ISO 10303-50:2002

INTERNATIONAL ISO
STANDARD 10303-50
First edition
2002-05-01
Industrial automation systems and
integration — Product data representation
and exchange —
Part 50:
Integrated generic resource: Mathematical
constructs
Systèmes d'automatisation industrielle et intégration — Représentation
et échange de données de produits —
Partie 50: Ressources génériques intégrées: Constructions mathématiques

Reference number
©
ISO 2002
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Contents Page
1 Scope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
2 Normative references . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
3 Terms, definitions, and symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
3.1 Terms defined in ISO 10303-1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
3.2 Other terms and definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
3.3 Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
4 Mathematical functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
4.1 Introduction .8
4.2 Fundamental concepts and assumptions .8
4.2.1 Mathematical function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
4.2.2 Mathematical object or value . . . . . . . . . . . . . . . . . . . . . . . . . 9
4.2.3 Mathematical expression . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
4.2.4 Mathematical space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
4.2.5 Mathematical tuple . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
4.2.6 Function domains and ranges . . . . . . . . . . . . . . . . . . . . . . . . . 11
4.2.7 Spaces of one-tuples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
4.2.8 Array function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
4.2.9 Table function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
4.2.10 Matrix. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
4.2.11 Inputs and Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
4.2.12 Function evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
4.2.13 Function application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
4.3 Mathematical functions schema constant definitions . . . . . . . . . . . . . . . 14
4.3.1 schema prefix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
4.3.2 the elementary spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
4.3.3 the empty space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
4.3.4 real intervals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
4.3.5 tuple spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
4.3.6 empty values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
4.4 Mathematical functions schema type definitions . . . . . . . . . . . . . . . . . . 17
4.4.1 nonnegative integer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
4.4.2 positive integer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
4.4.3 zero or one . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
4.4.4 one or two . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
4.4.5 local names for simple types . . . . . . . . . . . . . . . . . . . . . . . . . 18
4.4.6 maths simple atom . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
4.4.7 maths atom . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
4.4.8 atom based tuple . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
4.4.9 atom based value. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
4.4.10 maths tuple . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
4.4.11 maths value . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
4.4.12 maths expression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
4.4.13 maths function select . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
4.4.14 input selector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
4.4.15 elementary space enumerators . . . . . . . . . . . . . . . . . . . . . . . . 22
4.4.16 ordering type . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
4.4.17 lower upper . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
4.4.18 symmetry type . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
4.4.19 elementary function enumerators . . . . . . . . . . . . . . . . . . . . . . . 25
4.4.20 open closed . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
4.4.21 space constraint type . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
4.4.22 repackage options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
4.4.23 extension options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
4.4.24 maths enum atom . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
4.4.25 dotted express identifier . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
4.4.26 express identifier . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
4.4.27 product space. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
4.4.28 tuple space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
4.4.29 maths space or function . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
4.4.30 real interval . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
4.5 Mathematical functions schema entity definitions . . . . . . . . . . . . . . . . . 36
4.5.1 quantifier expression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
4.5.2 dependent variable definition . . . . . . . . . . . . . . . . . . . . . . . . . 37
4.5.3 bound variable semantics . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
4.5.4 free variable semantics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
4.5.5 complex number literal . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
4.5.6 logical literal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
4.5.7 binary literal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
4.5.8 maths enum literal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
4.5.9 real tuple literal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
4.5.10 integer tuple literal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
4.5.11 atom based literal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
4.5.12 maths tuple literal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
4.5.13 maths variable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
4.5.14 maths real variable. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
4.5.15 maths integer variable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
4.5.16 maths boolean variable . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
4.5.17 maths string variable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
4.5.18 function application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
