Turing’s Legacy Minds & Machines

Alan Turing Alan Turing was a British mathematician who was most famous for his work in theoretical computer science; During World War II, Turing helped break German codes using mechanical computers In 1952, he British government thought Turing’s homosexuality was a crime, and forced him to go through hormonal treatment. In 1954, age 41, Turing died from eating an apple laced with cyanide; probably a suicide In 1999, Turing was listed as one on the top 100 most important people of the 20th century On September 10, 2009, the British government apologized for their treatment of Alan Turing.

Turing’s Legacy Turing’s legacy consists of 2 parts: Turing Machines (1936) Turing Test (1950)

Turing Machines

Formal Logic The housemaid or the butler did it If the housemaid did it, the alarm would have gone off 1. H  B A. The alarm did not go off 2. HA A. 3. ~A A. … therefore … 4. ~H 2, 3 MT 5. B 1, 4 DS The butler did it!

Algorithms An algorithm is a systematic, step-by-step procedure: Steps: Algorithms take discrete steps Precision: Each step is precisely defined Systematicity: After each step it is clear which step to take next Examples: Cookbook recipe Filling out tax forms (ok, maybe not) Long division

Computations Computations are where the ideas of formal logic and algorithms come together. A computation is a symbol-manipulation algorithm. Example: long division. Not every algorithm is a computation Example: furniture assembly instructions

Computers A ‘computer’ is something that computes, i.e. something that performs a computation, i.e. something that follows a systematic procedure to transform input symbol strings into output symbol strings. As long as the procedure is effective, humans can take the role of a computer by following that procedure. Indeed, some 60 years ago, a ‘computer’ was understood to be a human being! By mechanizing this process, we obtain computers as we now know them.

The Decision Problem Suppose I give you an argument, with the premises and conclusion expressed in formal logic. As always in logic, we want to know if the conclusion follows form the premises, i.e. whether the argument is valid. Is there some kind of systematic algorithm to test for validity? Can we effectively decide whether some argument is valid or not?

The Scope and Limits of Computation I In 1936, Turing wrote a paper in which he tried to give an answer to the decision problem. Turing thought the answer to the decision problem was negative: that there is no procedure to tell all valid arguments from all invalid arguments. To prove this, Turing needed to generalize over all possible computations. Turing tried to do this by trying to find the basic elements of any such process.

The Scope and Limits of Computation II Take the example of multiplication: we make marks on any place on the paper, depending on what other marks there already are, and on what ‘stage’ in the algorithm we are (we can be in the process of multiplying two digits, adding a bunch of digits, carrying over). So, when going through an algorithm we go through a series of stages or states that indicate what we should do next (we should multiply two digits, we should write a digit, we should carry over a digit, we should add digits, etc).

The Scope and Limits of Computation III The stages we are in vary widely between the different algorithms we use to solve different problems. However, no matter how we characterize these states, what they ultimately come down to is that they indicate what symbols to write based on what symbols there are. Hence, all we should be able to do is to be able to discriminate between different states, but what we call them is completely irrelevant. Moreover, although an algorithm can have any number of stages defined, since we want an answer after a finite number of steps, there can only be a finite number of such states.

The Scope and Limits of Computation IV Next, Turing reasoned that while one can write as many symbols as one wants at any location on the paper, one can only write one symbol at a time, and symbols have a discrete location on the paper. Therefore, at any point in time the number of symbols on the paper is finite, hence we can number them, and hence we should be able to do whatever we did before by writing the symbols in one big long string of symbols.

The Scope and Limits of Effective Computation V Moreover, to get to some location (whether to read or write a symbol), we just need to be able to go back and forth, one symbol at a time, along this one big symbol string. Finally, there can only be finitely many symbols, or else there would have to be two symbols that are so much alike that we can no longer perceptually discriminate between them.

