web mathematic

CHAPTER 5: FEATURE BASED MATHEMATICAL SEARCH ENGINE

Overview

The chapter begins with an extensive summary of the algorithms proposed by Ma et al., detailing their innovative approaches to mathematical searching within various disciplines including mathematics, computer science, and engineering. Following this, a detailed account of their contributions and implementations is presented, highlighting evaluations and comparisons with EgoMath, a competitor search engine known for its semantic understanding of mathematical expressions.

5.1 Original Algorithm

The core algorithm developed by Ma et al. relies heavily on the Presentation MathML format, a standard for representing mathematical notation on the web. This format is crucial as it allows for the effective encoding of complex mathematical expressions, thereby facilitating retrieval operations. The algorithm's reliance on this format affords improved precision in formula searching.

Key Presentation MathML Tags

• <mo>: Represents mathematical operators and parentheses, essential for structuring expressions.

• <mi>: Denotes identifiers such as function names, coefficients, and variables that play critical roles in mathematical definitions.

• <mn>: Indicates numeric literals (e.g., integers, rational numbers), providing the foundation for quantitative aspects of formulas.

• <msup>, <msub>: Used for superscripts and subscripts, allowing for the representation of powers and indexed variables.

• <mrow>: Groups sub-expressions and denotes overall structure, which is critical for maintaining the integrity of complex formulas.

It’s important to note that semantic meanings in mathematics may not always align with their presentations, leading to potential extraction errors that can occur if the algorithm misinterprets the structure or content of a formula.

Features Extracted from Formulas

Structural Features

Structural features are obtained from nodes containing semantic attributes, requiring a minimum depth of ≥ 1 within the formula tree. The structural meaning is derived by concatenating parent node names to ensure contextual relevance.

Example with formula: R 5e^{2x - 1}

• Tags: mo:R, −, mn:5, 2, 1, mrow, msup, mi:e, x.

• Semantic features extracted: R (mo), e (mi, exponential function), msup, − (mo).

Semantic Features

Semantic features are extracted via a pre-order traversal of the formula tree, emphasizing nodes containing identifiers and operators while excluding benign structural elements like <mrow>. The original paper's claims regarding numeric constants highlighted that these constitute less meaningful information for formula search, which led to a methodological change—number constants are supplanted with specific keywords (e.g., var) to enhance the searchability and relevance of the features extracted.

Feature Vector Transformation

Extracted features undergo transformation into a feature vector utilizing the tf-idf (term frequency-inverse document frequency) weighting scheme, which is instrumental in prioritizing significant features in the context of search results:

• Term Frequency (tf): tf(feature, formula) = # of occurrences of feature in a formula / max {# of occurrences of any feature in a formula}

• Inverse Document Frequency (idf): idf(feature, formula set) = log(# of formulae in a formula set / # of formulae containing the feature)This transformation creates a normalized feature vector, allowing effective clustering methodologies to be applied subsequently.

Similarity Measurement

The algorithm employs cosine similarity to assess the similarity between a query formula and existing formulae represented by their respective normalized feature vectors. This comparison is pivotal in determining the most relevant mathematical expressions in response to user queries.

Evaluation of Clustering Algorithms

The original paper conducted evaluations on several clustering algorithms, specifically focusing on:

• K-means,

• Self-organized maps,

• Agglomerative hierarchical clustering.

These assessments provided insights into the performance of the original approach when applied to larger datasets, raising considerations for enhancement. The original evaluation encompassed 884 formulae that were meticulously categorized, employing both training and test samples to ensure robustness in the findings.

5.2 Modifications

Several issues with the original approach emerged during the analysis, particularly those affecting performance scalability in larger datasets. Suggestions for enhancements, including a modified version of the algorithms, were discussed to address these disparities and improve overall efficacy in mathematical search tasks.