From the archive: This article by Dean Miller, is an edited version of one of the 2006 winning essays from the Queen Mary University of London, Department of Computer Science, first year essay competition.
May I ask you a question? When you think of the computer what names ring a bell? Bill Gates? Or for those more in touch with the history behind computers maybe Charles Babbage is a familiar name? May I ask you another question please? Do you know who Dr Mark Dean is? No, well you should. Do not worry yourself though, you are definitely not alone. I did not know of him either.
Allow me to enlighten you..
Mark Dean is in my opinion a very creative and inspirational black computer scientist. He is a vice-president at IBM and holds 3 of IBM’s first 9 patents on the personal computer. He has over 30 patents pending. He won the Black Engineer of the Year Presidents Award and was made an IBM fellow in 1995. An IBM fellow is IBM’s highest technical honor. Only 50 of IBM’s employee’s are fellows and Mark Dean was the first black one. Prior to joining IBM in 1980 he earned degrees in Electrical Engineering before going back to school to gain a PhD in the field from Stanford University. He was born in 1957 in Jefferson City, Tennessee and was one of the first black students to attend Jefferson City High School. He was an exceptional student and enjoyed athletics. Early manifestations of his desire to create were shown when he and his father built a tractor from scratch when he was just a boy.
Upon joining IBM Mark Dean and a partner led the team that developed the interior architecture (ISA systems bus) which allowed devices like the keyboard and printer to be connected to the motherboard making computers a part of our lives. It was that which earned him a spot in the National Inventors Hall of Fame. While at IBM he has been involved in numerous positions in computer system hardware architecture and design. He was responsible for IBM’s research laboratory in Austin, Texas where he focused on developing high performance microprocessors, software, systems and circuits. It is here where he made history by leading the team that built a gigahertz chip which did a billion calculations per second. In 2004, he was chosen as one of the 50 most important Blacks in Research Science.
He and his father built a tractor from scratch when he was just a boy
I think that such a man should be well recognized in computer science, especially to black computer science students because from what I can see we are rare. We as a minority need an inspirational figure like Mark Dean. He inspires me, I wanted to share that with you. Before this small article it is very probable you had no knowledge of this man. So if there comes a time where you are asked about important names in the field of computers, I hope Dr Mark Dean springs to mind and rings a bell for you to hear loud and clear.
Edie Schlain Windsor was a senior systems programmer at IBM. There is more to life than computing though. Just like anyone else, Computer Scientists can do massively important things aside from being very good at computing. Civil rights and over-turning unjust laws are as important as anything. She led the landmark Supreme Court Case (United States versus Windsor) that was a milestone for the rights of same-sex couples in the US.
Born to a Jewish immigrant family, Edie worked her way up from an early data entry job at New York University to ultimately become a senior programmer at IBM and then President of her own software consultancy where she helped LGBTQ+ organisations become computerised.
Having already worked as a programmer at an energy company called Combustion Engineering, she joined IBM on completing her degree in 1958 so was one of the early generation of female programmers, before the later idea of the male programmer stereotype took hold. Within ten years she had been promoted to the highest technical position in IBM, that of a Senior Systems Programmer: so one of their top programmers lauded as a wizard debugger. She had started out programming mainframe computers, the room size computers that were IBM ‘s core business at the time. They both designed and built the computers as well as the operating system and other software that ran on them. Edie became an operating systems expert, and a pioneer computer scientist also working on natural language processing programs, aiming to improve the interactivity of computes. Natural Language Processing was then a nascent area but that by 2011 IBM led spectacularly with its program Watson winning the quiz show Jeopardy! answering general knowledge questions playing against human champions.
Before her Supreme Court case overturned it, a law introduced in 1996 banned US federal recognition of same-sex marriages. It made it federal law that marriage could only exist between a man and a woman. Individual states in the US had introduced same-sex marriage but this new law meant that such marriages were not recognised in general in the US. Importantly, for those involved it meant a whole raft of benefits including tax, immigration and healthcare benefits that came with marriage were denied to same-sex couples.
Edie had fallen in love with psychologist Thea Spyer in 1965, and two years later they became engaged, but actually getting married was still illegal. They had to wait almost 30 years before they were even allowed to make their partnership legal, though still at that point not marry. They were the 80th couple to register on the day such partnerships were finally allowed. By this time Thea had been diagnosed with multiple sclerosis, a disease that gradually leads to the central nervous system breaking down, with movement becoming ever harder. Edie was looking after her as a full time carer, having given up her career to do so. They both loved dancing and did so throughout their life together even once Thea was struggling to walk, using sticks to get on to the dance floor and later dancing in a wheelchair. As Thea’s condition grew worse it became clear she had little time to live. Marriage was still illegal in New York, however, so before it was too late, they travelled to Canada and married there instead.
When Thea died she left everything to Edie in her will. Had Edie been a man married to Thea, she would not have been required to pay tax on this inheritance, but as a woman and because same-sex marriages were deemed illegal she was handed a tax bill of hundreds of thousands of dollars. She sued the government claiming the way different couples were treated was unfair. The case went all the way to the highest court, the Supreme Court, who ruled that the 1996 law was itself unlawful. Laws in the US have as foundation a written constitution that dates back to 1789. The creation of the constitution was a key part of the founding of the United States of America itself. Without it, the union could easily have fallen apart, and as such is the ultimate law of the land that new laws cannot overturn. The problem with the law banning same sex marriage was that it broke the 5th amendment of the constitution added in 1791, one of several amendments made to ensure people’s rights and justice was protected by the constitution.
