Pointfree Factorization of Operation Refinement

Abstract

Abstract.The standard operation refinement ordering is a kind of "meet of opposites": non-determinism reduction suggests "smaller" behaviour while increase of definition suggests "larger" behaviour. Groves' factorization of this ordering into two simpler relations, one per refinement concern, makes it more mathematically tractable but is far from fully exploited in the literature. We present a pointfree theory for this factorization which is more agile and calculational than the standard set-theoretic approach. In particular, we show that factorization leads to a simple proof of structural refinement for arbitrary parametric types and exploit factor instantiation across different subclasses of (relational) operation. The prospect of generalizing the factorization to coalgebraic refinement is discussed.

A calculational approach to path-based properties of the Eisenstein–Stern and Stern–Brocot trees via matrix algebra

“…meaning that m can be written as the sum of two squares: (a + c) 2 + (b + d) 2 . Now, let the rational with path P be x y .…”

“…As stated in the opening quote by José N. Oliveira, conciseness facilitates reasoning and the identification of patterns. Indeed, Oliveira's work in pointfree calculational techniques and algebraic methods in programming [2,3,4] is an excellent example of how conciseness leads to shorter documents and elegant theories. As Oliveira writes in [3]:…”

Transforming Data by Calculation

“…By pointfree transform [61] ("PF-transform" for short) we essentially mean the conversion of predicate logic formulae into binary relations by removing bound variables and quantifiers -a technique which, initiated by De Morgan in the 1860s [62], eventually led to what is known today as the algebra of programming [11,5]. As suggested in [61], the PF-transform offers to the predicate calculus what the Laplace transform [41] offers to the differential/integral calculus: the possibility of changing the underlying mathematical space in a way which enables agile algebraic calculation.…”

“…As suggested in [61], the PF-transform offers to the predicate calculus what the Laplace transform [41] offers to the differential/integral calculus: the possibility of changing the underlying mathematical space in a way which enables agile algebraic calculation.…”

“…Theories "refactored" via the PF-transform become more general, more structured and simpler [59][60][61]. Elegant expressions replace lengthy formulae and easy-to-follow calculations replace pointwise proofs with lots of "· · ·" notation, case analyses and natural language explanations for "obvious" steps.…”