There are several Pi Day’s (14 March: 3.14; 22 July: 22/7) so we should look at how on earth you compute a number like Pi (3.1.4159….). It has an infinite number of digits containing no repeating pattern so you can never tie it down exactly. One of my favourite ways for calculating pi was first devised by the Indian mathematician Mādhava of Sangamagrāma 600 years ago. He worked out an algorithm for working out Pi based on the maths of infinite series that he had also worked out.
Pi is one of the most useful numbers in all of maths. In school you come across it when working out the area or circumference of a circle, but it crops up all over the place including in practical computer science situations. Digital music, for example, relies on it deep down. Remember that the next time you stream your favourite music!
So how, 600 years ago did Mādhava manage to work out a much more accurate version of Pi than anyone before him? He had worked out that certain sequences of infinite numbers wouldn’t get bigger and bigger but would just get closer and closer to some specific number. In particular, he worked out one such sequence linked to pi.
π / 4 = 1 – 1/3 + 1/5 – 1/7 + 1/9 – …
Writing this a slightly different way it gives us a way of calculating pi itself
π = 4 – 4/3 + 4/5 – 4/7 + 4/9 – …
With an infinite number of terms, this gives an accurate value for pi. We can’t add an infinite number of numbers together though. Instead, we can use it to get a good answer. To get an approximation to pi we just follow an algorithm where we gradually add / subtract the next term. Each new calculation then gives us a better estimate of what pi is.
So to start with we just take the first term which says
π = 4 (very approximately)
That isn’t very good as it doesn’t get any digits right! Pi is closer to 3 than to 4. So its not looking hopeful! That doesn’t matter though as it is just a starting point. When we subtract the next term it gets a bit better
π = 4 – 4/3 = 2.6666…
Hmm. Now we have overshot the other way. However, we are closer to the real value of pi than we were. So don’t lose heart, keep going and add the next term
π = 4 – 4/3 + 4/5 = 3.46666…
And another term …
π = 4 – 4/3 + 4/5 – 4/7 = 2.895 …
And another term …
π = 4 – 4/3 + 4/5 – 4/7 + 4/9 = 3.339…
and so on.
The important thing to notice is that after each term included we get a more accurate answer, and we can keep adding terms for as long as we are happy to do the calculations. Mādhava (or his followers) obviously liked doing calculations so kept going until he had worked out pi accurate to 10 decimal places (3.1415926535…) : a new world record at the time beating the previous best of 6 decimal places by a Chinese astronomer Zhao Youqin using a different algorithm, That record had been set 80 years earlier but was smashed by 4 decimal places. This new record lasted for another 96 years. In doing these calculations Mādhava was acting as a ‘computer’ in the original meaning of the word: a human following an algorithm to do computation.
His algorithm is what computer scientists call an iterative algorithm. This kind of algorithm is used quite a lot by computer scientists as it gives a general way of getting a good enough (if not perfect) answer to a problem that otherwise is hard (or impossible) to get a perfect answer to in a reasonable time. You start with a good guess and then gradually refine the answer until you are happy that it is accurate enough. These algorithms can be straightforward to code as it is just running a loop doing calculations that refine the answer. Mādhava was happy with 10 decimal places of accuracy but he could have kept going. The trouble is this is a very slow algorithm. As we saw with the first few iterations above, it takes a long time even to home in on the first digit being 3! Every new digit took a lot of extra work to get right. When calculating machines and then computers were invented it became easier to use slow algorithms like this, but even with a faster computer it is still better to have a faster algorithm. Now far faster algorithms have been invented and the world record at the time of writing gives pi accurate to 105,000,000,000,000 decimal places!
Mādhava would have needed to really like doing calculations (and have discovered the secret to eternal life) to have calculated pi that accurately. 600 years ago his world record for pi was still an amazing achievement.
Scientific fraud is worryingly common, though rarely talked about. It has been happening for years, but now Artificial Intelligence programs could supercharge it. If they do that could undermine Science itself.
Investigators of scientific fraud have found that large numbers of researchers have manipulated their results, invented data, or even produced nonsensical papers in the hope that no one will look closely enough to notice. Often, no one does. The problem is that science is built on the foundation of all the research that has gone before. If we can no longer trust that past research is legitimate, the whole system of science begins to break down. AI has the potential to supercharge this process.
We’re not at that point yet, luckily. But there are concerning signs that generative AI systems like ChatGPT and DALLE-E might bring us closer. By using AI technology, producing fraudulent research has never been easier, faster, or more convincing. To understand, let’s first look at how scientific fraud has been done in the past.
How fraud happens
Until recently, fraudsters would need to go through some difficult steps to get a fraudulent research paper published. A typical example might look like this:
Step 1: invent a title
Fraudsters look for a popular but very broad research topic. We’ll take an example of a group of fraudsters known as the Tadpole Paper Mill. They published papers about cellular biology. To choose a new paper to create, the group would essentially use a simple generator, or algorithm, based on a template. This uses a simple technique first used by Christopher Strachey to write love letters in an early “creative” program in the 1950s.
For each “hole” in the template a word is chosen from a word list.
Pick the name of a molecule
Either a protein name, a drug name or an RNA molecule name
Next, the fraudsters create the text of the paper. To do this, they often just plagiarise and lightly edit previous similar papers, substituting key words in from their invented title perhaps. To try to hide the plagiarism, they automatically swap out words, replacing them with synonyms. This often leads to ridiculous (and kind of hilarious) replacements, like these found in plagiarised papers:
“Big data” –> “Colossal information”
“Cloud computing” –> “Haze figuring”
“Developing countries” –> “Creating nations”
“Kidney failure” –> “Kidney disappointment”
Step 3: add in the results
Lastly, the fraudsters need to create results for the fake study. These usually appear in papers in the form of images and graphs. To do this, the fraudsters take the results from several previous papers and recombine them into something that looks mostly real, but is just a Frankenstein mess of other results that have nothing to do with the current paper.
