How to make AXIOM into a scratchpad

Scratchpad [GrJe71] was a computer algebra system developed in the early 1970s. Like M&M (Maple [CGG91ab] and Mathematical [W01S92]) and other systems today, Scratchpad had one principal representation for mathematical formulae based on “expression trees”. Its user interface design was based on a pattern-matching paradigm with infinite rewriterule semantics, providing what we believe to be the most natural paradigm for interactive symbolic problem solving. Like M&M, however, user programs were interpreted, often resulting in poor performance relative to similar facilities coded in standard programming languages such as FORTRAN and C. Scratchpad development stopped in 1976 giving way to a new system design ([ JenR79], [JeTr81]) that evolved into AXIOM [JeSu92]. AXIOM has a strongly-typed programming language for building a library of parameterized types and algorithms, and a type-inferencing interpreter that accesses the library and can build any of an infinite number of types for interactive use. We suggest that the addition of an expression tree type to AXIOM can allow users to operate with the same freedom and convenience of untyped systems without giving up the expressive power and run-time efficiency provided by the type system. We also present a design that supports a multiplicity y of programming styles, from the Scratchpad pattern-matching paradigm to functional programming to more conventional procedural programming. The resulting design seems to us to combine the best features of Scratchpad with current AXIOM and to offer a most attractive, flexible, and user-friendly environment for interactive problem solving. Section 2 is a discussion of design issues contrasting AXIOMwith other symbolic systems. Sections 3 and 4 is an assessment of AXIOM’s current design for building libraries and interactive use. Section 5 describes a new interface design for AXIOM, its resulting paradigms, and its underlying semantic model. Section 6 compares this work with others.