We forniulate recursive characterizations of the class of elementary functions and the class of functions computable in polynomial space that do not require any explicit bounded acheme. More specifically, we use functions where the input variables can occur in different kinds of positions -normal and safe -in the vein of the Bellantoni and Cook's characterization of the polytime funetiona.
We give a recursion-theoretic characterization of the complexity classes NC k for k ≥ 1. In the spirit of implicit computational complexity, it uses no explicit bounds in the recursion and also no separation of variables is needed. It is based on three recursion schemes, one corresponds to time (time iteration), one to space allocation (explicit structural recursion) and one to internal computations (mutual in place recursion). This is, to our knowledge, the first exact characterization of the NC k by function algebra over infinite domains in implicit complexity. Research supported by the project Teorias e linguagens de programação para computações com recursos limitados within the Programa PESSOA 2005
This paper gives an implicit characterization of the class of functions computable in polynomial space by deterministic Turing machines -PSPACE. It gives an inductive characterization of PSPACE with no ad-hoc initial functions and with only one recursion scheme. The main novelty of this characterization is the use of pointers (also called path information) to reach PSPACE. The presence of the pointers in the recursion on notation scheme is the main difference between this characterization of PSPACE and the well-known Bellantoni-Cook characterization of the polytime functions -PTIME.
A two-sorted term system characterizing NC implicitly is described. The term system is defined over the tree algebra T, the free algebra generated by 0, 1 and * , and the recursion scheme uses pointers over tier 0. This differs from previous characterizations of NC, where tier 1 pointers were used or full parameter substitution over tier 0 was allowed.
We describe the functions computed by boolean circuits in NC k by means of functions algebra for k ≥ 1 in the spirit of implicit computational complexity. The whole hierarchy defines NC. In other words, we give a recursion-theoretic characterization of the complexity classes NC k for k ≥ 1 without reference to a machine model, nor explicit bounds in the recursion schema. Actually, we give two equivalent description of the classes NC k , f ≥ 1. One is based on a tree structureà la Leivant [Lei98], the other is based on words. This latter puts into light the role of computation of pointers in circuit complexity. We show that transducers are a key concept for pointer evaluation.
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