Two function algebras defining functions in NCk boolean circuits

Abstract

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.

Pointers in Recursion: Exploring the Tropics

“…As above, we use the mutual version of these recursion schemes, with the same tiering discipline. Note that, unlike previous characterizations of sub-polynomial complexity classes [3,4,10], our tropical composition and recursion schemes are only syntactical refinements of the usual composition and primitive recursion schemes -removing the syntactical sugar yields indeed the classical schemes.…”

“…Later on, Bellantoni and Cook's purely syntactical approach proved also useful for characterizing other complexity classes. Leivant and Marion [14,13] used a predicative version of the safe recursion scheme to characterize alternating complexity classes, while Bloch [3], Bonfante et al [4] and Kuroda [10], gave characterizations of small, polylogtime, parallel complexity classes. An important feature of these results is that they use, either explicitly or not, a tree-recursion on the input.…”

“…Actual computation input are thus implicit in our function algebras: the fuel of the computational machinery is only pointer arithmetics. This proposal takes inspiration partially from the Rational Bitwise Equations of [4].…”

Cyclic Implicit Complexity

“…Parallel cut might also play a fundamental role for potential circular proof theoretic characterisations of circuit complexity classes, like ALOGTIME or NC. In such a setting, one would expect a conditional rule that implements a divide and conquer searching mechanism (see, e.g., [Lei98,LM00,BKMO16]).…”

ComplexityParser: An Automatic Tool for Certifying Poly-Time Complexity of Java Programs

“…The use of tiering techniques to certify program complexity was kick-started by the seminal works of Bellantoni-Cook [4] and Leivant-Marion [22], that provide sound and complete characterizations of the class of functions computable in polynomial time FP. Tiering was later adapted to several other complexity classes such as FPSPACE [23], NC [8,28,21], or L [28].…”