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Writing Constraints & Objectives

Introduction

We've already used constraints and objectives in our previous tutorials, and now we will begin writing some of our own! Being able to write your own constraints and objectives is an important step in becoming an advanced Penrose developer.

You are already equipped to create beautiful diagrams with what you have already learned from the previous tutorials. This tutorial will cover how you can extend the existing Penrose system, contributing to the platform for many other users.

We will start by discussing how are constraints and objectives are done in Penrose, and then we will go through several examples to apply our conceptual understanding.

Diagramming From A Technical Perspective

Making a diagram can be encoded as an optimization problem. Broadly speaking, optimization is the search for the best solution to a problem subject to certain rules. For example, finding the best way to arrange your day with all the tasks that you need to finish during a certain timeframe is an optimization. There are many different optimization techniques, and Penrose utilizes numerical optimization.

Numerical optimization uses functions that are called energy functions to quantify how good our current solution is. An energy function outputs a numerical value (the energy), hence numerical optimization. A key thing to remember here is we want to minimize the energy, the lower the better. Under the hood, all constraint functions are implemented as energy functions. In Penrose, a "good" diagram is one that satisfied all the objectives and constraints written in the program. Therefore, a diagram can be quantified with the energy of all its constraints and objective functions.

The energy of a diagram can take on a range of values, and there are 3 specific values we care about:

  • Global minimum: A diagram with a global minimum energy means it is a really good diagram, and it cannot be improved by making local changes.
  • Local minima: A diagram with a local minima energy is "pretty good".
  • Maxima: A diagram with a maxima energy is a bad diagram. Remember, we want to minimize energy, so hitting a maxima through the optimizing process is not good.

Often in the process of diagramming, there is not just one good diagram, but many solutions — that is, there are many local minima of the energy function. Given a Style program, which defines an energy function for your family of diagrams, Penrose looks for a local minimum of the energy function by using numerical optimization.

Lastly, we write energy functions in a particular way using autodiff helper functions, where autodiff stands for auto differentiation. This is because Penrose takes the energy function's gradient \nabla, i.e. take the derivatives of the function, to find better and better solutions. For more on optimization, here's a wonderful introduction video.

In short, we write energy functions with a specific set of operations for Penrose to optimize, allowing it to find the best diagram for us.

Conceptual: How To Come Up With Constraints?

Let's say we want to create a diagram that represents the mathematical idea of contaiment. If we wanted a circle A to be contained in another circle B, this is probably what comes to mind:

So we have a mental picture of what containment means to us, but there is no way for us to transfer this mental picture directly to the Penrose system and say, "Okay this is what we want when we write isContained(A, B)". Therefore we need to pause for a second, and really think about what it means for a circle to be contained in another circle mathematically.

There are generally 3 scenarios for the containment relationship between 2 circles.

We have completely contained, overlapping but not contained, and completely disjoint. Disjoint means none of the points in circle A is also in circle B, i.e. they do not overlap at all.

The three scenarios are visually obvious to us. We are shown 2 circles, and we can immediately identify their relationship. While Penrose does not have eyes, it speaks math! So, let's try looking at these circles a different way.

Recall the general equation for a circle where pp is some point, cc is the center and rr is the radius: pc=r||p-c||=r. This equation is in vector form since Penrose has built-in vector support so we prefer working with them when possible.

The center coordinate and radius are the information we have about any circle, and we will use this information to determine two circle's containment relationship.

Another piece of information we will be using is the distance dd between the radii. Notice how the distance gets progressively bigger as AA and BB become more disjoint. When A,BA, B are disjoint, we see that dd's value is the greatest.

Let's see a scenario when a circle is perfectly contained in another one.

Perfect Containment Example

Circle 1 contains circle 2 if and only if circle 1's radius is greater than the distance between their centers, plus circle 2's radius, i.e. r1>d+r2r_1 > d+r_2. This diagram shows the most illustrative case when Circle 2 is just contained, which we can understand by intuitively reasoning about the directions of change for each degree of freedom:

  • If r2r_2 is any smaller, Circle 2 remains contained. If r2r_2 is any larger, clearly Circle 2 is no longer contained.
  • If dd is any smaller, then Circle 2 remains contained; if dd is any larger, then Circle 2 is no longer contained.
  • If r1r_1 is any larger, then Circle 2 remains contained; if r1r_1 is any smaller, then Circle 2 is no longer contained.

So, by rearranging the containment expression, we arrive at the energy expression d(r1r2)<0d - (r1 - r2) < 0.

Here's a short proof. Read on if you are still a bit hesitant. Let rdifference=rBrAr_{difference}=r_B-r_A.

