Fundamental Boolean algebra:
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| Sprache: | English |
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London [u.a.]
Blackie & Son
1967
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| Beschreibung: | XX, 320 S. graph. Darst. |
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| 100 | 1 | |a Kuntzmann, Jean |e Verfasser |4 aut | |
| 240 | 1 | 0 | |a Algèbre de Boole |
| 245 | 1 | 0 | |a Fundamental Boolean algebra |c J. Kuntzmann. Transl. by Scripta Technica Ltd. Engl. transl. ed. by B. Girling |
| 264 | 1 | |a London [u.a.] |b Blackie & Son |c 1967 | |
| 300 | |a XX, 320 S. |b graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
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| 650 | 0 | 7 | |a Boolesche Algebra |0 (DE-588)4146280-4 |2 gnd |9 rswk-swf |
| 689 | 0 | 0 | |a Boolesche Algebra |0 (DE-588)4146280-4 |D s |
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Datensatz im Suchindex
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| adam_text | CONTENTS
List of symbols xix
CHAPTER 1
THE ELEMENTS OF BOOLEAN ALGEBRA
1.1 Boolean quantity
1.1.1 Simple Boolean quantity 1
1.1.2 Examples of Boolean quantities 1
1.1.3 Order relation 1
1.1.4 Duality 2
1.1.5 General Boolean quantity 2
1.1.6 Spatial representation of a general Boolean quantity 3
1.1.7 Binary representation of general Boolean quantities. Affixes 5
1.1.8 Total order relation for the affixes 6
1.1.9 Partial order relation for the values of a general Boolean quantity 6
1.1.10 Duality 7
1.2 Boolean functions
1.2.1 Boolean functions 7
1.2.2 Spatial interpretation of simple or general Boolean functions 8
1.2.3 Dual of a Boolean function 9
1.3 Elementary study of some simple functions
1.3.1 Number of Boolean functions of n variables 9
1.3.2 Simple Boolean functions of one variable 9
1.3.3 Properties of the complement 10
1.3.4 Interpretation of the complement 10
1.3.5 Complement of a general Boolean variable and of a Boolean function 11
1.3.6 Boolean sum 11
1.3.7 Interpretation of the sum 12
1.3.8 Sum of three quantities and of n quantities 13
1.3.9 Boolean products 14
1.3.10 Examples of products 15
1.3.11 Combined properties of the sum and the product 16
1.3.12 Product of three quantities and of n quantities 16
1.3.13 Complement of a sum or product 17
1.3.14 Duality between the sum and the product 17
1.3.15 Simple Boolean functions of two simple variables 18
1.3.16 Non equivalence operation 19
1.3.17 Majority function 20
1.3.18 Lagrange s U function 21
1.3.19 The q function, symmetric functions 21
1.3.20 Extension of the elementary operations to general Boolean variables 22
1.3.21 Particular case of the sum and product 22
1.3.22 Mixed operations 23
1.3.23 Additive condensate, multiplicative condensate 23
1.4 0 Boolean variables and functions
1.4.1 0 Boolean quantity 23
1.4.2 Order relation 24
ix
X CONTENTS
1.4.3 Boolean bounds of a general 0 Boolean quantity 24
1.4.4 Duality for a 0 Boolean quantity 24
1.4.5 Bounds and duality 25
1.4.6 0 Boolean function of a general Boolean variable 25
1.4.7 Interpretation as an incomplete function 25
1.4.8 Duality for a 0 Boolean function 26
1.4.9 Number of 0 Boolean functions 27
1.4.10 Boolean bounds of a 0 Boolean function 27
1.4.11 Compatible 0 Boolean functions 28
1.4.12 Very incomplete functions 29
1.4.13 Example 30
1.4.14 Finding the Boolean functions compatible with a given 0 Boolean
function which are independent of certain variables 30
1.4.15 Extension of a 0 Boolean function to the case of a 0 Boolean variable 32
1.4.16 Sum, product and complement of 0 Boolean quantities 32
1.4.17 Bounds of a complement, a sum, a product 32
1.4.18 Special features of the functions thus defined 33
1.4.19 Equivalence function 33
CHAPTER 2
SYSTEMATIC FORMS
2.1 First and second forms of Lagrange
2.1.1 General 0 Boolean monomial or product expression II 34
2.1.2 Inferiority relation 34