4.5.19 maths space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
4.5.20 elementary space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
4.5.21 finite integer interval. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
4.5.22 integer interval from min . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
interval to max . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
4.5.23 integer
4.5.24 finite real interval . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
4.5.25 real interval from min . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
4.5.26 real interval to max . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
4.5.27 cartesian complex number region . . . . . . . . . . . . . . . . . . . . . . . 49
4.5.28 polar complex number region . . . . . . . . . . . . . . . . . . . . . . . . . 50
4.5.29 finite space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
4.5.30 uniform product space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
4.5.31 listed product space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
4.5.32 extended tuple space. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
4.5.33 function space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
4.5.34 maths function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
4.5.35 finite function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
4.5.36 constant function. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
4.5.37 selector function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
4.5.38 elementary function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
4.5.39 restriction function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
iv © ISO 2002 — All rights reserved

4.5.40 repackaging function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
4.5.41 reindexed array function. . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
4.5.42 series composed function . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
4.5.43 parallel composed function . . . . . . . . . . . . . . . . . . . . . . . . . . 64
4.5.44 explicit table function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
4.5.45 listed real data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
4.5.46 listed integer data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
4.5.47 listed logical data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
4.5.48 listed string data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
4.5.49 listed complex number data . . . . . . . . . . . . . . . . . . . . . . . . . . 71
4.5.50 listed data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
4.5.51 externally listed data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
4.5.52 linearized table function . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
4.5.53 standard table function . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
4.5.54 regular table function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
4.5.55 triangular matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
4.5.56 strict triangular matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
4.5.57 symmetric matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
4.5.58 symmetric banded matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
4.5.59 banded matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
4.5.60 basic sparse matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
4.5.61 homogeneous linear function . . . . . . . . . . . . . . . . . . . . . . . . . 85
4.5.62 general linear function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
4.5.63 b spline basis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
4.5.64 b spline function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
4.5.65 rationalize function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
4.5.66 partial derivative function . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
4.5.67 partial derivative expression . . . . . . . . . . . . . . . . . . . . . . . . . 92
4.5.68 definite integral function . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
4.5.69 definite integral expression . . . . . . . . . . . . . . . . . . . . . . . . . . 95
4.5.70 abstracted expression function . . . . . . . . . . . . . . . . . . . . . . . . 97
4.5.71 expression denoted function . . . . . . . . . . . . . . . . . . . . . . . . . . 98
4.5.72 imported point function . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
4.5.73 imported curve function . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
4.5.74 imported surface function . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
volume function . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
4.5.75 imported
4.5.76 application defined function . . . . . . . . . . . . . . . . . . . . . . . . . . 102
4.5.77 mathematical description . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
4.6 Mathematical functions schema function definitions . . . . . . . . . . . . . . . 104
4.6.1 all members of es. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
4.6.2 any space satisfies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
4.6.3 assoc product space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
4.6.4 atan2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
4.6.5 bool . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
4.6.6 check sparse index domain . . . . . . . . . . . . . . . . . . . . . . . . . . 110
4.6.7 check sparse loc range . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
4.6.8 check sparse index to loc . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
4.6.9 compare basis and coef . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
4.6.10 compare list and value . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
4.6.11 compare values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
4.6.12 compatible complex number regions . . . . . . . . . . . . . . . . . . . . . 115
4.6.13 compatible es values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
v
4.6.14 compatible intervals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
4.6.15 compatible spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
4.6.16 composable sequence. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
4.6.17 convert to literal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
4.6.18 convert to maths function . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
4.6.19 convert to maths value . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
4.6.20 convert to operand . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
4.6.21 convert to operands . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