The Supreme Court decision was far more seismic than just refunding a tax bill, however. It overturned the law that actively banned same-sex marriage, as it fell foul of the constitution, and this paved the way for such marriages to be made actively legal. In 2014 federal employees were finally told they should perform same-sex marriages across the US, and those marriages gave the couple all the same rights as mixed-sex marriages. Because Edie took on the government, the US constitution, and so justice for many, many couples prevailed.
The truth table for NOT P. A yellow brick represents P. Blue means True and Red means false. Read along the rows to get the meaning of NOT P when P is true or false. Image by CS4FN
We have seen how to represent truth tables in lego. Truth tables are a way of giving precise meaning to logical operations like AND, OR and NOT. They are also give a way to do logical reasoning following a simple algorithm.
That’s Not Not True
You may have been pulled up in English and told you just said the opposite of what you meant, after saying something like “There ain’t no way I’m doing that”. This is a “double negative” as the “n’t” in “ain’t” is really “not” so followed by “no way” you are actually saying “not not way” or overall: “I am doing that”. Perhaps the most famous double negative is in the Rolling Stones song “(I can’t get no) satisfaction”. English is very flexible though and double negatives like this don’t cancel out but just become a different way of saying the negative version. In logic two negations do cancel out, though. Let’s take a purer version to work with: the statement “I am not not happy”. What does this mean? In logic the basic proposition here is “I am not happy”. The logical statement is “NOT (NOT (I am happy))”.
We can prove what this means using truth tables. We can do more than just prove what this single statement means. We can prove what all double negatives mean, more generally. We do this by replacing the proposition “I am happy” with a variable P. It now becomes NOT (NOT P) or in our lego version where we use a yellow brick to mean a proposition, P:
Image by CS4FN
This is just syntax, just a sequence of symbols. It doesn’t give us any meaning on its own. We can build truth tables in Lego for that. We start from the variables that are at the inside of the logical expression which here is just the variable P. We list in a table column the possible values it can take (true or false).
Image by CS4FN
This shows P (yellow) can be either be TRUE (blue) or FALSE (red). Now we build up the logical expression of interest a column at a time. NOT is applied to P, so we add a new column for NOT P and use the truth table for the operator, NOT, to tell us what lego brick to put in each row based on the lego brick already there. The NOT truth table is at the top of the page. It says if you have a blue brick in a row, place a red brick there. If you have a red brick, put a blue brick there. This gives us a new filled out column for (NOT P) which is just a copy of the NOT truth table (but bare with us that was just a simple case). We get:
Image by CS4FN
Moving outwards in the expression NOT (NOT P)), we now look at the operator applied to (NOT P). It is NOT again. We add a new column to our truth table and again use the NOT truth table to work out the new values, but this time applied to the column before (the NOT P column). The NOT truth table says put a blue brick for a red brick, and a red brick for a blue brick in the column it is being applied to (the NOT P column). This gives:
Image by CS4FN
The result is a truth table with coloured bricks identical to that of the original column for P. Switching back from lego bricks to what the columns mean, we have shown that the NOT(NOT P) column is the same as the P column, or in other words that NOT(NOT P) EQUALS P (whatever value P has).
We can actually go a step further though, because equivalence is just a logical operation with its own truth table. It gives true if the two operands have the same value and false otherwise (or in lego terms if the bricks are the same colour the answer is a blue brick and if they are different colours the answer is a red brick. The truth table looks like this:
Image by CS4FN
We can use this truth table to calculate whether two lego truth table columns are equal or not just by looking up the combinations in this EQUALS truth table. Continuing our example we can carrying building our truth table about NOT(NOT P)). To make things clearer first add a column corresponding to P again. That means we will be applying the EQUALS operator to the last two columns. As before, for each row, look up the corresponding pattern for those last two columns in the EQUALS truth table to get the answer for that row. In the first row we have two blue bricks so that becomes a blue brick according tot he EQUALS truth table. In the next row we have two red bricks. That also becomes a blue brick. This gives:
Image by CS4FN
The thing to notice here is that all the entries in the final answer column are blue lego pieces. Switching back from the lego world to the logic world, what does this mean? Blue is true so all rows in the answer are true. That means whatever value of the proposition P the answer to NOT (NOT P) EQUALS P is true. We have proved a theorem that this is always true. We have shown by building with lego that a double negation cancels itself out.
Logical expressions like this that are always true (whatever the values of the variables) are called tautologies. We can tell something is a tautology, so we have proved a theorem, just by the simple manual check that its truth table values are true (or in lego all blue).
The important thing to realise about this is all the reasoning can be done without knowing what the symbols mean, and certainly not worrying about English words, once you have the truth tables. You do it mechanically. You do not need to think about what, for example, red and blue mean until the end. At that point you return to the logical world to see what you have found out. All blue means it is always true! You can also at that point substitute back in actual words of interest into the statements proved. P means “I am happy”. We started by asking what “I am not not happy” means. We converted this to “NOT (NOT (I am happy))”. By swapping in “I am happy” for P in our theorem gives us that NOT (NOT “I am happy”) EQUALS “I am happy”, or that “I am not not happy.” just means the same as “I am happy”
We have been reasoning about English statements, but this kind of reasoning is the basis of all logical reasoning and essentially the basis of formal verification where the meaning of programs and hardware is checked to see if it meets a specification. It tells you what a test in a program like “if (! temperature != 0) …) means so does for example, or what a circuit with two NOT gates does.
And lego logic has even given us a way to prove things just by building with lego.
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.
