net of conclusion ! 2 B which does not have any cut at depth inferior or equal to 1, and has only exponential cuts at depth 2, by Lemma 24, applied here at depth 2, T (4) can then be reduced to its normal form in time O (|T (4) | 2 ). Moreover recall that |T (4) | 2 ≤ |T (3) | 2 which is polynomially bounded by n. So on the whole the computation has been carried out in time polynomial in n. Proof.…”

“…So we have shown that T can be reduced to T (4) in a number of steps which is polynomially bounded with respect to R w , hence with respect to n.…”

“…These only make the size decrease, by Lemma 5, and so the number of steps is bounded by |T (3) | 2 . We obtain in this way a proof-net T (4) of conclusion ! 2 B, which:…”

“…Note that in what precedes we have bounded the size of intermediary proof-nets obtained by reduction until reaching T (4) . The bounds given are polynomially bounded by |R w |, hence by n. As already argued in the proof of Proposition 20 there exists a polynomial Q such that: any reduction step on a proof-net R can be simulated on a Turing machine in time Q (|R|).…”