4.6.22 convert to operands prcmfn . . . . . . . . . . . . . . . . . . . . . . . . . . 128
4.6.23 definite integral check . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
4.6.24 definite integral expr check . . . . . . . . . . . . . . . . . . . . . . . . . . 129
4.6.25 derive definite integral domain . . . . . . . . . . . . . . . . . . . . . . . . 130
4.6.26 derive elementary function domain . . . . . . . . . . . . . . . . . . . . . . 132
4.6.27 derive elementary function range . . . . . . . . . . . . . . . . . . . . . . . 135
4.6.28 derive finite function domain . . . . . . . . . . . . . . . . . . . . . . . . . 137
4.6.29 derive finite function range . . . . . . . . . . . . . . . . . . . . . . . . . . 138
4.6.30 derive function domain . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
4.6.31 derive function range . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
4.6.32 domain from . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
4.6.33 dot count . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
4.6.34 dotted identifiers syntax . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
4.6.35 drop numeric constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
4.6.36 enclose cregion in pregion . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
4.6.37 enclose pregion in cregion . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
4.6.38 enclose pregion in pregion . . . . . . . . . . . . . . . . . . . . . . . . . . . 154
4.6.39 equal cregion pregion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160
4.6.40 equal maths functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162
4.6.41 equal maths spaces. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163
4.6.42 equal maths values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166
4.6.43 es subspace of es . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168
4.6.44 expression is constant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169
4.6.45 extract factors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169
4.6.46 extremal position check . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170
4.6.47 factor1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171
4.6.48 factor space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172
variables of . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172
4.6.49 free
4.6.50 function applicability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173
4.6.51 function is 1d array . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174
4.6.52 function is 1d table . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175
4.6.53 function is 2d table . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176
4.6.54 function is array . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177
4.6.55 function is table . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177
4.6.56 has values space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178
4.6.57 list selected components . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180
4.6.58 make abstracted expression function . . . . . . . . . . . . . . . . . . . . . 180
4.6.59 make atom based literal . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181
4.6.60 make b spline basis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181
4.6.61 make b spline function. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182
4.6.62 make banded matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182
4.6.63 make basic sparse matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . 183
4.6.64 make binary literal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184
4.6.65 make boolean literal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184
vi © ISO 2002 — All rights reserved

4.6.66 make cartesian complex number region . . . . . . . . . . . . . . . . . . . 185
4.6.67 make complex number literal . . . . . . . . . . . . . . . . . . . . . . . . . 185
4.6.68 make constant function . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186
4.6.69 make cos expression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186
4.6.70 make definite integral expression . . . . . . . . . . . . . . . . . . . . . . . 187
4.6.71 make definite integral function . . . . . . . . . . . . . . . . . . . . . . . . 187
4.6.72 make elementary function . . . . . . . . . . . . . . . . . . . . . . . . . . . 188
4.6.73 make elementary space . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188
4.6.74 make environment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189
4.6.75 make expression denoted function . . . . . . . . . . . . . . . . . . . . . . 189
4.6.76 make extended tuple space . . . . . . . . . . . . . . . . . . . . . . . . . . 190
4.6.77 make finite function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190
4.6.78 make finite integer interval . . . . . . . . . . . . . . . . . . . . . . . . . . 191
4.6.79 make finite real interval . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191
4.6.80 make finite space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192
4.6.81 make function application . . . . . . . . . . . . . . . . . . . . . . . . . . . 192
4.6.82 make function space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193
4.6.83 make general linear function . . . . . . . . . . . . . . . . . . . . . . . . . 194
4.6.84 make int literal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194
4.6.85 make integer interval from min . . . . . . . . . . . . . . . . . . . . . . . . 195
4.6.86 make listed complex number data . . . . . . . . . . . . . . . . . . . . . . 195
4.6.87 make listed data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196
4.6.88 make listed integer data . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196
4.6.89 make listed product space . . . . . . . . . . . . . . . . . . . . . . . . . . . 197
4.6.90 make listed real data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197
4.6.91 make logical literal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198
4.6.92 make maths enum literal . . . . . . . . . . . . . . . . . . . . . . . . . . . 198
4.6.93 make maths real variable . . . . . . . . . . . . . . . . . . . . . . . . . . . 199
4.6.94 make maths tuple literal . . . . . . . . . . . . . . . . . . . . . . . . . . . 199
4.6.95 make mult expression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200
4.6.96 make parallel composed function . . . . . . . . . . . . . . . . . . . . . . . 200
4.6.97 make partial derivative expression . . . . . . . . . . . . . . . . . . . . . . 201
4.6.98 make partial derivative function . . . . . . . . . . . . . . . . . . . . . . . 201
