Trajectory Databases - Mobility Data - page 50

instant

all pro-
jections
to range

all time
dependent
types for
these

int
real
bool
string

moving ( int )
moving ( real )
moving ( bool )
moving ( string )

range ( int )
range ( real )
range ( bool )
range ( string )

point
points
line
region

moving ( point )
moving ( points )
moving ( line )
moving ( region )

points
line
region

all projections to
domain

periods

Figure 3.2 Type system.

the domain of α . So, for example, range ( real ) is a set of real-valued intervals;
periods is in fact another name for range ( instant ). The types range ( α ) together
with the spatial types points , line ,and region are sufficient to represent the
projections into the ranges for all types moving ( α ). Further, the values of all
types moving ( α ) can be projected on the time axis resulting in a periods value.
Hence design rule D3 is fulfilled.
The design of operations proceeds in three steps:

1. Carefully define a set of operations on the static types.
2. By a technique called lifting , make these operations time dependent.
3. Add some specific operations for the time-dependent types.

Lifting means to make a static operation time dependent by allowing any
(combination) of its arguments to be time dependent. For example, consider
the equality and intersection operations on two points. By lifting, the following
signatures are available. 3

= : point × point → bool intersection : point × point → point

mpoint × point → mbool mpoint × point → mpoint

point × mpoint → mbool point × mpoint → mpoint

mpoint × mpoint → mbool mpoint × mpoint → mpoint

Lifted versions of these two operations are used in the last query of Section 3.2.1 .

3.2.3 Implementation

In the model described so far, the semantics of time-dependent types, that is,
of types moving ( α ), have been simply defined as partial functions, disregarding

3 We generally abbreviate the formally defined notation moving ( α )by m α .