Characterizingco-NLby a group action

Abstract

In a recent paper, Girard proposes to use his recent construction of a geometry of interaction in the hyperfinite factor in an innovative way to characterize complexity classes. We begin by giving a detailed explanation of both the choices and the motivations of Girard's definitions. We then provide a complete proof that the complexity class co-NL can be characterized using this new approach. We introduce as a technical tool the non-deterministic pointer machine, a concrete model to computes algorithms.Comment: To appear in Mathematical Structures in Computer Scienc

Logarithmic space and permutations

“…Using the crossed product construction between a von Neumann algebra and a group acting on it, he proposed to characterize complexity classes as sets of operators obtained through the internalization of outer automorphisms of the type II 1 hyperfinite factor. The authors showed in a recent paper [2] that this approach succeeds in defining a characterization of the set of co-NL languages as a set of operators in the type II 1 hyperfinite factor. The proof of this result was obtained through the introduction of non-deterministic pointer machines, which are abstract machines designed to mimic the computational behavior of operators.…”

“…1 The construction L 2 (G, H) is a generalization of the well-known construction of the Hilbert space of square-summable functions: in case G is considered with the discrete topology, the elements are functions f : G → H such that g∈G f (g) 2 < ∞. 2 Recall that in the algebra L (H) of bounded linear operators on the Hilbert space H (we denote by 〈·, ·〉 its inner product), there exists an anti-linear involution (·) * such that for any ξ, η ∈ H and A ∈ L (H), 〈Aξ, η〉 = 〈ξ, A * η〉. This adjoint operator coincides with the conjugate-transpose in the algebras of square matrices.…”

“…We will denote by N 0 the image of R in T through ι 0 . Notice that this tensor product algebra is isomorphic to R. The idea of Girard is then to use the action of the group S of finite permutations of N onto T, defined by: 3 A detailed explanation of this representation can be found in Appendix A, in the authors' previous work [2] or in the second author's PhD thesis [18]. 4 Something that will be proven useful in Section 5, for it will help us to determine in which direction the integer is read.…”

“…We previously introduced [2] non-deterministic pointers machines that were designed to mimic the computational behavior of operators: they do not have the ability to write, their input tape is cyclic, and they are "universally non-deterministic", i.e. rejection is meaningful whereas acceptation is the default behavior.…”

“…In this section, we will briefly explain the encoding of pointer machines as operators. The non-deterministic case, which was already defined in our previous work [2], will be given in Section 5.1. The specifics of the deterministic case, which is a novelty, concludes in Section 5.2 this section.…”