4.6.99 make polar complex number region . . . . . . . . . . . . . . . . . . . . . 202
4.6.100 make rationalize function . . . . . . . . . . . . . . . . . . . . . . . . . . . 202
real interval from min. . . . . . . . . . . . . . . . . . . . . . . . . . 203
4.6.101 make
4.6.102 make real interval to max . . . . . . . . . . . . . . . . . . . . . . . . . . . 203
4.6.103 make real literal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204
4.6.104 make regular table function . . . . . . . . . . . . . . . . . . . . . . . . . . 204
4.6.105 make reindexed array function . . . . . . . . . . . . . . . . . . . . . . . . 205
4.6.106 make repackaging function . . . . . . . . . . . . . . . . . . . . . . . . . . 205
4.6.107 make selector function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206
4.6.108 make series composed function . . . . . . . . . . . . . . . . . . . . . . . . 207
4.6.109 make sin expression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207
4.6.110 make standard table function . . . . . . . . . . . . . . . . . . . . . . . . . 208
4.6.111 make strict triangular matrix . . . . . . . . . . . . . . . . . . . . . . . . . 208
4.6.112 make string literal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209
4.6.113 make unary minus expression . . . . . . . . . . . . . . . . . . . . . . . . . 210
4.6.114 make uniform product space . . . . . . . . . . . . . . . . . . . . . . . . . 210
4.6.115 max exists . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211
4.6.116 max included . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211
4.6.117 member of . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212
4.6.118 min exists . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217
4.6.119 min included . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217
4.6.120 no cyclic domain reference . . . . . . . . . . . . . . . . . . . . . . . . . . 218
4.6.121 no cyclic space reference. . . . . . . . . . . . . . . . . . . . . . . . . . . . 219
4.6.122 nondecreasing. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220
4.6.123 number superspace of . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220
4.6.124 number tuple subspace check . . . . . . . . . . . . . . . . . . . . . . . . . 221
4.6.125 one tuples of . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221
4.6.126 parallel composed function composability check. . . . . . . . . . . . . . . 222
4.6.127 parallel composed function domain check . . . . . . . . . . . . . . . . . . 222
4.6.128 parse express identifier . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223
4.6.129 partial derivative check . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224
4.6.130 real max . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225
4.6.131 real min . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225
4.6.132 regular indexing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226
4.6.133 remove first . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227
4.6.134 repackage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227
4.6.135 shape of array . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228
4.6.136 simplify function application . . . . . . . . . . . . . . . . . . . . . . . . . 229
4.6.137 simplify generic expression . . . . . . . . . . . . . . . . . . . . . . . . . . 243
4.6.138 simplify maths space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249
4.6.139 simplify maths value . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 250
4.6.140 singleton member of . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251
4.6.141 space dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252
4.6.142 space is continuum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252
4.6.143 space is singleton . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253
4.6.144 stripped typeof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254
4.6.145 subspace of . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254
4.6.146 subspace of es . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 260
4.6.147 substitute . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261
4.6.148 values space of . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263
Annex A (normative) Short names of entities . . . . . . . . . . . . . . . . . . . . . . . 266
Annex B (normative) Information object registration . . . . . . . . . . . . . . . . . . . 269
Annex C (informative) Computer-interpretable listings . . . . . . . . . . . . . . . . . . . 270
Annex D (informative) EXPRESS-G diagrams . . . . . . . . . . . . . . . . . . . . . . . 271
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283
Figures
Figure 1 — Schema relationships of the mathematical functions schema . . . . . . . . . . xi
Figure D.1 — EXPRESS-G diagram of the mathematical functions schema (1 of 10) . . . 272
Figure D.2 — EXPRESS-G diagram of the mathematical functions schema (2 of 10) . . . 273
Figure D.3 — EXPRESS-G diagram of the mathematical functions schema (3 of 10) . . . 274
Figure D.4 — EXPRESS-G diagram of the mathematical functions schema (4 of 10) . . . 275
Figure D.5 — EXPRESS-G diagram of the mathematical functions schema (5 of 10) . . . 276
Figure D.6 — EXPRESS-G diagram of the mathematical functions schema (6 of 10) . . . 277
viii © ISO 2002 — All rights reserved

IS O 10303-50:2002(E)
Figure D.7 — EXPRESS-G diagram of the mathematical functions schema (7 of 10) . . . 278
Figure D.8 — EXPRESS-G diagram of the mathematical functions schema (8 of 10) . . . 279
Figure D.9 — EXPRESS-G diagram of the mathematical functions schema (9 of 10) . . . 280
Figure D.10 — EXPRESS-G diagram of the mathematical functions schema (10 of 10). . 281
Tables
Table 1 — Mathematical symbology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
Table 2 — Orderings indicated by ordering type . . . . . . . . . . . . . . . . . . . . . . . 23
Table A.1 — Short names of entities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 266
Foreword
ISO (the International Organization for Standardization) is a worldwide federation of national
standards bodies (ISO member bodies). The work of preparing International Standards is nor-
mally carried out through ISO technical committees. Each member body interested in a subject
for which a technical committee has been established has the right to be represented on that
committee. International organizations, governmental and non-governmental, in liaison with
ISO, also take part in the work. ISO collaborates closely with the International Electrotechnical
Commission (IEC) on all matters of electrotechnical standardization.