A new paper is born
Using that simple formula, fraudsters have produced thousands of fabricated articles in the last 10 years. Even after a vast amount of effort, the dedicated volunteers who are trying to clean up the mess have only caught a handful.
However, committing fraud like this successfully isn’t exactly easy, either: the fraudsters still need to come up with a research idea, write the paper themselves without copying too much from previous research, and make up results that look convincing—at least at first glance.
AI: Adding fuel to the fire
So what happens when we add modern generative AI programs into the mix? They are Artificial Intelligence programs like ChatGPT or DALL-E that can create text or pictures for you based on written requests.
Well, the quality of the fraud goes up, and the difficulty of producing it goes way down. This is true for both text and images.
Let’s start with text. Just now, I asked ChatGPT-4 to “write the first two paragraphs of a research paper on a cutting edge topic in psychology.” I then asked it to “write a fake results table that shows a positive relationship between climate change severity and anxiety”. I won’t copy the whole thing—in part because I encourage you to try this yourself to see how it works (not to actually create a fake paper!)—but here’s a sample of what it came up with:
“As the planet faces increasing temperatures, extreme weather events, and environmental degradation, the mental health repercussions for populations worldwide become a crucial area of investigation. Understanding these effects is vital for developing strategies to support communities in coping with the psychological challenges posed by a changing climate.”
As someone who has written many psychology research papers, I would find the results very difficult to identify as AI-generated—it looks and sounds very similar to how people in my field write, and it even generated Python code to analyse the fake data. I’d need to take a really close look at the origin of the data and so on to figure out that it’s fraudulent.
But that’s a lot of work required from me as a fraud-buster. For the fraudster, doing this takes about 1 minute, and would not be detected by any plagiarism software in the way previous kinds of fraud can be. In fact, this might only be detected if the fraudsters make a sloppy mistake, like leaving in a disclaimer from the model as in the paper caught which included the text
“[Please note that as an AI language model, I am unable to generate specific tables or conduct tests, so the actual resutls should be included in the table.]”!
Generative AIs are not close to human intelligence, at least not yet. So, why are they so good at producing convincing scientific research, something that’s commonly seen as one of the most difficult things humans can do? Two reasons play a big part: (1) scientific research is very structured, and (2) there’s a lot of training data. In any given field of research, most papers tend to look pretty similar—an introduction section, a method describing what the researchers did, a results section with a few tables and figures, and a discussion that links it back to the wider research field. Many journals even require a fixed structure. Generative AI programs work using Machine Learning – they learn from data and the more data they are given the better they become. Give a machine learning program millions of images of cats, telling it that is what they are, and it can become very good at recognising cats. Give it millions of images of dogs and it will be able to recognise dogs too. With roughly 3 million scientific papers published every year, generative AI systems are really good at taking these many, many examples of what a scientific report looks like, and producing similar sounding, and similarly structured pieces of text. They do it by predicting what word, sentence and paragraph would be good to come next based on probabilities calculated from all those examples.
Trusting future research
Most research can still be trusted, and the vast majority of scientists are working as hard as they can to advance human knowledge. Nonetheless, we all need to look carefully at research studies to ensure that they are legitimate, and we should be on extra alert as generative AI becomes even more powerful and widespread. We also need to think about how to improve universities and research culture generally, so that people don’t feel like they need to commit scientific fraud—something that usually happens because people are desperate to get or keep a job, or be seen as successful and reap the rewards. Somehow we need to change the game so that fraud no longer pays.
What do you think? Do you have ideas for how we can prevent fraud from happening in the first place, and how can we better detect it when it does occur? It is certainly an important new research topic. Find a solution and you could do massive good. If we don’t find solutions then we could lose the most successful tool human-kind has ever invented that makes all our lives better.
Greg Michaelson is an Emeritus professor of computer science at Heriot-Watt University in Edinburgh. He is also a novelist and a short story writer. He wrote this story for CS4FN.
“Come on!” called Alice, taking the coat off the peg. “We’re going to be late!”
“Do I have to?” said Henry, emerging from the front room.
“Yes,” said Alice, handing him the coat. “Of course you have to go. Here. Put this on.”
“But we’re playing,” said Henry, wrestling with the sleeves.
“Too bad,” said Alice, straightening the jacket and zipping it up. “It’ll still be there when we get back.”
“Not if someone knocks it over,” said Henry, picking up a small model dinosaur from the hall table. “Like last time. Why can’t we have electric games like you did?”
“Electronic games,” said Alice, doing up her buttons. “Not electric. No one has them anymore. You know that.”
“Were they really digital?” asked Henry, fiddling with the dinosaur.
“Yes,” said Alice, putting on her hat. “Of course they were digital.”
“But the telephone’s all right,” said Henry.
“Yes,” said Alice, checking her makeup in the mirror. “It’s analogue.”
“And radio. And record players. And tape recorders. And television,” said Henry.
“They’re all analogue now,” said Alice, putting the compact back into her handbag. “Anything analogue’s fine. Just not digital. Stop wasting time! We’ll be late.”
“Why does it matter if we’re late?” asked Henry, walking the dinosaur up and down the hall table.
“They’ll notice,” said Alice. “We don’t want to get another warning. Put that away. Come on.”
“Why don’t the others have to go?” asked Henry, palming the dinosaur.
“They went last Sunday,” said Alice, opening the front door. “You said you didn’t want to go. We agreed I’d take you today instead.”
“Och, granny, it’s so boring…” said Henry.
They left the house and walked briskly to the end of the street. Then they crossed the deserted park, following the central path towards the squat neo-classical stone building on the far side.
“Get a move on!” said Alice, quickening the pace. “We really are going to be late.”
