Conway and Sloane’s excellent tome, Sphere Packings, Lattices and Groups, is now due a fourth edition. Maryna Viakovska’s proofs of optimality of the E8 and Leech lattices should go in the fourth edition as well…

(I thank Avi Wigderson for telling me about it.) Serge Vlăduţ just arxived a paper with examples of lattices in $latex R^n$ such that the kissing number is exponential in $latex n$. The existence of such a lattice is a very old open problem and the best record was an example where the kissing number is $latex n^{log n}$. (This is based on a 1959 construction of Barnes and Wall and can be found in the famous book by Conway and Sloane.) Noga Alon found in 1997 collections of spheres so that each one touches subexponentially many others. Vlăduţ’s result is very exciting!

Over the last couple of months, I have been experimenting with the use of SAT solvers. This research recently came to fruition, with Tomas Rokicki and me announcing a new discovery with the intention of writing up the methodology in a paper.

Academic papers, however, typically feature a polished proof which omits all of the insights and attempts and failures which led to its discovery. This post attempts to remedy that by presenting a chronological itemisation of the journey; the technical details will be largely reserved for the paper.

30th January: graph theory supervisions

The previous term I had been supervising a lecture course on Graph Theory for the university. One of the questions on the final example sheet was the following:

11. Let G be a graph in which every edge is in a unique triangle and every non-edge is a diagonal of a unique 4-cycle. Show that |G| ∈ {3, 9, 99, 243, 6273, 494019}.

The usual method of solving this problem is to show that the graph must be regular, and therefore strongly regular; the resulting finite set of possibilities drops out from considering the eigenvalues of the adjacency matrix and observing that their multiplicities must be integers.

It is interesting to ask whether there actually exist suitable graphs with these numbers of vertices. For 3 vertices, a triangle obviously works; for 9 vertices, the toroidal 3-by-3 grid graph is a solution. There is also a known example with 243 vertices, related to the remarkable ternary Golay code. For 99, 6273, or 494019 vertices, however, it is unknown whether there is such a graph.

I mentioned in the supervision that John Conway had offered $1000 for a resolution of whether the 99-vertex graph exists, still unclaimed. One of my supervisees asked whether it could be amenable to computer search, and I responded that it might be within reach of a SAT solver: an algorithm for determining whether a set of Boolean equations has a solution.

The 99-graph problem is of a similar size to the Boolean Pythagorean Triples Problem, recently defeated by Marijn Heule and an army of SAT solvers. The UNSAT certificate (proof generated by the machine) is 200 terabytes in length, making it the largest proof of any mathematical problem in history. (The 13-gigabyte Erdös discrepancy proof, also found by SAT solvers, pales in comparison!)

At this point, I thought I would have a go at attacking the 99-graph problem myself. I’d done roughly as much exploration of the necessary structure of the graph as could be accomplished within a single side of A4 paper, and felt it was in a form amenable to SAT solving. I read several academic papers about SAT solvers, and closely monitored the competition results, to reach the conclusion that the state-of-the-art technique for approaching large SAT problems proceeds along lines similar to the following:

Initially simplify the problem using lingeling -S;

Split the problem space using march-cc into thousands of separate ‘cubes’;

Share the cubes amongst parallel instances of the incremental SAT solver iglucose.

I started to develop a framework, which I called MetaSAT, for automating this process. Coincidentally, at the same time, someone else also became interested in SAT solvers…

26th January: Oscar Cunningham’s LLS

In his Odyssean epic The Art of Computer Programming, Donald Knuth dedicates a large section to SAT solvers with a particular emphasis on applying them to Conway’s Game of Life. This left the realms of theory and entered practice when Oscar Cunningham, who studied mathematics in Cambridge at roughly the same time as I did (and taught me how to play frisbee), wrote a Python script to search for patterns using Knuth’s ideas.

Logic Life Search, or LLS, is a sophisticated Python search program capable of using one of many different SAT solvers as backends to find patterns in various cellular automata.