InternationalStandardsaredraftedinaccordancewiththerulesgivenintheISO/IECDirectives,
Part 3.
ThemaintaskoftechnicalcommitteesistoprepareInternationalStandards. DraftInternational
Standards adopted by the technical committees are circulated to the member bodies for voting.
PublicationasanInternationalStandardrequiresapprovalbyatleast75%ofthememberbodies
casting a vote.
Attention is drawn to the possibility that some of the elements of this document may be the
subject of patent rights. ISO shall not be held responsible for identifying any or all such patent
rights.
ISO10303–50waspreparedbyTechnicalCommitteeISO/TC184,Industrialautomationsystems
and integration, Subcommittee SC4, Industrial data.
This International Standard is organized as a series of parts, each published separately. The
structure of this International Standard is described in ISO 10303-1.
Each part of this International Standard is a member of one of the following series: description
methods,implementationmethods,conformancetestingmethodologyandframework,integrated
generic resources, integrated application resources, application protocols, abstract test suites,
application interpreted constructs, and application modules. This part is a member of the inte-
grated generic resources series. The integrated generic resources and the integrated application
resources specify a single conceptual product data model.
A complete list of parts of ISO 10303 is available from the Internet:

Annexes A and B form a normative part of this part of ISO 10303. Annexes C and D are for
information only.
x © ISO 2002 — All rights reserved

Introduction
ISO10303isanInternationalStandardforthecomputer-interpretablerepresentationofproduct
information and for the exchange of product data. The objective is to provide a neutral mech-
anism capable of describing products throughout their life cycle. This mechanism is suitable
not only for neutral file exchange, but also as a basis for implementing and sharing product
databases, and as a basis for archiving.
This part of ISO 10303 is a member of the integrated resource series. This part of ISO 10303
specifies the mathematical functions schema.
This part of ISO 10303 specifies EXPRESS data representations for a large class of mathemat-
ical functions, expressions, and arrays. They are intended to be used to communicate product
property data and related engineering analysis data. Familiarity with the branches of math-
ematics commonly used in engineering applications is assumed. The central concept is that
mathematical functions and arrays are the abstract data objects of interest for expressing any
property which requires more than a single number as a value, for describing deterministic rela-
tionshipsbetweenproperties, andfordocumentingbehaviouralresponsesofproductstovarying
conditions.
The relationships of the schema in this part of ISO 10303 to other schemas that define the
integrated resources of this International Standard are illustrated in Figure 1 using the EX-
PRESS-G notation. EXPRESS-G is defined in annex D of ISO 10303-11. The documents
containing the specifications for these related schemas are identified in note 1 at the beginning
of clause 4.
ISO13584 generic expressions schema
mathematical functions schema
e
ISO13584 expressions schema
e
label
support resource schema
text
? e
curve
dimension of
point
surface
geometry schema
volume
? e
external reference schema
externally defined item
? e
Figure 1 – Schema relationships of the mathematical functions schema
INTERNATIONAL STANDARD
Industrial automation systems and integration —
Product data representation and exchange —
Part 50 :
Integrated generic resource: Mathematical constructs
1 Scope
This part of ISO 10303 specifies the resource constructs for the explicit representation of math-
ematical structures and data related to properties of a product.