2.1.3 A theorem on the number of monomials inferior to a simple Boolean
function 35
2.1.4 General 0 Boolean polynomial on sum of products expression 211 35
2.1.5 Operations on polynomials 36
2.1.6 Relative complement of a general Boolean variable 36
2.1.7 Fundamental Lagrange theorem for a general 0 Boolean function 37
2.1.8 Transformation from a polynomial to the Lagrangian form 38
2.1.9 Operations on the Lagrangian form 39
2.1.10 0 Boolean monal or sum expression £ 39
2.1.11 0 Boolean polynal or product of sums expression IIS 40
2.1.12 Operations on polynals 40
2.1.13 Lagrange monal 41
2.1.14 Second Lagrange form 41
2.1.15 Transformation from one Lagrangian form to the other 42
2.2 Prefix Notation
2.2.1 Definition of prefixed notation 43
2.2.2 Expressions in prefixed notation 43
2.2.3 Necessary and sufficient condition for a sequence of operations in two
factors written in prefixed notation to be valid 44
2.2.4 Number of expressions containing n operational signs 45
2.3 Representation of an expression
2.3.1 Representation of an operator 46
2.3.2 Combination of operators representing an expression 46
2.3.3 Concept of level 46
2.4 Canonical and quasi canonical expressions
2.4.1 The U operator 47
2.4.2 Canonical expression in U 47
2.4.3 Canonical expression in Z 48
2.4.4 Canonical expressions utilizing sums, products and complements 48
2.4.5 Relationships with the Lagrangian forms 50
CONTENTS xi
2.4.6 Canonical expressions utilizing implication and complement 50
2.4.7 Canonical expression utilizing the NOR function and the complement 50
2.4.8 Canonical expression utilizing the majority function 51
2.4.9 General property of canonical expressions 51
2.4.10 Quasi canonical expressions 52
2.4.11 Quasi canonical expression involving implication 52
2.4.12 Case of an even number of variables 52
2.4.13 Other examples of quasi canonical expressions 53
2.4.14 Final remark 54
2.5 Lexicographic polynomial form
2.5.1 Ordering of the letters in the monomials of the Lagrangian form 55
2.5.2 Local lexicographic and lexicographic monomial 56
2.5.3 Properties of the local lexicographic monomials 56
2.5.4 Local lexicographic polynomial form of a function 56
2.5.5 Transformation of a local lexicographic polynomial form of a simple
function 57
2.5.6 Transformation of a local lexicographic polynomial form of a general
function 57
2.5.7 Irreducible local lexicographic monomial of a polynomial function 57
2.5.8 Irreducible local lexicographic polynomial form 58
2.5.9 Complement of a function written in the local lexicographic form 59
2.5.10 Proof of the theorem on the complement 59
2.5.11 Sum and product of local lexicographic forms 60
2.5.12 Order relation between functions written in the irreducible local lexico¬
graphic polynomial form 61
2.5.13 Study of the local lexicographic polynomial forms of functions com¬
patible with a 0 Boolean function 61
2.5.14 Modification of the lexicographic order in a particular case (abbrevia¬
tion) 62
2.5.15 Abbreviation of a complement, a sum, a product 64
2.5.16 Non uniqueness of the procedure of Sec. 2.5.13 if abbreviations are
permitted 65
2.5.17 Numerical information on the lexicographic forms 66
2.6 Lexicographic expressions
2.6.1 Lexicographic expressions 67
2.6.2 Lexicographic expressions for the U and NOR operators 68
2.6.3 The case of 0 Boolean functions 69
2.7 Sums of orthogonal monomials
2.7.1 Family of orthogonal monomials 70
2.7.2 Weight of a sum of orthogonal monomials 70
2.7.3 Derivation of a sum of orthogonal monomials, starting from a sum of
monomials 71
2.7.4 Orthogonal lexicographic polynomial form 72
2.7.5 Irreducible sum of orthogonal monomials 72
2.7.6 Maximum number of orthogonal monomials in an irreducible form
(simple function) 73
2.7.7 Note on local lexicographic forms 73
2.7.8 Product of two sums of orthogonal monomials 74
2.8 Covering of the cube
2.8.1 A note on the intersections of monomials 74