This drove me to attempt to integrate these ideas into the MetaSAT project, so that I could search larger spaces than is possible with a single SAT solver. What I ended up doing was creating a simplified Golly-compatible version of LLS, called GRILLS. Whilst LLS is strictly more powerful, GRILLS has a more user-friendly graphical interface, and allows for loading results from previous runs:

I departed slightly from Donald Knuth’s binary-tree approach of counting neighbours, instead opting for a ‘split-radix’ binary-ternary tree. This reduced the number of variables needed by the SAT solver by 40%, without affecting the number of clauses, thereby making the problem slightly easier to digest (at least in principle).

The pinnacle of its success was being able to find an already-known c/4 orthogonal spaceship. Whilst showing that SAT solvers could replicate existing discoveries, there was no evidence that they could find anything new — in other words, they seemed to be trailing behind discoveries found with special-purpose CA search programs.

31st January: Tomas Rokicki’s c/3 spaceship

This changed at the end of the month. Tomas Rokicki had also implemented a Perl script to search for spaceships in cellular automata, and came across a new discovery. It is tiny, completely asymmetric, fits in a 13-by-13 box, and remains to be named anything other than its systematic name of xq3_co410jcsg448gz07c4klmljkcdzw71066.

He then turned to attempting to find knightships: elusive ships which move parallel to a knight in chess, rather than orthogonally or diagonally. Prior to 2010, oblique spaceships had never been built, and the only examples since then were huge slow constructions. The smallest two, Chris Cain’s parallel HBK and Brett Berger’s waterbear, are approximately 30000 cells in length. The waterbear is the only one which moves at a respectable speed, and its creator produced a video of it:

Finding a small knightship has been considered one of the holy grails of cellular automata. The closest attempt was Eugene Langvagen’s 2004 near-miss, which has a slight defect causing its tail to burn up:

In six generations, it almost translates itself by (2, 1), but differs in two cells from its original state. Very soon, this error propagates throughout the entire pattern and destroys it. In the 14 years following that, no-one has been able to find a perfect knightship.

Tom’s search, however, did yield increasingly long and increasingly promising partials. He found four further 2-cells-away near-misses, but no 1-cell-away or perfect solution, and statistical analysis (modelling the distribution of errors as a Poisson distribution) suggested that we would need another 2 years of searching to find a result.

9th February: a phoenix from the ashes of pessimism

On 3rd February, Oscar Cunningham cast doubts on the efficacy of his search program for finding a knightship, remarking the following:

I’m under the impression that programs called -find are just much better at finding ships than programs called -search. Is that true? If so then there’s not much point looking for a knightship with LLS.

He was referring to David Eppstein’s gfind search program, which conducts a depth-first row-by-row brute-force backtracking search. This, and its descendants, have been responsible for the vast majority of new spaceships discovered in cellular automata. David Eppstein wrote a paper detailing the methodology, which I suggest reading before proceeding with this post. (If you’re impatient, then take a cursory glance now and read the paper properly later.)

This gave me an idea: what if somehow Tom Rokicki’s partial-extending approach could be hybridised with the row-by-row tree-searching approach in gfind? I sent an e-mail to Tom to this effect:

Good news: I’ve realised that the methodologies of knightsearch and gfind
are entirely compatible. In particular:

(a) Represent the search pattern as a two-dimensional lattice of unknown
cells by interleaving rows of different generations. (The way to do this for
a (2, 1)c/6 knightship is to take Z^3 and quotient by (2, 1, 6), giving a
two-dimensional lattice of unknown cells. Unlike in gfind, the ‘rows’ are
exactly perpendicular to the direction of travel, even for knightships.)

Specifically, we map cell (x, y, t) to lattice coordinates (2x-y, 3y+t).

