I guess I’m alone in this, since Matt Parker’s ‘guess the mean of the digits in all the entries’ competition received the most entries of any competition meaning he also won a prize.
Bored with ‘guess the mean of all the entries’ style competitions, I decided to come up with something that flips this concept on its head.
My competition was to come up with a function for mapping a set of $N$ integers onto a single value, and whoever’s function takes a pre-determined set of integers I’m thinking of closest to a particular predetermined value I’m thinking of is the winner. I was slightly sloppy in the way I defined the question (I used $N$ to refer to both the set of numbers, and the size of the set) but everyone responded beautifully – with only some people feeling the need to point out my error – in what became a virtual festival of making up your own notation.
I received 32 entries to the competition, not all of which were actually valid entries – several had the bright idea of using ‘the number you’re thinking of’ as a variable in their function, which is 100% cheating. Here are some of my favourites, in increasing order of amazingness, culminating with the winning entry (scroll down to the bottom if you just want to see that).
The set of numbers I was thinking of was in fact The Numbers, from the TV series Lost – in which a seemingly random set of integers play a mysterious yet crucial role in the plot. The numbers are 4, 8, 15, 16, 23 and 42. As well as being one character’s winning lottery numbers, and popping up in various other places, for a whole chunk of season 2 the numbers must be typed into a computer every 108 minutes, or else… something bad will happen.
I actually quite enjoyed it but I have notoriously bad taste in TV and films.
If you haven’t seen Lost, and you have a spare 122 hours of your life you’re happy to not get back, go for it.
My set of numbers is therefore as follows: 4, 8, 15, 16, 23 and 42 (in that order). The target number for the function was chosen to be zero, for reasons I can’t explain (mostly panic).
On to the entries!
Several examples fell into this category – including:
$f(x_0, x_1, \ldots, x_N) = N$
Simply counting the number of numbers (minus one). While we appreciate your lovely and obfuscating zero-indexing, this gives the answer $5$.
$f(x,y) = x^2 – y$
Since this function is only defined on two inputs, we took the first two, giving $ 4^2 – 8 = 8$.
$f(x_1, x_2, \ldots, x_N) = x_N^2 – 1$
Taking the last of the numbers and squaring it, then subtracting one. This gives $42^2 – 1 = 1763$.
Various statistical functions, including the square of the mean, the upper quartile, the lower quartile and the ‘anti-mode‘ (the least common value, and if there’s more than one value equally uncommon, the mean of these – since the set is all distinct, this is just the mean LOL!).
These gave the values $240.25, 19.5, 11.5$ and $18$.
“The first prime not a factor of any of the numbers”
This is $11$.
“Return a random element of the set”
Since the target value isn’t in the set, this won’t work but it’s a nice try!
\[f: x_1 \ldots x_N \rightarrow \frac{e^{\min(x_i)} \cdot \pi^{\max(x_i)}}{N}\]
Almost like Euler’s identity in how it effortlessly combines mathematical constants, but sadly returns a result in the region of $7 \times 10^{21}$.
Many entries fell into this category:
$f(x_1, x_1, \ldots, x_N) = \frac{\displaystyle\sum_{i=1}^N (-1)^i i x_N}{N}$
If this was meant to be an alternating sum of multiples of successive entries, which is how I initially interpreted it, it also doesn’t sum to zero. And you should be more careful with notation, because this is exactly what was written.
An alternating sum of multiples of the largest entry, which given $i$ runs from $1$ to $6$ means you end up with a total of three times that entry, divided by 6 – giving $x_N/2 = 42/2 = 21$.
$f(n_1, n_1, \ldots, n_N) = \frac{\displaystyle\prod_{i=1}^N n_i}{N}$
The product of all the entries divided by the number of entries, which is $1,236,480$ (very much not zero).
The $\big(\displaystyle\prod_{i=1}^N x_i\big)^{th}$ prime number, modulo $\displaystyle\sum_{i=1}^N x_i$
This one returns 67.
$\frac{1}{M} = \frac{1}{x_1} +\frac{1}{x_2} + \cdots +\frac{1}{x_N}$
For reference, $M$ was the name I gave to my target value, and this was described as the ‘parallel resistance sum’. It works out to give $M = 1.7499207463\ldots$.
