A. M. Turing

The Chemical Basis of Morphogenesis

It is suggested that a system of chemical substances, called morphogens, reacting together and diffusing through a tissue, is adequate to account for the main phenomena of morphogenesis. Such a system, although it may originally be quite homogeneous, may later develop a pattern or structure due to an instability of the homogeneous equilibrium, which is triggered off by random disturbances. Such reaction-diffusion systems are considered in some detail in the case of an isolated ring of cells, a mathematically convenient, though biologically unusual system. The investigation is chiefly concerned with the onset of instability. It is found that there are six essentially different forms which this may take. In the most interesting form stationary waves appear on the ring. It is suggested that this might account, for instance, for the tentacle patterns on Hydra and for whorled leaves. A system of reactions and diffusion on a sphere is also considered. Such a system appears to account for gastrulation. Another reaction system in two dimensions gives rise to patterns reminiscent of dappling. It is also suggested that stationary waves in two dimensions could account for the phenomena of phyllotaxis. The purpose of this paper is to discuss a possible mechanism by which the genes of a zygote may determine the anatomical structure of the resulting organism. The theory does not make any new hypotheses; it merely suggests that certain well-known physical laws are sufficient to account for many of the facts. The full understanding of the paper requires a good knowledge of mathematics, some biology, and some elementary chemistry. Since readers cannot be expected to be experts in all of these subjects, a number of elementary facts are explained, which can be found in text-books, but whose omission would make the paper difficult reading.

Systems of logic based on ordinals

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The chemical basis of morphogenesis

It is suggested that a system of chemical substances, called morphogens, reacting together and diffusing through a tissue, is adequate to account for the main phenomena of morphogenesis. Such a system, although it may originally be quite homogeneous, may later develop a pattern or structure due to an instability of the homogeneous equilibrium, which is triggered off by random disturbances. Such reaction-diffusion systems are considered in some detail in the case of an isolated ring of cells, a mathematically convenient, though biologically unusual system. The investigation is chiefly concerned with the onset of instability. It is found that there are six essentially different forms which this may take. In the most interesting form stationary waves appear on the ring. It is suggested that this might account, for instance, for the tentacle patterns on Hydra and for whorled leaves. A system of reactions and diffusion on a sphere is also considered. Such a system appears to account for gastrulation. Another reaction system in two dimensions gives rise to patterns reminiscent of dappling. It is also suggested that stationary waves in two dimensions could account for the phenomena of phyllotaxis. The purpose of this paper is to discuss a possible mechanism by which the genes of a zygote may determine the anatomical structure of the resulting organism. The theory does not make any new hypotheses; it merely suggests that certain well-known physical laws are sufficient to account for many of the facts. The full understanding of the paper requires a good knowledge of mathematics, some biology, and some elementary chemistry. Since readers cannot be expected to be experts in all of these subjects, a number of elementary facts are explained, which can be found in text-books, but whose omission would make the paper difficult reading. 1. A Model of the Embryo. Morphogens. In this section a mathematical model of the growing embryo will be described. This model will be a simplification and an idealization, and consequently a falsification. It is to be hoped that the features retained for discussion are those of greatest importance in the present state of knowledge.

Computability and λ-definability

Several definitions have been given to express an exact meaning corresponding to the intuitive idea of ‘effective calculability’ as applied for instance to functions of positive integers. The purpose of the present paper is to show that the computable functions introduced by the author are identical with the λ-definable functions of Church and the general recursive functions due to Herbrand and Godel and developed by Kleene. It is shown that every λ-definable function is computable and that every computable function is general recursive. There is a modified form of λ-definability, known as λ- K -definability, and it turns out to be natural to put the proof that every λ-definable function is computable in the form of a proof that every λ- K -definable function is computable; that every λ-definable function is λ- K -definable is trivial. If these results are taken in conjunction with an already available proof that every general recursive function is λ-definable we shall have the required equivalence of computability with λ-definability and incidentally a new proof of the equivalence of λ-definability and λ- K -definability. A definition of what is meant by a computable function cannot be given satisfactorily in a short space. I therefore refer the reader to Computable pp. 230–235 and p. 254. The proof that computability implies recursiveness requires no more knowledge of computable functions than the ideas underlying the definition: the technical details are recalled in §5.

