PROCEEDINGS OF A SYMPOSIUM IN APPLIED MATHEMATICS

OF THE AMERICAN MATHEMATICAL SOCIETY

Held in New York City April 5-7, 1966

J. T. SCHWARTZ EDITOR

http://dx.doi.org/10.1090/psapm/019

VOLUME XIX

PROCEEDINGS OF SYMPOSIA IN APPLIED MATHEMATICS

MATHEMATICAL ASPECTS OF COMPUTER SCIENCE

AMERICAN MATHEMATICAL SOCIETY PROVIDENCE, RHODE ISLAND

1967

Prepared by the American Mathematical Society with the support of the Air Force Office of Scientific Research of the Office of Aerospace Research, under Contract No. AF 49(638)-1667 and the Institute for Defense Analyses.

Library of Congress Catalog Number 67-16554 Copyright © 1967 by the American Mathematical Society

Reprinted 1988

Printed in the United States of America All rights reserved except those granted to the United States Government. Otherwise, this book or parts thereof, may not be reproduced in any form

without permission of the publishers.

The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability. @

CONTENTS

A review of automatic theorem-proving 1 J. A. Robinson

Assigning meanings to programs 19 Robert W. Floyd

Correctness of a compiler for arithmetic expressions 33 John McCarthy James Painter

Context-free languages and Turing machine computations 42 J. Hartmanis

Computer analysis of natural languages 52 Susumu Kuno

The use of computers in the theory of numbers 111 P. Swinnerton-Dyer

A machine calculation of a spectral sequence 117

M. E. Mahowald

M. D. MacLaren

Numerical hydrodynamics of the atmosphere 125 C. E. Leith

The calculation of zeros of polynomials and analytic functions 138 J. F. Traub

Mathematical theory of automata 153 Michael O. Rabin

Linearly unrecognizable patterns 176 Marvin Minsky Seymour Papert

Author Index 219

Subject Index 221

AUTHOR INDEX

Roman numbers refer to pages on which a reference is made to an author or a work of an author.

Italic numbers refer to pages on which a complete reference to a work by the author is given. Boldface numbers indicate the first page of the articles in the book.

Aanderaa, S., 156,173 Abbott, R., 58,107 Adams, F. J., 124 Aparo, E., 151,152 Arakawa, A., 134,137

Backus, J . W., 121,124 Baer, F., 126,136 Bauer, F. L., 151,152 Bagrinovsky, K., 132,137 Bar-Hillel, Y., 42, 44,51,100,106,107,107 Bartlett,J. M., 82,110 Beatty,J., 162,173 Bessinger, J. B., 52,107 Bever, T. G., 98,100,108 Birch, B. J., 115,116 Bledsoe, W.W., 199,217 Blum, 178 Bobrow, D. G.> 55,102,107,108 Bolinger, D., 96,108 Browning, I., 199,2/7 Biichi,J.R., 162,163,173

Carson, D., 9,17,18 Cheatham, T. E., Jr., 61,108 Chinlund,M.,9,17,18 Chomsky, N., 42, 44,50, 55, 65, 67, 74,

75, 81,108

Davis, M., 1, 9,17 Dejan, F., 157,173 Doner, J. E., 169,173 Dreyfus, H. L., 106,107,108

Eggan, L. C , 157,174 Elgot, C. C , 163,164,173,174 Ellsaesser, H. W., 126,136 Fjortoft, R., 134,137 Floyd, R.W., 19,43,51 Fodor,J.A.,93,94, 100,108

Forsythe, G. E., 140,152 Friedman, J., 88,108

Gamba, A., 211,217 Ginsburg,S., 51, 162,174 Godel, K., 3,18 Green, B. F., Jr., 101,108 Greibach, S., 51, 52, 58,108 Griffiths, T., 55, 64,108 Gross, L., 55, 90,108 Grzegorczyk, A., 173,174 Guiraud, P., 52,108

Hadamard, J., 151,152 Hartmanis, J., 42, 42, 50, 51 Hays, D., 61,108 Henrici, P., 151,152 Herbrand, J., 3,18 Hinman, P. G., 9,17,18 Householder, A. S., 151,152 Howard, L. N., 135,137 Huffman, D., 204

