Unterausdrücke

Unterausdrücke, or subexpressions, are parts of a larger expression that themselves qualify as valid expressions. They can be viewed as subtrees of the expression’s parse tree or as contiguous segments of the syntax. A subexpression may be the entire expression, but in many contexts a smaller, proper subexpression is distinguished that is strictly contained within the larger one.

For example, in the arithmetic expression (a + b) * c, the subexpressions include a, b, a + b, c, and the whole expression (a + b) * c. In f(x) + g(y), the subexpressions are x, f(x), y, g(y), and f(x) + g(y). Parentheses and operator precedence determine which parts form subexpressions, especially when operations are nested.

Subexpressions are used in a variety of areas. In algebra and calculus, they facilitate manipulation, simplification, and differentiation by allowing rules to apply to parts of an expression. In computer science, subexpressions underpin evaluation strategies, symbolic computation, and pattern matching. In compiler design, identifying common subexpressions enables optimizations such as common subexpression elimination, which reduces redundant calculations.

Formally, expressions are built according to a grammar, and subexpressions correspond to the subtrees rooted at any node in the expression’s syntax tree. This view supports systematic rewriting, transformation, and analysis of expressions, making Unterausdrücke a fundamental concept in mathematics, logic, and programming.