(b) The ‘state’ of the search is a 6-tuple of consecutive rows of length W.
The ‘search grid’ is a width-W height-(K+7) grid of lattice points, where
the first 6 rows are populated by the state, the next row is the ‘target’,
and the following K rows are for lookahead purposes. There are a few columns
of forced-dead cells on either side of the search grid. Pass this problem to
an incremental SAT solver to find all possibilities for the 7th row which
are consistent with the remaining K rows (I think gfind uses K = 1 or 2; SAT
solvers can clearly go much deeper). Each time the incremental SAT solver
gives a solution, provide the clause given by the negation of the 7th row
so that the next run gives a new possible 7th row. Repeat until UNSAT. The
‘child states’ are the 6-tuples obtained by taking the last 5 elements of
the parent 6-tuple and appending one of the possible 7th rows.

(c) Run a breadth-first search starting with a small partial, using (b) each
time a new 6-tuple appears. We can canonise 6-tuples by doing a horizontal
shift to centre the non-zero columns into the middle of the search grid.
This means that the knightship needs only be *locally* thin, and is thereby
permitted to drift in various directions (similar to in knightsearch). The
narrowness of [the bitwise disjunction of the members of] a 6-tuple may
be a good heuristic to decide which nodes to attempt to extend first.

(d) If you hit a dead-end, you can always rerun the already-found tuples
using higher values of W. Hence, the width need not be known in advance
when commencing the search. (If W increases particularly high, then it
may be necessary to decrease K to avoid overloading the SAT solver.)

Within the next few days, I implemented the aforementioned search program. It used the same backend (iglucose with named pipes for communication) as Tom’s search, but was otherwise very different.

19th February: adaptive widening

Over the coming weeks, the search program grew in complexity, gaining various features and heuristics. In programs such as gfind, the search-width is set as a constant, and it either finds a solution or exhausts the search space to no avail. I implemented adaptive widening, where the latter causes the width to increment and the completed search to act as a backbone for the new (higher width) search.

Another approach, borrowed from cryptanalysis, is meet-in-the-middle: separately grow search trees of fronts and backs, and compare them to see whether they can join to form a complete solution.

25th February: save-and-restore

One of the later changes I made, which logically should have been the first feature, is to save progress to disk and restore it. That way, I no longer needed to start the search afresh whenever I added a new optimisation to the program. I spent the next week running a width-35 search, saving the tree to disk every hour.

6th March: a last attempt in desperation

I resumed the search at width-0 to allow adaptive widening to take over, as smaller search widths lead to faster SAT solving. I also decided to adjoin one of Tom Rokicki’s partial results to the search tree to make things more interesting: this way, it would be able to piggy-back from his independent research. I watched the search width gradually increase as each completed without success, and as the night was drawing to a close it looked as though width-24 would finish as the queue gradually emptied:

