On Restricting Real-Valued Genotypes in Evolutionary Algorithms

Real-valued genotypes together with the variation operators, mutation and crossover, constitute some of the fundamental building blocks of Evolutionary Algorithms. Real-valued genotypes are utilized in a broad range of contexts, from weights in Artificial Neural Networks to parameters in robot control systems. Shared between most uses of real-valued genomes is the need for limiting the range of individual parameters to allowable bounds. In this paper we will illustrate the challenge of limiting the parameters of real-valued genomes and analyse the most promising method to properly limit these values. We utilize both empirical as well as benchmark examples to demonstrate the utility of the proposed method and through a literature review show how the insight of this paper could impact other research within the field. The proposed method requires minimal intervention from Evolutionary Algorithm practitioners and behaves well under repeated application of variation operators, leading to better theoretical properties as well as significant differences in well-known benchmarks

Real Interactive Proofs for VPSPACE

We study interactive proofs in the framework of real number complexity as introduced by Blum, Shub, and Smale. The ultimate goal is to give a Shamir like characterization of the real counterpart IP_R of classical IP. Whereas classically Shamir\u27s result implies IP = PSPACE = PAT = PAR, in our framework a major difficulty arises from the fact that in contrast to Turing complexity theory the real number classes PAR_R and PAT_R differ and space resources considered alone are not meaningful. It is not obvious to see whether IP_R is characterized by one of them - and if so by which. In recent work the present authors established an upper bound IP_R is a subset of MA(Exists)R, where MA(Exists)R is a complexity class satisfying PAR_R is a strict subset of MA(Exists)R, which is a subset of PAT_R and conjectured to be different from PAT_R. The goal of the present paper is to complement this result and to prove interesting lower bounds for IP_R. More precisely, we design interactive real protocols for a large class of functions introduced by Koiran and Perifel and denoted by UniformVSPACE^0. As consequence, we show PAR_R is a subset of IP_R, which in particular implies co-NP_R is a subset of IP_R, and P_R^{Res} is a subset of IP_R, where Res denotes certain multivariate Resultant polynomials. Our proof techniques are guided by the question in how far Shamir\u27s classical proof can be used as well in the real number setting. Towards this aim results by Koiran and Perifel on UniformVSPACE^0 are extremely helpful