  • We know rdifference<0r_{difference}<0 when rA>rBr_A > r_B, i.e. radius of the circle A (that we want to be contained) is greater than the radius of circle B. In that case, A cannot be contained by B. Then we have drdifference>dd-r_{difference}>d.
  • We have rdifference=0r_{difference} = 0 when rA=rBr_A = r_B, i.e. the radii of the two circles are equal, and they can be contained in each other if and only if distance d=rB=rAd=r_B=r_A , then drdifference=dd-r_{difference}=d.
  • We have rdifference>0r_{difference}>0 when rA<rBr_A < r_B, i.e. AA is a smaller circle than BB. In that case AA is perfectly containable, and drdifference<dd-r_{difference} < d.
  • As shown above, we can conclude that as circle AA becomes more contained within circle BB, the value of drdifferenced-r_{difference} decreases accordingly.

Concrete: How We Write Constraints

The syntax for writing a constraint works in a way that allows Penrose to use a particular technique called automatic differentiation, autodiff for short. The Penrose system uses autodiff to find the optimized diagram. For more on autodiff, read here.

1. The Autodiff Code is Built in Typescript

Unlike the previous tutorials where we were working with file triples in the custom Penrose language, this part of Penrose is fundamentally built in Typescript. As you will see, there are still some key rules and tricks to follow when writing constraints for autodiff.

2. Autodiff Functions

In order to perform automatic differentiation, Penrose needs to keep track of the operations that are performed on all numbers in an internal graph-based format. As a result, we have to make use of pre-defined Penrose operators and shift away from native Typescript operations such as + and -.

We have unary, binary, trinary, n-ary, and composite operations. The terms unary, binary, etc. refer to the number of objects that are passed into the operation. For example, instead of a + b, we now write add(a, b), which is a binary operator because it takes two inputs, a and b.

The composite operations are accessed through ops.<function-name>. For example, to access the dist operator to find the distance between two points expressed in vector form (say A and B), we'd write ops.dist(A, B).

The full list of autodiff functions can be found here.

3. Special Number Types

All numbers are required to be a special type called ad.Num in order to be valid inputs for autodiff functions. We can convert between normal numbers and ad.Num using the functions:

  • varOf: number -> ad.Num or constOf: number -> ad.Num to convert from a number to a ad.Num. These two functions are interchangeable.
  • numOf: ad.Num -> number to convert from a ad.Num to a number.

The type number represents our common, constant numerical values like 1, 2, 3, .... For example, if we need to do 5 + 3 , the equivalent autodiff expression is add(constOf(5), constOf(3)) or add(varOf(5), varOf(5)).

4. Write Functional Code

We must write these functions in straight-line functional style (i.e. no imperative style, no mutating state, no for-loops or if statements). We do this so that the Penrose system can work its magic. We need to avoid writing things like x = x + 1 which translates to let x = add(x, constOf(1)) in autodiff code, and instead use constant intermediate variables like this:

const x0 = constOf(1);
const x1 = add(x0, constOf(1));

5. Zero-Based Inequality to Energy Function

For every constraint function we write, we take in shapes and output either a number or a tensor as a penalty. In short, Penrose will try to minimize the outputs of all the constraint functions used for the diagram.

When we write a constraint, for example, we want to constrain one circle s1 to be smaller than another circle s2 by some amount offset. In math, we would require that r1 - r2 > offset. An inequality constraint needs to be written in the form p(x) > 0 since we penalize the amount the constraint is greater than 0. So, this constraint is written as r1 - r2 - offset > 0, or p(r1, r2) = r1 - r2 - offset.

Some general rules on writing energy function:

Let's say I want the constraint f(x) <= c to be true.

  1. Translate it to the zero-based inequality f(x) - c <= 0.
  2. Translate the inequality constraint into an energy (aka penalty) E(x) = f(x) - c, it is greater than 0 if and only if the constraint is violated. The more the constraint is violated, the higher the energy (i.e. if f(x) is much larger than c then the energy E(x) is also larger than 0).

6. Negative Outputs of Energy Functions

Previously, we've talked about how we convert everything to zero-based inequality, so what happens when the energy function outputs a negative value? It simply means that the constraint is satisfied.

6. Accessing a Value of a Shape's Field

One common operation is to access the parameter of a shape via shapeName.propertyName.contents, which will return a ad.Num. For example, if you have a circle c as input, and you want its radius, c.r.contents will give you something like 5.0 (of type ad.Num).

Constraints Example: minSize & maxSize

We will go through simplified examples of minSize and maxSize constraints that are specifically for circles only.

minSize: ([shapeType, props]: [string, any]) => {
const limit = 20;
return sub(constOf(limit), props.r.contents);
};

If you have never used Typescript before, we are defining an arrow function. It takes two parameters, a shapeType, which is of type string, and props, which has type any. Generally speaking, function declarations will follow this pattern:

functionName: (param1: param1Type, param2: param2Type): returnType => {
/* function body */
};

You can read more about function declarations in Typescript here.