2.8.2 Application to covering of the cube 74
2.8.3 Maximum number of monomials required for an irredundant covering
of the cube 75
2.8.4 Minimum number of monomials required for a covering of the cube 75
2.8.5 Covering of the cube adapted to an incomplete general function t (X) 76
2.8.6 Expression for O(A ) in terms of the monomials of an adapted covering 77
Xli CONTENTS
CHAPTER 3
RESIDUE FIELDS: POST ALGEBRA
3.1 Residue fields
3.1.1 Classes modulo p 78
3.1.2 Properties of classes 78
3.1.3 Addition of classes 79
3.1.4 Properties of addition 79
3.1.5 Multiplication of classes 80
3.1.6 Properties of multiplication 80
3.1.7 Study of division when the modulus is prime 81
3.1.8 Relationship with Boolean algebra when p = 2 81
3.1.9 Expressions for the Boolean sum and complement operations in Galois
notation 82
3.1.10 Note on the calculus modulo 2 82
3.1.11 Some examples of applications 83
3.1.12 Lagrange s theorem 83
3.1.13 Applications 84
3.1.14 Absence of an analogous theorem in the ring of classes if the modulus is
not prime 85
3.1.15 Possible applications of the case p=£2(p prime) 86
3.2 Post algebra
3.2.1 Definition 87
3.2.2 Properties of the operations 87
3.2.3 Duality 88
3.2.4 Examples of Post algebra 88
3.2.5 Shift 89
3.2.6 Lagrange s theorem 89
CHAPTER 4
PRIME MONOMIALS: IRREDUNDANT BASES
4.1 Consensus
4.1.1 Monomial irredundant from below to a sum of monomials 90
4.1.2 Consensus 91
4.1.3 Characteristic property of the consensus 93
4.1.4 Property of the maximum monomials less than a sum of monomials 94
4.1.5 Case of two monomials 94
4.1.6 Lemmas on the consensus of two and three monomials 94
4.1.7 Marked monomials 98
4.1.8 Description of an algorithm for forming consensuses based on ordering
of the variables 99
4.1.9 Properties of this algorithm 100
4.2 Prime monomials
4.2.1 Prime monomials of a Boolean function 101
4.2.2 Another method of determining the prime monomials. Circuit of a
consensus 102
4.2.3 Consensus relations between prime momomials 104
4.2.4 Prime monomials of a product 105
4.2.5 Prime monomials of a complement 106
4.2.6 Obtaining the prime monomials by starting from the Lagrangian form 106
4.2.7 Formation of functions having a large number of prime monomials 108
4.2.8 Upper bound of the number of prime monomials of a function 109
4.2.9 Representation of Boolean functions by their prime monomials 110
CONTENTS Xiii
4.2.10 Prime monomials and order relations 111
4.2.11 Prime monomial of a simple 0 Boolean function 111
4.2.12 Case of a very incomplete Boolean function 111
4.3 Irredundant bases of simple 0 Boolean functions
4.3.1 Base. Complete base. Irredundant base 112
4.3.2 Remarks on the number of elements of an irredundant base 113
4.3.3 Obligatory prime monomial. Forbidden prime monomial of the first kind 113
4.3.4 A theorem on the non obligatory prime monomials 114
4.3.5 Forbidden prime monomials of the second kind 115
4.3.6 Obtaining all the irredundant bases by covering of the points 115
4.3.7 Finding the irredundant bases by covering the monomials 119
4.3.8 Finding the irredundant bases of a simple Boolean function by con¬
sensus relations 120
4.3.9 Finding the irredundant bases of a simple 0 Boolean function by means
of consensus relations 121
4.3.10 Comparison of the methods from the point of view of the number of
operations 122
4.3.11 Minimization problem in the case of a simple function (complete or
incomplete) 124
4.3.12 Monomials which are not useless but play no part in a representation of
minimum cost 125
4.4 Prime monomials and bases of general functions
4.4.1 Simple function associated with a general function 126
4.4.2 Properties 126
4.4.3 Application to general Boolean monomials 127
4.4.4 Consensus of general Boolean monomials 128
4.4.5 Prime monomial of a general function 130
4.4.6 Base of a general function 131
4.4.7 Finding the irredundant bases 132
4.4.8 Minimization problems 136