Increasing head search width to 24...
280700 head edges traversed (qsize ~= 47966).
280800 head edges traversed (qsize ~= 47616).
280900 head edges traversed (qsize ~= 47369).
281000 head edges traversed (qsize ~= 47200).
281100 head edges traversed (qsize ~= 46993).
281200 head edges traversed (qsize ~= 46516).
281300 head edges traversed (qsize ~= 46180).
281400 head edges traversed (qsize ~= 45471).
281500 head edges traversed (qsize ~= 44632).
281600 head edges traversed (qsize ~= 44364).
281700 head edges traversed (qsize ~= 44036).
281800 head edges traversed (qsize ~= 43808).
281900 head edges traversed (qsize ~= 43231).
282000 head edges traversed (qsize ~= 43021).
282100 head edges traversed (qsize ~= 42759).
282200 head edges traversed (qsize ~= 42272).
282300 head edges traversed (qsize ~= 41795).
282400 head edges traversed (qsize ~= 41256).
282500 head edges traversed (qsize ~= 40706).
282600 head edges traversed (qsize ~= 40428).
282700 head edges traversed (qsize ~= 39523).
282800 head edges traversed (qsize ~= 38931).
Saving backup backup_head_odd.pkl.gz
Saved backup backup_head_odd.pkl.gz
Saving backup backup_tail_even.pkl.gz
Saved backup backup_tail_even.pkl.gz
Backup process took 53.0 seconds.
282900 head edges traversed (qsize ~= 38000).
283000 head edges traversed (qsize ~= 37268).
283100 head edges traversed (qsize ~= 37023).
283200 head edges traversed (qsize ~= 36598).
283300 head edges traversed (qsize ~= 35869).
Increasing tail search width to 11...
...adaptive widening completed.
283400 head edges traversed (qsize ~= 35024).
283500 head edges traversed (qsize ~= 33950).
283600 head edges traversed (qsize ~= 32596).
283700 head edges traversed (qsize ~= 31262).
283800 head edges traversed (qsize ~= 29728).
527600 tail edges traversed (qsize ~= 149949).
283900 head edges traversed (qsize ~= 28707).
284000 head edges traversed (qsize ~= 26488).
284100 head edges traversed (qsize ~= 24744).
Saving backup backup_head_even.pkl.gz
Saved backup backup_head_even.pkl.gz
Saving backup backup_tail_odd.pkl.gz
Saved backup backup_tail_odd.pkl.gz
Backup process took 61.7 seconds.
284200 head edges traversed (qsize ~= 21792).
284300 head edges traversed (qsize ~= 19753).
284400 head edges traversed (qsize ~= 17345).
284500 head edges traversed (qsize ~= 14687).
284600 head edges traversed (qsize ~= 14041).
527700 tail edges traversed (qsize ~= 128776).
284700 head edges traversed (qsize ~= 12097).
284800 head edges traversed (qsize ~= 9503).
284900 head edges traversed (qsize ~= 8984).
285000 head edges traversed (qsize ~= 8388).
285100 head edges traversed (qsize ~= 6719).
285200 head edges traversed (qsize ~= 6670).
285300 head edges traversed (qsize ~= 6606).
285400 head edges traversed (qsize ~= 6548).
285500 head edges traversed (qsize ~= 6409).
285600 head edges traversed (qsize ~= 6463).
285700 head edges traversed (qsize ~= 6405).
285800 head edges traversed (qsize ~= 5271).
285900 head edges traversed (qsize ~= 5047).
286000 head edges traversed (qsize ~= 5030).
286100 head edges traversed (qsize ~= 5027).
286200 head edges traversed (qsize ~= 5089).
286300 head edges traversed (qsize ~= 5066).
286400 head edges traversed (qsize ~= 5056).
286500 head edges traversed (qsize ~= 5142).
286600 head edges traversed (qsize ~= 5241).
286700 head edges traversed (qsize ~= 5326).
286800 head edges traversed (qsize ~= 5344).
286900 head edges traversed (qsize ~= 5394).
287000 head edges traversed (qsize ~= 5298).
287100 head edges traversed (qsize ~= 4950).
287200 head edges traversed (qsize ~= 4618).
287300 head edges traversed (qsize ~= 4601).
287400 head edges traversed (qsize ~= 4646).

But then, something remarkable happened. It spewed out a series of long partials, which had extended from about two-thirds of the way along Tom Rokicki’s input partial and overtaken it! I then saw something which caught my eye:

Near the bottom of the terminal screen (highlighted in red above), I saw that the partial solution had a slanted fissure! It looked as though one partial had completed and another had begun immediately below and to the right of it! It seemed too good to be true, but I tried running it through ikpx2golly to convert it into a runnable Life pattern. Sure enough, the back partial burned away to leave, emerging unscathed from the ash, a perfectly self-contained knightship:

I checked the stdout logs of the search program, and it transpires that it had also found the completion; it just so happened that it was hidden under the barrage of following pseudo-partials that I didn’t notice it until several minutes later.

7th March: Hello, world!

The first thing to calculate was the attribution: since it grew from Tom Rokicki’s partial, there was the issue of determining who ‘discovered’ it. Fortunately, the row-by-row approach used by the search program made this exactly calculable: Tom found the front 62% of the pattern, and I found the rear 38%.

