Some complexity issues on the simply connected regions of the two-dimensional plane

This paper studies the computational complexity of subsets of the plane R2. We propose a general framework in which continuous problems in computational complex analysis can be studied in the context of discrete complexity theory (i.e., the NP theory). This framework is based on the bit-operation model used in recursive analysis [Pour-El and Richards, 1989] and complexity theory of real functions of Ko and Friedman [1982]. It is an extension of the polynomial-time measure theory studied in Chapter 5 of Ko [1991]. The fundamental notion in this study is the class of bounded subsets of the plane R2 whose membership problem is polynomial-time solvable. We define two such notions: the polynomial-time approximable sets and the polynomial-time recognizable sets. Informally, a subset S ~ R2 is polynomial-time approzimable if there is a machine M which, on a given point z c R2 and an integer n, determines whether z is in S within time polynomial in n and admitting errors only on a set E ~ R2 of measure 2-”. A subset S ~ R2 is polynomial-time recognizable if there is a machine M which on a given point z G R2 and an integer n, determines whether z is in S within time polynomial in n and admitting errors only on points z that are within a distance 2-n of the boundary of S.3 To demonstrate that these two notions of polynomialtime computable sets are natural and interesting, we