In the last post, I illustrated how the 0-1 integer linear programming problem can be reduced to solving a multivariate system of linear and quadratic polynomials over the complex numbers, demonstrating that the latter problem is NP-hard. Let’s see what other problems we can prove are NP-hard by reduction from 0-1 integer programming.

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Suppose we are given a system of polynomial equations in \(n\) variables, where each polynomial has degree at most \(d\), and the coefficients are integers given as binary. We want to determine whether this system of equations has any solutions over the complex numbers. In other words, we want to determine whether the algebraic set specified by the system of polynomial equations is nonempty.

What computational complexity class class does this problem fall into? Can we do it in polynomial time if we take the degree \(d\) to be a fixed constant?

As it turns out, this problem is NP-hard, even for \(d = 2\). The proof is straightforward and requires no algebraic geometry or other “hard” math—you just have to know the trick.

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JDK-8178701: Compile error with switch statement on protected enum defined in parent inner class

I discovered this bug when I wrote some code that compiled in Eclipse, committed it, and then got an email a few minutes later from our Jenkins continuous integration server saying that the build failed. From the error message, I managed to track it down to a specific section of code that compiled in Eclipse but gave a compile error in javac.

This isn’t the first time I’ve ran into a Java compiler or standard library bug while developing CertSAFE, nor is it the first time that I’ve submitted a bug report via the Oracle web form. However, it is the first time that I’ve had a report accepted and published as a verified OpenJDK bug.

I’m always happy when I find a compiler bug, because it makes me feel better about bugs in my code to know that the platform developers screw up too.

Here is a description of an abstract data type: a confluently persistent set data type containing a subset of the integers in a range \([1..n]\) and supporting \(O(\log(n)^c)\) time creation, \(O(\log(n)^c)\) time size query, \(O(n \log(n)^c)\) time conversion to a duplicate-free list, and \(O(\log(n)^c)\) time merging, for some constant \(c\). Here, “creation” takes as arguments the range of the element domain \([1..n]\) and an optional single element in this range, and “merging” takes two sets with the same domain and returns a new set representing the union of the two input sets. In Haskell, the type signatures for these operations might look like this:

data MergeableSet = ...
type Elem = Int

empty :: (Elem, Elem) -> MergeableSet
singleton :: (Elem, Elem) -> Elem -> MergeableSet
size :: MergeableSet -> Int
toList :: MergeableSet -> [Elem]
union :: MergeableSet -> MergeableSet -> MergeableSet

Seems fairly reasonable, right? I’m going to show that it is likely that no such data structure exists.

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rulesgen is a simple Haskell program I wrote that allows you to generate randomized text, similar to SCIgen. You can do all sorts of entertainingly nasty things with it…

Source code and documentation for rulesgen are available on GitHub.

I just noticed you can do this:

{-# LANGUAGE GADTs, RankNTypes #-}

module Data.Foldable.Mono((*$*)) where

import Data.MonoTraversable(Element, MonoFoldable(..))
    -- ^ from the mono-traversable package

(*$*) :: MonoFoldable mono =>
    (forall t. Foldable t => t (Element mono) -> a) -> mono -> a
f *$* o = f (Foldabilized o)

data Foldabilized a where
    Foldabilized :: MonoFoldable mono =>
        mono -> Foldabilized (Element mono)
instance Foldable Foldabilized where
    foldr f z (Foldabilized o) = ofoldr f z o
    -- (Similar implementations for the other methods can be included
    -- here for efficiency.)

And then use it like this:

import Data.Foldable.Mono
import qualified Data.Text.Lazy as T

testText = T.pack "foo quux bar"
example1 = maximum *$* testText
    -- ^ equals 'x'
example2 = mapM_ print *$* testText
    -- ^ prints "'f'\n'o'\n'o'\n..."

Notice that those are the polymorphic Foldable functions maximum and mapM_, not Text-specific functions. I don’t know if this has any real-world applications, but it’s kind of neat…

Update: As pointed out by lfairy on Reddit, the FMList type works kind of like this.

The capsule (also called a “pill”) is one of a few common types of shape used for collision detection in games and other applications. A 3-D capsule is a cylinder capped by hemispheres. The 2-D equivalent, a rectangle capped by semicircles, is technically called a stadium, but they are commonly just called capsules as well.

