A Step-by-Step Solution Methodology for Mathematical Expressions

In this paper, we propose a methodology for the step-by-step solution of problems, which can be incorporated into a computer algebra system. Our main aim is to show all the intermediate evaluation steps of mathematical expressions from the start to the end of the solution. The first stage of the methodology covers the development of a formal grammar that describes the syntax and semantics of mathematical expressions. Using a compiler generation tool, the second stage produces a parser from the grammar description. The parser is used to convert a particular mathematical expression into an Abstract Syntax Tree (AST), which is evaluated in the third stage by traversing al its nodes. After every evaluation of some nodes, which corresponds to an intermediate solution step of the related expression, the resulting AST is transformed into the corresponding mathematical expression and then displayed. Many other algebra-related issues such as simplification, factorization, distribution and substitution can be covered by the solution methodology. We currently focuses on the solutions of various problems associated with the subject of derivative, equations, single variable polynomials, and operations on functions. However, it can easily be extended to cover the other subjects of general mathematics.

Extended Positional Grammars

Much recent research is focusing on formal methods for the definition and implementation of visual programming environments. Extended positional grammars naturally extend context-free grammars for string languages to grammars for visual languages by considering new relations in addition to string concatenation. Thanks to this analogy, most results from LR parsing can be extended to extended positional grammars while preserving their well-known efficiency. XPGs include mechanisms for handling contextual information enabling us to model a broader class of visual languages, which includes the diagrammatic notations used in software engineering. Moreover, the XPG grammar formalism can be effectively used for modeling both visual and textual notations in a seamless way. The XPG model is the underlying formalism of the VLDesk system for the automatic generation of visual programming environments. VLDesk inherits and extends to the visual field, concepts and techniques of compiler generation tools like YACC.

Attribute Grammars and Their Applications

Attribute grammars are a framework for defining semantics of programming languages in a syntax-directed fashion. In this chapter, we define attribute grammars, and then illustrate their use for language definition, compiler generation, definite clause grammars, design and specification of algorithms, and so forth. Our goal is to emphasize its role as a tool for design, formal specification and implementation of practical systems, so our presentation is example rich.