The following are within the scope of this part of ISO 10303:
— multi-dimensional tables;
— mathematical expressions;
— mathematical functions;
— mathematical spaces.
The following are outside the scope of this part of ISO 10303:
— context of application;
— physical units;
— non-mathematical semantics.
2 Normative references
The following normative documents contain provisions which, through reference in this text,
constitute provisions of this part of ISO 10303. For dated references, subsequent amendments
to, or revisions of, any of these publications do not apply. However, parties to agreements
based on this part of ISO 10303 are encouraged to investigate the possibility of applying the
most recent editions of the normative documents indicated below. For undated references, the
latesteditionofthenormativedocumentreferredtoapplies. MembersofISOandIECmaintain
registers of currently valid International Standards.
ISO/IEC 8824-1:1998, Informationtechnology— AbstractSyntaxNotationOne(ASN.1): Spec-
ifi cation of basic notation
ISO 10303-1:1994, Industrial automation systems and integration — Product data representa-
tion and exchange— Part 1: Overview and fundamental principles
ISO 10303-11:1994, Industrial automation systems and integration — Product data representa-
tion and exchange— Part 11: Description methods: The EXPRESS language reference manual
ISO 10303-41:2000, Industrial automation systems and integration — Product data representa-
tion and exchange — Part 41: Integrated generic resource: Fundamentals of product description
and support
ISO 10303-42:2000, Industrial automation systems and integration — Product data representa-
tion and exchange — Part 42: Integrated generic resource: Geometric and topological represen-
tation
ISO 13584-20:1998, Industrial automation systems and integration — Parts Library — Part 20:
Logical resource: Logical model of expressions
3 Terms, definitions, and symbols
3.1 Terms defined in ISO 10303-1
For the purpose of this part of ISO 10303, the following terms defined in ISO 10303-1 apply:
— integrated resource
3.2 Other terms and definitions
For the purposes of this part of ISO 10303, the following definitions apply:
3.2.1
actual function domain
mathematical space containing precisely the tuples of inputs to the function which are related
to some tuple of outputs
NOTE Seethedefinitionsfor“mathematicalfunction”and“functiondomain”. Seealso4.2.1and4.2.6.
3.2.2
actual function range
mathematicalspacecontainingpreciselythetuplesofoutputsfromthefunctionwhicharerelated
to some tuple of inputs
NOTE See the definitions for “mathematical function” and “function range”. See also 4.2.1 and 4.2.6.
3.2.3
array function
function whose domain is either a Cartesian product of finite intervals of integers or the one-
tuples from such a Cartesian product
NOTE An element of such a domain is a subscript tuple. Evaluation of the function at such an element
produces the “array entry” for that subscript tuple.
3.2.4
bound variable
variable which has been specifically referenced by a quantifier in a quantifier expression
NOTE A bound variable is no longer available for substitution by a constant value. The semantics of
the quantifier expression uses all possible values of the variable rather than one (perhaps undetermined)
value.
EXAMPLE 1 The variable x in the statement8x(x=x).
EXAMPLE 2 The variable x in the set declarationfxjx =xg.
R
EXAMPLE 3 The variable x in the definite integral ln(x)dx.
EXAMPLE 4 The variable x in the function definition f(x)‘x+2.
3.2.5
Cartesian product space
mathematical space consisting of all ordered tuples whose components are members of the cor-
responding factor spaces of the product
NOTE Some care must be taken to be explicit about which spaces are the factors entering into the
3 3
Cartesian product. For example, R ‚R is the space of ordered pairs of ordered triples from space R,
not the space of ordered sextuples from space R. When the factor spaces of the explicit factors are to be
treated as the factor spaces of the Cartesian product, that is, the member tuples are to be concatenated
rather than entupled, a subscript ‘a’ (indicating the “associative” product) is appended to the Cartesian
3 3 6
product symbol. Thus, R ‚ R ‘R .
a
3.2.6
compatible spaces
mathematical spaces whose intersection is not determined to be empty by the algorithm repre-
sented by the EXPRESS function compatible spaces
NOTE Two spaces are compatible if their intersection is not “obviously” empty. In practice, “obvious”
is defined by what is detected by the algorithm. The purpose is to rule out some obvious nonsense while
permitting ignorance.