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Henry really hadn’t paid enough attention at school. He knew that Turing Machines were named for Alan Turing, the first Martyr of the Digital Age. And he knew that a Turing Machine could work out sums, a bit like a school child doing arithmetic. Only instead of a pad of paper and a pencil, a Turing Machine used a tape of cells. And instead of rows of numbers and pluses and minuses on a page, a Turing Machine could only put one letter on each cell, though it could change a letter without having to actually rub it out. And instead of moving between different places on a piece of paper whenever it wanted to, and maybe doodling in between the sums, a Turing Machine could only move the tape left and right one cell at a time. But just like a school child getting another pad from the teacher when they ran out of paper, the Turing Machine could somehow add another empty cell whenever it got to the end of the tape.
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When they reached the building, they mounted the stone staircase and entered the antechamber through the central pillars. Just inside the doorway, Alice gave their identity cards to the uniformed guard.
“I see you’re a regular,” she said approvingly to Alice, checking the ledger. “But you’re not,” sternly to Henry.
Henry stared at his shoes.
“Don’t leave it so long, next time,” said the guard, handing the cards back to Alice. “In you go. They’re about to start. Try not to make too much noise.”
Hand in hand, Alice and Henry walked down the broad corridor towards the central shrine. On either side, glass cases housed electronic equipment. Computers. Printers. Scanners. Mobile phones. Games consoles. Laptops. Flat screen displays.
The corridor walls were lined with black and white photographs. Each picture showed a scene of destitution from the Digital Age.
Shirt sleeved stock brokers slumped in front of screens of plunging share prices. Homeless home owners queued outside a state bank soup kitchen. Sunken eyed organic farmers huddled beside mounds of rotting vegetables. Bulldozers shovelled data farms into land fill. Lines of well armed police faced poorly armed protestors. Bodies in bags lay piled along the walls of the crematorium. Children scavenged for toner cartridges amongst shattered office blocks.
Alice looked straight ahead: the photographs bore terrible memories. Henry dawdled, gazing longingly into the display cases: Gameboy. Playstation. X Box…
“Come on!” said Alice, sotto voce, tugging Henry away from the displays.
At the end of the corridor, they let themselves into the shrine. The hall was full. The hall was quiet.
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Henry was actually quite good at sums, and he knew he could do them because he had rules in his head for adding and subtracting, because he’d learnt his tables. The Turing Machine didn’t have a head at all, but it did have rules which told it what to do next. Groups of rules that did similar things were called states, so all the rules for adding were kept separately from all the rules for subtracting. Every step of a Turing machine sum involved finding a rule in the state it was working on to match the letter on the tape cell it was currently looking at. That rule would tell the Machine how to change the symbol on the tape, which way to move the tape, and maybe to change state to a different set of rules.
————————-
On the dais, lowered the Turing Machine, huge coils of tape links disappearing into the dark wells on either side, the vast frame of the state transition engine filling the rear wall. In front of the Turing Machine, the Minister of State stood at the podium.
“Come in! Come in!” he beamed at Alice and Henry. “There’s lots of space at the front. Don’t be shy.”
Red faced, Alice hurried Henry down the aisle. At the very front of the congregation, they sat down cross legged on the floor beneath the podium.
“My friends,” began the Minister of State. “Welcome. Welcome indeed! Today is a special day. Today, the Machine will change state. But first, let us be silent together. Please rise.”
The Minister of State bowed his head as the congregation shuffled to its feet.
———————–
According to Henry’s teacher, there was a different Turing Machine for every possible sum in the world. The hard bit was working out the rules. That was called programming, but, since the end of the Digital Age, programming was against the law. Unless you were a Minister of State.
————————
“Dear friends,” intoned the Minister of State, after a suitable pause. “We have lived through terrible times. Times when Turing’s vision of equality between human and machine intelligences was perverted by base greed. Times when humans sought to bend intelligent machines to their selfish wills for personal gain. Times when, instead of making useful things that would benefit everybody, humans invented and sold more and more rarefied abstractions from things: shares, bonds, equities, futures, derivatives, options…”
————————
The Turing Machine on the dais was made from wood and brass. It was extremely plain, though highly polished. The tape was like a giant bicycle chain, with holes in the centre of each link. The Machine could plug a peg into a hole to represent a one or pull a peg out to represent a zero. Henry knew that any information could be represented by zeros and ones, but it took an awful lot of them compared with letters.
————————-
“… Soon there were more abstractions than things, and all the wealth embodied in the few things that the people in poor countries still made was stolen away, to feed the abstractions made by the people in the rich countries. None of this would have been possible without computers…”
————————-
The state transition unit that held the rules was extremely complicated. Each rule was a pattern of pegs, laid out in rows on a great big board. A row of spring mounted wooden fingers moved up and down the pegs. When they felt the rule for the symbol on the tape cell link, they could trigger the movement of a peg in or out of the link, and then release the brakes to start up one revolution of the enormous cog wheels that would shift the tape one cell left or right.
————————-
“…With all the computers in the world linked together by the Internet, humans no longer had to think about how to manage things, about how best to use them for the greatest good. Instead, programs that nobody understood anymore made lightening decisions, moving abstractions from low profits to high profits, turning the low profits into losses on the way, never caring how many human lives were ruined…”
————————-
The Turing Machine was powered by a big brass and wooden handle connected to a gear train. The handle needed lots of turns to find and apply the next rule. At the end of the ceremony, the Minister of State would always invite a member of the congregation to come and help him turn the handle. Henry always hoped he’d be chosen.
——————————
“…Turing himself thought that computers would be a force for untold good; that, guided by reason, computers could accomplish anything humans could accomplish. But before his vision could be fully realised, he was persecuted and poisoned by a callous state interested only in secrets and profits. After his death, the computer he helped design was called the Pilot Ace; just as the pilot guides the ship, so the Pilot Ace might have been the best guide for a true Digital Age…”
——————————
Nobody was very sure where all the cells were stored when the Machine wasn’t inspecting them. Nobody was very sure how new cells were added to the ends of the tape. It all happened deep under the dais. Some people actually thought that the tape was infinite, but Henry knew that wasn’t possible as there wasn’t enough wood and brass to make it that long.