$f(x) =\displaystyle\sum_{k=1}^n \sin (kx_k)$
I like it, but it gives $-1.168\ldots$ – close, but no cigar.
Several entries took a chance on me, and hoped they could guess what number I was thinking of, defining their function as the constant function giving a single value regardless of the inputs. Entries of this form, in increasing order of size, included $\pi, 4, 7, 20$ and $37$, sadly none of which win.
Ask a stupid question, and you’ll get the following answers. We tried to calculate some of these in the ten minutes we had to judge the competition, but for some of them we merely established they were definitely not zero and gave up.
$f(x_1, x_2, \ldots, x_N) = x_1 + \frac{1}{x_2+ \frac{1}{\cdots + \frac{1}{x_{N-1} + \frac{1}{x_N}}}}$
A continued fraction. We calculated the answer but all that’s written on here is “4.something”. Not equal to zero.
“Arrange the numbers in order and multiply the difference between them in pairs.”
I enjoy a good bit of tedious busy work, and I got $3724$.
$f(n_1, n_2, \ldots, n_x) = (x+2) \cdot \sqrt{n_1 \cdot n_x \cdot\displaystyle\prod_{\textrm{all } x} n_x}$
Interesting notation switcheroo from what everyone else seems to be using. This was a pain to calculate too and my bit of paper says ‘TOO BIG’.
“The median of all the answers to this competition + 1”
Given that we had 32 different entries, some of which we couldn’t properly calculate, but there was literally no chance the median answer was $-1$, the only thing written on the paper is the word ‘NO’.
“Sum the number of letters of the positive integers, minus the number of letters of the negative integers”
It’s lovely! Our bit of paper says “about 40ish?” but if anyone wants to work it out, go for it.
“For each element of N, go to the Wikipedia page for N(i) and find the N(i)th letter on the page; typecast this letter into I32 and take the sum of these values”
A bit of Googling has allowed me to establish that this is a way of converting letters into numbers, but since I assume none of the standard letters convert to a negative number, this will definitely not equal zero.
$f(x_1, x_2, \ldots, x_N) = e^{e^{\displaystyle\sum_{i=1}^N x_i}}$
We had a go at calculating this in Google Sheets, which was our terrible calculating tool of choice under pressure, and the value was actually too big for it to handle. In some ways, this could be the least correct answer.
Given that I named the number I was thinking of $M$, I received a small selection of entries from people who didn’t want to play like normal people, at least one of which just specified $f(N)=M$ – this is OBVIOUSLY not what you were meant to do. Others tried to define a function in terms of $N$ and $M$, including the following:
\[f(x_1, x_2, \ldots, x_N) = \begin{cases}
\Bigg(\frac{\displaystyle\sum_{i=1}^N \displaystyle\sum_{j=1}^{x_i} x_i^j}{N!}\Bigg)^{\displaystyle\frac{1}{x_{\max{3,N}}}}&& \textrm{if }N\textrm{ is coprime with }M\\
6 && \textrm{otherwise}
\end{cases}\]
For the record, while typing this in LaTeX I’ve hit a record of four consecutive close braces. Sadly $6$, the number of entries in my list, is not coprime with $0$, so this gives the answer $6$ (and is disqualified anyway for using $M$ in the definition).
$f(N,M) = 0. \big(\displaystyle\sum_{i=1}^{N-2} x_i\big) + M$
This was presumably hoping to get as small a number as possible in the first part, then add it to M and hope nobody was closer. Turns out not only is this cheating, it was also not close enough, because we did actually get an exact winner, and it was glorious. See below.
$f(n_1, n_2, n_3, \ldots, n_N) = n_2 + n_3 + \ldots + n_N \mod n_1$
This function takes all but the first number, adds them together and then takes the result modulo the first number, aka remainder on division. In the case of The Numbers, the result is:
$f(4, 8, 15, 16, 23, 42) = (8 + 15 + 16 + 23 + 42) \mod 4 \equiv 0$
Amazingly, through pure chance, the sum of these five numbers (104) is a multiple of 4, and so the answer is zero! We have a winner. Congratulations to the winner, who forgot to write their name on the sheet, but was identified and awarded their prize – a wind-up robot, as per the rules of the Competition Competition, worth strictly less than £1.