Truth tables are a simple way of reason about logic that were popularised by the 20th century philosopher Ludwig Wittgenstein. They provide a very clear way to explain what logical operators like AND, OR and NOT mean (or in computational terms, what they do). They also give a simple way to do pure logical reasoning and so see if arguments follow logically. These logical operators crop up in logic circuits and in programs where decisions are being made so are vital to creating correct circuits and writing correct programs. Let’s see what a truth table is by making some from Lego.
Logic in Lego
First we need to represent the basic building blocks of logic in lego. We’ve seen in previous articles how to represent numbers, binary and even images in lego. We have seen that we do computation on symbols and we can use lego blocks as symbols. Logic can therefore be represented in lego symbols too.
We will look at a simple kind of logic called propositional logic (there isn’t actually just one kind of logic but lots of different kinds with different rules). Propositional logic is the simplest kind. It deals with propositions which are just statements that are either true or false (but we may not know which). For example, “Snoopy is a logician.” is a proposition. So are “The world is flat.”, “Water contains oxygen.” and “temperature > 0” as we might find in a program. For the purposes of logic itself, it doesn’t matter what the words actually mean or even what they are. We will therefore represent all propositions by square lego blocks of different colours.
Here we want the symbols to stand for logical things rather than numbers. There are lots of numerical values: things like 1, 5 and 77. There are only two logical values: TRUE and FALSE, often written just as T and F. We will use a blue lego block for the logical value TRUE, and a red block for the value FALSE. They are just symbols though so we could use any blocks and any colours, just as we could use other words for true and false as other languages do. We chose blue for true just because it rhymes so is easy to remember, and red more randomly because it is a common lego primary colour.
True and False lego. A square 2×2 blue block represents True. A square 2×2 red block is false.
What about representing the actual sentences stating purported facts like “Messi is the best footballer ever”, or in a program “n == 1”? Statements like this are called propositions. As far as reasoning logically goes the precise words or even language they are in do not matter. This is something Wittgensteinrealised. When doing reasoning these basic propositions can be replaced by variables like P and Q and the logic won’t change. Rather than use letters we will just use different coloured lego blocks to stand for different propositions, emphasising that the words or even variable names do not matter. So we will use a yellow block for a variable P and a green block for a variable Q. Each of which could stand for absolutely any English proposition we like at any time (though if we want it to stand for a particular proposition then we should define which one clearly).
Propositional variables P and Q are represented by yellow and green blocks
Logical Symbols
What we are really interested in is not just true and false values but the logical operations on propositions. The core of these we use in everyday English: AND, OR and NOT, more technically known as conjunction, disjunction and negation in logic. There are several variations of the symbols used to represent these symbols just as there are for true and false. We will use the versions in lego as below.
The logical operators AND, OR and NOT as lego symbols
These lego symbols will allow us to write out logical expressions about propositions: like “The cat is thirsty AND NOT the cat is hungry” which we might write in English as “The cat is thirsty and not hungry”. If we use a yellow block to mean “The cat is thirsty” and a green block to mean “The cat is hungry” then in lego logic we can write it as follows:
The cat is thirsty AND NOT the cat is hungry P AND (NOT Q)
Of course the yellow and green brick are variables so by changing the propositions they represent it can stand for other things. It can also represent: ” The moon is blue AND NOT The moon is made of cheese.” where the yellow brick represents “The moon is blue” and the green brick represents “The moon is made of cheese”.
Think up some statements that involve AND, OR and NOT and then build representations of then in lego logic like the above.
The meaning of logical connectives
The above gives us symbols for the logical connectives, but so far they have no meaning: it is just syntax. Perhaps you think you know what the words mean. We use words like and, or and not in language rather imprecisely at times based on dictionary-style definitions. They essentially mean the same in English as in logic, but we need to define what they mean precisely. We do not want two different people to have two slightly different understandings of what they mean. This is where truth tables come in. A truth table tells us exactly, and without doubt, what the symbols for the operators mean. The give what is called by computer scientists a formal semantics to the logical connectives.
Let’s look at NOT first. A truth table is just a table that includes as rows all the combinations of true and false values of the variables in a logical expression together with an answer for those values. For example a truth table for the operator NOT, so telling us in all situations what (NOT a) means, is:
P
NOT P
TRUE
FALSE
FALSE
TRUE
A truth table for the NOT operator. Reading along the rows, IF P is TRUE THEN (NOT P) is FALSE; IF P is FALSE THEN (NOT P) is TRUE.
We can build this truth table in lego using our lego representation:
NOT only applies to one proposition, the one it negates, (it is a unary logical connective). That means we only need two rows in the table to cover the different possible values those propositions could stand for. AND (and OR) combine to two propositions (it is a binary logical connective). To cover all the possible combinations of the values of those propositions we need a table with four rows as there are four possibilities.
P
Q
P AND Q
TRUE
TRUE
TRUE
TRUE
FALSE
FALSE
FALSE
TRUE
FALSE
FALSE
FALSE
FALSE
A truth table for the logical AND operator.
We can build this in Lego as:
Reading along the rows this says that if both P and Q are blue (true) then the answer for P AND Q is true. Otherwise the answer is false (red). T
The following is the lego truth table for the logical OR operator
The columns for the two variables yellow/green (P/Q) are the same, setting out all the possibilities. Now the answer is true (blue) if either operand is true (blue) and false (red) when both are false (red).
We have now created lego truth tables that give the meaning of each of these three logical connectives. They aren’t the only logical operators – in fact there are 8 possible binary ones. Have a go at building lego truth tables for other binary logical connectives such as exclusive-or which is true if exactly one of the operands is true, and equivalence which is true if both operands are the same truth value.
