## More problems that seem easy but are actually NP-hard

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.

## How hard is determining whether an algebraic set is nonempty?

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.

## A plausible-sounding data structure that probably doesn’t exist

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:

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

## Propositional logic proofs are awful

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.
Continue reading Propositional logic proofs are awful

## The shortest proof for every theorem

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.