]]>
What’s The Calculus Story all about?
It’s an introduction to calculus for an unusually wide readership, mainly through the story of how the subject developed. And this turns out to be something of an adventure, largely because of the way infinity comes in at almost every twist and turn.
How is the book different to previous ones you’ve written?
My previous book, 1089 and All That, was a fairly light-hearted look at maths in general, and became something of a bestseller, with 11 foreign translations. The new book takes the subject a bit further, and is, if anything, even more ambitious, because it tries to explain not only what calculus is, but how to actually start doing it.
Calculus is all about the rate at which things change, and this is how we often get to understand, through the laws of physics, how the world really works. But the subject contains many results, too, that can be enjoyed purely for their own sake, usually because they are surprising in some way.
What do you hope the book will achieve?
It is always easier to embark on calculus if you have some idea from the very beginning of the subject as a whole – some ‘big picture’, if you like. Without that, you can easily get bogged down in comparatively minor detail, lose direction, and – even worse – lose interest.
My main hope, then, is that The Calculus Story provides that big picture, and in an unusually readable way.
And if anyone were to read the book as preparation for a university or college course, they would – in my view – really hit the ground running.
Why did you choose to write the book now?
I didn’t. The book has been gently brewing, in a way, since 1962, when I first met calculus, at the age of 16. For once you’ve met it, your mathematical life is never quite the same again – calculus just opens so many new doors.
What’s the most interesting fact you learned while writing the book?
If you drill a hole of given depth straight through the centre of a sphere, the amount of material left over is independent of the radius of the sphere!
Have you ever used calculus in ‘everyday life’?
No. Calculus underpins much of modern life, but in a rather hidden way, through the laws of physics, chemistry, biology and economics. It tends to be ‘under the bonnet’, so to speak.
The most likely way that calculus might be used explicitly in a truly ‘everyday’ context is to solve some optimisation problem.
Who do you think the book would best suit as a Christmas present?
You’d better ask the New Scientist. They have just selected it as one of their choices for Christmas, claiming that (a) it will fit in a stocking and (b) it has ‘something for everyone’!
The Calculus Story, on Oxford University Press
The Calculus Story, on Amazon
The new live DVD from science comedy trio Festival of the Spoken Nerd, Just for Graphs, is out now, and we’ve been sent a copy to review. We got together a pile of appropriately nerdy science fans to watch (left), and here’s what we thought.
The latest Festival of the Spoken Nerd DVD/download is from the Nerds’ 2015 show Just for Graphs – it toured the UK in late 2015, and had a hugely successful Edinburgh Fringe run in August 2015. The show is themed around graphs, plots, charts and diagrams – as mathematicians, we were sad to see they’re not entirely using graphs made from nodes and arcs (although a few of those do make it in the show!)
As well as plenty of classic diagrams (Venn, Euler and otherwise) there’s also plots – Steve Mould plots the birth of his child, while Matt Parker plots a function that plots itself – and Helen Arney brings musical interludes and live demos, including an electrifying demonstration of how the graph of electrical voltage in a speaker cable can be transmitted through not just wires but people, and at one point a large section of the audience.
The show also contains some nostalgia for retro technology (which raised some cheers from us), and an interesting new way of plotting a graph of the pressure inside a tube of gas – by setting the gas on fire, of course.
While watching it on a screen doesn’t quite have the same ambience as seeing it live in a theatre, you still get a sense of how the live audience experiences it and the show is full of visual spectacles, which do come across well on screen. I was part of the production team for some of the tour shows and Fringe run, and it was just like being there for real (only smaller and more pixelated).
The show is full of science references and deliciously geeky jokes, and without spoiling too many of the punchlines/conclusions, if you’re into maths or science and want to be entertained, it’s a graph a minute.
Just for Graphs is available on DVD, as a digital download and for some reason on VHS.
]]>The closing talk of the HLF’s main lecture programme (before the young researchers and laureates head off to participate in scientific interaction with SAP representatives to discuss maths and computer science in industry) was given by Fields Medalist Steve Smale.