Irons, E. T., 61,108

Jaonv,J. I., 156,174

Kasami, T., 108 Katz, J. J., 93, 94,100,108 Kay, M., 61,107 Kesten, H., 168 Kleene,S.C, 156,174 Konig, J., 151,152 Krohn,K., 158,174 Kuno, S., 52, 55, 56, 58, 89,107,109 Kuo, H. L., 135,137 Kurahara,Y., 127,137

Lakoff, G., 52, 65,109 Langevin, R. A., 57,106,109

219

220

Lee, D. T., 134,137 Leith, C , 125, 136 Lewis, P . M . , 42,50, 51 Lilly, D., 133, 137 Liulevicius, A., 117, 124 Lochak, D., 90,109 Loveland, D . W . , 9 , 1 8

MacLane,S. , 118,124 MacLaren, M . E . , 117 Mahowald, M. E., 117, 117, 118, 124 M a r c h u k , G . L , 132, 137 Matthews, G. H., 82,109 McCarthy, J., 20, 32, 33, 33, 36, 41 McCulloch,W.S. , 2 / 7 Mcllroy, M. D., 9 ,17,18 McNaughton, R., 157, 163,174 Mealy, G.H.,174 Meltzer, B., 15,18 Mendelson, E., 18 Mezei,J., 169,174 Miller, G. A., 55, 75,108 Miller, R . E . , 162, 173,174 Minsky, M., 176, 178, 217 Mintz, Y., 125,136 Moore, E. F., 160, 161,174 Mordell, L. J., 112 Muller, D. E., 163, 169,174 Myhill ,J. , 158, 174

Nerode,A., 158,174 Nilsson,N., 2 /7 Novikoff,A.B. J., 199,2/7

Oettinger, A. G., 52, 55, 92, 106, 107,109 Owens, M . F . , 57, 109

Page, C. V., 174 Painter, J., 33 Papert, S., 176 Paull, M . C . , 162, 174 Paz, A., 164, 168, 174 Perles, M., 42, 44, 51, 168, 174 Peters, P . S., Jr., 82, 109 Petrick, S. R., 55, 58, 64, 86, 109 Phillips, N. A., 127, 136, 137 Pitts, W., 27 7 P la th ,W. ,52 , 109 Platzman, G., 126, 136 Post, E., 42, 51 Postal, P., 55, 109 Prawitz, D., 7, 9, 18 Putnam, H., 1,9, 18

AUTHOR INDEX

Rabin, M. O., 153, 155, 158, 159, 161, 163, 164,167, 168,171,774

Redko, V. N. , 156, 174 Reynolds, J. C., 121, 124 Rhodes, J., 158,774 Richardson, L. F., 125, 133, 136, 137 Richtmyer, R. C., 134,757 Ritchie, R. W., 82, 109 Robinson, G. A., 17,78 Robinson, J. A., 1, 8, 9, 13, 15, 18, 61, 109 Rosenbaum, P., 90, 98, 100, 108, 109 Rosenblatt, F., 178,27 7 Rutishauser, H., 151, 752

Sakai, I., 61, 709 Salomaa,A., 156,164,774 Samelson, K., 151, 752 Sa t t l ey ,K . ,61 , 708 Schroder, E., 151,752 Schutzenberger, M. P., 157, 158, 173, 174 Scott, D., 155, 158, 159, 161, 171,174 Sebastiao e Silva, J., 151, 752 Selfridge,0. G.,277 Shamir, E., 42, 44, 57, 168, 174 Shapiro, P. A., 109 Shepherdson, J. C., 158, 174 Simmons, R. F., 101, 109 Skolem, T., 3, 78 Smagorinsky, J., 125, 127, 133, 136 Spiegel, E. A., 135,137 Stearns, R . E . , 42, 50, 57 Swinnerton-Dyer, H. P. F., I l l , 776

Tate, J., 115,776 Thatcher, J . W., 169, 171, 174 Traub, J . F., 138, 138, 140, 141, 142, 144,

148,151, 752

Unger,S. H., 162, 174

von Neumann, J., 134, 137

Walker, D. E., 82, 106, 110, 770

Wang, H., 17,78 Wolfowitz, J., 168, 174 Woods, W. A., 106, 770 Wos, H., 17, 78 Wright, J. G., 157, 169, 171, 174