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“…But almost sixty years after Turing’s needless death, his beloved universal machines had bankrupted the nations of the world one by one, reducing their peoples to a lowest common denominator of abject misery. Of course, the few people that benefited from the trade in abstractions tried to make sure that they weren’t affected but eventually even they succumbed…”
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Nobody seemed to know what the Turing Machine on the dais was actually computing. Well, the Minister of State must have known. And Turing had never expected anyone to actually build a real Turing Machine with real moving parts. Turing’s machine was a thought experiment for exploring what could and couldn’t be done by following rules to process sequences of symbols.
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“…For a while, everything stopped. There were power shortages. There were food shortages. There were medical shortages. People rioted. Cities burned. Panicking defence forces used lethal force to suppress the very people they were supposed to protect. And then, slowly, people remembered that it was possible to live without abstractions, by each making things that other people wanted, by making best use of available resources for the common good…”
——————————
The Turing Machine on the dais was itself a symbol of human folly, an object lesson in futility, a salutary reminder that embodying something in symbols didn’t make it real.
——————————
“…My friends, let us not forget the dreadful events we have witnessed. Let us not forget all the good people who have perished so needlessly. Let us not forget the abject folly of abstraction. Let the Turing Machine move one step closer along the path of its unknown computation. Let the Machine change its state, just as we have had to change ours. Please rise.”
The congregation got to their feet and looked expectantly at the Minister of State. The Minister of State slowly inspected the congregation. Finally, his eyes fixed on Henry, fidgeting directly in front of him.
“Young man,” he beamed at Henry. “Come. Join me at the handle. Together we shall show that Machine that we are all its masters.”
Henry looked round at his grandmother.
“Go on,” she mouthed. “Go on.”
Henry walked round to the right end of the dais. As he mounted the wooden stairs, he noticed a second staircase leading down behind the Machine into the bowels of the dais.
“Just here,” said the Minister of State, leading Henry round behind the handle, so they were both facing the congregation. “Take a good grip…”
Henry was still clasping the plastic dinosaur in his right hand. He put the dinosaur on the nearest link of the chain and placed both hands on the worn wooden shaft.
And turn it steadily…”
Henry leant into the handle, which, much to his surprise, moved freely, sweeping the wooden fingers across the pegs of rules on the state transition panel. As the fingers settled on a row of pegs, a brass prod descended from directly above the chain, forcing the wooden peg out of its retaining hole in the central link. Finally, the chain slowly began to shift from left to right, across the front of the Machine, towards Henry and the Minister of State. As the chain moved, the plastic dinosaur toppled over and tumbled down the tape well.
“Oh no!” cried Henry, letting go of the handle. Utterly nonplussed, the Minister of State stood and stared as Henry peered into the shaft, rushed to the back of the Machine and hurried down the stairs into the gloom.
A faint blue glow came from the far side of the space under the dais. Henry cautiously approached the glow, which seemed to come from a small rectangular source, partly obscured by someone in front of it.
“Please,” said Henry. “Have you seen my dinosaur?”
“Hang on!” said a female voice.
The woman stood up and lit a candle. Looking round, Henry could now see that the space was festooned with wires, leading into electric motors driving belts connected to the Turing Machine. The space was implausibly small. There was no room for a finite tape of any length at all, let alone an infinite one.
“Where are all the tape cells?” asked Henry, puzzled.
“We only need two spare ones,” said the woman. “When the tape moves, we stick a new cell on one end and take the cell off the other.”
“So what’s the blue light?” asked Henry.
“That’s a computer,” said the woman. “It keeps track of what’s on the tape and controls the Turing Machine.”
“A real digital computer!” said Henry in wonder. “Does it play games?”
“Oh yes!” said the woman, turning off the monitor as the Minister of State came down the stairs. “What do you think I was doing when you showed up? But don’t tell anyone. Now, let’s find that dinosaur.”
Our Turing Machine so far has an Infinite Tape, a Tape Head and a Controller. The Tape holds data values taken from a given set of 4×4 bricks. It starts in a specific initial pattern: the Initial Tape. There is also a controller. It holds different coloured 3×2 bricks representing an initial state, an end state, a current state and has a set of other possible states (so coloured bricks) to substitute for the current state.
Why do we need a program?
As the machine runs it changes from one state to another, and inputs from or outputs to the tape. How it does that is governed by its Program. What is the new state, the new value and how does the tape head move? The program gives the answers. The program is just a set of instructions the machine must blindly follow. Each instruction is a single rule to follow. Each program is a set of such rules. In our Turing Machines, these rules are not set out in an explicit sequence as happens in a procedural program, say. It uses a different paradigm for what a program is. Instead at any time only one of the set of rules should match the current situation and that is the one that is followed next.
Individual Instructions
A single rule contains five parts: a Current State to match against, a Current Value under the Tape Head to match against, a New State to replace the existing one, and a New Value to write to the tape. Finally, it holds a Direction to Move the Tape Head (left or right or stay in the same place). An example might be:
Current State: ORANGE
Current Value: RED
New State: GREEN
New Value: BLUE
Direction: RIGHT
But what does a rule like this actually do?
What does it mean?
You can think of each instruction as an IF-THEN rule. The above rule would mean:
IF
the machine is currently in state ORANGE AND
the Tape Head points to RED
THEN (take the following actions)
change the state to GREEN,
write the new value BLUE on the tape AND THEN
move the tape head RIGHT.
This is what a computer scientist would call the programming language Semantics. The semantics tell you what program instructions mean, so what they do.
Representing Instructions in Lego
We will use a series of 5 bricks in a particular order to represent the parts of the rule. For example, we will use a yellow 3×2 brick in the first position of a rule to represent the fact that the rule will only trigger if the current state is yellow. A blue 2×2 brick in the second position will mean the rule will also only trigger if the current value under the tape head is blue. We will use a grey brick to mean an empty tape value. The third and fourth position will represent the new state and new value if the rule does trigger. To represent the direction to move we will use a 1×2 Red brick to mean move Right, and a 1×2 yeLLow brick to mean move Left. We will use a black 1×2 brick to mean do not move the tape head (mirroring the way we are also using black to mean do nothing in the sense of the special end state). The above rule would therefore be represented in Lego as below.
