Peter Landin: Elegance from Logic

Celebrating LGBTQ+ Greats

Thousands of programming languages have been invented in the many decades since the first. But what makes a good language? A key idea behind language design is that they should make it easy to write complex algorithms in simple and elegant ways. It turns out that logic is key to that. Through his work on programming language design, Peter Landin as much as anyone, promoted both elegance and the linked importance of logic in programming.

Pride flag with lambda x.x (identity) superimposed — Pride image by Image by Pete Linforth from Pixabay. Composite by PC

Peter was an eminent Computer Scientist who made major contributions to the theory of programming languages and especially their link to logic. However, he also made his mark in his stand against war, and support of the nascent LGBTQ+ community in the 1970s as a member of the Gay Liberation Front. He helped reinvigorate the annual Gay Pride marches as a result of turning his house into a gay commune where plans were made. It’s as a result of his activism as much as his computer science that an archive of his papers has been created in the Oxford Bodleian Library.

However, his impact on computer science was massive. He was part of a group of computing pioneers aiming to make programming computers easier, and in particular to move away from each manufacturer having a special programming language to program their machines. That approach meant that programs had to be rewritten to work on each different machine, which was a ridiculous waste of effort! Peter’s original contribution to programming languages was as part of the team who developed the programming language, ALGOL which most modern programming languages owe a debt to.

ALGOL included the idea of recursion, allowing a programmer to write procedures and functions that call themselves. This is a very mathematically elegant way to code repetition in an algorithm (the code of the function is executed each time it calls itself). You can get an idea of what recursion is about by standing between two mirrors. You see repeated versions of your reflection, each one smaller than the last. Recursion does that with problem solving. To solve a problem convert it to a similar but smaller version of the same problem (the first reflection). How do you solve that smaller problem? In the same way, as a smaller version of the same problem (the second reflection)… You keep solving those similar but smaller problems in the same way until eventually the problem is small enough to be trivial and so solved. For example, you can program a factorial method (multiplying all numbers from 1 to n together),in this way. You write that to compute factorial of a number, n, it just calls itself and computes the factorial of (n-1). It just multiply that result by n to get the answer. In addition you just need a trivial case eg that factorial of 1 is just 1.

factorial (1) = 1
factorial (n) = n * factorial (n-1)

Peter was an enthusiastic and inspirational teacher and taught ALGOL to others. This included teaching one of the other, then young but soon to be great, pioneers of Programming Theory, Tony Hoare. Learning about recursion led Hoare to work out a way, using recursion, to finally explain the idea that made his name in a simple and elegant way: the fast sorting algorithm he invented called Quicksort. The ideas included in ALGOL had started to prove their worth.

The idea of including recursion in a programming language was part of the foundation for the idea of functional programming languages. They are mathematically pure languages that use recursion as the way to repeat instructions. The mathematical purity makes them much easier to understand and so write correct programs in. Peter ran with the idea of programming in this way. He showed the power that could be derived from the fact that it was closely linked to a kind of logic called the Lambda Calculus, invented by Alonso Church. The Lambda Calculus is a logic built around mathematical functions. One way to think about it is that it is a very simple and pure way to describe in logic what it means to be a mathematical function – as something that takes arguments and does computation on them to give results. This Church showed was a way to define all possible computation just as Turing’s Turing machine is. It provides a simple way to express anything that can be computed.

Peter showed that the Lambda Calculus could be used as a way to define programming languages: to define their “semantics” (and so make the meaning of any program precise).

Having such a precise definition or “semantics” meant that once a program was written it would be sure to behave the same way whatever actual computer it ran on. This was a massive step forward. To make a new programming language useful you had to write compilers for it: translators that convert a program written in the language to a low level one that runs on a specific machine. Programming languages were generally defined by the compiler up till then and it was the compiler that determined what a program did. If you were writing a compiler for a new machine you had to make sure it matched what the original compiler did in all situations … which is very hard to do.

So having a formal semantics, a mathematical description of what a compiler should do, really makes a difference. It means anyone developing a new compiler for a different machines can ensure the compiler matches that semantics. Ultimately, all compilers behave the same way and so one program running on two different manufacturer’s machines are guaranteed to behave the same way in all situations too.

Peter went on to invent the programming language ISWIM to illustrate some of his ideas about the way to design and define a programming language. ISWIM stands for “If you See What I Mean”. A key contribution of ISWIM was that the meaning of the language was precisely defined in logic following his theoretical work. The joke of the name meant it was logic that showed what he meant, very precisely! ISWIM allowed for recursive functions, but also allowed recursion in the definition of data structures. For example, a List is built from a List with a new node on the end. A Tree is built from two trees forming the left and right branches of a new node. They are defined in terms of themselves so are recursive.