The Scope and Limits of Effective Computation VI Turing thus obtained the following basic components of effective computation: A finite set of states A finite set of symbols A big symbol string to work on An ability to move along this symbol string An ability to read a symbol An ability to write a symbol What is this? A Turing Machine!

Turing Machines Demo

Representations, Efficiency, and Computational Power In our example, we used a ‘unary’ representation of numbers (i.e. the number four was represented as ‘1111’), but we could also have used some other representation, such as ‘4’ or ‘IV’. The choice in representation obviously has an effect on the nature of the program that is needed to do the ‘right’ thing (now you need rules for encountering a ‘4’ instead of a ‘I’). Depending on the problem, some representations lead to computations that are simpler (I always wondered how the Romans did long division!) and more efficient than others. However, if one is able to eventually get the answer using some representational scheme then, no matter how inefficient that representational scheme is, the computational ‘power’ of that scheme is equal to that of some other scheme that gets the same answer.

0’s and 1’s As it turns out, we can always use a string of bits, i.e. a string of 0’s and 1’s to represent objects, without losing any computational power. This is indeed how the modern ‘digital computer’ does things. That is, at the machine level, it’s all 0’s and 1’s. Of course, the 0’s and 1’s are just abstractions here; they refer to some kind of physical dichotomy, e.g. hole in punch card or not, voltage high or low, quantum spin up or down, penny on piece of toilet paper or not, etc. The fact that one can decide to use anything to represent anything is exactly why there can be mechanical computers, electronic computers, bio-chemical computers, DNA computers, optical computers, quantum computers and, as demonstrated by the Turing machine, computers made of one (very) big roll of toilet paper and (a lot of) pennies!

The Church-Turing Thesis Many definitions have been proposed to capture the notion of a ‘computation’ other than Turing-Machines. It turns out that all proposed definitions up to this date are equivalent in the sense that whatever one is able to compute using one computational method, one is able to compute using any other proposed method as well. The Church-Turing thesis states that Turing-machines capture the notion of effective computation: whatever is computable, Turing-machines can compute. The Church-Turing thesis shows the amazing computational power of Turing machines: Turing machines can compute what this very laptop computes, and they can do so using only 0’s and 1’s!

Universal Turing Machines One of Turing’s great achievements was his finding that one can make a Universal Turing Machine, which is a Turing Machine U that can simulate the behavior of any Turing Machine M by giving a description of that machine M and the input I that M would work on to machine U (Turing used this also as part of his proof that the decision problem is not solvable). This led to the notion of stored programs (programs as part of the data), and thus to programmable, all-purpose, computers. What was used as part of a proof of a rather abstract mathematical/logical theorem turned out to revolutionize our lives!

Computing Things The symbols that computations manipulate are representations of things. By manipulating those representations we come to know something about the things that those representations represent. Thus, things become computable: ‘I can compute the product of 2 numbers’. I can use a mechanical device to help me manipulate those representations This mechanical device we now call a computer

Computationalism Computationalism is the view that cognition is (syntactic) computation (i.e. computers can think, have beliefs, be intelligent, self-conscious, etc.) The idea behind Computationalism is that the brain is, like a computer, an information-processing device: it takes information from the environment (perception), it stores that information (memory/knowledge), infers new information from existing information (reasoning, learning), and makes decisions based on this information (action). Computationalism fits well with the earlier finding that one can obtain powerful information-processing capacities using very simple resources: Early views on the brain supposed that neurons firing or not would constitute 0’s and 1’s

The Turing Test

Computing Machinery and Intelligence “I propose to consider the question, 'Can machines think?' This should begin with definitions of the meaning of the terms 'machine 'and 'think'. The definitions might be framed so as to reflect so far as possible the normal use of the words, but this attitude is dangerous. If the meaning of the words 'machine' and 'think 'are to be found by examining how they are commonly used it is difficult to escape the conclusion that the meaning and the answer to the question, 'Can machines think?' is to be sought in a statistical survey such as a Gallup poll. But this is absurd. Instead of attempting such a definition I shall replace the question by another, which is closely related to it and is expressed in relatively unambiguous words. The new form of the problem can be described' in terms of a game which we call the 'imitation game'. “

