Complexity of mathematical expressions and its application in automatic answer checking

Abstract

The complexity of a mathematical expression is a measure that can be used to compare the expression with other mathematical expressions and judge which one is simpler. In the paper, we analyze three effect factors for the complexity of a mathematical expression: representational length, computational time, and intelligibility. Mainly, the paper introduces a binary-lambda-calculus based calculation method for representational complexity and a rule based calculation method for algebraic computation complexity. In the process of calculating the representation complexity of mathematical expressions, we transform the de bruijn notation into the binary lambda calculus of mathematical expressions that is inspired by compressing symmetry strings in Kolmogorov complexity theorem. Furthermore, the application of complexity of mathematical expressions in MACP, a mathematics answer checking protocol, is also addressed. MACP can be used in a computer aided assessment system in order to compute correct answers, verify equivalence of expressions, check user answers whether in a simplification form, and give automatic partial grades.