3.2.7
computable function
mathematical function for which the relationship between tuples of inputs and tuples of out-
puts is expressible by means of an algorithm which takes the inputs and produces the related
outputs
NOTE All the mathematical functions of interest in the intended applications of this part of ISO 10303
are computable. Nevertheless, the possibility of representing non-computable functions is not ruled out.
3.2.8
expression
languageconstruct composed ofconstants, variables, operators, quantifiers, andgrouping mark-
ers, organized in accordance with the language’s grammar and denoting some object in the
domain of the language
3.2.9
free variable
variable which has not been bound by a quantifier in the expression in which it is used
NOTE A free variable denotes an undetermined member of some set of possible values. There is an
implicit assumption that it may be substituted by any constant denoting one of those values.
3.2.10
function abstraction
quantifieroperationwhichcreatesanexpressiondenotingamathematicalfunctionfromanother
expression and a list of variables
NOTE The variables become bound and identify the inputs to the function. The algorithm to be used
to produce an output is implied by the expression. The ”abstraction” comes from the change in the
semantics of what the new expression denotes. The expression going into the operation denotes some
object. Theresultingexpressiondenotesahigher-orderobject, namelyafunctionwhichproducesobjects
of the original type from inputs of the types associated with the variables named as inputs.
3.2.11
function application
operation which creates an expression by applying a function to appropriate inputs
NOTE The resulting expression denotes the object which would be the first (and, usually, only) output
of the function when it is applied to the given inputs and evaluated.
3.2.12
function domain
mathematical space specifying all nominally valid tuples of inputs to the mathematical func-
tion
NOTE See the definitions for “mathematical function” and “actual function domain”. See also 4.2.1
and 4.2.6.
3.2.13
function evaluation
process of ascertaining either the outputs a function associates to given inputs, or an error
indication if the function does not associate any outputs to the given inputs
NOTE In the case of a computable function, the process amounts to carrying out the algorithm on the
inputs to produce the outputs, or an error indication.
3.2.14
function range
mathematical space specifying all permissible tuples of outputs from the mathematical func-
tion
NOTE See the definitions for “mathematical function” and “actual function range”. See also 4.2.1 and
4.2.6.
3.2.15
mathematical function
relationship between tuples of mathematical objects called inputs and tuples of mathematical
objects called outputs in which each tuple of inputs is related to at most one tuple of outputs,
together with a mathematical space specifying the function domain and a mathematical space
specifying the function range
NOTE The phrase “mathematical function” in this part of ISO 10303 implies a specification of the
domain and range in addition to the pairing of inputs and outputs. See 4.2.1 and 4.2.6 for the reasons
this definition was chosen for this part of ISO 10303.
4 © ISO 2002 — All rights reserved

3.2.16
mathematical object; mathematical value
abstract object from the domain of mathematics
NOTE The terms mathematical object and mathematical value are formally synonymous in this part
ofISO10303. Thereisatendencytofavortheuseofmathematical value forsimpleobjectslikenumbers,
characters, and enumeration values, and to favor object for complex objects like spaces, functions and
expressions, but no formal distinction is being made.
3.2.17
mathematical space; mathematical set
instance of the fundamental aggregate object type in the domain of mathematics
NOTE No topological, metric, ordering, or vector structure is implied or required. In this part of
ISO 10303, mathematical space is synonymous with mathematical set. See 4.2.4.
3.2.18
matrix
table function whose domain contains pairs of integers
3.2.19
table function
array function with the additional constraint that all the integer interval factor spaces of the
domain either start at zero or at one
NOTE Theimplicationoftheconstraintisthatonlytherelativevaluesofthesubscriptsaresignificant.
Hence, one can use pure relat
...