Going back to our minSize function, we see several things in play:

  • Input: The function takes in the shape, which is represented by a string of its name, "Circle" in this case, and a prop which is an Object containing the properties of the circle. All Shape objects that are passed into objective functions or constraints must have the type [string, any]. For example, this is Penrose's definition of a Circle object.
  • Numbers: Instead of directly using constant numbers like 20, we have to return constOf(limit) in order to pass it as a valid input for the autodiff function.
  • Operations: Instead of using the subtraction operator - like we normally do, we have to use the autodiff function sub .
  • Accessing the Shape Property: We access the shape's property value by shapeName.propertyName.contents , where propertyName = r for radius in this case.
  • Logic: We want the input circle to have a minimum size (at least r = 20) as the function name suggests, so we want to express our returned answer in terms of energy, where energy > 0 is bad (the constraint is unsatisfied), and energy <= 0 is good (the constraint is satisfied). For example, with a small circle of r = 1, we will return 19 (not good), whereas with a big circle of r = 30, we will return -10, a negative number that satisfies the constraint.
maxSize: ([shapeType, props]: [string, any], limit: ad.Num) => {
return sub(props.r.contents, div(limit, constOf(2)));
};

The function maxSize is very similar to minSize with the addition of another input parameter limit that is used to limit the circle's diameter. If you're curious about the full implementations of minSize and maxSize in Penrose, you can find the code here.

Objectives Example: Circle Repel

Unlike constraints, which are binary in that they are either satisfied (<=0) or unsatisfied (>0), objective functions should output the "badness" of the inputs (as a number or Tensor), and have local minima where we want the solution to be.

Now we look at an objective that makes two circles repel, encouraging the two circles to stay far away from each other.

repel: ([t1, s1]: [string, any], [t2, s2]: [string, any]) => {
const repelWeight = 10e6;
let res = inverse(ops.vdistsq(s1.center.contents, s2.center.contents));
return mul(res, constOf(repelWeight));
};

Let's look at this code together step by step:

  • Input: The function takes inputs similar to the constraint functions we just looked at, where for convenience, we substituted in t for shapeType. We add a parameter, weight, which is present because repel() typically needs to have a weight multiplier since its magnitude is small.
  • Operations: Here we use 3 autodiff functions:
    • inverse, which returns 1 / v
    • mul performs multiplication
    • ops.vdistsq returns the squared Euclidean distance between vectors v and w. Remember we useops to access composite functions that work on vectors.
  • Logic: We will convert the math done in the second line of the function to its corresponding mathematical equation.
1CACB2=1d2\frac{1}{||C_A - C_B||^2 } = \frac{1}{d^2}

So essentially, the repel function takes in two circles and returns the inverse of the distance between them squared, i.e. we plug in the distance dd between the circles as an input to the f(x)=1x2f(x)=\frac{1}{x^2} function.

Graph of 1/x^2 from Desmos

If you look at the graph of f(x)=1x2f(x)=\frac{1}{x^2}, notice how the output increases as dd decreases, i.e. the higher penalty value we return. We block the negative horizontal range since we cannot have negative distances.

We then return the value above multiplied by the repelWeight, and let's take a look at the graph of 10e6x2\frac{10e6}{x^2}.

Graph of 10e6/x^2 from Desmos

If you compare the two graphs above, you can see how we expanded the range of extreme high outputs by multiplying 1x2\frac{1}{x^2} with a big constant.

Exercises

  • Write a constraint that makes 2 circles disjoint from each other. Remember disjoint means that the two circles do not overlap at all.
  • Write a new disjoint function that allows padding, i.e. the minimum distance between two circles will be the padding value.

Exercise Solutions

/* d(c1, c2) + r1 + r2 >= 0 */
disjoint: ([t1, s1]: [string, any], [t2, s2]: [string, any]) => {
const res = add(t1.r.contents, t2.r.contents);
return sub(res, ops.vdist(t1.center.contents, t2.center.contents));
};
/* d(c1, c2) + r1 + r2 >= padding */
disjointPadding: (
[t1, s1]: [string, any],
[t2, s2]: [string, any],
padding: number
) => {
const res = add(add(t1.r.contents, t2.r.contents), constOf(padding));
return sub(res, ops.vdist(t1.center.contents, t2.center.contents));
};

More Reading: link.

Takeaways

In this tutorial, we took a leap from being users of the Penrose system to being developers contributing to the Penrose system. In particular, we learned the following things:

  • Diagramming can be encoded as an optimization problem, and Penrose uses numerical operations.
  • Constraints and objectives are implemented as energy functions, and the outputs of an energy function is called energy.
  • The lower the energy, the better! A diagram with low energy for all of its constraints and objectives is a good diagram.
  • To write an energy function, we use autodiff functions with special number types, and we write everything in functional style.