Younger, D. H., 64, 7/0

Zeeman, E. C., 197,27 7

SUBJECT INDEX

accident discrimination, 106 adjunctions, 75 algebraization of finite automata, 157,172 ALGOL, 26 algorithm, unification, 7 aliasing, 133 alternating description, 45 ambiguities, 57 ambiguity tolerence, 106 analysis by synthesis, 82 analytic functions, zeros of, 138, 150 analyzable, 72 antecedent, 21 assembly language, 34 asymptotic error constant, 144 attribute-value, 102 automata

finite, 153,162 incomplete, 162 nondeterministic, 155 probabilistic, 164, 167

automatic syntactic analysis, 55

base component, 67 baseball, 101 basic algorithm, 138, 139

variations of, 145 behavior of error, 144 Bernoulli recurrence, 142 binary trees, 170 Boolean functions, 178, 181, 182, 191,

201,212 bounded automaton, linear, 53 branching rules, 69

categorical symbols, 67 clash, 10 companion matrix, 144 complement(s), 42, 43, 47, 48 complete, 21 complex

conjugate zeros, 148 symbols, 67

compiler, 37

computational linguistics, 52 semantics, 92

computer implementation, 151 languages, 42

configuration space representation, 132 conjugate zeros, complex, 148 connectivity, 200, 206, 207

algorithm, 209 consciousness, fringe, 106 consequent, 21 consistent, 21 constituent analyzer, immediate, 55 context-free

grammar, 43, 45, 53 languages, 42, 45, 47, 48, 55 grammar with erasing, 53

context-sensitive grammar without erasing, 53

convection, 134 convective processes, 130 counterexample, 21 cubic equation, 111 cut-points, isolated, 166 cyclical transformational rules, 78

data base, 101 deduction, 11 deep structure, 65 deletions, 75 0-derivation, 54 derived phrase-marker, 67 diagnostic variables, 128 diameter limited perceptron, 213 diameter-restricted predicates, 177 disagreement set, 8 disambiguation, 94 discrimination, accident, 106 distinguishers, 93 dominant zero(s), 138, 139, 144

equimodular, 138, 148

eddy diffusion, 130

221

222 SUBJECT INDEX

edges, 20 efficiency of parsing algorithms, 63 elementary transformations, 75 embeddings, 53 entrance, 20 equimodular dominant zeros, 138, 148 equivalence problem, 159 erasure, 83 error

behavior of, 144 constant, asymptotic, 144

essence, 106 Ext, 118

finite algebra, 159 automata, 153, 162

algebraization of, 157 representation, 125 state

grammars, 53 machine, 53

tree-automaton, algebraization of, 172 first-order predicate calculus, 1 flowchart, 20 fractional time steps, method of, 132 fringe consciousness, 106 fundamental theorem of logic, 5

G polynomials, 140, 142, 143, 144, 149, 151 recursion formula for, 148

generalized phrase-marker, 67 generative power of transformational

grammars, 82 geometric property, 197 global

convergence, 142, 148 geometric properties, local vs., 177

graded algebras, 118 grammar with and without erasing,

context sensitive, 53 grammatical markers, 13 group invariance theorem, 185, 193

H polynomials, 146, 147 Haar measure theory, 185 "have" or "par t of" relationship, 99 Herbrand universe, 4 homotopy

groups, 118 theory,117

hydrostatic equilibrium, 128 hyper-resolvent, 13

/ polynomials, 149 immediate constituent analyzer, 55 incomplete automata, 162 induction hypothesis, 38 infinite sequences, 162 instantaneous description, 44 interpretation, 19, 20 intersections, 42 invariance theorem, group, 185 isolated cut-points, 166 iteration

function(s), 141, 142, 143, 146, 148 of arbitrary order, 142 second-order, 141

Newton-Raphson, 140, 143

J polynomials, 149

language, 43 latent clash, 13 lexical

formative, 73 rule, 74 symbols, 68

lexicon, 67 limitations of diameter-limits,

perceptron, 215 linear

bounded automaton, 53 representation, 180, 186 threshold function(s), 179, 181, 183, 192 truncation, 126

linguistics, computational, 52 list

elements, 121 structures, 121

local vs. global geometric properties, 177 logic, fundamental theorem of, 5

machine translation, 106 mapping problem, 127 masks, 181 method of fractional time steps, 132 M I T R E syntactic analysis procedure, 82 multi-layer perceptrons, 211 multiple zeros, 138, 147, 148