4.5 Covering of the faces and edges
4.5.1 Consensus of order p 136
4.5.2 Relation between the consensuses of order p and p— 1 138
4.5.3 Finding the consensuses of order p 139
4.5.4 Property relative to the j dimensional faces 141
4.5.5 Covering the g dimensional faces of a 0 Boolean function 142
4.5.6 Covering of minimum cost of the ^ dimensional faces 143
4.5.7 Examples 143
4.5.8 Consensus of order p for a family of general monomials 144
4.5.9 Covering of the ^ dimensional faces of a general function 145
4.5.10 Covering of minimum cost of the ^ dimensional faces of a general
function 145
4.5.11 Example 145
CHAPTER 5
DISTANCES, SYMMETRIES
5.1 Distances on the hypercube
5.1.1 Distance between two vertices of the hypercube 147
5.1.2 Properties of the distance 147
5.1.3 Distance from a vertex to a simple monomial 148
5.1.4 Property of two intersecting monomials 148
5.1.5 Distance between two monomials 149
5.2. Connectivity
5.2.1 Connected simple Boolean function 150
5.2.2 Connected constituent of a simple function 151
Xiv CONTENTS
5.2.3 Monomials and connectivity 151
5.2.4 Finding the connected constituents of a function written as a sum of
monomials 152
5.2.5 Numerical information on the number of connected constituents 154
5.3 Vertices contiguous to a Boolean function
5.3.1 Definition 155
5.3.2 Property of the vertices contiguous to a function 155
5.3.3 Obtaining the vertices contiguous to a function given as a sum of
monomials 155
5.4 Types of simple Boolean functions
5.4.1 Types of simple Boolean functions 156
5.4.2 List of types of functions of 0, 1, 2, 3 variables 156
5.4.3 Number of functions depending effectively on n variables 158
5.4.4 Permutation and symmetry group of functions of n variables 159
5.4.5 Effect of a cycle on the vertices of the cube 160
5.4.6 Transitivity classes of a product of cycles 161
5.4.7 Functions preserved by a product of cycles 164
5.4.8 Relationship between the number of permutations symmetries which
preserve a function and the number of functions belonging to its type 164
5.4.9 Determination of the number of types of functions having certain
properties 164
5.4.10 Determination of the number of types of functions having i points in
their representative set for n = 3 165
5.4.11 Numerical results on the numbers of types of functions 167
5.4.12 Information on the types of Boolean function of four variables 167
CHAPTER 6
DETAILED STUDY OF FUNCTIONS
6.1 Linear functions
6.1.1 Linear functions 169
6.1.2 Relationship with the parity keys 169
6.2 Monotonic functions
6.2.1 Increasing functions and those that are monotonic with respect to one 170
variable
6.2.2 Increasing functions, monotonic functions 171
6.2.3 Properties of functions which are increasing, or increasing in a 171
6.2.4 Upper envelope increasing in a of a function 172
6.2.5 Expression for, and properties of, the upper envelope increasing in a 173
6.2.6 Lower envelope, increasing in a, of a function 174
6.2.7 Expression for, and properties of, the lower envelope increasing in a 174
6.2.8 Increasing upper envelope of a function 176
6.2.9 Properties of, and expression for, the increasing upper envelope 176
6.2.10 Increasing lower envelope of a function 178
6.2.11 Properties of, and expression for, the increasing lower envelope 178
6.2.12 Properties of increasing functions and their bounds 179
6.2.13 Condition for compatibility of a 0 Boolean function with an increasing
Boolean function 179
6.2.14 Characteristic vertices of an increasing function 180
6.2.15 Characteristic vertices of a function increasing with respect to certain
variables 182
6.2.16 Extension of an increasing 0 Boolean function to a 0 Boolean variable 183
6.2.17 Possibility of extending identities only involving increasing functions to a
0 Boolean variable 183
6.2.18 Maximal number of prime monomials of an increasing function 184
CONTENTS XV
6.3 Increasing function attached to a given function