Building on his ideas around functional programming, Peter also invented something he called the SECD machine (named after its components: a Stack, Environment, Control and Dump). It effectively implements the Lambda calculus itself as though it is a programming language.ISWIM provided a very simple but useful general-purpose low level language. It opened up a much easier way to write compilers for functional programming languages for different machines. Just one program needed to be written that compiled the language into SECD. Then you had a much simpler job of writing a compiler to convert from the low level SECD language to the low level assembly language of each actual computer. Even better, once written, that low level SECD compiler could be used for different functional programming languages on a single machine. In SECD, Peter also solved a flaw in ALGOL that prevented functions being fully treated as data. Functions as Data is a powerful feature of the best modern programming languages. It was the SECD design that first provided a solution. It provided a mechanism that allowed languages to pass functions as arguments and return them as results just as you could do with any other kind of data without problem.

In the later part of his life Peter focussed much more on his work supporting the LGBTQ+ community having decided that Computer Science was not doing the good for humanity he once hoped. Instead, he thought it was just supporting companies making profit, ahead of the welfare of people. He decided that he could do more good as part of the LGBTQ+ community. Since his death there has been an acceleration in the massing of wealth by technology companies, whereas support for diversity has made a massive difference for good, so in that he was prescient. His contributions have certainly, though, provided a foundation for better software, that has changed the way we live in many ways for the better. Because of his work they are less likely to cause harm because of programming mistakes, for example, so in that at least he has done a great deal of good.

by Paul Curzon, Queen Mary University of London

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A visit to the Turing Machine: a short story

by Greg Michaelson

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.

From the cs4fn archive.

Burning City — Image by JL G from Pixabay

“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.

————————-

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.

A stone looking like a scared face — Image by Dean Moriarty from Pixabay

————————-

“…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.

——————————

“…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…”

——————————

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.

——————————

“…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.”

Related Magazines …

Issue 14 – Alan Turing – The genius who gave us the future

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EPSRC supports this blog through research grant EP/W033615/1,

Lego Computer Science: Turing Machines Part 3: the program

by Paul Curzon, Queen Mary University of London

We have so far built the hardware of a Lego Turing Machine. Next we need the crucial part: software. It needs a program to tell it what to do.

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:

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 Turing machine instruction
Current state: orange
Currnent value red
New state green
New value blue
Direction red — 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 Turing Machine Program
Red-Blue -> Red-Blue-Red
Red-Grey -> Orange-Blue-Yellow
Orange-Blue -> Orange-Blue-Yellow
Orange-Grey -> Black-Grey-Red — 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 Turing Machine with Tape, Controller and Program. — 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…

Lego Computer Science

Image shows a Lego minifigure character wearing an overall and hard hat looking at a circuit board, representing Lego Computing — Image by Michael Schwarzenberger from Pixabay

Part of a series featuring featuring pixel puzzles,
compression algorithms, number representation,
gray code, binary and computation.

Related Magazines …

Issue 14 – Alan Turing – The genius who gave us the future

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.

Lego Computer Science: Turing Machines Part 2: the controller

by Paul Curzon, Queen Mary University of London

Last time we started to build a working computer out of Lego: a Turing Machine. So far we have seen that we can make the memory of a Turing Machine (its Infinite Tape) in Lego. We can also create a movable Tape Head that marks the position where data can be read from and written to the tape (see image).

Controlling states

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…

Lego Computer Science

Part of a series featuring featuring pixel puzzles,
compression algorithms, number representation,
gray code, binary and computation.

Related Magazines …

Issue 14 – Alan Turing – The genius who gave us the future

Lego Computer Science: Turing Machines Part 1: the tape

by Paul Curzon, Queen Mary University of London

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... — 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 Data Values 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…).

Lego Computer Science

Part of a series featuring featuring pixel puzzles,
compression algorithms, number representation,
gray code, binary, logic and computation.

Related Magazines …

Issue 14 – Alan Turing – The genius who gave us the future

Why the Romans were pants at maths

Paul Curzon, Queen Mary University of London

The Romans were great at counting and addition but they were absolutely pants at multiplication. It wasn’t because they were stupid. It was because they hadn’t invented a good way to represent numbers, and that meant they needed really convoluted algorithms.

The Roman system is based on an earlier really simple way of writing numbers. You just put a line for each thing you’ve counted. Its probably the way shepherds kept count of sheep, drawing a line for each sheep. Those lines turned into the Roman letter I. To add 1 to a number you just add another I. You count: I, II, III, and so on and it makes counting easy.

Roman number blocks image by Clker-Free-Vector-Images from Pixabay

This system is called unary – whereas binary involves counting with two symbols, 1 and 0, in unary you only have one symbol to count with. Addition in unary is easy too at least for small numbers. Take the first number and add on the end all the Is for the second and you’ve got the answer number. This is exactly the way we all start doing addition on our fingers.To add 2+3, hold up 2 fingers (II) then hold up another three fingers (III) and you have the answer (IIIII).

This is fine for small numbers but it gets a bit tedious as the numbers increase (and you run out of fingers!) Comparing numbers is easy in principle – do you have the same number of Is, but hard in practice for large numbers. We can’t keep all those Is in our head so a large number is hard to think about. To get round this the Romans invented new letters to stand for groups of Is. This is what we do when we tally numbers making a crossbar for every fifth number we count. It helps us keep track of larger numbers. The Romans invented a whole bunch of symbols to help: so for example in the Roman numeral system, V stands for 5 (IIIII), X stands for 10, L for 50, C for 100, D for 500 and M for 1000. They had invented a new way to represent numbers.

This makes it much easier to write and compare larger numbers. Now when counting and you get up to 5 you just replace all those Is with a V and then carry on adding Is: VI, VII, VIII, VIIII. Then you get to VIIIII (10) so replace it all with an X, starting again adding a new lot of Is: XI, XII, XIII, XIIII, XV, and so on. Counting large numbers is now a bit more involved – the algorithm involves more than just adding an I on the end, but it is much more convenient. The addition algorithm has now become more complicated, though it is still fairly simple too. Take any two numbers to add like VII and VIII and string them together: VIIVIII. Now group together the same letters: VVIIIII. Anywhere you have enough to replace symbols with the next character do so. VV can be replaced by X and IIIII can be replaced by V to give XV in the above. Keep making replacements until you can make no more. Put the symbols in order from largest to smallest symbol and you have your answer.

Now the Romans were obviously a bit lazy as bored with writing even four Is in a row they sometimes introduced a new set of abbreviations, so that IIII became IV and VIIII became IX. Putting a smaller symbol (like I) before a larger one (like X) instead of after meant subtract it to get the number. so IX means “one less than 10” or 9. Counting just got a tiny bit more complicated to get the advantage of writing fewer symbols. Addition now needs a more convoluted algorithm though. There are several ways to do it. The easiest is actually just to change the numbers to add to the simpler form (so IV goes back to IIII). You them do the addition that way, and convert back at the end. Addition just got that little bit harder, and all because of a change in representation.

Worse, doing any more complicated maths is even harder still using the Roman number representation. See if you can work out how to multiply Roman numbers. The Roman number system doesn’t help at all. The only really easy way is to just repeatedly add ( so III x VI is VI + VI + VI). That just isn’t practical for large numbers. Try it on XXIII x LXV1. There are other possible ways including one that is actually based on the binary multiplication algorithms computers use – multiplying and dividing repeatedly by 2. See if you can work out how to do it. Whatever way you do it, its clear that the number system the Romans chose made maths hard for them to do!

A good representation makes maths easy. A bad one makes it much harder to do

Luckily, Indian and Arabian scholars understood that the representation they used mattered. They invented, and spread, the Hindu-Arabic numbers and decimal system we use today. What is special about it is that rather than introducing new symbols for bigger and bigger numbers, the position of a symbol is used instead. As we go from nine to ten we go back to the start of our symbols, from 9 back to 0, but stick a 1 in a new 10s column to count how many 10s we have. Counting is still pretty easy but suddenly not only is the algorithm for addition straightforward but we can come up with fairly simple algorithms for multiplication and division too. They are the algorithms you learn at school – though as with any algorithm making sure you follow the steps exactly and don’t miss steps is hard for a human (unlike for a computer). That is why we tend to find learning maths hard at first and it gets easier the more we practice.

In fact Romans needing to do serious maths probably used a variation of an abacus representing numbers with stones. They would do a calculation on the abacus and then convert the answer back into the Roman number system. And guess what. The Roman Abacus uses columns to represent larger numbers in a very similar way to the Hindu-Arabic system. The Romans understood that representation matters too.

Sometimes things are hard to do just because we make them hard! The secret of coming up with good algorithms is often to come up with a good representation first. In programming too, if you come up with a good way to represent data, a good data structure, you can often then make it much easier to write an efficient program.

This article was first published on the original CS4FN website.

EPSRC supports this blog through research grant EP/W033615/1.

Balls, beams and quantum computers – performing calculations with patterns of light

by Jane Waite, Queen Mary University of London

Have you played the seaside arcade game where shiny metal balls drops down to ping, ping off little metal pegs and settle in one of a series of channels? After you have fired lots of balls, did you notice a pattern as the silver spheres collect in the channels? A smooth glistening curve of tiny balls forming a dome, a bell curve forms. High scores are harder to get than lower ones. Francis Galton pops up again*, but this time as a fellow Victorian trend setter for future computer design.

Photo credit: Galton Box by Klaus-Dieter Keller, Public Domain, via Wikimedia Commons, via the Wikipedia page for the Galton board

Francis Galton invented this special combination of row after row of offset pins and narrow receiving channels to demonstrate a statistical theory called normal distribution: the bell curve. Balls are more likely to bounce their way to the centre, distributing themselves in an elegant sweep down to the left and right edges of the board. But instead of ball bearings, Galton used beans, it was called the bean machine. The point here though is that the machine does a computation – it computes the bell curve.

Skip forward 100 years and ‘Boson Samplers’, based on Galton’s bean machine, are being used to drive forward the next big thing in computer design, quantum computers.

“Stripes beams lights” image by David from Pixabay

Instead of beans or silver balls computer scientists fire photons, particles of light through minuscule channels on optical chips. These tiny bundles of energy bounce and collide to create a unique pattern, a distribution though one that a normal digital computer would find hard to calculate. By setting it up in different ways, the patterns that result can correspond to different computations. It is computing answers to different calculations set for it.

Through developing these specialised quantum circuits scientists are bouncing beams of light forwards on the path that will hopefully lead to conventional digital technology being replaced with the next generation of supercomputers.

Watch the bell curve appear in real time.

This article was first published on the original CS4FN website and a copy is also on page 17 of issue 20 of the CS4FN magazine, Ada Lovelace: the computer scientist without a computer. You can download a free PDF copy below, along with all of our free material at our downloads site.

*Francis Galton appears earlier in Issue 20, you can read more about him on page 15 of the PDF. Although a brilliant mathematician he held views about people that are unacceptable today. In 2020 University College London (UCL) changed the name of its Galton Lecture Theatre, which had been named previously in his honour, to Lecture Theatre 115.

Related Magazine …

CS4FN Issue 20 – Ada Lovelace: the computer scientist without a computer

EPSRC supports this blog through research grant EP/W033615/1.

Hidden Figures: NASA’s brilliant calculators

Full Moon with a blue filter — Full Moon image by PIRO from Pixabay

NASA Langley was the birthplace of the U.S. space program where astronauts like Neil Armstrong learned to land on the moon. Everyone knows the names of astronauts, but behind the scenes a group of African-American women were vital to the space program: Katherine Johnson, Mary Jackson and Dorothy Vaughan. Before electronic computers were invented ‘computers’ were just people who did calculations and that’s where they started out, as part of a segregated team of mathematicians. Dorothy Vaughan became the first African-American woman to supervise staff there and helped make the transition from human to electronic computers by teaching herself and her staff how to program in the early programming language, FORTRAN.

The women switched from being the computers to programming them. These hidden women helped put the first American, John Glenn, in orbit, and over many years worked on calculations like the trajectories of spacecraft and their launch windows (the small period of time when a rocket must be launched if it is to get to its target). These complex calculations had to be correct. If they got them wrong, the mistakes could ruin a mission, putting the lives of the astronauts at risk. Get them right, as they did, and the result was a giant leap for humankind.

See the film ‘Hidden Figures’ for more of their story (trailer below).

– Paul Curzon, Queen Mary University of London

from the archive

Magazines …

Front cover of CS4FN issue 29 - Diversity in Computing

See more in ‘Celebrating Diversity in Computing‘

We have free posters to download and some information about the different people who’ve helped make modern computing what it is today.

Screenshot showing the vibrant blue posters on the left and the muted sepia-toned posters on the right

Or click here: Celebrating diversity in computing

Subscribe to be notified whenever we publish a new post to the CS4FN blog.

This page is funded by EPSRC on research agreement EP/W033615/1.

Lego computer science: What is computation? (simple cellular automata)

Continuing a series of blogs on what to do with all that lego scattered over the floor: learn some computer science…what is computation? Using binary.