Nomenclature was the next thing to consider. Again, the knightship has a long systematic name, but something more catchy was in order. The name ‘seahorse’ was suggested by Dave Greene as a clever pun; the pattern resembles a seahorse, and ‘horse’ is an informal term for a knight in a game of chess. A pseudonymous user named Apple Bottom suggested the alternative name of Sir Robin, after the character in Monty Python and the Holy Grail. As this discovery is a knightship, considered a ‘holy grail’ in CA searching, and was found by a program written in Python, that name stuck.

Even though I wanted to hold back the publication until this article was published, it was beyond my control at this point. The discovery soon found its way onto Hacker News where it accumulated 727 up-votes and a horde of comments.

8th March: The discovery gets digested

After 48 hours, the surprise gradually settled and the cellular automata community became accustomed to this new discovery and using it as a component of larger constructions. In particular, Martin Grant found a configuration capable of eating Sir Robins, and there has been talk of trying to create a knightpuffer (arrangement of Sir Robins which leave behind sporadic debris).

Soon afterwards, on the 10th March, Dave Greene published an article on Sir Robin.

What’s next?

It is not too difficult to prove that a pattern in Conway’s Life can translate itself by (a, b) in p generations only if |a| + |b| <= p/2. For all periods p up to 7, we now have such examples, but there is currently no known (2,1)c/7 knightship. It may be possible to find such a pattern using these search methods, but both Tom Rokicki and I are busy with a backlog of other things. I’m preparing a paper as part of my PhD research to submit to NIPS before the deadline in May, which means I won’t have time in the next 10 weeks to commit to searching for spaceships.

Consequently, you are just as likely as anyone else to find a period-7 knightship — 10 weeks is nearly twice as long as the total chronology of this post, and it’s unclear how many more CPU-hours and ideas would be sufficient to yield this gem.

As for SAT solvers more generally, I am investigating whether they can be applied to my own area of research (topological data analysis, which might just be combinatorial enough to be susceptible to these ideas). If not, there’s always that 99-vertex graph problem…

I’d like to open with congratulations to the talented students who made the leaderboard in the second round of the British Mathematical Olympiad. Some highlights worth mentioning:

Three of the four team members for the European Girls’ Mathematical Olympiad made the top 10 in the leaderboard, which I think is record-breakingly high. This bodes very well for the competition taking place in Florence this April. Good luck to Emily, Alevtina, Naomi and Melissa!

Joseph Myers has made an eighth release of his open-source software for organising mathematical olympiads, as used in the EGMO.

Congratulations to Agnijo Banerjee, Nathan Creighton and Harvey Yau on their double-perfect-scores in the BMO, and best of luck in the forthcoming Romanian Master of Mathematics competition.

Weird Maths, by Agnijo Banerjee and David Darling

I thought I should mention that the aforementioned Agnijo Banerjee, who is amongst the readership of Complex Projective 4-Space, recently coauthored a book about some of the more bizarre and pathological aspects of mathematics. I’ve been meaning to mention this for a while, but waited until it was launched at the beginning of this month. If you’re interested in this excellent book, which you should be, then download the trailer (PDF) or visit the website.

The other author, David Darling, is a prolific science writer who was apparently born in the same county as I was.

You are probably familiar with the Riemann hypothesis. This concerns the behaviour of the Riemann zeta function, which is defined on the complex plane by analytic continuation of the following series:

The behaviour of the zeroes of the function outside the strip is well-known: they are precisely at the even negative integers. The Riemann hypothesis states that within this strip, the only zeroes have real part exactly equal to ½. It is customary to include, at this point, an entirely gratuitous plot of the real part of the zeta function near the origin:

Whilst interesting enough for the Clay Mathematics Institute to offer a prize of USD for its resolution, the generalised Riemann hypothesis has far more far-reaching implications. This states that for every Dirichlet L-function, all of its zeroes within the critical strip have real part exactly equal to ½. Since the Riemann zeta function is the simplest example of a Dirichlet L-function, the ordinary Riemann hypothesis is subsumed by the generalised Riemann hypothesis.