6.3.1 Increasing 0 Boolean function attached to a 0 Boolean function by
splitting the variables into two 185
6.3.2 Existence of ffl 186
6.3.3 Obtaining the Boolean functions compatible with jfl 187
6.3.4 Finding the lower bound of ff 188
6.3.5 Expression for the upper bound 189
6.3.6 Another expression for the upper bound 190
6.3.7 Transformation from ^° to 0° for a Boolean function 190
6.3.8 Order relation 190
6.3.9 Variables not appearing in f 191
6.3.10 Relation between the bases of $ and those of jfi 191
6.4 Odd functions
6.4.1 Definition 192
6.4.2 Odd symmetric functions 193
6.4.3 Structure of odd functions 193
6.4.4 Even functions 193
6.4.5 Over odd and under odd functions 194
6.4.6 Over odd function as a sum of odd functions 194
6.4.7 Odd, under odd and over odd increasing functions 195
6.4.8 Property of odd increasing functions 196
6.4.9 Structure of over odd increasing functions 196
6.4.10 Structure of odd increasing functions 197
6.4.11 Maximum number of prime monomials in an over odd (or odd, or under
odd) increasing function 197
6.4.12 Type S and type p functions 197
6.5 Families of Boolean functions
6.5.1 Reduction operation 198
6.5.2 Composition operation 198
6.5.3 Commutativity of reduction and composition 199
6.5.4 Family of Boolean functions 199
6.5.5 Intersection and union of two families 199
6.5.6 A family maximal in another 200
6.5.7 Characteristic property of a family maximal in another 200
6.5.8 Finite families 200
6.5.9 Infinite families 201
6.5.10 Linear families 201
6.5.11 Reduction of a non linear function 203
6.5.12 2 and n families 205
6.5.13 The So 2i ^o. l. no, rii and n0,i families 205
6.5.14 ME and Mn families 206
6.5.15 MXX and MH0 families 207
6.5.16 Reduction of a non increasing function 207
6.5.17 Family of odd functions 208
6.5.18 Odd increasing family Ml 208
6.5.19 Reduction of a function which is not over odd or not odd 209
6.5.20 5 and p families 210
6.5.21 Reduction of a function not of type S 210
6.5.22 Over odd and under odd increasing families 211
6.2.23 Reduction of an over odd increasing function not belonging to MS 211
6.5.24 Sub families of MS 212
6.5.25 Families MSi, Mpi 212
6.5.26 A generator of MS, 213
6.5.27 Maximal sub families of MS) 214
6.5.28 Maximal sub families of MS 215
6.5.29 MSn, Mpn families 215
6.5.30 Maximal sub families of MSn 215
Xvi CONTENTS
6.5.31 Classes x, p, y, 8 216
6.5.32 Odd functions of class a.: the family /« 216
6.5.33 Odd functions: the/family 218
6.5.34 Increasing functions: the M family 218
6.5.35 The families Mo, Mi, Moi 219
6.5.36 aX, all families 219
6.5.37 Families aS, ap 220
6.5.38 Families aSu «.p, 221
6.5.39 Maximal sub families of aS) 222
6.5.40 The family a 222
6.5.41 Families ajSS, ajtfl 223
6.5.42 Families S, p 224
6.5.43 Families Sjt p} 224
6.5.44 Maximal sub families of S] 225
6.5.45 Families a.fj and ay 225
6.5.46 The family Q. 227
6.5.47 Summary of results 229
6.5.48 Diagrams of the families if the constants 0 and 1 are at our disposal 229
6.5.49 Diagram of families in which every function is accompanied by its com¬
plement 229
6.6 Threshold functions
6.6.1 Definition 229
6.6.2 Properties of threshold functions 230
6.6.3 Inclusion relation for threshold functions 231
6.6.4 Finding a threshold function compatible with a given increasing 0
Boolean function 232
6.6.5 Successive eliminations 232
6.6.6 Example 232
6.6.7 Minimization of T in the case of the example 233
6.6.8 Another example 234
6.6.9 Window functions 236
6.7 Symmetric functions and total comparison functions
6.7.1 Symmetric functions 237
6.7.2 Number of symmetric functions 238
6.7.3 Increasing symmetric functions 238
6.7.4 Subsets comparable with respect to a function 239
6.7.5 Relation with increasing and decreasing 239
6.7.6 Relation with the bounds 240
6.7.7 Effect of duality 240
6.7.8 Total comparison functions 240
6.7.9 Reduction to total comparison functions 240
