A Calculus of Ideas: A Mathematical Study of Human Thought by Ulf Grenander PDF

By Ulf Grenander

ISBN-10: 981438318X

ISBN-13: 9789814383189

This monograph experiences a proposal test with a mathematical constitution meant to demonstrate the workings of a brain. It provides a mathematical thought of human suggestion in keeping with development conception with a graph-based method of considering. the tactic illustrated and produced via vast machine simulations is expounded to neural networks. dependent usually on introspection, it truly is speculative instead of empirical such that it differs noticeably in perspective from the traditional knowledge of present cognitive technology.

Readership: Graduate scholars in utilized arithmetic, information and cognitive technological know-how in addition to postgraduates and researchers in those fields.

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Hence the substitutions do indeed form a bijactive map: a permutation within modalities. 2 The above transformations constitute configuration homomorphisms. Also form subgroups of S over the modalities m1 , m2 , . . Sm1 ,m2 ,... = Sm1 × Sm2 × · · · . 10) A set T of thoughts, T ⊂ MIND is called a thought pattern if it is invariant with respect to the modality group S. It is called a (restricted) thought pattern over the modalities m1 , m2 , . . if it is invariant with respect to the similarities over these modalities.

1) we shall assume its probability p = P (thought) 1) to be proportional to Q(gi ) for each node i in σ; 2) to be proportional to A[bj (gi ), bj (gi )] for each connected pair of bonds bj (gi ), bj (gi ) in σ; 3) to be proportional to κn where κn is a complexity parameter for size n of thought. To illustrate this consider the following picture with five elementary ideas “apple”, “sweet”, “sour”, “good” and “bad”” with Q and A values as indicated. 1: 42 July 2, 2012 3. 3; so that the thought σ(apple, sweet, good) is much more likely than the others.

M and so on for the edges (also called connections) of the graph (for the down-bonds of a single generator). 6) we have n = 3, m = 2. A central concept in Pattern Theory is that of regularity. In the following we shall use two types of regularity: DEFINITION: A configuration thought = σ(g1 , g2 , . . , gi , . . 7) and a weaker concept: DEFINITION: A configuration, or thought, c = σ(g1 , g2 , . . , gi , . . , gn ) is said to be REGULAR if any connected jth down-bond βj (gi ) satisfies the bond relation ρ : gi ∈ βj (gi ).

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A Calculus of Ideas: A Mathematical Study of Human Thought by Ulf Grenander

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