A single instruction for a Lego Turing Machine. Image by CS4FN
Notice we are using the same colour to represent different things here. The representation is the colour combined with the size of brick and position in the rule. So a Red brick can mean a red state (a red 3×2 brick) or a red value (a red 2×2 brick) or move right (a red 1×2 brick).
Lego programs
That is what a rule, so single Turing Machine instruction, looks like. Programs are just a collection of such rules: so a series of lines of bricks.
Suppose we have a Turing machine with two states (Red and Orange) and two values on the tape (Blue or Empty), then a complete program would have 4 rules, one for each possible combination. We have given one example program below. If there were more states or more possible data values then the program would be correspondingly bigger to cover all the possibilities.
A 4 instruction Turing Machine Program for a Turing Machine with two states (Red, Orange) and two data values (Blue, Empty). Image by CS4FN
A Specific Turing Machine
Exactly what it does will depend on its input – the initial tape it is given to process, as well as the initial state and where the tape head initially points to. Perhaps you can work out what the above program does given a tape with an empty value followed by a series of three blue bricks (and then empty data values off to infinity (the blank value is the only value that is allowed to appear an infinite number of times on an initial tape) and the Head pointing to the rightmost blue brick value. The initial state is red. See the Lego version of this specific machine below.
A full Turing Machine ready to execute. Image by CS4FN
Note something we have glossed over. You also potentially need an infinite number of bricks of each value that is allowed on the tape. We have a small pile, but you may need that Lego factory we mentioned previously, so that as the Turing Machine runs you always have a piece to swap on to the machine tape when needed. Luckily, for this machine a small number of bricks should be enough (as long as you do not keep running it)!
What does this Turing Machine do? We will look at what it does and how to work it out in a future article. In the meantime try and work out what it does with this tape, but also what it does if the tape has more or less blue bricks in a row on it to start with (with everything else kept the same).
Note that, to keep programs smaller, you could have a convention that if no rule fits a situation then it means the program ends. Then you could have fewer rules in some programs. However, that would just be shorthand for there being extra rules with black new states, the tape being left alone, and the tape head moves right. In real programming, it is generally a good idea to ALWAYS be explicit about what you intend the program to do, as otherwise it is an easy way for bugs to creep in, for example, because you just forgot to say in some case.
Alan Turing invented Turing Machines before any computer existed. At the time a “computer” was a person who followed rules to do calculations (just like you were taught the rules to follow to do long multiplication at primary school, for example). His idea was therefore that a human would follow the rules in a Turing Machine program, checking the current state and value under the tape head, and changing the state, the value on the tape and the movement of the head. A person provides the power and equivalent of a robotic arm that follows the underlying Turing Machine algorithm: the Turing Machine algorithm that if followed causes each Turing Machine’s program to execute.
If a human animating the machine was good enough for Turing, it is good enough for us, so that is how our Lego Turing Machines will work. Your job will be to follow the rules and so operate the machine. Perhaps, that is exactly what you did to work out what the program above does!
Next we will look at how to work out what a Turing Machine does. Then it will be time to write, then run, some Turing Machine programs of your own…
EPSRC supports this blog through research grant EP/W033615/1, The Lego Computer Science post was originally funded by UKRI, through grant EP/K040251/2 held by Professor Ursula Martin, and forms part of a broader project on the development and impact of computing.
How does the machine decide where and when to move the Tape Head, though? It has a Controller. The key part of the controller is that it holds a Current State of the machine. Think of traffic lights for what we mean by the state of a machine. In the UK traditional traffic lights have a Red state, an Amber state, a Green state and a Red-Amber state. Each means a different thing (such as “Stop” and “Go”). The controller of the lights moves between these different internal states. With a traffic light, the current internal state is also shown to the world by the lights that light up! Machine states do not have to be visible to the outside world, however. In fact, they only are if the person who designs the interface makes them visible. For most machines, only some of their internal state is made visible. In our Turing Machine we will be able to see the states as they will be visible in the controller. However, the output of a Turing Machine is the state of the tape, so if we wanted the states to really be visible we would write a version on to the tape. You can then imagine the tape triggering external lights to come on or off, or change colour as a simple form of actual output. This is what Computer Scientists call memory-mapped peripherals – where to send data (output) to a peripheral device (a screen, a panel of lights, a printer, or whatever, you write to particular locations in memory, and that data is read from there by the peripheral device. That is going beyond the pure idea of a Turing Machine though, where the final state of the machine when it stops is its output.
Representing States
How do we represent states in Lego? Any finite set of things (symbols) could be used to represent the different states (including numbers or binary codes, for example). We will use different coloured 3×2 blocks. Each colour of block will stand for a different state that the machine is in. The controller will have a space that holds the brick representing the Current State. It will also have space for a set of places for the blocks representing the other allowable states of this Turing Machine. As the machine runs, the state will change as represented by swapping one of these state bricks for another.
Different Turing Machines can allow a different number of possible states the machine could be in, so this part of the controller might be bigger or smaller depending on the machine and what it needs to do its job. Again think of traffic lights, in some countries, and on pedestrian crossings there are only two states, a Red state (stop) and a Green state (go). Its controller only needs two states so we would only need two different coloured bricks.
A Turing Machine Controller with current state red, end state black and three other possible states (green, orange and blue). Image by CS4FN
Initial States
The current state will always start in some initial state when the machine first starts up. It is useful to record in the controller what state that is so that each time we restart the machine anew it can be reset. We will just put a block in the position next to the current state to indicate what the initial state should be. We won’t ever change it for a given machine.
End States
One of the states of a Turing Machine is always a special End State. We will always use a black brick to represent this. Whatever is used has to be specified at the outset, though. When not in use we will keep the end state brick next to the initial state brick. Once the machine finishes operations it will enter this End State, or put another way, if the black brick ever becomes the current state brick the machine will stop. From that point on the machine will do nothing. Some machines might never reach an end state, they just go on forever. Traffic lights just cycle round the states forever, for example, never reaching an end state. Other machines do end though. For example, a kettle controller stops the machine when the water has boiled. An addition Turing Machine might end when it has output the answer to an addition. To do another addition you would start it up again with new information on the tape indicating what it was to add.
We have now created the physical part of the Turing Machine. All we need now is a Program to tell it what to do! Programs come next in Part 3…
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This blog is funded by EPSRC on research agreement EP/W033615/1.
The Lego Computer Science posts were originally funded by UKRI, through grant EP/K040251/2 held by Professor Ursula Martin, and forms part of a broader project on the development and impact of computing.
It it possible to make a working computer out of lego and you do not even have to pay for an expensive robot Mindstorm kit…but only if you are willing to provide the power yourself.
A machine in theory
In fact, Alan Turing, grandfather of Computer Science worked out how to do it before the War and before any actual computer existed. His version also needed humans to power it. Now we call it a Turing Machine and it is a theoretical model of what is computable by any machine.
The Tape
To make a working Turing Machine you first need to build an infinitely long Tape that can hold Symbols, representing data values, along it at fixed intervals. That is easy (as long as you have a lego factory). You just need to create a long line of flat pieces, say 2 studs wide. Each 2×2 square on it is then a position on the Tape.
An infinite tape out of Lego (relies on having a Lego factory at the right-hand end churning out new tape if and when it is needed… Image by CS4FN
Be lazy
Of course you can’t actually make it infinitely long, but you can make it longer every time you need some more of it (so no problem if you do have a lego factory to churn out extra bricks as needed!) This approach to dealing with infinite data structures where you just make it bigger only when needed is now called lazy programming by computer scientists and is an elegant way that functional programs deal with input that needs to represent an infinite amount of input…It is also the way some games (like Minecraft) represent worlds or even universes. Rather than create the whole universe at the start, things over the horizon, so out of sight, are only generated if a player ever goes there – just-in-time world generation! Perhaps our universe is like that too, with new galaxies only fleshed out as we develop the telescopes to see them!
Fill it with data
The Tape has a set of Data Symbols that can appear on it that act as the DataValues of the machine. Traditional computers have symbols 0 and 1 underpinning them, so we could use those as our symbols, but in a Turing Machine we can have any set of symbols we like: ten digits, letters, Egyptian hieroglyphs, or in fact any set of symbols we want to make up. In a lego Turing Machine we can just use different coloured blocks as our symbols. If our tape is made of grey pieces then we could use red and blue for the symbols that can appear on it. Every position on the tape will then either hold a red block or a blue block. We could also allow EMPTY to be a symbol too in which case some 2×2 slots could be empty to mean that.
A tape containing data where the allowed symbols are EMPTY, RED and BLUE. Image by CS4FN
To start with
Any specific Turing Machine has an Initial Tape. This is the particular data that is on the tape at the start, before it is switched on. As the machine runs, the tape will change.
The tape with symbols on it takes the place of our computer’s memory. Just as a modern computer stores 1s and 0s in memory, our Lego Turing Machine stores its data as symbols on this tape.
The Head
A difference is that modern computers have “random access memory” – you can access any point in memory quickly. Our tape will be accessed by a Tape Head that points to a position on the tape and allows you to read or change the data only at the point it is at. Make a triangular tape head out of lego so that it is clear which point on the tape it is pointing at. We have a design choice here. Either the Tape moves or the Head moves. As the tape could be very long so hard to move we will move the Head along beside it, so create a track for the Head to move along parallel to the tape. It will be able to move 2 studs at a time in either direction so that each time it moves it is pointing to a new position on the tape.
An infinite tape with Head (yellow) pointing at position 4 on the tape. Image by CS4FN
We have memory
We now have the first element in place of a computer, then: Memory. The next step will be to provide a way to control the tape head and how data is written to and read from the tape and so computation actually happen. (For that you need a controller which we cover in Part 2…).
EPSRC supports this blog through research grant EP/W033615/1, The Lego Computer Science post was originally funded by UKRI, through grant EP/K040251/2 held by Professor Ursula Martin, and forms part of a broader project on the development and impact of computing.
Conjure up a stereotypical image of a scientist and they likely will have a white coat. If not brandishing test tubes, you might imagine them working with mice scurrying around a maze. In future the scientists may well be doing a lot of programming, and the mice for their part will be scurrying around in their own virtual world wearing Virtual Reality goggles.
Scientists have long used mazes as away to test the intelligence of mice, to the point it has entered popular culture as a stereotypical thing that scientists in white lab coats do. Mazes do give ways to test intelligence of animals, including exploring their memory and decision making ability in controlled experiments. That can ultimately help us better understand how our brains work too, and give us a better understanding of intelligence. The more we understand animal cognition as well as human cognition, the more computer scientists can use that improved understanding to create more intelligent machines. It can also help neurobiologists find ways to improve our intelligence too.
Flowers for Algernon is a brilliant short story and later novel based on the idea, there using experiments on mice and humans to test surgery intended to improve intelligence. In a slightly different take on mice-maze experiments, Douglas Adams, in ‘The Hitchhikers Guide to the Galaxy’, famously claimed that the mice were actually pan-dimensional beings and these experiments were really incredibly subtle experiments the mice were performing on humans. Whatever the truth of who is experimenting on who, the experiments just took a great leap forward because scientists at Northwestern University have created Virtual Reality goggles for their mice.
For a long time researchers at Northwestern have used a virtual reality version of maze experiments, with mice running on treadmills with screens around them projecting what the researchers want them to see, whether mazes, predators or prey. This has the advantage of being much easier to control than using physical mazes, and as the mice are actually stationary the whole time , just running on a treadmill, brain-scanning technology can be used to see what is actually happening in their brains while facing these virtual trials. The problem though is that the mice, with their 180 degree vision, can still see beyond the edges of the screens. The screens also give no sense of 3 dimensions, when like us the mice naturally see in 3D. As the screens are not fully immersive, they are not fully natural and that could affect the behaviour of the mice and so invalidate the experimental results.
That is why the Northwestern researchers invented the mousey VR googles, the idea being that they would give a way to totally immerse the mice in their online world, and so improve the reliability of the experiments. In the current version the goggles are not actually worn by the mice, as they are still too heavy. Instead, the mouse’s head is held in place really close to them, but with the same effect of total immersion. Future versions may be small enough for the mice to wear them though.
The scientists have already found that the mice react more quickly to events, like the sight of a predator, than in the old set-up, suggesting that being able to see they were in a lab was affecting their behaviour. Better still, there are new kinds of experiment that can be done with this set up. In particular, the researchers have run experiments where an aerial predator like an owl appears from above the mice in a natural way. Mounting screens above them previously wasn’t possible as it got in the way of the brain scanning equipment. What does happen when a virtual owl appears? The mice either run faster or freeze, just as in the wild. This means that by scanning their brains while this is happening, how their perception of the threat works can be investigated, as well as how decision-making is taking place at the level of their brain activity. The scientists also intend to run similar experiments where the mouse is the predator, for example chasing a virtual fly too. Again this would not have been possible previously.
That in any case is what we think the purpose of these new experiments is. What new and infinitely subtle experiments it is allowing the pan-dimensional mice to perform on us remains to be seen.
Computer science research in part involves inventing new algorithms or improving new ones. But what does that mean. Let’s explore some mazes to explore algorithms.
What does computer science research involve? It is very varied: from interviewing people to find out what the real problems that need solving in their lives or jobs are; to running experiments to find out what works and what doesn’t; to writing programs to solve problems.
Improving algorithms
A core part of much research is coming up with new and better algorithms that solve particular problems. The kind of algorithm could be anything from a new more secure cryptographic protocol, or a better way to rank the results of a search engine, to a new more effective machine learning algorithm that is less likely to make things up, or perhaps can better explain how it came to its conclusions.
What does it mean to come up with a better algorithm though? Once a problem is solved, isn’t it solved? Let’s explore a simple problem to see. Let’s explore mazes. Solve the simple maze puzzle above before you go on. Find a route that gets the mouse to the cheese.
Wandering around mazes, finding algorithms
If you’ve ever been in a hedge maze in the garden of some stately home, or a corn field maze, the chances are you just dived in and wandered rather aimlessly. Perhaps you tried to remember which way you went at each junction, to avoid going down the same dead-ends more than once. How about solving the paper version of a maze puzzle like the one above? Now perhaps you looked ahead to spot dead-ends to avoid tracing wrong paths with your pencil.
Probably what you are doing is at least a little random. You could, in theory at least, end up going back over the same paths, never taking the right one and and never getting to the middle. Could we come up with an algorithm that guarantees to solve mazes? To be an algorithm it would need to guarantee you ended up finding a path to the centre of the maze if you followed the steps of the algorithm precisely. It should also work for any maze, or at least all mazes of a particular kind. Ideally, the algorithm gives you a path that can then be followed by anyone without them having to run the algorithm themselves. They can just follow the path generated by the algorithm for that maze.
Wall-following
In fact lots of maze algorithms have been invented. Perhaps the one most people have heard of, if they know of any maze algorithm, is called Wall-following. It is very simple to do, You just pick a wall at the entrance either to the left or right and then follow it, If in a garden maze, keep your hand on the hedge as you walk round. If doing a paper puzzle, draw the path sticking to the chosen wall. Try it on the following simple maze.
Image by CS4FN
Simply connected
The wall-following algorithm will guarantee to get you to the centre of the maze, and back out again too, but only as long as the maze is what is called simply connected. That just means the maze is constructed from a single hedge (or one unbroken drawn line) not a series of unconnected hedges. If you look at both examples above you will see I created them by just drawing a single wiggly line.
If a maze is simply connected then it cannot have looping paths, so no going round in circles for ever. It will also only have one entrance/exit. That shows the first aspect of inventing algorithms that is important. They often only work for some situations, not all. You must be sure you know what situations they do and don’t work.
Often the earliest algorithms invented to solve a problem are like wall-following: they only work for simple situations. Other people then come along and find new algorithms that can cover more problems (here more mazes). Can you tweak the wall-following maze algorithm to work even if there are multiple exits from the maze, for example? As it stands our algorithm could just take you from the entrance straight out of another exit without exploring much of the maze at all! See the end for one simple way to tweak the algorithm. What if there are paths that take you round in circles? Can you come up with an algorithm to deal with that?
Some times the improvements invented just involve tweaking an existing algorithm as with dealing with multiple exits in a maze. Some times a whole new algorithm is needed.
Faster, higher, stronger?
Even for a simple constrained version of the problem, like simply connected mazes, people can invent better algorithms. What does better mean for a maze? Well one way you might have a better algorithm is if it is faster in coming up with a solution. Another is that the solution it comes up with is faster. For a maze that means a shorter (ideally the shortest) path to the centre. Wall following may get you in to the centre (and out again) but you probably will have discovered a very long path that takes you in and out of lots of dead-ends needlessly. You do find a path to the centre, but it may be a very long path. Can you come up with an algorithm that finds shorter paths?
We will explore an algorithm that does next.
More to come…
Some solutions
The result of wall following on our simple maze
Image by CS4FN
One way to deal with multiple exits
To deal with a maze that has multiple exits, so multiple breaks in the outer wall, tweak the wall-following algorithm as follows. First mark the exit you use to enter the maze, so you know when you return to it. If you come to any other exit then pretend there is a gate there and keep following the wall as though it were unbroken and there were no exit.