Speaking without slides, Smale shared with us some of his recent work in the crossover between mathematics and biology, but the central theme of his talk was one which goes to the heart of what mathematics is. Mathematics is embedded in science, and is used to describe and understand many aspects of scientific discovery.
The question was of whether mathematics is realistic or idealistic – do the mathematical models we use to solve problems and understand the universe give a realistic picture of how the world works, or is it all fantasy and we’re ignoring the fine details in order to get a model that works nicely? It’s a constant struggle, and Smale illustrated this with several historical examples.
Alan Turing’s theory of computation was an inspiring vision of how we can understand and use computers, and has influenced the whole field since. But was it realistic? For some applications, Turing’s approach is the correct model, but for others it fails. Modern study of NP-completeness in computability assumes an infinite amount of input, but obviously this doesn’t model the real world – it’s an idealisation.
Moving on to another giant of maths history, Smale turned to Isaac Newton. Newton’s work on physics, differential equations and mechanics was all an idealisation – to the extent that his calculations didn’t even include friction, which was added to the theory 100 years later.
John Von Neumann, who created the early models of quantum mechanics, introduced the concept of a Hilbert space – again, an idealisation. And even in other fields – Watson and Crick discovered the structure of DNA, but didn’t include the protein core of chromatin, later discovered to be a fundamental part of the more detailed structure.
This idealism is possible because this major work was often done without Newton, Turing or Von Neumann doing lab experiments – they used experimental data from other people’s work, but data which had already been ‘digested’ by the rest of the scientific community. Newton built on the work of Galileo, Copernicus and Kepler before him.
Smale’s current research is on the human body, and in particular the heart. How does it manage to beat in such a coordinated way, with all the myocytes, or muscle cells, acting together to create a regular heartbeat?
The synchronisation was compared to what happens in a crowd applauding for a long time, when the clapping falls into synchronisation almost accidentally. It’s a dynamical system, and can be understood through maths, much like the beating heart.
Smale is working towards a deeper understanding of the heart through mathematics, and in this case standing on the shoulders of Alan Turing. Turing’s final paper on morphogenesis – the biological processes that lead to stripes on a zebra, or the arrangements of seeds in plants – included some differential equations. These were again an idealisation of the reality, as they were based on what happens as the numbers of cells increases to infinity (obviously not the real situation – some biologists disliked Turing’s work for this reason, as they mainly worked with small quantities of cells).
The cells making up the heart are of the same cell type – they all behave in the same way. Smale is looking at how this happens, and outlined how if you consider the set of genes as a graph, with individual genes as the nodes, and directed edges indicating how each transcription factor controls the adjacent gene, elevating protein production, you can build a system of ordinary differential equations to describe how it works.
This system of equations can be seen to reach an equilibrium – a global basin of attraction where the levels of each gene work in exactly the right way to determine how the cell behaves. These basins define the cell types – so a liver cell, or a heart cell, knows exactly what quantities of each protein to produce based on these equations, which are hard-wired into the DNA. You can even consider stable periodic behaviour in the system, to understand the heartbeat’s regular cycle.
Of course, this is again all idealisation – but the workings of the heart are something that have been understood from the biological angle for some time, and now mathematics is providing new ways to model and understand it – which will hopefully lead to powerful medical advances we can implement in reality.
]]>Friends of the Aperiodical, nerd-comedy troupe Festival of the Spoken Nerd, are currently on tour around the UK. As part of their show, questionably titled You Can’t Polish a Nerd, Matt Parker attempts to calculate the value of $\pi$ using only a length of string and some meat encased in pastry. He’s previously done this on YouTube, and the idea was inspired by the Aperiodical’s 2015 Pi Approximation Challenge, and in particular my own attempt to approximate $\pi$ with a (more conventional) pendulum.
For our $\pi$ approximate-off, we wanted to derive values for $\pi$ using a suite of methods that mostly didn’t involve measuring the circumference and diameter of a circle. This included evaluating as many terms of some infinite series as the length of our room booking permitted, and a recreation of the famous Buffon’s Needle experiment. But surely the most satisfying method is to just swing a heavy ball on a bit of string.