Capsules are nice because they can form both spherical and elongated shapes in any direction. The animation above shows how Super Smash Bros. Melee uses spherical hitboxes that are “stretched” across frames into capsules to prevent fast-moving attacks from going through opponents without hitting them. (Marvel vs. Capcom 3 uses the same trick.) What really makes capsules useful is that they have a very simple mathematical description: a capsule is the set of all points less than a certain radius from a line segment. This means you can check whether two capsules intersect each other by just finding the shortest distance between the two line segments and checking whether it is less than the sum of the radii.

Calculating the distance between two line segments is a well-known problem. This StackOverflow answer gives the code to do that with floating-point arithmetic. Sometimes, though, approximating the correct answer with floating-point isn’t good enough. What if we want an exact intersection test for capsules using only integer arithmetic?

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Screen-space subsurface scattering (also known by the amusing acronym SSSSS) refers to a class of techniques for approximating subsurface scattering effects in 3D graphics as a fast postprocessing pass, rather than attempting to accurately model light passing through a translucent material. The original article on screen-space subsurface scattering used a series of blur passes modulated by the depth buffer to fake the detail-blurring effects of subsurface scattering in real time. The diffuse and specular components of reflection in the scene are rendered out to separate textures; only the diffuse layer is blurred, and then the sharp specular shading is composited in afterwards. This is designed to model the multi-layered behavior of many materials, like human skin in the original paper, where light rays either reflect off the surface immediately on contact or penetrate into the material and bounce around before exiting.

• One slider controls that simulated scattering radius by adjusting the distance between samples for the blur operation. If you turn this parameter up very high, you can get wave-like artifacts near sharp transitions in depth due to the way the depth buffer is factored into the blur operation. Increasing the number of Gaussian blur samples would reduce this effect at the cost of performance.
• The other slider controls how sharp a depth difference has to be before the shader will stop blurring across that area. If you turn this up to a large value, disconnected areas of the mesh will start blurring into each other, but if you turn it down too low the scattering effect will disappear completely.

In my last post, I gave an implementation for a program that searches through every proof in a particular proof notation for propositional logic, and collects all the theorems that it finds. The proof formalism used was Hindley–Milner, with types being the formulas of the theory and well-typed lambda terms the valid proofs (per the Curry–Howard correspondence).

The program’s output looked like this:

\a. DoubleNegElim (\b. a) : forall a. False -> a

On the right side of the colon, we have our type/theorem: for every proposition \(a\), \(\bot \rightarrow a\). The notation there is about as good as you’re going to get in ASCII text. But then we have the proof, on the left side of the colon. It’s an inscrutable lambda term encoding a natural deduction proof. The type inference engine is filling in the steps in the proof based only on which deductive rules were used, with the result that, even if you know how to read the notation, you really have to stare at it for a while to figure out why the proof works. While this admittedly lends an aura of elegance and mystery to the proceedings, I think I’d prefer a proof that is comprehensible by a human without significant head-scratching.

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Suppose we want to show that there is a number \(n\) such that no upper bound on the busy beaver number \(\Sigma(n)\) is provable from, say, the axioms of ZFC. One possible proof is as follows: Assume that no such \(n\) exists. That is, assume that for every \(n\), there is a proof in first-order logic that uses only the axioms of ZFC and derives a sentence \(\Sigma(n) \leq m\) for some \(m\). Then we could compute an upper bound for \(\Sigma(n)\) by exhaustively searching all of the possible valid ZFC proofs until we found one that deduced a theorem of the form \(\Sigma(n) \leq m\); by assumption, we must eventually find such a proof. But we know that we cannot compute an upper bound on \(\Sigma(n)\) for arbitrary \(n\); thus, we have derived a contradiction, and the original assumption must be false.

Enumerating proofs until you find one with the desired properties is one of those grand CS traditions that, like calling an \(O(n^{12})\) algorithm “efficient” while dismissing an \(O(1.001^n)\) algorithm as “intractable”, is handy for theoretical purposes but has essentially no bearing on the real world whatsoever. After all, no one would really write a program that loops through all possible proofs and tests each one.

Right?

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