The paper introduces a formalism for describing language processing systems, which is useful for designing complex programming systems and their documentation. It provides a mathematical basis for analyzing such systems and highlights areas where it can be extended. The formalism does not describe the amount of compilation or interpretation without precise language definitions, nor does it support incremental compilation or systems with programs in multiple languages. The authors acknowledge comments from J. Gray and J. Reynolds. The paper also includes several algorithms, including Algorithm 395 for computing the Student's t-distribution, Algorithm 396 for computing t-quantiles, Algorithm 397 for solving an integer programming problem, Algorithm 398 for tableless date conversion, Algorithm 399 for finding a spanning tree, and Algorithm 400 for modified Havie integration. These algorithms are described in detail, including their implementation and performance characteristics. The authors also provide references to various works in the field of computer science and mathematics, including studies on numerical integration, statistical tables, and algorithmic solutions to programming problems. The paper concludes with a discussion of the accuracy and efficiency of the algorithms, as well as their applications in various contexts.The paper introduces a formalism for describing language processing systems, which is useful for designing complex programming systems and their documentation. It provides a mathematical basis for analyzing such systems and highlights areas where it can be extended. The formalism does not describe the amount of compilation or interpretation without precise language definitions, nor does it support incremental compilation or systems with programs in multiple languages. The authors acknowledge comments from J. Gray and J. Reynolds. The paper also includes several algorithms, including Algorithm 395 for computing the Student's t-distribution, Algorithm 396 for computing t-quantiles, Algorithm 397 for solving an integer programming problem, Algorithm 398 for tableless date conversion, Algorithm 399 for finding a spanning tree, and Algorithm 400 for modified Havie integration. These algorithms are described in detail, including their implementation and performance characteristics. The authors also provide references to various works in the field of computer science and mathematics, including studies on numerical integration, statistical tables, and algorithmic solutions to programming problems. The paper concludes with a discussion of the accuracy and efficiency of the algorithms, as well as their applications in various contexts.