As many people learn in high school, the formula governing how long a pendulum takes to do one complete swing is \[ T = 2\pi\sqrt{\frac l g} \] where $l$ is the length of the pendulum and $g$ is the acceleration due to gravity (about $9.8ms^{-2}$). Slightly counterintuitively, the swing time doesn’t depend on the weight of the bob (or if you prefer, the mass of the weight), nor on how far back you swing it from. So just from timing a swing and measuring the string, you can solve the equation to get a numerical value for $\pi$.
But all that calculation sounds pretty boring. We can choose the length of the string, so why not choose it so that it cancels the $l$ and the 2, to give $T=\pi$? Conveniently that requires a length of $g/4 \approx 2.45$ metres (if you’ll excuse my abuse of dimensionality), which is just about doable in a tallish room or a stairwell or a theatre stage. When we tried this timing ten swings to reduce the impact of imperfect human reaction time, we ended up with a value of 3.133. (I speculate that the overwhelming source of inaccuracy is the difficulty in measuring the length of the string from the precise point of pivot at the top to the precise centre of mass of the bob/weight/pie.) On his video, Matt, using a fancy slow-mo camera, got a value of 3.128 seconds per swing. The Aperiodical has yet to take its annual pilgrimage to the Festival, so it remains to be seen how successfully the feat is accomplished live.
You might think that using a pendulum in an attempt to find $\pi$ without using circles is cheating a bit: surely a pendulum’s swing time depends on $\pi$ because it moves along the arc of a circle? Well, no. In fact the opposite is true. I lied a bit when I said the time doesn’t depend on how far back you pull the pendulum. The formula above is only an approximation for small initial angles, when a pendulum approximates simple harmonic motion. The full formula is this:
So in fact the more of a circle you make your pendulum trace out, the worse your approximation will be. Circles kill pi.
If you’d like to see Matt attempt this live on stage, the Spoken Nerd show is still on tour until the end of November, and as well as swinging food, it includes songs, live experiments, comedy and many nerdy references.
Calculating π with a pendulum, by Matt Parker on YouTube
Aperiodical’s π approximation challenge, on YouTube
Festival of the Spoken Nerd tour
]]>
At the start of his HLF lecture on Asymptotic Group Theory on Thursday morning, Fields medalist Efim Zelmanov described the ‘group’ as: “the great unifying concept in mathematics,” remarking “if you go for a trip, and you are allowed to take only two or three mathematical concepts with you, give serious consideration to this one.” Very loosely defined, a group is a set of things (its ‘elements’) that you can ‘multiply’ together, with this multiplication behaving in certain helpful ways. Think of numbers being added, functions composed together or rotations and reflections of a shape being carried out one after the other. I doubt any mathematician would accuse Zelmanov of overstating their importance in mathematics.
In his talk he discussed residually finite groups. These are groups which are infinite in size but still just a little bit finite-y. In technical terms, the group has a set of homomorphisms with finite kernels having trivial intersection. Although the group is too large to see all at once, as Zelmanov put it, we have “photos from all sides of the group”. He contrasted this to “hopelessly infinite groups”, for which no such photo album is possible.
A common way to look at a group is to find a set of ‘generators’: these are elements of the group which you can multiply together to create any element of a group (the elements ‘generate’ the entire group). Some infinite groups can’t be generated from a finite set — consider trying to find a set of rational numbers that you can multiply together to create any rational number. Those that can be generated from a finite set are unexcitingly called ‘finitely generated’. Of course, finite groups are also finitely generated.
Zelmanov considered under what circumstances finitely generated groups can be proved to be finite. One immediate way this won’t happen is if one of the generators is not periodic: if you keep multiplying it by itself you keep getting new elements forever, never ‘looping back’ to the original generator. (Imagine starting with 1 and continually adding 1…) The Burnside problem asks whether there are any other ways to make a finitely-generated, yet infinite, group. In 1991, Zelmanov proved that for residually finite groups, there aren’t. However, this isn’t the case for the ‘hopelessly infinite’ groups.
In his lecture Zelmanov, accompanied by his excellent hand-drawn slides, discussed this before moving on to related topics such as the growth of groups (if you start with a generating set, and create new elements by multiplying them together, how quickly does the set grow?) and ‘approximate groups’ (which, as the name suggests, are things that are like, but not quite, groups).
