On A. M. Turing's Intelligent Machinery, 1948
Quotations and Comments
The selected fragments are quoted after Collected Works of A.M.Turing, vol. Mechanical Intelligence edited by D.C.Ince, North-Holland 1992. The paper `Intelligent Machinery' is so introduced by the Editor.Turing says even more intesting things on theorem proving than those mentioned by the Editor of that volume; these will be quoted directly after Turing.
"This is one of the most startling of the artickes in terms of insight and prediction. In it, Turing predicts three main concepts: one of which has played a major part in computer science research, one which lies at the centre of commercial software development, and one which is a current burgeoning research area. [...] The concept that has fertilized a number of branches of computer science is the idea of computer-based theorem proving. Much of current artificial intelligence activity can be seen as the production of systems which carry out efficient deductions, based on facts culled from the environment in which a system is to be embedded. For example, expert systems are a highly successful product of the current boom in artificial intelligence. At their heart lies a database of rules which a human consultant employs in his work. The expert system takes rules and attempts to make a deduction using a form of theorem proving." To give an idea of the whole content, there is quoted below a chosen passage from the introductory Abstract, then the titles of all sections.
In the abstract preceding the article, among the motives of objecting that machines could show intelligent behaviour, Turing mentions - as (d) - the following one.The theorem of Gödel and related results (Gödel 1931, Church 1936, Turing 1937) have shown that if one tries to use machines for such purposes as determining the truth or falsity of mathematical theorems and one is not willing to tolerate an occasional wrong result, then any given machine will in some cases be unable to give an answer at all. On the other hand the human intelligence seems to be able to find methods of ever-increasing power for dealing with such problems `transcending the methods available to machines. [p.108]In the first section
-- Refutation of some objections
Turing refutes the one quoted above in the following way.The argument from Gödel and other theorems (objection d) rests essentially on the condition that the machine must not make mistakes. But this is not a requirement for intelligence. [p.4]When continuing this reply, he observes that even most intelligent people happen to obtain wrong results, hence there is no reason to claim any superiority of humans.
The next sections bear the following titles.
-- Varieties of machinery
Here the idea of the universal logical computing machine, ie, universal Turing machine, is sketched; also a man equipped with paper, pencil, rubber, and strictly following a set of instructions, is a universal machine.
-- Unorganized machines
-- Interference with machinery, modifiable and self-modifying machinery
-- Man as a machine
-- Education of machinery
-- Organizing unorganized machinery
-- The cortex as an unorganized machine
It is observed in this section that, starting from a primitive, unorganized stage, the brain can be organized in many different ways owing to appropriate training.
-- Experiments in organizing: pleasure-pain systems
-- The P-type unorganized machine
-- Discipline and initiative
-- Intelligence as an emotional concept
Here is a passage (from the last but one section) entitled `Intellectual, genetical and cultural searches'. It is highly interesting for three reasons. (i) A very general concept is proposed in it, so general that it may be considered as a new ontological category, namely the concept of search. (ii) At the same time, it implicitly defines intelligence: the greater is one's efficiency in finding the object searched, the greater is his intelligence. (iii) The logical process of proving theorems is also a kind of search, by which other kind of searches can be replaced to be made more efficient.
The abbreviation `UPCM', as used below, is to be read as a name for universal computing machine. Italics and the change of colour by the end of the first paragraph are added by the Forum Editor to hint at a sentence to be commented later.Intellectual, genetical and cultural searchesThe prospective participants of the Workshop are encouraged to consider the following problems which arise with interpreting the above text.
A very typical sort of problem requiring some sort of initiative consists of those of the form `Find a number n such that...'. This form covers a very great variety of problems. For instance problems of the form `See if you can find a way of calculating the function which will enable us to obtain the values for arguments ... to accuracy ... within a time ... using the UPCM ... ' are reducible to this form, for the problem is clearly equivalent to that of finding a program to put on the machine in question, and it is easy put the programs into correspondence with the positive integers in such a way that given either the number or the program the other can be easily found. We should not go far wrong for the time being if we assumed thatall problems were reducible to this form. It will be time to think again when something turns up which is obviously not of this form.