To become a Jedi Knight you must have complete control of your thoughts. As you feel the force you start to control your surroundings and make objects move just by thinking. Telekinesis is clearly impossible, but could technology give us the same ability? The study of brain-computer interfaces is an active area of research. How can you make a computer sense and react to a person’s brain activity in a useful way?
Imagine the game of Mindball. Two competitors face each other across a coffee table. A ball sits at the centre. The challenge is to push the ball to your opponent’s end before they push it down to you. The twist is you can use the power of thought alone.
Sound like science fiction? It’s not! I played it at the Dundee Sensation Science Centre many, many years ago where it was a practical and fun demonstration of the then nascent area of brain-computer interfaces.
Each player wears a headband containing electrodes that pick up your brain waves – specifically alpha and theta waves. They are shown as lines on a monitor for all to see. The more relaxed you are, the more you can shut down your brain, the more your brain wave lines fall to the bottom of the screen and start to flatline together. This signals are linked to a computer that drives competing magnets in the table. They pull the metal ball more strongly towards the most agitated person. The more you relax the more the ball moves away from you…unless of course your opponent can out relax you.
Of course it’s not so easy to play. All around the crowd heckle, cheering on their favourite and trying to put off the opponent. You have to ignore it all. You have to think of nothing. Nothing but calm.
The ball gradually edges away from you. You see you are about to win but your excitement registers, and that makes it all go wrong! The ball hurtles back towards you. Relax again. See nothing. Make everything go black around you. Control your thoughts. Stay relaxed. Millimetre by millimetre the ball edges away again until finally it crosses the line and you have won.
Its not just a game of course. There are some serious uses. It is about learning to control your brain – something that helps people trying to overcome stress, addiction and more. Similar technology can also be used by people who are paralysed, and unable to speak, to control a computer. The most recent systems, combining this technology with machine learning to learn what thoughts correspond to different brain patterns can pick up words people are thinking.
For now though it’s about play. It’s a lot of fun, just moving a ball apparently by telekinesis. Imagine what mind games will be like when embedded in more complex gaming experiences!
– Paul Curzon, Queen Mary University of London (updated from the archive)
The Formula 1 car screams to a stop in the pit-lane. Seven seconds later, it has roared away again, back into the race. In those few seconds it has been refuelled and all four wheels changed. Formula 1 pit-stops are the ultimate in high-tech team work. Now the Ferrari pit stop team have helped improve the hospital care of children after open-heart surgery!
Open-heart surgery is obviously a complicated business. It involves a big team of people working with a lot of technology to do a complicated operation. Both during and after the operation the patient is kept alive by computer: lots of computers, in fact. A ventilator is breathing for them, other computers are pumping drugs through their veins and yet more are monitoring them so the doctors know how their body is coping. Designing how this is done is not just about designing the machines and what they do. It is also about designing what the people do – how the system as a whole works is critical.
Pass it on
One of the critical times in open-heart surgery is actually after it is all over. The patient has to be moved from the operating theatre to the intensive care unit where a ‘handover’ happens. All the machines they were connected to have to be removed, moved with them or swapped for those in the intensive care unit. Not only that, a lot of information has to be passed from the operating team to the care team. The team taking over need to know the important details of what happened and especially any problems, if they are to give the best care possible.
A research team from the University of Oxford and Great Ormond Street Hospital in London wondered if hospital teams could learn anything from the way other critical teams work. This is an important part of computational thinking – the way computer scientists solve problems. Rather than starting from scratch, find a similar problem that has already been solved and adapt its solution for the new situation.
Rather than starting from scratch, find a similar problem that has already been solved
Just as the pit-stop team are under intense time pressure, the operating theatre team are under pressure to be back in the operating theatre for the next operation as soon as possible. In a handover from surgery there is lots of scope for small mistakes to be made that slow things down or cause problems that need to be fixed. In situations like this, it’s not just the technology that matters but the way everyone works together around it. The system as a whole needs to be well designed and pit stop teams are clearly in the lead.
Smooth moves
To find out more, the research team watched the Ferrari F1 team practice pit-stops as well as talking to the race director about how they worked. They then talked to operating theatre and intensive care unit teams to see how the ideas might work in a hospital handover. They came up with lots of changes to the way the hospital did the handover.
For example, in a pit-stop there is one person coordinating everything – the person with the ‘lollipop’ sign that reminds the driver to keep their brakes on. In the hospital handover there was no person with that job. In the new version the anaesthetist was given the overall job for coordinating the team. Once the handover was completed that responsibility was formally passed to the intensive care unit doctor. In Formula 1 each person has only one or two clear tasks to do. In the hospital people’s roles were less obvious. So each person was given a clear responsibility: the nurses were made responsible for issues with draining fluids from the patient, anaesthetist for ventilation issues, and so on. In Formula 1 checklists are used to avoid people missing steps. Nothing like that was used in the handover so a checklist was created, to be used by the team taking on the patient.
These and other changes led to what the researchers hoped would be a much improved way of doing handovers. But was it better?
Calm efficiency saves the day
To find out they studied 50 handovers – roughly half before the change was made and half after. That way they had a direct way of seeing the difference. They used a checklist of common problems noting both mistakes made and steps that proved unusually difficult. They also noted how well the teams worked together: whether they were calm and supported each other, planned what they did, whether equipment was available when needed, and so on.
They found that the changes led to clearly better handovers. Fewer errors were made both with the technology and in passing on information. Better still, while the best performance still happened when the teams worked well, the changes meant that teamwork problems became less critical. Pit-stops and open-heart surgery may be a world apart, with one being about getting every last millisecond of speed and the other about giving as good care as possible. But if you want to improve how well technology and people work together, you need to think about more than just the gadgets. It is worth looking for solutions anywhere: children can be helped to recover from heart surgery even by the high-octane glitz of Formula 1.
Paul Curzon, Queen Mary University of London (Updated from the archive)
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