]]>The text reads:
An orchestra of 120 players takes 40 minutes to play Beethoven’s 9th Symphony. How long would it take for 60 players to play the Symphony? Let P be the number of players and T the time playing.
Well, once you’re done laughing, we’ve done some investigative journalism and found the origin of this question. And it turns out it’s quite nice!
I wrote this!! How did you get this??? I am a maths teacher in Nottingham UK. Wrote this 10 years ago. Here is the original whole worksheet pic.twitter.com/jYX55GSBKz
— Claire Longmoor (@LongmoorClaire) October 11, 2017
The question is from a worksheet developed by maths teacher Claire Longmoor (who is, based on current evidence, brilliant) ten years ago. Claire put together a selection of example questions with relationships in direct and inverse proportion, and deliberately included the orchestra question as an example of something where it doesn’t work that way. It’s a nice activity to help reinforce the difference, and in context the question works nicely.
Other examples on the sheet include a bricklaying example with creditably diverse gender representation, a car with terrifyingly low fuel efficiency, good cow names and a delightful insight into the bygone world of fruit picking.
]]>Paul and I have spent this week blogging from the Heidelberg Laureate Forum, an international event for PhD/postdoc students and top-level maths and computer science researchers.
It was a long week of extravagant dinners, incredible talks and press conferences, (maths) celeb spotting, branded conference freebies, hilarious quotes and exceptional hospitality. Oh, and blogging. Here’s a round-up of what we wrote, in case you’ve missed it this week, as well as some of the other posts the rest of the HLF blog team wrote.
Blog posts by Katie
Blog posts by Paul
A few more posts may appear on the blog over the coming week or so, and we’ll post them across here as well.
Katie and Paul were only part of the blog team at the HLF – they were joined by Math With Bad Drawings’ Ben Orlin, a maths teacher, prolific blogger and author; Constanza (Coni) Rojas-Molina, who draws amazing sketch summaries of talks, blogs at The RAGE of the Blackboard and researches mathematical physics at the University of Bonn; Nana Liu, a quantum computing researcher there to cover the event’s ‘hot topic’ session on Quantum Computing; and Alaina Levine, a freelance writer, speaker and consultant based in Arizona. They were joined by (not pictured) Tobias Maier (biologist) and Markus Possel (physicist), who both blog in English and German.
Here are some of our favourite posts from the rest of the blog team:
You can see the rest of the HLF blog where it’s hosted at Spektrum SciLogs.
]]>The Open University and UK Mathematics Trust have teamed up to launch Perplex, a mobile app containing mathematical puzzles and games. It’s available for iPhone and Android, and can also be played directly on their website.
The description promises 8 main puzzles and over 40 daily challenges to keep you occupied, and with the mobile versions it’s also possible to share your scores and compete against your friends. It looks to be a fairly pretty game, with fun cartoony graphics and (thankfully muteable) music. It seems like exactly the kind of thing my friends and I got hooked on to distract ourselves from work while we were at uni.
The puzzle games cover various different areas of maths – including colouring problems, arithmetic challenges, and variants of classic river-crossing puzzles. Some will be pretty familiar already to fans of Henry Dudeney puzzles or Simon Tatham’s games, but for anyone who hasn’t seen this kind of thing before it should be a nice introduction – and even if you have, they’re still fun to play.
Sadly, the game has limited content, at least for now – the ‘daily challenges’ will presumably be added daily, and further puzzles are promised, but as it stands there’s only one version of each type of puzzle, and your challenge is to complete each in the most efficient way to earn all three stars. Given that it’s a free game, I’d say that’s still good value, but it would surely be easy enough to generate more versions of each?
Now if you’ll excuse me, I still haven’t got three stars on the bubbles one…
Perplex, on the iTunes app store
Perplex, on the Android app store
Perplex, on the Open University website
A wonderful potted history of the theory of communication was capably presented by 2002 Nevanlinna Prize winner Madhu Sudan, who talked us through from the earliest mathematical thinking on the subject through to the present day, and his team’s work. It was also almost a love letter to one of his mathematical heroes, the father of information theory, Claude Shannon.