6.7.10 Case of symmetric functions 241
6.7.11 Case of threshold functions 242
CHAPTER 7
EQUATIONS AND RELATED TOPICS
7.1 Boolean equations
7.1.1 First fundamental property 244
7.1.2 Second fundamental property 244
7.1.3 Solution of a simple Boolean equation by passing to the Lagrangian
form on both sides 245
7.1.4 Uniqueness theorem for Boolean equations 245
7.1.5 Parametric solution 245
7.1.6 Expression for the solution when it is unique 249
7.1.7 Study of a particular system 250
CONTENTS XVii
7.2 Parametrization
7.2.1 Parametrization of a simple function 251
7.3 Elimination. Boolean resultant
7.3.1 Boolean resultant 252
7.3.2 Exact Boolean resultant 253
7.3.3 Inequality between the resultants 253
7.3.4 Mode of elimination by using the theory of equations 254
7.3.5 Inverse problem to that of parametrization 254
7.3.6 Solution based on the Lagrangian form 254
7.4 Independence of functions
7.4.1 Definition 256
7.4.2 Maximal number of independent functions 256
7.4.3 Properties of systems of n independent functions of n variables 257
7.4.4 Establishment of independence for a system of n functions of n variables 257
7.4.5 Properties of functions belonging to a system of n independent functions
of n variables 258
7.4.6 Expression for any function in terms of a system of n independent
functions 259
7.4.7 Transformation of a function by a mapping of the hypercube onto itself 259
7.4.8 Systems of p functions of n variables 260
7.4.9 Representation of a given function of n simple variables by means of p
given functions 261
7.4.10 Framing of a function relative to p given functions 262
7.4.11 Representation of p functions of n simple variables by means of k
functions 262
7.5 Decomposition of functions
7.5.1 Simple disjoint decomposition 263
7.5.2 Example 264
7.5.3 Disjoint decomposition 265
7.5.4 Semi disjoint decomposition of a function 266
CHAPTER 8
REALIZATION OF BOOLEAN FUNCTIONS BY MEANS OF
GIVEN OPERATORS
8.1 Representation of incomplete functions by means of sums of given functions or q
operators
8.1.1 Nature of the problem 267
8.1.2 Preliminary list 268
8.1.3 Restricted list 268
8.1.4 Finding the representations 269
8.1.5 Structure of a combination of sums and products 269
8.1.6 Lower bound of the cost of a function 270
8.1.7 Upper bound of the cost 271
8.1.8 Properties of the expressions for functions monotonic in a (the cost being
measured by the number of variables) 271
8.1.9 Representations of signature 11211 and XIIE of incomplete Boolean
functions 272
8.1.10 Supression of a factor 274
8.1.11 Continuation of the example in Sec. 8.1.9 276
8.1.12 Generation of incomplete functions, starting from x V yz = q 277
8.1.13 Practical generation 277
8.1.14 Evaluation of the number of operators 279
8.2 Representation of incomplete functions by means given of increasing operators
8.2.1 A general principle 280
8.2.2 Suppression of rows 281
XVlii CONTENTS
8.2.3 Property of rows in the table 282
8.2.4 Representation of incomplete increasing functions by the majority
operation 282
8.2.5 Determination of an odd function greater than a given under odd 0
Boolean function 283
8.2.6 Realization of an incomplete increasing function by majority operators,
using two supplementary variables 286
8.2.7 Realization of an increasing function by means of threshold operators 287
8.2.8 Systematic method of investigation 288
8.2.9 Representation of an arbitrary function by means of increasing operators 289
8.3 Realization of functions by means of monotonic operators
8.3.1 Principle of the method 290
8.3.2 Modification of the method 290
8.3.3 Suppression of columns 291
8.3.4 Suppression of rows 292
8.3.5 Example 292
8.3.6 Parity NOR structure 293
8.3.7 Arbitrary NOR structure 294
8.3.8 Cost of such a formula 295
8.4 Method of superfluous variables
8.4.1 Introduction of superfluous variables into an incomplete function 296
8.4.2 Case of explicit variables 296