What is a Dirichlet L-function?

A Dirichlet L-function is specified by a positive integer k and a completely multiplicative function satisfying:

χ(n) = 0 if and only if gcd(n, k) ≠ 1;

Periodicity: χ(n + k) = χ(n) for all integers n;

Complete multiplicativity: χ(m) χ(n) = χ(mn) for all integers m, n.

Given such a Dirichlet character χ (there are finitely many choices, namely φ(k), for any given k), the corresponding L-function is defined as:

When k = 1, the unique Dirichlet character is the constant function 1, whence we recover the original Riemann hypothesis.

Knottedness is in NP, modulo GRH

This is the title of a very interesting paper by Greg Kuperberg. It has been proved that an unknotted knot diagram can be disentangled into a simple closed loop in a number of Reidemeister moves polynomial in the number of crossings in the original diagram. It follows that unknottedness is in NP, as an optimal disentangling sequence is a polynomial-time-checkable certificate of unknottedness.

Kuperberg’s paper proves that unknottedness is in co-NP, i.e. knottedness is in NP. However, the proof relies on the unproven generalised Riemann hypothesis in a very interesting way. Specifically, if the fundamental group of the complement of the knot is non-Abelian — as is the case for any non-trivial knot — then it has a non-Abelian representation over SU(2). If the generalised Riemann hypothesis holds, it is possible to find a reasonably small prime p for which there is such a modulo-p representation; in this case, the images of the generators under this representation (expressed as 2-by-2 matrices with entries modulo p) form a certificate for knottedness. It then suffices to check that they do not all commute, and that they satisfy the relations of the knot group presentation.

ab + bc + ca

Which positive integers cannot be expressed in the form ab + bc + ca, where a, b, c are positive integers? There are either 18 or 19 such numbers, the first 18 of which are enumerated in A025052:

There may be a 19th such number, which must necessarily be greater than 10^11, and can only exist provided the generalised Riemann hypothesis is false. In particular, Jonathan Borwein proved that the existence of this 19th number requires a Dirichlet L-function to have a Siegel zero, which would contradict GRH.

The positive integers inexpressible in this form with a, b, c distinct positive integers are called idoneal numbers, and there are either 65 or 66 of them. The 65 known examples are sequence A000926:

Idoneal numbers have an equivalent definition of being the positive integers D such that any number uniquely expressible in the form , with and coprime, is necessarily either a prime, a prime power, or twice a prime power.

Deterministic Miller-Rabin

The fastest known deterministic algorithm for determining if a number is prime is AKS, which takes time . This is, unfortunately, rather slow.

The Miller-Rabin probabilistic test tells you whether a number is prime in time . If the number is prime, it correctly claims that it is prime. If it is composite, however, it can lie with probability 1/4. In practice, running this test with many randomly-chosen initial parameters will give a result with sufficiently high probability that it is useful in practice (e.g. in cryptography).

Assuming GRH, it can be shown that applications of Miller-Rabin are sufficient to guarantee primality, and therefore that there exists a deterministic algorithm with running time , noticeably faster than AKS. The Shanks-Tonelli algorithm for finding square-roots modulo a prime will also run deterministically in polynomial time as a result of the same idea.

Artin’s conjecture on primitive roots

This states that every integer a which is neither −1 nor a perfect square is a primitive root modulo infinitely many primes. In 1967, Hooley proved this conditional on the Generalised Riemann Hypothesis, establishing that the set of primes for which a is a primitive root has positive density (among the set of all primes).

Something amazing has happened. A couple of years ago, we closely followed the progress of AlphaGo, the distributed DeepMind algorithm which defeated Lee Sedol in four out of five games of the ancient board game Go. This has since been defeated by a variant, AlphaGo Zero, which is considerably stronger — after 20 days of training, it was able to win 100 out of 100 games against the version of AlphaGo which played against Sedol. After a further 20 days of training, it won 89 out of 100 games against a stronger instantiation of AlphaGo, namely the one which defeated the world champion Ke.

