New terms metric based on substitutions

Many works are available in the literature to define dissimilarity metrics between expressions represented in the form of first-order logic, which is a powerful representation language. Generally speaking, first-order terms are basic logic expressions such that the first step in defining a valid metric between first-order expressions is to define a metric between terms. In this work, we introduce a new metric between terms which is an extension of the metric based on substitutions proposed by Alan Hutchinson. Our approach breaks the limitation of Alan Hutchinson's metric which is only suitable for ground terms and enhances the power of reflecting the dissimilarity in ground terms. In fact, both function symbols and variable symbols are the source of difference between terms. As a consequence, the new metric considers difference caused the two factors as J. Ramon et al. do, but in a different way which is based on substitutions. It is also defined in the form of 2-tuples, the first element of which is used for estimating difference caused by function symbols and another for estimating difference caused by variable symbols. Besides, some experimental results are also shown in the paper, which illustrates the effects and improvement compared with Alan Hutchinson's metric.