8.4.3 Substitution theorem 297
8.4.4 Case where the function is required to be increasing with respect to
explicit variables 297
8.4.5 Case where the function is required to be increasing with respect to all
the variables 298
8.4.6 Relationship with the method of Sec. 8.2.1 298
8.4.7 Application to three level NOR representations of Boolean functions 299
8.4.8 Case of an incomplete general function 301
8.4.9 Application to II£n and EIIZ representations 302
8.4.10 Direct sum 302
8.4.11 Finding the direct sums 303
8.4.12 Extremal direct term 303
8.4.13 Function privileged with respect to the sum and product 304
8.4.14 Passage to a privileged function by splitting the variables 305
8.4.15 Direct decomposition with respect to the NOR operator 306
8.4.16 Obtaining NOR operator representations by splitting the variables 306
8.5 Serial structures
8.5.1 Generation of linear functions on the basis of the non equivalence
operation 307
8.5.2 Serial structure 308
8.5.3 Possibilities of serial structures with operations in two variables 308
8.5.4 Other examples of serial structures 310
8.5.5 On a generalization of serial structures enabling any function to be
obtained 310
8.5.6 Obtaining functions of n+ 1 variables by means of serial operators in n
variables (n 2) 311
8.5.7 Study of functions of five variables, realizable by means of three input
serial operators 311
Recent references 315
Index 317
|
| any_adam_object | 1 |
| author | Kuntzmann, Jean |
| author_facet | Kuntzmann, Jean |
| author_role | aut |
| author_sort | Kuntzmann, Jean |
| author_variant | j k jk |
| building | Verbundindex |
| bvnumber | BV001933169 |
| classification_rvk | SK 150 |
| ctrlnum | (OCoLC)630412267 (DE-599)BVBBV001933169 |
| discipline | Mathematik |
| format | Book |
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| id | DE-604.BV001933169 |
| illustrated | Illustrated |
| indexdate | 2024-07-09T15:37:28Z |
| institution | BVB |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-001260057 |
| oclc_num | 630412267 |
| open_access_boolean | |
| owner | DE-91G DE-BY-TUM DE-703 DE-355 DE-BY-UBR DE-11 DE-210 |
| owner_facet | DE-91G DE-BY-TUM DE-703 DE-355 DE-BY-UBR DE-11 DE-210 |
| physical | XX, 320 S. graph. Darst. |
| publishDate | 1967 |
| publishDateSearch | 1967 |
| publishDateSort | 1967 |
| publisher | Blackie & Son |
| record_format | marc |
| spelling | Kuntzmann, Jean Verfasser aut Algèbre de Boole Fundamental Boolean algebra J. Kuntzmann. Transl. by Scripta Technica Ltd. Engl. transl. ed. by B. Girling London [u.a.] Blackie & Son 1967 XX, 320 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Boolesche Algebra (DE-588)4146280-4 gnd rswk-swf Boolesche Algebra (DE-588)4146280-4 s DE-604 Girling, Brian Sonstige oth HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=001260057&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Kuntzmann, Jean Fundamental Boolean algebra Boolesche Algebra (DE-588)4146280-4 gnd |
| subject_GND | (DE-588)4146280-4 |
| title | Fundamental Boolean algebra |
| title_alt | Algèbre de Boole |
| title_auth | Fundamental Boolean algebra |
| title_exact_search | Fundamental Boolean algebra |
| title_full | Fundamental Boolean algebra J. Kuntzmann. Transl. by Scripta Technica Ltd. Engl. transl. ed. by B. Girling |
| title_fullStr | Fundamental Boolean algebra J. Kuntzmann. Transl. by Scripta Technica Ltd. Engl. transl. ed. by B. Girling |
| title_full_unstemmed | Fundamental Boolean algebra J. Kuntzmann. Transl. by Scripta Technica Ltd. Engl. transl. ed. by B. Girling |
| title_short | Fundamental Boolean algebra |
| title_sort | fundamental boolean algebra |
| topic | Boolesche Algebra (DE-588)4146280-4 gnd |
| topic_facet | Boolesche Algebra |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=001260057&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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