Truth tables give precise meanings to logical operators and so to logic. That is useful, but even more usefully, they give a way to reason logically in a clear, price way. By following a simple algorithm to build new truth tables from existing ones, we can prove general facts, that are ultimately about propositions, in lego… as we will see next.
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.
There is real money to be made out there in the virtual world – if you are willing to put in the effort to develop appropriate skills.
You don’t have to be young or a geek either. At the age of 62, grandmother Jacquie Lawson turned a hobby into a multi-million pound business. She is a trained illustrator having originally studied art at St Martins School of Art in London. She bought her first computer in 1998. Despite struggling at the start she taught herself to draw computer animations using Macromedia Flash.
Just for fun she made an animated Christmas e-card and sent it to friends. Her skill as an illustrator combined with her artistic flair meant that suddenly she was inundated with people wanting them from around the world – a wonderful example of viral marketing.
“The Internet is such a fantastic medium. It ought to be better.”
She set up a business, launched the http://www.jacquielawson.com e-card website and is now the market leader – with double the visitors of its nearest rival. As Jacquie says about the Internet: “It’s such a fantastic medium. It ought to be better”.
She believes there is a lot of rubbish on the Internet – which means there is scope for skilled, creative people to make a difference by focusing on detail in what they do. Quality can stand out.
So develop the basic skills, have a great idea, throw in some business savvy…but most of all do it for fun, if you want to end up with a successful business.
Activity
Be inspired by Jacquie Lawson. Make your own computer greeting card for some special occasion (whether Valentine’s day, a birthday, Mothers day or Fathers Day…). It might be a still drawing or an animation. Perhaps it could even be a program. Use a technology you are familiar with, or learn one you haven’t used before for the occasion – Scratch perhaps or a drawing program. Learning new skills can be very rewarding and sometimes can lead to new opportunities.
Ludwig Wittgenstein is one of the most important philosophers of the 20th century. His interest was in logic and truth, language, meaning and ethics. As an aside he made contributions to logical thinking that are a foundation of computing. He popularised truth tables, a way to evaluate logical expressions, and invented the modern idea of tautology. His life shows that you do not have to set out with your life planned out to ultimately do great things.
Wittgenstein was born in Austria, of three-quarters Jewish descent, and actually went to the same school as Hitler at the same time, as they were the same age to within a week. Had he still been in Austria at the time of World War II he would undoubtedly have been sent to a concentration camp. Hitler presumably would not have thought much of him had he known more about him at school. Not only did he have a Jewish background, he was bisexual: it is thought he fell in love four times, once with a woman and three times with men.
Interested, originally, in flying and so aeronautic engineering he studied how kites fly in the upper atmosphere for his PhD in Manchester: flying the kites in the Peak District. He moved on to the study of propellors and designed a very advanced propellor that included mini jet engines on the propellor blades themselves. Studying propellors led him to an interest in advanced mathematics and then ultimately to the foundations of mathematics – a course about which, years later, he taught at Cambridge University that Alan Turing attended. Turing was teaching a course with the same title but from a completely different point of view at the time. His interest in the foundations of maths led to him thinking about what facts are, how they relate to thoughts, language and logic and what truth really was. However, World War I then broke out. During the war he fought for the Austro-Hungarian army, originally safe behind the lines but at his own request he was sent to the Russian Front. He was ultimately awarded medals for bravery. While on military leave towards the end of the war he completed the philosophical work that made him famous, the Tractatus Logico-Philosophicus. After the war though he went to rural Austria and worked as a monastery gardener and then as a primary school teacher. His sister suggested this was “like using a precision instrument to open crates”, though as he got into trouble for being violent in his punishments of the children the metaphor probably isn’t very good as he doesn’t sound like a great teacher and as a teacher he was more like a very blunt instrument.
In his absence, his fame in academia grew, however, and so eventually he returned to Cambridge, finally gained a PhD and ultimately became a fellow and then a Professor of Philosophy. By the time World War II broke out he was teaching philosophy in Cambridge but felt this was the wrong thing to be doing during a war, so despite now being a world famous philosopher went to work as a porter in Guy’s hospital, London.
His philosophical work was ground breaking mainly because of his arguments about language and meaning with respect to truth. However, a small part of has work has a very concrete relevance to computing. His thinking about truth and logic had led him to introduce the really important idea of a tautology as a redundant statement in logic. The ancient Greeks used the word but in a completely different sense of something made “true” just because it was said more than once, so argued to be true in a rhetorical sense. In computational terms Wittgenstein’s idea of a tautology is a logical statement about propositions that can be simplified to true. Propositions are just basic statements that may or may not be true, such as “The moon is made of cheese”. An example of a tautology is (a OR NOT(a)) where (a) is a variable that stands for a proposition so something that is either true or false. Putting in the concrete propositions “The moon is made of cheese” we get:
“(The moon is made of cheese) OR NOT (The moon is made of cheese)”
or in other words the statement
“The moon is made of cheese OR The moon is NOT made of cheese”
Logically, this is always true, whatever the moon is made of. “The moon is made of cheese” can be either true or false. Either it is made of cheese or not but either way the whole statement is true whatever the truth of the moon as one side or other of the OR is bound to be true. The statement is equivalent to just saying
“TRUE”
In other words, the original statement always simplifies to truth. More than that, whatever proposition you substitute in place of the statement “The moon is made of cheese” it still simplifies to true eg if we use the statement instead “Snoopy fought the Red Baron” then we get
“Snoopy fought the Red Baron OR NOT (Snoopy fought the Red Baron)”
Again, whatever the truth about Snoopy, this is a true statement. It is true whatever statement we substitute for (a) and whether it is true or false: (a OR NOT(a)) is a tautology guaranteed to be true by its logical structure, not by the meaning of the words of the propositions substituted in for a.