natural languages, 42 Newton-Raphson iteration, 140, 143 nondeterministic automata, 155 nonlinear

transfer processes, 133 truncation, 126

SUBJECT INDEX 223

nonterminal vocabulary, 53 numerical example, 138, 139, 141, 149

object language, 35, 36, 40 order-limited perceptron, 196 ordering of rules, 74

parallel computations, serial vs., 177 parity

function, 187, 189 theorem, 193

parse, 101 parsing algorithms, efficiency of, 63 perceptron (s), 203

diameter limited, 213 limitations of diameter-limits, 215 multi-layer, 211 order-limited, 196

phrase structure grammars, 53 phonological component, 67 phrase-marker, 54

derived, 67 generalized, 67

polynomials, 139 post-cyclical transformations, 80 Post's correspondence problem, 42 power sum, 142, 145 predicate(s), 187,206

calculus, first-order, 1 predictive analyzer, 55 preterminal string, 74 probabilistic automata, 164, 167 processing time for sentence recognition,

upper bounds of, 64 productions, 52, 121 prognostic variables, 128 projection rule, 94 proof, 1 push-down

automata, 42, 46, 47 store machine, 53

query-language statement, 103 question-answering, 101

system based on transformational grammar, 105

quotient (s), 50

recognition problems of transformational

grammars, 81 procedures, 55

recursion formula for G polynomials, 148 matrix, 131

relationship, "have" or "part of", 99 resolvent, 10, 13 restricted rewriting systems, 53 retriever, 101 reverse

transformational components, 84 transformations, 84

rewriting rules, 52 systems, 52

restricted, 53 rules, ordering of, 74 second-order iteration function, 141 sectional

features, 72 rules, 69

selective top-to-bottom algorithm, 55 semantic

definition, 21 markers, 93 theory, 91

sentence generation of a transformational

grammar, simulation of, 90 recognition, upper bounds of processing

time for, 64 serial

computation, 206 vs. parallel computations, 177

shaper test, 56 simulation of sentence generation of a

transformational grammar, 90 smallest zero, 146 source language, 34, 36, 40 spec list, 103 spectral sequence(s), 118 stability theorems, 167 standard-form grammar, 55 state

grammars, finite, 53 machine, finite, 53 vectors, 36

Stiefel manifolds, 117 stopping rule, 42, 48 store machine, pushdown, 53 strict subcategorization rules, 69 structural

change, 75 description, 54 index, 75

224

structure grammars, phrase, 53 student, 101 subcategorization rules, 69

strict, 69 subdominant zeros, 138 substitutions, 75 surface

grammar, 84 structure, 65

syntactic analysis procedure, M I T R E , 82 features, 67

syntax analyzer, 121 syntax-directed compilers, 55 synthesis, analysis, 82

tag, 21 terminal

string, 54 vocabulary, 53

theorem-proving problem, 1 theory of numbers, 111 thermal convection, 135 threshold function (s), linear, 179,181,

183, 192 transfer processes, nonlinear, 133 transformational

component, 67 reverse, 84

grammars, 52 generative power of, 82 question-answering system based

on,105 recognition problems of, 81 simulation of sentence generation

of, 90 rules, 75

cyclical, 78 transformations

elementary, 75 post-cyclical, 80 reverse, 84

SUBJECT INDEX

translator, 101 tree-automata, 169, 171 tree(s)

binary, 170 valued, 172

truncation, linear and nonlinear, 126

turbulent flow, 132 Turing machine, 42, 44, 45, 48, 53 two-dimensional turbulence, 134 two-stage

algorithm, 139 process, 151

undecidable, 42, 43, 48, 49 unification algorithm, 7 upper bounds of processing time for

sentence recognition, 64

valued tree, 172 variations of basic algorithm, 145 verifiable consequent, strongest, 25 verification, 21

condition, 21 viscosity

artificial, 133,134 coefficient, 133

vocabulary nonterminal, 53 terminal, 53

wave-number space representations, 126,132

well-ordered set, 30, 31

zero(s) of analytic functions, 138, 150 dominant, 138, 139, 144 equimodular dominant, 138, 148 multiple, 138, 147, 148 smallest, 146 of smallest magnitude, 150 subdominant, 138

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