Privacy Law

Other Laws of the Program

More operators, more laws

$\#(d)+1 = \#(d\boxplus x)$

$\#(d)+1=\#(d\boxminus x)$

Log Transactions Laws

Free Algebra $F\langle X\rangle$

• $$\begin{aligned} F\langle X\rangle & = \emptyset\\ & \mid (d:F\langle X\rangle)\boxplus (x:X)\\ & \mid (d:F\langle X\rangle)\boxminus (x:X)\\ \end{aligned}$$ E.g. $(\emptyset \boxplus x)\boxminus y):F\langle x,y\rangle$
• Laws $R$:  equations you want to be true, e.g. $$d=(d\boxplus x)\boxminus x$$.
• Test: Quotient $$F\langle X\rangle/\langle R\rangle$$ is nontrivial.
• Church-Rosser, Todd-Coxeter, Schreier-Sims, Tait-Martin-Lof, ....

Consistency testing

by algebra

• Free algebra $F\langle X\rangle$: the algebra of all formulas using variables $X$ and the given operations.  E.g. polynomials, words in groups, $(d\boxplus x)\boxminus x)$, &c.
• Laws $R$:  equations you want to be true, e.g. $d=(d\boxplus x)\boxminus x$.
• Test: Show $$F\langle X\rangle/\langle R\rangle$$ is nontrivial.
• Church-Rosser, Todd-Coxeter, Schreier-Sims, Tait-Martin-Lof, ....

Algebraist

Programmer

Not solved...

Algebra rewriting is "solved" on homogeneous algebra, e.g.

$$*:A\times A\to A$$

Most applications are heterogeneous, e.g.

$$*:A\times B\to C$$

Operads overshot the generality end up with topology questions not algebra.

Equals is different things to different people.

• Bank manager: databases are equal if have the same clients.
• IT manager: databases are equal if one is a backup of the other.

Equality you begin with

E.g. IT/Software company

A solution

IT keeps two notions of equal.

• $d=f$ for interchangeable databases.
• $d\equiv f$ for equal client data.

Manager axioms are true under $\equiv$ only.

Implement $\equiv$ by a function $$\pi:DBSnaps\to DB\qquad \pi(d,actions)=d$$ that makes technical IT data private to management system.

Equals is different things to the same people.

• Bank manager: databases are equal if have the same clients.
• IT manager: databases are equal if one is a backup of the other.
• Bank manager: storing multiple copies of the same thing is not as cheap as storing fewer!

Equality you begin with

E.g. IT/Software company

How to refine equality

• Admit you have a problem. $$S=T\Longleftrightarrow S\subset T \vee T\subset S$$ is not a open to interpretation.  Set theory is not capable of performing interesting equality.
• Appeal to a higher power...
  • Higher Inductive Types
  • Higher Categories (next talk)

Leibniz

$$x=y\Longleftrightarrow (\forall P)(P(x)\Leftrightarrow P(y))$$

Coarser equality

looks at fixed $P_i$

Ex. Manger's database

$$\begin{aligned} DB[X] & = 0:DB \\ & \mid (d:DB)\boxplus (x:X)\\ & \mid (d:DB)\boxminus (x:X)\\ & / \text{refl}(d): (d=_{DB}d)\\ & / \text{priv}(x): (d=_{DB} (d\boxplus x)\boxminus x)\\ & / duplicate(d):(d=_{BD} copy(d))\end{aligned}$$

Ex. IT's database

$$\begin{aligned} DB[X] & = 0:DB \\ & \mid (d:DB)\boxplus (x:X)\\ & \mid (d:DB)\boxminus (x:X)\\ & / \text{refl}(d): (d=_{DB}d)\\ & / duplicate(d):(d=_{BD} copy(d))\end{aligned}$$

Ex. Full database

$$\begin{aligned} DB[X] & = 0:DB \\ & \mid (d:DB)\boxplus (x:X)\\ & \mid (d:DB)\boxminus (x:X)\\ & / \text{refl}(d): (d=_{DB}d)\end{aligned}$$

Testing =

So equals is not absolute but variable.

New problem: 

• Given level $i$
• Want level $j$

Basic equality refinement

Less basic equality test

Less basic equality test

Non-basic equality test

Breakit up

When frontal assualt fails, go the other direction!

1. Relax equality until it is easy.

2. Do the successive refinement with groupoid isomorphism.