On boolean functions and connected sets

McKenzie, Ralph; Mycielski, Jan; Thompson, David

doi:10.1007/bf01694182

Cited by 12 publications

(5 citation statements)

References 6 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…These results have inspired similar studies of the power of expression of other languages either related or quite different from inequalities of the form (4), see [6], [8].…”

mentioning

confidence: 63%

Book Review: Perceptrons, An Introduction to Computational Geometry

Mycielski¹

1972

Bull. Amer. Math. Soc.

Self Cite

View full text Add to dashboard Cite

This book is a very interesting and penetrating study of the power of expression of perceptrons and some other mathematical problems concerning memory and learning. This subject is still quite new and hence at a stage of development in which the most important discoveries are being done. It seems to differ from the theory of automata in its greater relevance to our ideas about the organization of the brain and the construction of "such" machines. The main positive result which motivates the research presented in this book is the perceptron learning theorem (proved in Chapter 11). This theorem was discovered as a property of Rosenblatt's perceptrons (the history is given in the book) but in fact it belongs to general linear approximation theory and can be stated as follows. Let if be a real Hubert space, F + 9 F~ s H,F + n F~ = 0 and for every xeF + uF~we have ö ^ ||x|| ^ M, where ô > 0. We assume moreover that F + and F~ are linearly separable with resolution <5, i.e. there exists ana* e H with ||a*|| = 1 such that (a*,x)^<5 ifxeF + and (a* 9 x)£-S ifxeF". Given any sequence x l9 x 2 ,...eF + uF" and any s ^ 0 we define by induction a sequence of vectors a 0 ,a u ...eH and a sequence of integers fe 1 ,fc 2 , We put a 0 = 0 and (1) a i+l = a t + k t+1 x i+l9 where k i+l is the unique integer with minimal absolute value such that if x £+1 eF + then {a i9 x i+1) + fc i+1 ||x i+1 || 2 > 6. and if x i+1 eF~ then (a i9 x i+1) + fe f+1 ||x f+1 || 2 <-e. Then the sequence k u k 2 ,... has finitely many terms different from 0 and moreover f |/c,| S (M 2 + 2e)/ô 2. i=l (In the book e = 0 and M = 1, but the proof is almost the same for all s ^ 0 and M ^ Ô.

show abstract

“…These results have inspired similar studies of the power of expression of other languages either related or quite different from inequalities of the form (4), see [6], [8].…”

mentioning

confidence: 63%

Book Review: Perceptrons, An Introduction to Computational Geometry

Mycielski¹

1972

Bull. Amer. Math. Soc.

Self Cite

View full text Add to dashboard Cite

show abstract

“…Many interesting f's cannot be represented by a program (T,K,F) of moderate size, say with T having no more than m2 elements, see [2,6,7]. E.g., if f(x) indicates the parity of the number of l's for every sequence x = (x(l),..., x(m)), then 1 must have at least 2m -1 elements.…”

Section: Miscellaneous Remarksmentioning

confidence: 99%

“…Here again our algorithm could be applied. This task suggests a modification of our algorithm in which the decisions of stopping the growth of a branch of T and assigning to its end a value F comes earlier than specified in clauses (a) and (b), namely when the fraction of counterexamples to (*) is small enough; 6. The 0's and l's of the coordinates of the xi's may be produced by some threshold functions applied to some continuous parameters.…”

Section: Miscellaneous Remarksmentioning

confidence: 99%

See 1 more Smart Citation

Organization of Memory

Ehrenfeucht¹,

Mycielski²

1973

Proc. Natl. Acad. Sci. U.S.A.

Self Cite

View full text Add to dashboard Cite

A new learning algorithm is presented that may have applications in the theory of natural and artificial intelligence.In order to construct machines that could display an intelligence somewhat like that of animals or man, or to understand how the brain functions, one must develop a model of the organization of memory. Several models have been proposed [1,[3][4][5]9, and other references therein]. Here we propose still another model, which is new in that it tends to explain how a brain learns to interpret correctly (or act purposefully upon) the immense amount of information which it obtains continuously from the senses.For simplicity, we assume at first that the brain serves only to answer questions that admit a "yes" or ''no"y answer and that the questions are sequences of 0's and l's of a fixed length m. Our questions depict the totality of data (parameters) that the brain gets at a given time (ours is a discrete time model) and, hence, m is very large (see miscellaneous remarks, below: remarks 1 and 2 concerning the possible meanings of m). At the beginning the brain does not know what to say and answers arbitrarily, but it gets a "reward" when the answer is right and a "punishment" when the answer is wrong. Thus, it gets post facto information what is the right answer and attempts to produce correct answers to the questions that follow. By the problem of organization of memory, we understand the problem of defining an algorithm to answer new questions on account of past experience. Various statistical estimation procedures and classical methods of interpolation and approximation of functions seemingly would apply to this problem. But as information theory showsj, they loose their power when the dimension m is large, say m > 100. Our algorithm is free from this defect, because we exploit the following natural assumption: Few of the parameters are important for solving any given question (see remarks 6 and 7 below concerning the validity and the interpretation of this assumption), although different parameters are needed to solve different questions. The brain probably develops a method of selecting the important parameters of any given question; so does our algorithm. We shall not attempt in this paper to define a neural network capable of performing our algorithm nor theorize on the question how the nervous tissue could do it since, although easy, this would seem to us premature (see EXPERIMENTS below).

show abstract

“…Then {Q £ «co: <a>, +, S> 1= ff} is the set of relations defined by #• in {co, +>. This kind of definability was first studied by J. Mycielski [15], who proved that the set of relations C = {Q ^ 2 co: Q is finite and connected} is not definable by any first-order sentence in {co, +>. (By connected we mean that 2 co is regarded as the set of points in the cartesian plane whose coordinates are nonnegative integers, and a chess king can visit all points in Q without leaving Q.…”

mentioning

confidence: 99%