We’ve been focussing on representing data so far but data on its own doesn’t do a lot. It is when you combine it with computation that things get exciting and suddenly you have something that can change the world. But what is computation? We will start to explore computation using something called cellular automata. They are just one simple way to do computation (of many).

We have seen that a data representation is just a way of storing information using symbols. It just gives meaning to otherwise arbitrary symbols. Those 1s and 0s, red blocks and blue blocks, Xs, Vs and Is could mean anything. Indeed at different times they mean different things: sometimes a particular group of 1s and 0s stand for a number, sometimes the colour of a pixel, sometimes a letter. So symbols become interesting when we give them meanings (and that is an important point to remember).

Computation is also about symbols, but about manipulating them using sets of rules. What do the rules do? Given one or more symbols they tell you to swap those symbols for new symbols. To do computation you just repeatedly apply a given set of rules, starting with some starting symbols and the symbols change and then change again and then change again …

Elementary Cellular Automata

Cellular automata are just a particular kind of rules that apply to grids of symbols (called cells). They were invented by one of the great original computer scientists, John von Neumann along with Stanislaw Ulam in the 1940s.

Elementary cellular automata, which we will look at here, are a simple version where you just have a row of cells (so a row of symbols). There are only two symbols allowed, usually 1 and 0. We will of course use lego blocks as our two symbols instead: a red brick for 1 and a blue brick for 0. A particular row of red and blue bricks is called the state. The rules change the colour of the bricks in the row and so change the state of the cellular automata. Here is an example state of such a ‘machine’ where the rows are 16 bricks (symbols) long (essentially the memory of the machine will be 16 bits long):

An initial state in lego bricks: a pattern of red and blue bricks. — An example cellular automaton state consisting of 16 symbols. Traditionally cellular automata have symbols 0 and 1. We use a red block to mean a 1 and a blue block to mean a 0. Image by Paul Curzon

Rules

One rule RED-B:LUE-BLUE -> RED — A rule that says if we have RED-BLUE-BLUE then change the middle cell to a RED block.
Image by Paul Curzon

Now if we are going to do computation, we need rules (essentially a program) to apply that changes the state. The rules of an elementary cellular automaton like this are applied to each lego brick, changing it to a new lego brick. To do so they take the brick on either side into account though. Each rule therefore looks at three bricks at a time and changes (or not) the middle brick.

We can write out the rules using lego bricks too – saying what to do for each pattern of three lego blocks. So we could have a rule that if we have a triple RED-BLUE-BLUE then we change the middle of that triple to RED so that the triple becomes RED-RED-BLUE instead. In lego we could represent this rule as shown right, where we show the new value for the middle cell that changes. (Notice we are now using lego bricks, so symbols, to represent rules: a rule is just data too!)

Now a vital thing about rules for computation is that you MUST give a rule for every possibility. Our above rule only tells us what to do for one of the eight possibilities of those triples of bricks that might occur in each position. We must give 8 different rules, so that whatever pattern we come across we have a rule that says what to do.

Here is one possible set of 8 rules we could use:

A set of rules to define how a cellular automaton will behave.
Image by Paul Curzon

Altogether, there are 256 different possible sets of rules like this.

Notice that we have ordered our rules using a binary pattern of the triples counting from 0 to 7 as a way to make sure we have covered every possible pattern exactly once and to make it easier to find the right rule. We could write then in any order of course. It would make no difference to what the rules do.

Doing Computation

Now to do computation we just apply the rules we have chosen to every position in an initial state – an initial pattern of red and blue blocks. We start at one end of the row and apply the set of rules in each position finding the one that matches the pattern at that position. Once we find the rule that matches that position, we note the new middle block accordingly, then move on to the next position. Once we get to the end of the row, we know what the whole new state for the automaton will be: we have done one step of computation. For the cell at either end of the row, assume its adjacent value off the end is 0 (so blue for us). At every position the rule applies to the original triple of bricks in the current state, not ones changed by rules applied to other positions: the rules are applied to every position at the same time.

The easiest way to do this with lego is to line up a row of red and blue lego blocks as the initial state and apply our rules as above to get a new pattern of red and blue lego blocks placed below it. That new pattern is the new state of the machine, Here is a step as applied to our random state we gave above.

