We introduce a new mathematical basis for similarity space. For the first time, we describe the relationship between distance and similarity from set theory. Then, we derive generally valid relations for the conversion between similarity and a metric and vice versa. We present a general solution for the normalization of a given similarity space or metric space. The derived solutions lead to many already used similarity and distance functions, and combine them into a unified theory. The Jaccard coefficient, Tanimoto coefficient, Steinhaus distance, Ruzicka similarity, Gaussian similarity, edit distance and edit similarity satisfy this relationship, which verifies our fundamental theory.

In this article, we introduce novel fixed point results for Boyd–Wong-type contraction mappings within the framework of similarity spaces, which have broad practical applications. The development of these results extends the classical theory of Boyd–Wong contractions by exploiting the unique structure and properties of similarity spaces. We provide an in-depth examination of the derived contractions, establishing conditions under which fixed points exist and are unique. In the latter part of the paper, we illustrate the applicability and effectiveness of the proposed results with representative examples.

In this paper, we introduce an advanced Fuzzy Record Similarity Metric (FRMS) that improves approximate record matching and models human perception of record similarity. The FRMS utilizes a newly developed similarity space with favorable properties combined with a metric space, employing a bag-of-words model with general applications in text mining and cluster analysis. To optimize the FRMS, we propose a two-stage method for approximate string matching and search that outperforms baseline methods in terms of average time complexity and F measure on various datasets. In the first stage, we construct an optimal Q-gram count filter as an optimal lower bound for fuzzy token similarities such as FRMS. The approximated Q-gram count filter achieves a high accuracy rate, filtering over 99% of dissimilar records, with a constant time complexity of $\approx \mathcal{O}\left(1\right)$. In the second stage, FRMS runs for a polynomial time of approximately $\approx \mathcal{O}\left(n^4\right)$ and models human perception of record similarity by maximum weight matching in a bipartite graph. The FRMS architecture has widespread applications in structured document storage such as databases and has already been commercialized by one of the largest IT companies. As a side result, we explain the behavior of the singularity of the Q-gram filter and the advantages of a padding extension. Overall, our method provides a more accurate and efficient approach to approximate string matching and search with real-time runtime.