The Quantity Lattice for Engineering Design

We propose a new mathematical engine for reasoning with abstraction to support engineering design. The new Qua,|i|y Lattice (qL) will built on the foundation of the existing Quantity Lattice by Reid Simmons [Simmons, 1986], the Bounder inequality reasoner by Elisha Sacks [Sacks, 1987], and the Minima hybrid real/sign algebra by Brian Williams [Williams, 1991]. The new qL supports representation and manipulation of evolving designs that involve abstraction at any of several levels, including "qualitative" (signs only), "semi-quantitative" (bounded quantities), and fully quantitative (real numbers). The qL seamless, allowing abstraction levels to be mixed freely aa needed within a single design. The qL also is tunable, allowing trade-offs between solution time and solution quality, so that effort can be applied where it is most needed or effective in an evolving design. Supporting mixed abstraction levels and localised computational effort enables a design methodology including iterstive refinement and tight control over the exploration of design alternatives. The qL is applicable to design problems which can be modeled using systems of linear and nonlinear equations and inequalities, including any differential equations which can be solved or approximated to yield such relations. This is appropriate for a wide range of problem domains, including both synthetic and analytical tasks.