270
contraction on a quasimetric, 133
Egli–Milner monotonic, 126
fixed point of, 125
Hoare monotonic, 126-128, 135
non-empty, 126
non-expanding on a gum, 129
non-expanding on a quasimetric,
133
orbit of, 127
Smyth monotonic, 126
strictly contracting on a (d-)gum,
129
strictly contracting on orbits, 129
A b
1
-contraction, 131
2
n-dimensional level mapping, 207
n-layer feedforward network, 189, 190
n-layer recurrent network, 190
Negation, xx, 8, 10, 20, 44, 56, 80
Negation as (finite) failure, xx
Negative literal, 8
Neg P , 144
Neg
∗ , 145
P
Neighbourhood, 235
base, 235
basic, 235
system, 235
Net, 237
convergence of, 67, 237
converges to, 67, 237
eventually (satisfies a property),
237
greatest limit of, 75
limit of, 237
subnet of, 237
tail of, 237
Network function, 189
Network made recurrent, 190
Neural network, 185-191
RBF, 204
sigmoidal, 201
vector-based, 207
Neural-Symbolic Cycle, 186
Neural-symbolic integration, 185-188
Node in a neural network, 188
Index
Non-empty multivalued mapping, 126
Non-expanding function in a d-gum,
99
Non-expanding function in a quasi-
metric, 106
Non-expanding multivalued mapping
on a gum, 129
Non-expanding multivalued mapping
on a quasimetric, 133
Non-termination, 10
None, 9
Normal logic program, 24
Occurs negatively in body, 24
Occurs positively in body, 24
ω-chain, xxiii, 1-6
ω-complete partial order, 2-6, 14, 135
ω-continuous function, 5, 6, 40, 42, 57,
135
ω-continuous multivalued mapping,
135
ω-cpo, 2
ω-locally hierarchical program, 141
ω-orbit of a multivalued mapping,
127, 133-135
Ontology, 223, 224
Open ball about a point, 89
Open cover of a topological space, 239
subcover of, 239
Open set, 66, 79, 82, 89, 163, 234
Operational semantics, xxi
Orbit of a function, 99
Orbit of a multivalued mapping, 127
eventually increasing, 127
increasing, 127
tight, 127
Order isomorphism, 230
Order-continuous function, 5
Ordered semigroup with identity, 89
Ordered set, 1
Ordering, 1
dual, 2, 119
Egli–Milner, 126
Hoare, 126
knowledge, 10, 16, 37
contraction on a quasimetric, 133
Egli–Milner monotonic, 126
fixed point of, 125
Hoare monotonic, 126-128, 135
non-empty, 126
non-expanding on a gum, 129
non-expanding on a quasimetric,
133
orbit of, 127
Smyth monotonic, 126
strictly contracting on a (d-)gum,
129
strictly contracting on orbits, 129
A b
1
-contraction, 131
2
n-dimensional level mapping, 207
n-layer feedforward network, 189, 190
n-layer recurrent network, 190
Negation, xx, 8, 10, 20, 44, 56, 80
Negation as (finite) failure, xx
Negative literal, 8
Neg P , 144
Neg
∗ , 145
P
Neighbourhood, 235
base, 235
basic, 235
system, 235
Net, 237
convergence of, 67, 237
converges to, 67, 237
eventually (satisfies a property),
237
greatest limit of, 75
limit of, 237
subnet of, 237
tail of, 237
Network function, 189
Network made recurrent, 190
Neural network, 185-191
RBF, 204
sigmoidal, 201
vector-based, 207
Neural-Symbolic Cycle, 186
Neural-symbolic integration, 185-188
Node in a neural network, 188
Index
Non-empty multivalued mapping, 126
Non-expanding function in a d-gum,
99
Non-expanding function in a quasi-
metric, 106
Non-expanding multivalued mapping
on a gum, 129
Non-expanding multivalued mapping
on a quasimetric, 133
Non-termination, 10
None, 9
Normal logic program, 24
Occurs negatively in body, 24
Occurs positively in body, 24
ω-chain, xxiii, 1-6
ω-complete partial order, 2-6, 14, 135
ω-continuous function, 5, 6, 40, 42, 57,
135
ω-continuous multivalued mapping,
135
ω-cpo, 2
ω-locally hierarchical program, 141
ω-orbit of a multivalued mapping,
127, 133-135
Ontology, 223, 224
Open ball about a point, 89
Open cover of a topological space, 239
subcover of, 239
Open set, 66, 79, 82, 89, 163, 234
Operational semantics, xxi
Orbit of a function, 99
Orbit of a multivalued mapping, 127
eventually increasing, 127
increasing, 127
tight, 127
Order isomorphism, 230
Order-continuous function, 5
Ordered semigroup with identity, 89
Ordered set, 1
Ordering, 1
dual, 2, 119
Egli–Milner, 126
Hoare, 126
knowledge, 10, 16, 37