Pointfree expression and calculation: from quantification to temporal logic

Abstract

Pointfree formulation means suppressing domain variables to focus on higherlevel objects (functions, relations). Advantages are algebraic-style calculation and abstraction as formal logics pursue by axiomatization. Various specific uses are considered, starting with quantification in the wider sense (∀, ∃, , etc.). Pointfree style is achieved by suitable functionals that prove superior to pointwise conventions such as the Eindhoven notation. Pointwise calculations from the literature are reworked in pointfree form. The second use considered is in describing systems, with generic functionals capturing signal flow patterns. An illustration is the mathematics behind a neat magician's trick, whose implementation allows comparing the pointfree style in Funmath, LabVIEW, TLA + , Haskell and Maple. The third use is making temporal logic calculational, with a simple generic Functional Temporal Calculus (FTC) for unification. Specific temporal logics are then captured via endosemantic functions. The example is TLA + . Calculation is illustrated by deriving various theorems, most related to liveness issues, and discovering results by calculation rather than proving them afterwards. To conclude, various ramifications, style and abstraction issues are discussed, in relation to engineering mathematics in general and to categorical formulations.

A Precise and Reliable Multivariable Chain Rule

“…Since students may encounter both kinds of operators in various forms in courses, on the web, and in their later work, learning how to use them properly with clear, safe calculation rules eliminates confusion and enhances understanding. Also, exploiting the functional style offers many opportunities [3] in other areas of mathematics.…”