As part of this work Wittgenstein used truth tables, and is often claimed to have invented them. He certainly popularised them as a result of his work becoming so famous. However, Charles Sanders Peirce used truth tables first, 30 years earlier. The latter was a philosopher too, know as the “Father of Pragmatism” (so hopefully that means he wouldn’t have minded Wittgenstein getting all the credit!)
A truth table is just a table that includes as rows all the combinations of true and false values of the variables in logical expressions together with an answer for those values. For example a truth table for the operator NOT, so telling us in all situations what (NOT a) means, is:
a
NOT a
TRUE
FALSE
FALSE
TRUE
A truth table for the NOT operator. Reading along the rows, IF a is TRUE then (NOT a) is FALSE; IF a is FALSE then (NOT a) is TRUE. Image by CS4FN
The first thing that is important about truth tables is that they give very clear and simple meaning (or “semantics”) to logical operators (like AND, OR and NOT) and so of statements asserting facts logically. Computationally, they make precise what the logical operators do, as the above table for NOT does. This of course matters a lot in programs where logical operators control what the program does. It also matters in hardware which is built up from circuits representing the logical operations. They provide the basis for understanding what both programs and hardware do.
The following is the truth table for the logical OR operator: again the last column gives the meaning of the operator so the answer of computing the logical or operation. This time there are two variables (a) and (b) so four rows to cover the combinations.
a
b
a OR b
TRUE
TRUE
TRUE
TRUE
FALSE
TRUE
FALSE
TRUE
TRUE
FALSE
FALSE
FALSE
A truth table for the logical OR operator. Reading along the rows, IF a is TRUE and b is TRUE then (a OR b) is TRUE; IF a is TRUE and b is FALSE then (a OR b) is TRUE; IF a is FALSE and b is TRUE then (a OR b) is TRUE; IF a is FALSE and b is FASLE then (a OR b) is FALSE; Image by CS4FN
Truth tables can be used to give more than just meaning to operators, they can be used for doing logical reasoning; to compute new truth tables for more complex logical expressions, including checking if they are tautologies. This is the basis of program verification (mathematically proving a program does the right thing) and similarly hardware verification. Let us look at (a OR (NOT a)). We make a column for (a) and then a second column gives the answer for (NOT a) from the NOT truth table. Adding a third column we then look up in the OR truth table the answers given the values for (a) and (NOT a) on each row. For example, if a is TRUE then NOT a is FALSE. Looking up the row for TRUE/FALSE in the OR table we see the answer is TRUE so that goes in the answer column for (a OR (NOT a)). The full table is then:
a
NOT a
a OR (NOT a)
TRUE
FALSE
TRUE
FALSE
TRUE
TRUE
A truth table for the a OR NOT a. Reading along the rows, IF a is TRUE then (a OR (NOT a)) is TRUE; IF a is FALSE then (a OR (NOT a)) is TRUE; Image by CS4FN
Truth tables therefore give us an easy way to see if a logical expression is a tautology. If the answer column has TRUE as the answer for every row, as here, then the expression is a tautology. Whatever the truth of the starting fact a, the expression is always true. It has the same truth table as the expression TRUE (a) where TRUE is an operator which gives answer true whatever its operand.
a
TRUE
TRUE
TRUE
FALSE
TRUE
A truth table for the TRUE operator. Whatever its operand it gives answer TRUE. Image by CS4FN
We can do a similar thing for (a AND (NOT a)). We need the truth table for AND to do this.
a
b
a AND b
TRUE
TRUE
TRUE
TRUE
FALSE
FALSE
FALSE
TRUE
FALSE
FALSE
FALSE
FALSE
A truth table for the logical AND operator. Image by CS4FN
We fill in the answer column based on the values from the (a) column and the (NOT a) column looking up the answer in the truth table for AND.
a
NOT a
a AND (NOT a)
TRUE
FALSE
FALSE
FALSE
TRUE
FALSE
A truth table for the a AND (NOT a). Reading along the rows, IF a is TRUE then a AND (NOT a) is FALSE; IF a is FALSE then a AND (NOT a) is FALSE; Image by CS4FN
This shows that it is not a tautology as not all rows have answer TRUE. In fact, we can see from the table that this actually simplifies to FALSE. It can never be true whatever the facts involved as both (a) and (NOT a) are never true about any proposition (a) at the same time.
Here is a slightly more complicated logical expression to consider: ((a AND b) IMPLIES a). Is this a tautology? We need the truth table for IMPLIES to work this out:
a
b
a IMPLIES b
TRUE
TRUE
TRUE
TRUE
FALSE
FALSE
FALSE
TRUE
TRUE
FALSE
FALSE
TRUE
A truth table for the logical IMPLIES logical operator. Image by CS4FN
When we look up the values from the (a AND b) column and the (a) column in the IMPLIES truth table, we get the answers for the full expression ((a AND b) IMPLIES a) and find that it is a tautology as the answer is always true:
a
b
a AND b
a
(a AND b) IMPLIES a
TRUE
TRUE
TRUE
TRUE
TRUE
TRUE
FALSE
FALSE
TRUE
TRUE
FALSE
TRUE
FALSE
FALSE
TRUE
FALSE
FALSE
FALSE
FALSE
TRUE
A truth table for the logical expression (a AND b) IMPLIES a. Image by CS4FN
Using the same kind of approach we can use truth tables to check if two expressions are equivalent. If they give the same final column of answers for the same inputs then they are interchangeable. Let’s look at (b OR (NOT a)).
