|
|
 |
|
|
|
Euclid
Domains
Domains (Data types) available in Euclid
This document presents the domains used
in Euclid together with the structure of their terms, their
respective functions and relations. Note that equality is
defined in ALL domains.
This document gives only a short summary
of each domain. Please refer to the Euclid manual for more
detailed information.
The real numbers are used much in the same
way as in CLP(R).
- Terms (Constants): {1,2,...},{1.01,...},{infty,pi,e}
- Functions (operators): -/1,+/2,-/2,//2,*/2,^/2,cos/1,sin/1,tan/1,cot/1,log/1,ln/1
- Relations (constraints): real/1,>/2,=/2,<=/2,\=/2
For example X^3>=3*cos(Y)
top of page
Euclid also knows about real intervals
-in the mathematical sense: a tuple of the end points and
the type of interval at each point (open or closed). The
relations between intervals include (/2 (before),
>/2 (after), meets/2, in/2
and many more. Intervals are represented as cc(X,Y)
for [x,y], oc(X,Y) for (X,Y] and so forth for
the other two cases.
- Terms (Constants): oc(X,Y),oo(X,Y),cc(X,Y),co(X,Y)
where X,Y are real numbers, such that X<=Y. ('o' stands
for open and 'c' for closed interval side.
- Functions (operators):
- Relations (constraints): interval/1,>/2,=/2,<=/2,meets/2,overlaps/2.
'>' and '<' stand for (strict) before/2 and after/2
For example oc(1,X) meets (Y,5)
top of page
The real vectors are handled as lists of
real elements
- Terms (Constants): [X,Y,...,Z],
where X,Y,...,Z are reals terms
- Functions (operators): -/1,+/2,-/2,//2,*/2,x/2,^/2,cos/1,sin/1,tan/1,cot/1,log/1,ln/1
(direct generalizations of the real ones)
- Relations (constraints): vector/2
(second argument is dimension)
For example vector(X,2),X^2=25,X=[5,M]
top of page
These objects are much the same as what
is known in mathematics as one-valued reals functions but
we will reserve the term function in Euclid to denote
a much more complex object (one that has an interval based
domain to take its values from). Real formulae are built
up from real terms with the addition of a placeholder variable
- the one we expect to substitute in the future. The formula
f(t)=3*cos(w t) where w is a real
variable can be represented in euclid as F=(3*cos(W*T))//T.
The variable W will take its value from the
real solver while T is only a placeholder for
real values. The domain of real formulae comes equipped
with the functors +, -, *, /, ^ ,limit/2, o/2, integral/1,
derivative/1. The functors limit/2 and ^/2
are cross--domain functions defined as FxR->R. It is also
interesting to note the function o/2 (composition)
which allows for change of placeholder variables (i.e. if
you know f(x) and x(t) you can calculate f(t) by x(t) o
f(x), or in Euclid X o F). integral/1
and derivative/1 give the standard integral
and total derivative respectively of the formula at hand.
The way to obtain values from formulae is through the @/2}
function defined as F x R -> R (i.e. f(2) would be given
by the real term F@2).
- Terms (Constants): KK//X, where
KK is a real symbolic term which may or may not contain
the real variable X.
- Functions (operators): -/1,-/2,+/2,*/2,derivative/1,integral/1,@/2(value
at), o/2(compose)
- Relations (constraints): formula/1
For example formula(X),X=(3*Z^2-cos(2*pi))//Z,X@2=12
top of page
A function is an ordered collection
of (interval,formula) pairs such that the interval of each
pair is before the interval of the previous
pair. If such a pair of intervals meets at a point both
formulae must have the same value for that point. This type
of function is some times referred to as a multi-branch
function.
- Terms (Constants): {[ int1:form1,...,intn:formn
]} where inti are intervals, and formi are formulae constrained
as explained above
- Functions (operators): #/2 (combine
branches), @/2 (value at), derivative/2, integral/2 (direct
generalizations of the real ones)
- Relations (constraints): function/1
For example :
formula(F),
F = {[
cc(-infty,2): (X^2-2)//X ,
cc(2,infty): (X^3-6)//X
]}.
It is natural for variables to be declared
at the beginning of each clause; this way the programmer
makes sure that there is no ambiguity (except intentionally)
on constraints (e.g. X+Y=Z could be answered as correct
if X,Y,Z belong to the reals, vectors, formulae or functions
-unless any of the variables has already been committed
to a single domain).
© 1995 Dr. K J Dryllerakis
Please read the Legal
Information concerning Euclid.
CLP(R) is a Trademark of IBM Watson Research Centre, Quintus
Prolog is a trademark of Quintus Corporation, USA. SunOS
is a trademark of Sun MicroSystems.
|
|
|
|
|
