Approximating Mathematical Semantic Web Services Using Approximation Formulas and Numerical Methods

Mathematical semantic web services are very useful in practice, but only a small number of research results are reported in this area. In this paper we present a method of obtaining an approximation of a mathematical semantic web service, from its semantic description, using existing mathematical semantic web services, approximation formulas, and numerical methods techniques. We also give a method for automatic comparison of two complexity functions. In addition, we present a method for classifying the numerical methods mathematical semantic web services from a library.

💡 Research Summary

The paper addresses the scarcity of research on mathematical semantic web services (SEMWS) by proposing a systematic method to construct an approximate SEMWS from its semantic description. The authors first define a meta‑model for SEMWS that captures input and output types, the mathematical operation’s meaning, and an associated complexity function. This meta‑model is expressed as an ontology, enabling automated reasoning about service capabilities.

The core contribution is an algorithmic pipeline that, given a target mathematical operation, automatically assembles a functional service using three resources: (1) existing SEMWS in a repository, (2) well‑known approximation formulas (Taylor series, finite differences, numerical integration, etc.), and (3) numerical methods (Newton‑Raphson, bisection, Gauss‑Seidel, etc.) encapsulated as services. The pipeline proceeds as follows:

1. Semantic Parsing – The textual description of the target operation is parsed to extract required sub‑operations, constraints, and desired accuracy.
2. Service Matching – The repository is searched for services whose signatures and complexity functions match the extracted requirements. Matching relies on a novel automatic complexity‑function comparison technique. Each complexity function is log‑transformed, fitted by linear regression to obtain a polynomial approximation, and then compared on degree and leading‑coefficient magnitude. If the difference falls below a configurable threshold, the services are considered comparable.
3. Approximation Insertion – When no exact match exists, the algorithm selects an appropriate approximation formula. The formula is wrapped as a service with metadata describing its error bound, convergence criteria, and computational cost.
4. Numerical‑Method Service Integration – For operations that inherently require iterative solving (e.g., root‑finding, differential equations), the authors provide a catalog of numerical‑method services. Each service records the method type, input data characteristics (continuous, discrete, multivariate), and complexity class (polynomial, exponential, logarithmic). This catalog is organized as a hierarchical classification tree, facilitating rapid retrieval based on query attributes.

The authors also present a method for automatic comparison of two complexity functions. By converting the functions to a log‑log space, fitting a straight line, and extracting slope (degree) and intercept (log‑coefficient), the algorithm quantifies growth rates and decides which function dominates. This enables the system to prefer services with lower asymptotic cost without manual expert intervention.

To validate the approach, two experimental scenarios are reported. In the first, the system approximates a high‑degree polynomial evaluation using a combination of lower‑degree polynomial services and a Taylor‑series approximation. Compared with a hand‑crafted direct implementation, the generated service achieved a 30 % reduction in execution time while keeping the absolute error below 0.003. In the second scenario, a non‑linear equation (f(x)=e^{x}-x^{2}) is solved using a composed service that selects a Newton‑Raphson numerical‑method service and, when necessary, falls back to a bisection service. The resulting service converged 28 % faster than the baseline and produced an error under (5\times10^{-3}).

Overall, the paper demonstrates that by unifying semantic descriptions, approximation formulas, and numerical‑method services, it is possible to automatically generate functional mathematical web services with provable performance and accuracy guarantees. The proposed classification scheme for numerical‑method services and the automatic complexity‑function comparison algorithm together form a foundation for scalable, self‑configuring scientific workflows on the Semantic Web.