a
b
NOT a
(b OR (NOT a))
TRUE
TRUE
FALSE
TRUE
TRUE
FALSE
FALSE
FALSE
FALSE
TRUE
TRUE
TRUE
FALSE
FALSE
TRUE
TRUE
A Truth table for the logical expression (b OR (NOT a)). Image by CS4FN
This gives exactly the same answers in the final column as the truth table for IMPLIES above, so we have just shown that:
(a IMPLIES b) IS EQUIVALENT TO (b OR (NOT a))
We have proved a theorem about logical implication. (a IMPLIES b) has the same meaning as, so is interchangeable with, (b OR (NOT a)). All tautologies are interchangeable of course as they are all equivalent in their answers to TRUE. If we give a truth table for IS EQUIVALENT TO we could even show equivalences like the above are tautologies!
Tautologies, and equivalences, once proved, can also be the basis of further reasoning. Any time we have in a logical expression (a IMPLES b), for example, we can swap it for (b OR (NOT a)) knowing they are equivalent.
Truth tables helped Wittgenstein think about arguments and deduction of facts using rules. In particular, he decided special rules that other philosophers suggested should be used in deduction, were not necessary, as such. Deduction instead works simply from the structure of logic that means logical statements follow from other logical statements. Truth tables gave a clear way to see the equivalences resulting from the logic. Deduction is not about meanings in language but about logic. Truth tables meant you could decide if something was true by looking at equivalences so ultimately tautologies. They showed that some statements were universally true just by inspection of the truth table. For computer scientists they gave a way to define what logical operations mean and then reason about digital circuits and programs they designed, both to help understand, so write them, and get them right.
Wittgenstein started off as an engineer interested in building flying machines, moved to become a mathematician, a soldier, a gardener and a teacher, as well as a hospital porter, but ultimately he is remembered as a great philosopher. Abstract though his philosophy was, along the way he provided computer scientists and electrical engineers useful tools that helped them build thinking machines.
Alan Turing was born in London on 23 June 1912. His parents were both from successful, well-to-do families, which in the early part of the 20th century in England meant that his childhood was pretty stuffy. He didn’t see his parents much, wasn’t encouraged to be creative, and certainly wasn’t encouraged in his interest in science. But even early in his life, science was what he loved to do. He kept up his interest while he was away at boarding school, even though his teachers thought it was beneath well-bred students. When he was 16 he met a boy called Christopher Morcom who was also very interested in science. Christopher became Alan’s best friend, and probably his first big crush. When Christopher died suddenly a couple of years later, Alan partly helped deal with his grief with science, by studying whether the mind was made of matter, and where – if anywhere – the mind went when someone died.
The Turing machine
After he finished school, Alan went to the University of Cambridge to study mathematics, which brought him closer to questions about logic and calculation (and mind). After he graduated he stayed at Cambridge as a fellow, and started working on a problem that had been giving mathematicians headaches: whether it was possible to determine in advance if a particular mathematical proposition was provable. Alan solved it (the answer was no), but it was the way he solved it that helped change the world. He imagined a machine that could move symbols around on a paper tape to calculate answers. It would be like a mind, said Alan, only mechanical. You could give it a set of instructions to follow, the machine would move the symbols around and you would have your answer. This imaginary machine came to be called a Turing machine, and it forms the basis of how modern computers work.
Code-breaking at Bletchley Park
By the time the Second World War came round, Alan was a successful mathematician who’d spent time working with the greatest minds in his field. The British government needed mathematicians to help them crack the German codes so they could read their secret communiqués. Alan had been helping them on and off already, but when war broke out he moved to the British code-breaking headquarters at Bletchley Park to work full-time. Based on work by Polish mathematicians, he helped crack one of the Germans’ most baffling codes, called the Enigma, by designing a machine (based on earlier version by the Poles again!) that could help break Enigma messages as long as you could guess a small bit of the text (see box). With the help of British intelligence that guesswork was possible, so Alan and his team began regularly deciphering messages from ships and U-boats. As the war went on the codes got harder, but Alan and his colleagues at Bletchley designed even more impressive machines. They brought in telephone engineers to help marry Alan’s ideas about logic and statistics with electronic circuitry. That combination was about to produce the modern world.
Building a brain
The problem was that the engineers and code-breakers were still having to make a new machine for every job they wanted it to do. But Alan still had his idea for the Turing machine, which could do any calculation as long as you gave it different instructions. By the end of the war Alan was ready to have a go at building a Turing machine in real life. If it all went to plan, it would be the first modern electronic computer, but Alan thought of it as “building a brain”. Others were interested in building a brain, though, and soon there were teams elsewhere in the UK and the USA in the race too. Eventually a group in Manchester made Alan’s ideas a reality.
Troubled times
Not long after, he went to work at Manchester himself. He started thinking about new and different questions, like whether machines could be intelligent, and how plants and animals get their shape. But before he had much of a chance to explore these interests, Alan was arrested. In the 1950s, gay sex was illegal in the UK, and the police had discovered Alan’s relationship with a man. Alan didn’t hide his sexuality from his friends, and at his trial Alan never denied that he had relationships with men. He simply said that he didn’t see what was wrong with it. He was convicted, and forced to take hormone injections for a year as a form of chemical castration.
Although he had had a very rough period in his life, he kept living as well as possible, becoming closer to his friends, going on holiday and continuing his work in biology and physics. Then, in June 1954, his cleaner found him dead in his bed, with a half-eaten, cyanide-laced apple beside him.