However, its superiority over the previous algorithm isn’t the most interesting aspect. What makes it interesting is that, unlike AlphaGo which both trained on human games and made use of hardcoded features (such as ‘liberties’), AlphaGo Zero is remarkably simple:

The algorithm has no external inputs, learning only from games against itself;

The input to the neural network is just 17 inputs, namely the parity of the turn, the indicator functions of white stones for the last 8 positions, and the indicator functions of black stones for the last 8 positions. (Storing the immediate history is a necessity due to ko rules.)

Instead of separate policy and value networks, the algorithm uses only one neural network;

Monte Carlo rollouts are ditched in favour of a feedback loop where the tree search evolves together with the network.

Read the Nature paper for more details. AlphaGo Zero was trained on just four tensor processing units (TPUs), which are fast hardware implementations of fixed-point limited-precision linear algebra. This is much more efficient (but less numerically precise) than a GPU, which is in turn much more efficient (but less flexible) than a CPU.

In the last post, we discussed Voevodsky’s homotopy type theory. One of the important notions is whether a space is contractible, this being the base case for the inductive definition of homotopy levels. It turns out that the algorithmic undecidability of whether a finite simplicial complex is contractible appears not to have been discussed much, the result instead buried in Appendix A of an arXiv paper. The strongest version of the result is:

It is undecidable to determine whether a finite 4-dimensional simplicial complex is contractible or not.

An analogous result about fundamental groups, also worth mentioning, can be proved immediately from the undecidability of the word problem in groups:

It is undecidable to determine whether a finite 2-dimensional simplicial complex is simply-connected or not.

(Proof: we can consider the presentation complex of any finitely-presented group G, where we take a single point, together with a loop for every generator, and a disc for every relator. This has fundamental group isomorphic to G. After two successive applications of barycentric subdivision, we obtain an abstract simplicial complex homotopy-equivalent to the original presentation complex.)

Clearly, the ‘2’ in the second theorem cannot be lowered; a 1-dimensional simplicial complex is a graph, and it is easy to verify whether or not it is a tree (equivalent to both contractibility and simply-connectedness). I believe it is unknown as to whether the ‘4’ in the first theorem can be lowered to 3 or even 2.

2017 has been an unfortunate year for Fields medallists. Maryam Mirzakhani, who won the Fields medal in 2014, passed away at the untimely age of 40. Two days ago, she was joined by Vladimir Voevodsky, 2002 Fields medallist, who was awarded the prize for introducing a homotopy theory on schemes, uniting algebraic topology and algebraic geometry.

He also defined the ‘univalence axiom’ which underpins the recent area of homotopy type theory. This is an especially elegant approach to the foundations of mathematics, with several particularly desirable properties.

Problems with set theory

The most common foundation of mathematics is first-order ZFC set theory, where the objects are sets in the von Neumann universe. It is extremely powerful and expressive: all naturally-occurring mathematics seems to be readily formalised in ZFC. It does, however, involve somewhat arbitrary and convoluted definitions of relatively simple concepts such as the reals. Let us, for instance, consider how the real numbers are encoded in set theory:

A real is encoded as a Dedekind cut (down-set) of rationals, which are themselves encoded as equivalence classes (under scalar multiplication) of ordered pairs of an integer and a non-zero natural number, where an integer is encoded as an equivalence class (under scalar addition) of ordered pairs of natural numbers, where a natural number is a hereditarily finite transitive well-ordered set, and an ordered pair (a, b) is encoded as the pair {{a, b}, {a}}.

Whilst this can be abstracted away, it does expose some awkward concepts: there is nothing to stop you from asking whether a particular real contains some other set, even though we like to be able to think of reals as atomic objects with no substructure.

