Chapter 6 — The Algebra of Transformations

Part II · Composition, Sequencing, Hysteresis · Week 5

K→𝒢→Q→Ξ

Order matters. Composing programs shrinks domains and, in general, does not commute: the same two operations in different orders can land on different states — or one order can be infeasible where the other completes. This is the algebra behind sequencing, deadlock, and the value of bundling.

Learning objectives

  • LOS 6.1–6.2 — Characterize when two transformations commute; construct failures in both domain and value.
  • LOS 6.3 — Define hysteresis for a transformation–reversal pair and decompose it exactly.
  • LOS 6.5 — State the Sequencing Theorem and identify feasible orders versus stranded ones.

The laboratory module

Module 6 — The Program Composer. Operations as draggable tiles: a composer (execution domain shrinking), a commutator checker, a hysteresis meter, and a bundle builder.

The worked failure (Example 6.4), on a two-attribute state \((k_1, k_2)\) = (liquidity, receivables), with \(L\) = list (gate \(k_1 \ge 6\), effect \(k_1 \mapsto k_1 - 4\)) and \(S\) = securitize (gate \(k_2 \ge 5\), cash \(+3\) private / \(+5\) listed):

  • Value asymmetry at \((7,8)\): \(L\)-then-\(S \to (8,3)\); \(S\)-then-\(L \to (6,3)\). The rated-issuer listing premium is earned in one order, forfeited in the other.
  • Domain asymmetry at \((5,8)\): \(L\) is gated out (\(5 < 6\)), so \(L\)-first fails; \(S\)-first opens \(L\)’s gate and completes at \((4,3)\).

One pair of ordinary operations, both failures, all arithmetic.

Guided experiments

  1. Find all feasible orders of a four-operation plan and the two that strand.
  2. Reproduce both failures of Example 6.4.
  3. Produce a negative structural residue and explain it in one sentence.
  4. Deadlock the financing cycle and close it with the bundle.

📓 Notebook · 📊 Workbook · 🎞 Slides (.pptx) · Open the Laboratory →

ch06 · 8/8 PASS

Exercises

B · Computations

  • 6.4 Reproduce both commutation failures of Example 6.4 and identify which clause of Proposition 6.5 each breaks. Hint: the value failure breaks the germ-independence clause; the domain failure breaks the gate-preservation clause.

Statements and hints surfaced here; full solutions in the Instructor’s Manual.