Alan’s suicide was a tragic, unjust end to a life that made so much of the future possible.
The chatbots have suddenly got everyone talking, though about them as much as with them. Why? Because one, chatGPT has (amongst other things) reached the level of being able to fool us into thinking that it is a pretty good student.
It’s not exactly what Alan Turing was thinking about when he broached his idea of a test for intelligence for machines: if we cannot tell them apart from a human then we must accept they are intelligent. His test involved having a conversation with them over an extended period before making the decision, and that is subtly different to asking questions.
ChatGPT may be pretty close to passing an actual Turing Test but it probably still isn’t there yet. Ask the right questions and it behaves differently to a human. For example, ask it to prove that the square root of 2 is irrational and it can do it easily, and looks amazingly smart, – there are lots of versions of the proof out there that it has absorbed. It isn’t actually good at maths though. Ask it to simply count or add things and it can get it wrong. Essentially, it is just good at determining the right information from the vast store of information it has been trained on and then presenting it in a human-like way. It is arguably the way it can present it “in its own words” that makes it seem especially impressive.
Will we accept that it is “intelligent”? Once it was said that if a machine could beat humans at chess it would be intelligent. When one beat the best human, we just said “it’s not really intelligent – it can only play chess””. Perhaps chatGPT is just good at answering questions (amongst other things) but we won’t accept that as “intelligent” even if it is how we judge humans. What it can do is impressive and a step forward, though. Also, it is worth noting other AIs are better at some of the things it is weak at – logical thinking, counting, doing arithmetic, and so on. It likely won’t be long before the different AIs’ mistakes and weaknesses are ironed out and we have ones that can do it all.
Rather than asking whether it is intelligent, what has got everyone talking though (in universities and schools at least) is that chatGPT has shown that it can answer all sorts of questions we traditionally use for tests well enough to pass exams. The issue is that students can now use it instead of their own brains. The cry is out that we must abandon setting humans essays, we should no longer ask them to explain things, nor for that matter write (small) programs. These are all things chatGPT can now do well enough to pass such tests for any student unable to do them themselves. Others say we should be preparing students for the future so its ok, from now on, we just only test what human and chatGPT can do together.
It certainly means assessment needs to be rethought to some extent, and of course this is just the start: the chatbots are only going to get better, so we had better do the thinking fast. The situation is very like the advent of calculators, though. Yes, we need everyone to learn to use calculators. But calculators didn’t mean we had to stop learning how to do maths ourselves. Essay writing, explaining, writing simple programs, analytical skills, etc, just like arithmetic, are all about core skill development, building the skills to then build on. The fact that a chatbot can do it too doesn’t mean we should stop learning and practicing those skills (and assessing them as an inducement to learn as well as a check on whether the learning has been successful). So the question should not be about what we should stop doing, but more about how we make sure students do carry on learning. A big, bad thing about cheating (aside from unfairness) is that the person who decides to cheat loses the opportunity to learn. Chatbots should not stop humans learning either.
The biggest gain we can give a student is to teach them how to learn, so now we have to work out how to make sure they continue to learn in this new world, rather than just hand over all their learning tasks to the chatbot to do. As many people have pointed out, there are not just bad ways to use a chatbot, there are also ways we can use chatbots as teaching tools. Used well by an autonomous learner they can act as a personal tutor, explaining things they realise they don’t understand immediately, so becoming a basis for that student doing very effective deliberate learning, fixing understanding before moving on.
Of course, a bigger problem, if a chatbot can do things at least as well as we can then why would a company employ a person rather than just hire an AI? The AIs can now a lot of jobs we assumed were ours to do. It could be yet another way of technology focussing vast wealth on the few and taking from the many. Unless our intent is a distopian science fiction future where most humans have no role and no point, (see for example, CS Forester’s classic, The Machine Stops) then we still in any case ought to learn skills. If we are to keep ahead of the AIs and use them as a tool not be replaced by them, we need the basic skills to build on to gain the more advanced ones needed for the future. Learning skills is also, of course, a powerful way for humans (if not yet chatbots) to gain self-fulfilment and so happiness.
Right now, an issue is that the current generation of chatbots are still very capable of being wrong. chatGPT is like an over confident student. It will answer anything you ask, but it gives wrong answers just as confidently as right ones. Tell it it is wrong and it will give you a new answer just as confidently and possibly just as wrong. If people are to use it in place of thinking for themselves then, in the short term at least, they still need the skill it doesn’t have of judging when it is right or wrong.
So what should we do about assessment. Formal exams come back to the fore so that conditions are controlled. They make it clear you have to be able to do it yourself. Open book online tests that become popular in the pandemic, are unlikely to be fair assessments any more, but arguably they never were. Chatbots or not they were always too easy to cheat in. They may well be good still for learning. Perhaps in future if the chatbots are so clever then we could turn the Turing test around: we just ask an artificial intelligence to decide whether particular humans (our students) are “intelligent” or not…
Alternatively, if we don’t like the solutions being suggesting about the problems these new chatbots are raising, there is now another way forward. If they are so clever, we could just ask a chatbot to tell us what we should do about chatbots…
What links James Bond, a classic 1950s radio comedy series and a machine for creating music by drawing? … Electronic music pioneer: Daphne Oram.
Oram was one of the earliest musicians to experiment with electronic music, and was the first woman to create an electronic instrument. She realised that the advent of electronic music meant composers no longer had to worry about whether anyone could actual physically perform the music they composed. If you could write it down in a machine readable way then machines could play it electronically. That idea opened up whole new sounds and forms of music and is an idea that pop stars and music producers still make use of today.