THE WORD PROBLEM IN SEMI-GROUPS WITH CANCELLATION

It will be shown that the word problem in semi-groups with cancellation is not solvable. The method depends on reducing the unsolvability of the problem in question to a known unsolvable problem connected with the logical computing machines introduced by Post (Post, [1]) and the author (Turing, [1]). In this we follow Post (Post, [2]) who reduced the problem of Thue to this same unsolvable problem.

The þ-function in λ-K-conversion

In the theory of conversion it is important to have a formally defined function which assigns to any positive integer n the least integer not less than n which has a given property. The definition of such a formula is somewhat involved: I propose to give the corresponding formula in λ- K -conversion, which will (naturally) be much simpler. I shall in fact find a formula þ such that if T be a formula for which T ( n ) is convertible to a formula representing a natural number, whenever n represents a natural number, then þ( T, r ) is convertible to the formula q representing the least natural number q , not less than r , for which T ( q ) conv 0. 2 The method depends on finding a formula Θ with the property that Θ conv λ u · u (Θ( u )), and consequently if M →Θ( V ) then M conv V ( M ). A formula with this property is, The formula þ will have the required property if þ( T, r ) conv r when T ( r ) conv 0, and þ( T, r ) conv þ( T, S ( r )) otherwise. These conditions will be satisfied if þ( T, r ) conv T ( r , λ x ·þ( T , S ( r )), r ), i.e. if þ conv {λ ptr · t ( r , λ x · p ( t , S ( r )), r )}(þ). We therefore put, This enables us to define also a formula, such that ( T, n ) is convertible to the formula representing the n th positive integer q for which T ( q ) conv 0.

A Formal Theorem in Church's Theory of Types

This note is concerned with the logical formalism with types recently introduced by Church [1] (and called (C) in this note) It was shewn in his paper (Theorem 26 α ) that if Y α stands for (a form of the “axiom of infinity” for the type α ), Y α can be proved formally, from Y ι and the axioms 1 to 7, for all types α of the forms ι′ , ι″ , …. For other types the question was left open, but for the purposes of an intrinsic characterisation of the Church type-stratification given by one of us, it is desirable to have the remaining cases cleared up. A formal proof of Y α is now given for all types α containing ι , but the proof uses, in addition to Axioms 1 to 7 and Y ι , also Axiom 9 (in connection with Def. 4), and Axiom 10 (in Theorem 9).

Practical Forms of Type Theory

Analysts agree that random information are an interesting new topic in the field of robotics, and security experts concu r. In fact, few futurists would disagree with the deployment of lambda calculus. In this work we disprove not only that the famous embedded algorithm for the improvement of model checking by Ivan Sutherland et al. is optimal, but that the same is true for cache coherence.

Visit to national cash register corporation of Dayton, Ohio

In recent years, much research has been devoted to the study of Boolean logic; however, few have developed the extensive unification of the location-identity split and co ntextfree grammar. After years of appropriate research into mode l checking, we verify the visualization of systems. In our research, we validate that the lookaside buffer and red-bla ck trees are never incompatible [54], [58], [59], [62], [68], [ 68], [68], [70], [95], [99], [114], [128], [129], [148], [152], [ 168], [168], [179], [188], [191].

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 di cult 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.” It is played with three people, a man (A), a woman (B), and an interrogator (C) who may be of either sex. The interrogator stays in a room apart front the other two. The object of the game for the interrogator is to determine which of the other two is the man and which is the woman. He knows them by labels X and Y, and at the end of the game he says either “X is A and Y is B” or “X is B and Y is A.” The interrogator is allowed to put questions to A and B thus: