中文摘要：

基因表达式编程（gene expression programming,GEP）解空间模型理论对算法性能的改进有现实指导意义。公开文献对GEP解空间模型的研究较少,鲜见针对GEP表现型的理论研究。基于此,提出一种基于表现型的GEP解空间模型。首先,通过定义GEP染色体表现型高度,给出单基因染色体和多基因染色体表现型高度确定上界的定理及证明,利用GEP算法自身函数发现的能力,探索出操作符集最小目数为1或2的GEP染色体表现型高度上界计算的通项公式,以保证GEP表现型解空间模型的确定有界性与可计算性。其次,以GEP表现型高度的确定上界定理为基础,构建基于表现型的GEP解空间模型,总结GEP表现型解空间模型的性质和定理。通过进一步定义GEP表现型的完全解空间概念,对最优解在GEP表现型解空间和完全解空间中的分布特征进行探索研究,获知在完全解空间中最优解随子空间序号的增长呈大比例增加的分布特征。基于表现型空间模型知识,提出限制GEP种群搜索空间的基本思想与控制策略,利用模型知识合理地解释公开文献中多种GEP改进算法的有效性。

英文摘要：

The theory of solution space model has practical significance to improve the performance of Gene Expression Programming （GEP） algorithms.There are few studies on the GEP solution space model,and the theoretical research on GEP phenotype is also scarce.To address this problem,a GEP solution space model based on phenotype was proposed.Firstly, by defining the height of GEP chromosome phenotype, a theorem and the proof of the upper bound of single gene chromosome and polygene chromosome manifestation were given.To ensure the boundedness and calculability of the GEP phenotype solution space model,the general formula of height upper bound of GEP chromosome phenotype with the minimum number of operators 1 or 2 was calculated,by using the ability of GEP algorithm to find out the function.Secondly,on basis of the definition for upper bound theorem of GEP phenotype height,the GEP solution space model based on phenotype was constructed, and the properties and theorems of GEP phenotype solution space model were summarized.By further defining the concept of the complete solution space of the GEP phenotype,the distribution of the optimal solution in the GEP phenotype solution space and the complete solution space were explored.It was found that the optimal solution of the subspace in the complete solution space largely increased in proportion to the order number of subspace.Based on the knowledge of phenotypic spatial model,the basic idea and control strategy of limiting the GEP population search space were put forward,and the effectiveness of various GEP improvement algorithms in the literature was explained by the theories of space model.