Most human communication has historically used speech or writing, but the digital age has changed this into the communication of bits and bytes of digital data. Of course, speech and writing can both also be encoded in this way, but communication theory as a subject is more concerned with the pure bandwidth of data for any type of communication.
There are many problems when you attempt to communicate any kind of information from one place to another. Communication is expensive, depending on how much data you are transferring; it can be noisy, as communication lines aren’t always perfect in transmitting the data; it sometimes needs to be interactive, so systems need to be able to communicate both ways; and it’s often contextual, and data transfer needs to be robust to people speaking different languages, or using different hardware/software.
Sudan gave a simple example of an early method (still used today in simple cases!) to encode a message before sending through a noisy system, so as to improve the likelihood of the message arriving intact – this is to simply repeat each character of the message three times. In his example, to prevent the message WE ARE NOT READY being catastrophically mis-received as WE ARE NOW READY (the opposite meaning!), you could send it in this form:
WWW EEE AAA RRR EEE NNN OOO TTT RRR EEE AAA DDD YYY
Now, even if some of your data is corrupted, as below:
WXW EEA ARA SSR EEE BNN OOO PTT RSR EEE AAS DFR IYY
Errors can still make it through – if, for example, as in the second-to-last group, two different errors happen, you can’t tell what the original letter was meant to be. In the fourth group there are two errors both going to the same incorrect letter, which would lead to an incorrect conclusion about that letter if you simply looked at the most common letter in the group.
So how do we improve this system? Of course, we could use groups of 5 or 10 characters – but as Sudan points out, increasing the number of repeats decreases the probability that all the symbols in a set might be corrupted, but it decreases the amount of information you’re actually able to communicate. For 100 characters of data sent, instead of sending a 100-letter message, you can now only send 10 or 20 letters. As the number of repeats increases to infinity, usefulness of your system drops to zero – and equally, as your message gets longer, the number of repeats you can afford drops so your likelihood of errors increases.
This fine balance between methods which are more likely to preserve your message with a higher certainty, and methods which will allow your system to send the largest amount of information, was long believed to be an unwinnable fight. That is, until the hero of Madhu Sudan’s talk came along – Claude Shannon. In the late 1940s Shannon worked out a theory of communication that changed the game completely.
He envisioned ways of encoding messages which would allow a good compromise, and worked out formulae to compare rates of communication for different encoders and decoders. Put simply, if your encoder and decoder are functions E and D which you can apply to a message M, you need:
M = D(E(M) + error) : with a high probability
If you know the probability that any given bit of data will be damaged by the transmission or storage method you’re using, you can find the kind of function you’ll need.
If the probability of a bit being flipped is 0, you can send the messages as they already are with no errors. If it’s as large as ½, you may as well send anything because the message received at the other end will be essentially random, so the rate of data transfer is 0.
Even with a probability of 0.4999, there would still be a non-zero rate of data transfer, and it’s possible to design systems that work under these conditions. Shannon’s work was revolutionary, and considered a major leap of faith, since at the time no computers existed, and even the functions he was imagining weren’t known yet.
He was the first to use a probabilistic approach to this kind of mathematics, and invented many deep concepts including that of entropy, of information and even coined the word ‘bit’ – a binary digit. His work was non-constructive, as the maths it has since been applied to was yet to be invented, but all subsequent technology has kept his ideas in mind.
Sudan also explained some later contributions from Chomsky (on language structure and human communication), Yao (on designing protocols for two-way communication, which won him a Gödel prize), and Schulman (on interactive coding schemes, useful for collaborative shared document editing).
He finished by mentioning some of his group’s research into communicating in situations where there is a large shared body of information, or ‘context’. For example, in giving his talk he’d assumed everyone in the room spoke English, and had given the talk in English on this basis.
However, if the shared context isn’t quite perfect, his system is still robust to this – if there’s a word he uses nobody else in the room knows, he can take a little extra time to define it. This will increase the length of the talk (and decrease the amount he can communicate in a given time), which must be taken into account. The systems he’s working on have this same kind of shared context, but must be robust to imperfectly shared context, and this has been explored in his recent work.
]]>