Another issue is that when we use ZFC in practice, we also have to operate within first-order predicate logic. So, whilst ZFC is considered a one-sorted theory (by contrast with NBG set theory which is two-sorted, having both sets and classes), our language is two-sorted: we have both Booleans and sets, which is why we need both predicate-symbols and function-symbols.

Type theory

In type theory, we have a many-sorted world where every object must have a type. When we informally write , we are thinking type-theoretically, that ‘the proposition φ(x) holds for for all reals x’. In set theory, this is actually shorthand for , namely ‘for every object x, if x is a real then the proposition φ(x) holds’.

This idea of types should be familiar from most programming languages. In C, for instance, ‘double’ and ‘unsigned int’ are types. It is syntactically impossible to declare a variable such as x without stating its type; the same applies in type theory. Functional programming languages such as Haskell have more sophisticated type-theoretic concepts, such as dependent types, allowing functions to be ‘first-class citizens’ of the theory: given types A and B, there is a type (A → B) of functions mapping objects of type A to objects of type B.

These parallels between type theory and computation are not superficial; they are part of a profound equivalence called the Curry-Howard isomorphism.

Types can themselves be objects of other types (in Python, this is familiar from the idea of a metaclass), and indeed every type belongs to some other type. Similar to the construction of the von Neumann hierarchy of sets, we have a sequence of nested universes, such that every type belongs to some universe. This principle, together with a handful of rules for building new types from old (dependent product types, dependent sum types, equality types, inductive types, etc.), is the basis for intuitionistic type theory. Voevodsky extended this with a further axiom, the univalence axiom, to obtain the richer homotopy type theory, which is sufficiently expressive to be used as a foundation for all mathematics.

Homotopy type theory

In type theory, it is convenient to think of types as sets and objects as elements. Homotopy type theory endows this with more structure, viewing types as spaces and objects as points within the space. Type equivalence is exactly the same notion as homotopy equivalence. More interestingly, given two terms a and b belonging to some type T, the equality type (a = b) is defined as the space of paths between a and b. From this, there is an inductive hierarchy of homotopy levels:

A type is homotopy level 0 if it is contractible.

A type is homotopy level n+1 if, for every pair of points a and b, the type (a = b) is homotopy level n.

They are nested, so anything belonging to one homotopy level automatically belongs to all greater homotopy levels (proof: the path spaces between two points in a contractible space are contractible, so level 1 contains level 0; the rest follows inductively). The first few homotopy levels have intuitive descriptions:

Homotopy level 0 consists of contractible spaces, which are all equivalent types;

Homotopy level 1 consists of empty and contractible spaces, called mere propositions, which can be thought of as ‘false’ and ‘true’ respectively;

Homotopy level 2 consists of disjoint unions of contractible spaces, which can be viewed as sets (the elements of which are identified with the connected components);

Homotopy level 3 consists of types with trivial homotopy in dimensions 2 and higher, which are groupoids;

Homotopy level 4 consists of types with trivial homotopy in dimensions 3 and higher, which are 2-groupoids, and so on.

More generally, n-groupoids correspond to types of homotopy level n+2. The ‘sets’ (types in homotopy level 2) are more akin to category-theoretic sets than set-theoretic sets; there is no notion of sets containing other sets. I find it particularly beautiful how the familiar concepts of Booleans, sets and groupoids just ‘fall out’ in the form of strata in this inductive definition.

Voevodsky’s univalence property states that for two types A and B, the type of equivalences between them is equivalent to their equality type: (A ≈ B) ≈ (A = B). Note that the equality type (A = B) is the space of paths between A and B in the ambient universe U, so this is actually a statement about the universe U. A universe with this property is described as univalent, and the univalence axiom states that all universes are univalent.

Further reading

I definitely recommend reading the HoTT book — it is a well-written exposition of a beautiful theory.

[P. S. My partner and I are attending a debate this evening with Randall Munroe of xkcd fame. Watch this (cp4)space…]