The crudest way of dealing with such a problem is to take the integers in order and to test each one to see whether it has the required property, and to go on until one is found which has it. Such a method will only be successful in the simplest cases. For instance in the case of problems of the kind mentioned above, where one is normally searching for a program, the number required will normally be somewhere between 21000 and 21.000.000. For practical work therefore some more expeditious method is necessary. In a number of cases the following method would be successful. Starting with a UCPM we first put a program into it which corresponds to building in a logical system (like Russell's Principia Mathematica). This would not determine the behaviour of machine completely: at various stages more than one choice as to the next step would be possible. We might arrange, however, to take all possible arrangement of choices in order, and go until the machine proved a theorem, which, by its form, could be verified to give a solution of the problem. This may be seen to be a conversion of the original problem into another of the same form. Instead of searching through values of the original variable n one searches through values of something else. In practice when solving problems of the above kind one will probably apply some very complex `transformation' of the original problem, involving searching through various variables, some more analogous to the original one, some more like a `search through all proofs'. Further research into intelligence of machinery will probably be very concerned with `searches' of this kind. We may perhaps call such searches `intellectual searches'. They might be very briefly defined as `searches carried out by brains for combinations with particular properties'.
It may be of interest to mention two other kinds of searches in this connection. There is the genetical or evolutionary search by which a combination of genes is looked for, the criterion being survival value. The remarkable success of this search confirms to some extent the idea that intellectual activity consists mainly of various kinds of search.
The remaining form of search is what I should like to call the `cultural search'. As I have mentioned, the isolated man does not develop any intellectual power. It is necessary for him to be immersed in an environment of other men, whose techniques he absorbs during the first twenty years of his life. He may then perhaps do a little research of his own and make a very few discoveries which are passed on to other men. From this point of view the search for new techniques must be regarded as carried out by the human community as a whole, rather than by individuals.
1. The first one should yield just illustrations useful in considering the next issues. Let some examples of search be given. Do you think that the following search problems fall under the schema in the first proposition of the text quoted above? (`number' means a natural number, incl. 0)
2. The arises an issue with Turing's conjecture (italicized) that all problems are reducible to the form of such search problems.
- Fermat problem. Is there any set of numbers w, x, y, z for which the following equation is satisfied?
(x+1)w+3+(y+1)w+3 = (z+1)w+3In Turing's version the statement of the problem would start from `Find a set' instead of `Is there any set'. According to Fermat's assertion, such a set will never be found, ie, a machine will never stop.
- Find a number not being the sum of three squares (of numbers). There is a number satisfying this condition, viz. 7. Thus we have an example of successful search.
- Lagrange theorem: Every number is the sum of the squares of four integrals. It follows that if a machine is told to find a number not being such a sum, it will not find it, hence it will never stop.
How is this statement related to the Church-Turing Thesis? Is the class of such problems - to Turing's mind - identical with the class of those which are solvable by a universal Turing machine? If so, then the above conjecture means that all problems (not only mathematical ones) are solvable by a Turing machine (if solvable at all). Did Turing really wish to assert this? This question is worth putting, even if evidence in Turing's texts is too scarce, since it urges us to take a stand on the question (those deeming that the intellectual omnipotence of Turing machine is a reasonable claim, would tend to ascribe that claim to Turing, as one supposed to be reasonable).
3. A next issue to be raised is related to the preceding one and to Turing's mention of a search based on an axiomatic system like that of Principia. When one resorts to an axiomatic system, this means that he has settled in the afirmative each question concerning the truth of the axioms. For example the axiom of infinity in Principia. Or, the axiom of induction in Peano arithmetic, the axiom of choice, tertium non datur, etc. Do such answers result from the search in a set (then, which set)?
4. The last cluster of questions may appear as involving rather subjective problems (ie, deriving from gaps in the knowledge of the one who puts it). However, there may be people less expert in Turing machines who would profit from posing such naive questions (to be, hopefully, answered by experts).
Let us return to the last example in 1, viz. Lagrange theorem. Since it is a true universal proposition, a machine told to look for a counterexample will never find it. Thus the machine will never stop. However such a reason of non-halting has to be different from that which consists in the unsolvability of the problem involved (which is, possibly, at stake with Fermat's claim), since this problem has been solved by Lagrange. If this observation is fitting, then one may wonder how to distinguish these two kinds of non-halting.
Moreover, the reader of popular introductions to Turing's results (as Turing - A Natural Philosopher by A.Hodges, p.24 in Polish version) learns that a machine sometimes makes a loop. How this behaviour is related to that which consists in never stopping? Does it also mean the impossibility of giving a solution?
Let it be still noted that in a pasage in Intelligent Machinery (p.4 as quoted near the beginning of this file) Turing admits of the posssibility of giving wrong answers by a machine. Giving an answer involves halting, but halting should mean the finding of a correct solution. However, Turing's defense of machines - that their fallibility does not speak against them as being akin to that of humans - seems to imply that a machine may answer wrongly; which, then, overt behaviour of it would be a sign of erring?
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