The mnesor theory is the adaptation of vectors to artificial intelligence. The scalar field is replaced by a lattice. Addition becomes idempotent and multiplication is interpreted as a selection operation. We also show that mnesors can be the foundation for a linear calculus.
Geometrical and physical quantities require the notions of point, direction and magnitude and also appropriate operations like addition and scalar multiplication [1]. The former is able to compose vectors or points, the latter to modify magnitude.
( )
The axiomatization of abstract vectors [2] comes up as follows: let be given a vector space
But ias far as information is concerned, we’d better talk about precision instead of magnitude and about ordering instead of direction. Hence, addition should be defined as an aggregation operation and scalar multiplication as a filtering operator. For that, scalar field will be replaced by a lattice.
As simple example of mnesors, we give the column tuple
Two column tuples are added as follows:
EU (for European Union) represents an element of the lattice of intergovernmental organizations, multiplication by € EU acts on a column tuple like a filter:
Only components belonging to the European Union are retained.
Because information structures and processes don’t necessarily commute, we don’t assume commutativity for mnesors. EXAMPLE:
Slowenia comes first in the former colomn list but second in the latter. The addition is not necessarily commutative.
We define a mnesor space as a two-sorted structure made up with a monoid
) . We assume an identity element for
The lattice here plays the same role as a scalar field and for that reason the elements are called granular. Scalar multiplication is replaced by the granular multiplication, which multiplies a mnesor by any lattice element and returns another mnesor. Granular multiplication is considered as a filtering operation. The definition properties follow next:
(unital property) A common assumption with the vector theory is to make addition reversible. We mean that there exists a reverse operation undoing
x + y and calculating
x from € z = x + y . In the vector theory this operation is the substraction ( € x = zy ). We here postulate that there exists a granular doing it (
Idempotence. The addition of mnesors is idempotent.
PROOF:
Applying (4) with EXAMPLE. In
the former is a prefix and the latter a suffix.
The next three propositions are equivalent :
evidently
Prefix relation and ordering. The relation "
x is a prefix for
As a result of a filtering operation, the column list gains in selectivity. The prefix ordering plays the role of a selectivity indicator.
The order is compatible with the addition and the granular multiplication.
For any mnesor
Empty mnesor.
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