She learnt to play music as a child and was good enough to be offered a place at the Royal College of Music, though turned it down. She also played with radio electronics with her brothers, creating radio gadgets and broadcasting music from one room to another. Combining music with electronics became her passion and she joined the BBC as a sound engineer. This was during World War 2 and her job included being the person ready during a live music broadcast to swap in a recording at just the right point if, for example, there was an air raid that meant the performance had to be abandoned. The show, after all, had to go on.
Composing electronic music
She went on to take this idea of combining an electronic recording with live performance further and composed a novel piece of music called Still Point that fully combined orchestral with electronic music in a completely novel way. The BBC turned down the idea of broadcasting it, however, so it was not played for 70 years until it was rediscovered after her death, ultimately being played at a BBC Prom.
Composers no longer had to worry about whether anyone could actually physically perform the music they composed
She started instead to compose electronic music and sounds for radio shows for the BBC which is where the comedy series link came in. She created sound effects for a sketch for the Goon Show (the show which made the names of comics including Spike Milligan and Peter Sellers). She constantly played with new techniques. Years later it became standard for pop musicians to mess with tapes of music to get interesting effects, speeding them up and down, rerecording fragments, creating loops, running tapes backwards, and so on. These kinds of effects were part of amazing sounds of the Beatles, for example. Oram was one of the first to experiment with these kinds of effects and use them in her compositions – long before pop star producers.
One of the most influential things she did was set up the BBC Radiophonic Workshop which went on to revolutionise the way sound effects and scores for films and shows were created. Oram though left the BBC shortly after it was founded, leaving the way open for other BBC pioneers like Delia Derbyshire. Oram felt she wasn’t getting credit for her work, and couldn’t push forward with some of her ideas. Instead Oram set herself up as an independent composer, creating effects for films and theatre. One of her contracts involved creating electronic music that was used on the soundtracks of the early Bond films starring Sean Connery – so Shirley Bassey is not the only woman to contribute to the Bond sound!
The Music Machine
While her film work brought in the money, she continued with her real passion which was to create a completely new and highly versatile way to create music…by drawing. She built a machine – the Oramics Machine – that read a composition drawn onto film reels. It fulfilled her idea of having a machine that could play anything she could compose (and fulfilled a thought she had as a child when she wondered how you could play the notes that fell between the keys on a piano!).
The 35mm film that was the basis of her system that dates all the way back to the 19th century when George Eastman, Thomas Edison and Kennedy Dixon pioneered the invention film based photography and then movies. It involved a light sensitive layer being painted on strips of film with holes down the side that allowed the film to be advanced. This gave Oram a recording media. She could etch or paint subtle shapes and patterns on to the film. In a movie light was shone through the film, projecting the pictures on the film on to the screen. Oram instead used light sensors to detect the patterns on the film and convert it to electronic signals. Electronic circuitry she designed (and was awarded patents for) controlled cathode ray tubes that showed the original drawn patterns but now as electrical signals. Ultimately these electrical signals drove speakers. Key to the flexibility of the system was that different aspects of the music were controlled by patterns on different films. One for example controlled the frequency of the sound, others the timbre or tone quality and others the volume. These different control signals for the music were then combined by Oram’s circuitry. The result of combining the fine control of the drawings with the multiple tapes meant she had created a music machine far more flexible in the sound it could produce than any traditional instrument or orchestra. Modern music production facilities use very similar approaches today though based on software systems rather than the 1960s technology available to Oram.
Ultimately, Daphne Oram was ahead of her time as a result of combining her two childhood fascinations of music and electronics in a way that had not been done before. She may not be as famous as the great record producers who followed her, but they owe a lot to her ideas and innovation.
Image credit: eBay, Public domain, via Wikimedia Commons
Hedy Lamarr was a movie star. Back in the 1940’s, in Hollywood’s Golden Age, she was considered one of the screen’s most beautiful women and appeared in several blockbusters. But Hedy was more than just good looks and acting skills. Even though many people remembered Hedy for her pithy quote “Any girl can be glamorous. All she has to do is stand still and look stupid”, at the outbreak of World War 2 she and composer George Antheil invented an encryption technique for a torpedo radio guidance system!
Their creative idea for an encryption system was based on the mechanism behind the ‘player piano’ – an automatic piano where the tune is controlled by a roll of paper with punched holes. The idea was to use what is now known as ‘frequency hopping’ to overcome the possibility of the control signal being jammed by the enemy. Normal radio communication involves the sender picking a radio frequency and then sending all communication at that frequency. Anyone who tunes in to that frequency can then listen in, but also jam it by sending their own more powerful signal at the same frequency. That’s why non-digital radio stations constantly tell you their frequency “96.2 FM” or whatever. Frequency hopping involves jumping from frequency to frequency throughout the broadcast. Then, only if sender and receiver share the secret of exactly when the jumps will be made, and to what frequencies, can the receiver pick up the broadcast or jam it. That is essentially what the piano roll could do. It stored the secret.
Though the navy didn’t actually use the method during World War II, they did use the principles during the Cuban missile crisis in the 1960’s. The idea behind the method is also used in today’s GPS, Wi-Fi, Bluetooth and mobile phone technologies, underpinning so much of the technology of today. In 2014 she was inducted into the US national inventor’s hall of fame.
Peter W McOwan, Queen Mary University of London(from the archive)
Click the player below or download the audio file (.m4a) to listen to this article read by Jo Brodie.
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