The Imitation Game or: The Turing Test Machine Interrogator Human

“I believe that in about fifty years’ time it will be possible to programme computers, with a storage capacity of about 109, to make them play the imitation game so well that an average interrogator will not have more than 70 per cent chance of making the right identification after 5 minutes of questioning” -Alan Turing (1950)

Turing’s Argument for AI Premise 1: Machines can pass the Turing Test Premise 2: Anything that passes the Turing Test is intelligent Conclusion: Machines can be intelligent

Can Machines pass the Turing Test? Turing thinks that machines will at some point in the future be able to pass the Turing Test, because Turing thinks that passing the test requires nothing more than some kind of information processing ability, which is exactly what computers do.

A Puzzle But wait, can’t we then just argue as follows: Intelligence requires nothing more than some kind of information processing ability, Computers can have this information processing ability Therefore, computers can be intelligent. Indeed, this is exactly how proponents of AI make the argument today. So why didn’t Turing make this very argument? Why bring in the game?

A Second Puzzle Also, why the strange set-up of the Turing-Test? Why did Turing ‘pit’ a machine against a human in some kind of contest? Why not have the interrogator simply interact with a machine and judge whether or not the machine is intelligent based on those interactions?

The Super-Simplified Turing Test Interrogator Machine

Answer: Bias The mere knowledge that we are dealing with a machine will bias our judgment as to whether that machine can think or not, as we may bring certain preconceptions about machines to the table. Moreover, knowing that we are dealing with a machine will most likely lead us to raise the bar for intelligence: it can’t write a sonnet? Ha, I knew it! By shielding the interrogator from the interrogated, such a bias and bar-raising is eliminated in the Turing-Test.

The Simplified Turing Test Interrogator Machine or Human

Level the Playing Field Since we know we might be dealing with a machine, we still raise the bar for the entity on the other side being intelligent. Through his set-up of the test, Turing made sure that the bar for being intelligent wouldn’t be raised any higher for machines than we do for fellow humans.

Is the Imitation Game a Test? Still, we are left with the first puzzle: if Turing wanted to argue that machines can be intelligent, why bring up any test at all? I believe that Turing never intended the test as part of any argument for machine intelligence. Instead, Turing used the test to make us aware of the point that we should level the playing field as far as attributing intelligence goes: if we attribute intelligence to humans based on their behavior then, just to be fair, we should do it for machines as well. Thus, the convoluted set-up wasn’t merely a practical consideration to eliminate bias: it was the whole point of his article! In fact, Turing hardly uses the word ‘test’ in his original article, and instead talks about it as the Imitation Game.

Pluto and Planets Asking how many planets there are in our solar system seems to be a factual matter: We believe there is a straightforward fact of the matter to this issue. If I say: “There are X planets in our solar system” then this statement is either true or false. How many planets there are is an empirical issue: observations will tell us how many there are However, as the case of Pluto demonstrated, things aren’t that easy. This issue isn’t just an empirical issue, but also one of interpretation. Maybe the same is true for machine intelligence!

Artificial Flight and Artificial Intelligence Imagine going back 100 years when the Wright Brothers had their first flight. We can imagine people say: “Well, but that’s not real flight. There is no flapping of the wings!” But over time, we realized that it is, from the standpoint of using concepts that help us think and make sense of the world around us, a good idea to consider airplanes as really flying. Maybe the same is true for intelligence!

In Turing’s Words The original question, “Can machines think?”, I believe to be too meaningless to deserve discussion. Nevertheless I believe that at the end of the century the use of words and general educated opinion will have altered so much that one will be able to speak of machines thinking without expecting to be contradicted. -Alan Turing (1950)