Applying the rules to a random starting state — Image by Paul Curzon

Calculating Number Sequences

We seem to be just replacing patterns by new patterns. Are we doing anything useful? Of course we could give some simple meaning to these patterns. Interpret the pattern as a binary number and what is happening? We are generating a number sequence. To see this use the above rules on a shorter pattern, starting with a single red lego block at the left hand end, with the rest blue. This is the binary for the number 1 (00001). Apply the rules and we get the number 2 (00010). Apply the rules again and the pattern of lego turns into the binary for 5 (00101), then 8 (01000) and then 20 (10100) and so on…

The series of transformations through binary patterns from applying the rules. — Image by Paul Curzon

We have created a machine that does a calculation on a number to create a new number. Let it run and it calculates the whole number sequence. Different rules will compute different number sequences: some perhaps more interesting than others.

Images from numbers

If you think numbers are a bit boring, then instead just give a different meaning to the patterns – as giving the colour of pixels, with each new state giving the next row of an infinite lego pixel picture. Now our rules are generating art. Each rule set will compute a different image as will different starting states (again some images generated will be more interesting than others). Here is what our above rules generate if we start with a single red brick in the centre:

The top of a Sierpiński triangle as generated in lego bricks from out rules. — The image generated by our rule if we see it as rules to generate the next line of a lego art image.
Image by Paul Curzon

This is actually a fractal pattern called the Sierpiński triangle. It contains the same triangular pattern over and over again, and If you create a massive version of it on a large lego board you will see that each triangle has the same pattern within it. It is a beautiful recursive pattern.

Sierpiński triangle Image from Wikimedia CC BY-SA 3.0

Apply the rules and create a Lego pixel version yourself.

Explore the different rules

Stephen Wolfram has exhaustively explored all the elementary cellular automata, categorising them and describing their properties. However, that is no reason not to explore them yourself, whether with lego, on graph paper or by writing a program to apply the rules for you.

Of course you do not have to stick to only automata with 2 symbols. Add more symbols / colours of lego blocks (so you will need lots more rules in each set) and explore some more.

There is one cellular automaton, so one rule set (with only two symbols) that is very intriguing. It turns out that, rather than just generate a particular number sequences or pattern as the one above does, it can do absolutely any computation – it is a general purpose machine that can do anything that a modern computer can do…but that is another story.

Lego Computer Science

Part of a series featuring featuring pixel puzzles,
compression algorithms, number representation,
gray code, binary, logic and computation.

This post was 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.

Executable Biology

Computing cancer using computational modelling

(From the archive)

Cells Mitosis — Image by Colin Behrens from Pixabay

Can a robot get cancer? Silly question. Our bodies are made of cells. Robots aren’t. Cells are the basic building blocks of life and come in lots of different forms from long thin nerve cells that allow us to sense the world, to round blood cells that carry oxygen around our bodies. Cancer occurs when cells go rogue and start reproducing in an uncontrolled way. A computer can’t get cancer, but you can allow virtual diseases to attack virtual cells inside a computer. Doing that may just help find cures. That is what Jasmin Fisher, who leads a research group at Microsoft Research in Cambridge, has devoted her career to.

Becoming a medic isn’t the only way to help save lives!

Computational Modelling is changing the way the sciences are done. It is the idea that you can run experiments on virtual versions of things you are investigating. A computer model is essentially just a program that simulates the phenomena of interest. For example, by writing a program that simulates the laws of Physics, you can use it to run virtual Physics experiments about the motion of the planets, say. If your virtual planets do follow the paths real planets do, then you have evidence the laws are right. If they don’t your laws (or the models) need to change. You can also make predictions such as when an eclipse will happen. If you are right it suggests the laws you coded are good descriptions of reality. If wrong, back to the drawing board.

Jasmin has been pioneering this idea with the stuff of life and death. She focusses on modelling cells and the specific ways that we think cancer attacks them. It gives a way of exploring what is going on at the level of the molecules inside cells, and so how well new medicines might, or might not, work. Experiments can be done quickly and easily on the programmed models by running simulations. That means the real experiments, taking up expensive lab time, can focus on things that are most likely to be successful. Jasmin’s work has helped researchers design more effective actual experiments because they start with a better understanding of what is going on. One of the most important questions she is studying is how cells end up becoming what they are, and how this differs between normal cells and cancer cells. Understand this and we will be much closer to understanding how to stop cancer.

– Paul Curzon, Queen Mary University of London

Related Magazines …

This story was originally published here and is an article from CS4FN, a free computer science magazine from Queen Mary University of London which is sent to subscribing UK schools. To find out more please visit our About page. The article was also published in issue 23, The Women Are (Still) Here, on p3.

EPSRC supports this blog through research grant EP/W033615/1.

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