This document presents a rough guide
to the syntax of Euclid. The conventions followed when
moving from constraint normal forms to computer language
statements are these employed by Prolog and CLP(R).
A working knowledge of Prolog and/or CLP(R) is assumed.
A Quick Reference to the terminology
used:
A Euclid program is a collection
of constraint clauses. Each constraint clause is
a statement about the problem in hand. Constraint clauses
have a standard form which is based on the constraint
normal form. Clauses do not offer the full expresionality
of the logic but they offer easy computational manipulation
(and as can be later seen the expressive power is not
that limited).
A constraint clause is a logical
statement of the form:
Head :- {C1,...,Cn},Body
where Head is an atom, C1,...,Cn
are D-domain constraints and Body is a conjunction
of atoms.
A D-domain constraint is a relation
defined (internally) on a specific domain(Euclid
contains a number of different domains). Syntactically
it is an atom with some restrictions on the terms it contains.
 |
Example
1 |
1 gravity_force_field(M,G,F):-
2 reals([M,G]),
3 formula(F),
4 F=(M*G)//X.
5 object_dynamics(F,M,A):-
6 formulae([F,A]),
7 real(M),
8 (1/M)*F=A.
|
Here the program is made of two constraint
clauses (lines 1-4 and lines 5-8). There are no body atoms
since all relations in lines 2,3,4,6,7 and 8 are constraints.
The constraints reals([...]) and formulae([..])
constraint the variables in their lists to the respective
domains (should any of the variables be already tagged
from a different domain the constraint simply fails).
Line 4 is a constraint from the domain of formulæ
equating domain variable F with the domain term
(M*G)//X a formula on the variable X (which as
it happens does not appear)
A euclid term is either a constant
or a variable or a function containing
euclid terms. All terms are divided into classic, D-domain
or mixed (general) terms.
D-domain terms are special symbols
(variables, constants and functions) defined for each
external domain. Their meaning (interpretation) is fixed
at the creation of the language. In order to avoid using
a great number of different symbols variables are (internally)
tagged and constants take their meaning from the current
context (e.g. 0 can be a constant for both reals and vectors).
Any term not made of D-domain elements
(variables, constants or functions) is a classical
term. Of course it is entirely possible (and many
times highly desirable) to have mixed terms made
up from classic or D-domain variables or constants and
classic function symbols.
In Euclid variables are tagged; they
belong to a specific domain. They start their life belonging
to the Herbrand domain. These variables can take any symbolic
value and are open to all interpretations. In the course
of a computation some variables can migrate from Herbrand
to other domains: after that stage their values are limited
to domain terms (constructs from domain variables,constants
and functions) and their interpretation fixed by the language
to the intended one. Thus if X is a real variable it can
only be substituted by a real constant (e.g. 5), or a
more complex domain term (e.g. 5+cos(Y) where
Y is a real variable).
Examples:
- 3*X^2-5.0*cos(2*pi) is a
real term. cos is a d-domain function and pi
a d-domain constant (as are ofcourse all the real numbers.
In this example, the variable X will be tagged
as belonging to the domain of the reals.
- job(3,Z+5) is a mixed term;
job is a classical function symbol but 3,+ are d-domain
symbols (from the domain of reals).
- cos([1,2]) is an illegal
term. This term is not part of the language. In the
current version of Euclid, the interpreter will simply
fail upon execution of the statement containing the
term. Strictly speaking a syntax error should be given
(the checking happens at runtime when more knowledge
of the context is available).
For each external domain attached to
the language a number of constant, function and relation
symbols are introduced. Since variables are tagged there
is no need to introduce domain variables as symbols. The
relation `=' is defined on all external domains
(semantic equality) and the definition of the same symbol
on the domain of terms plays the role of the standard
unification (syntactic equality).
Currently Euclid contains the domains
of Reals, Real Intervals, Real Vectors, Real Formulae
and Real Symbolic Functions. A more detailed
presentation of the domains contains all the symbols
(and their respective meaning) defined for each domain.
These symbols are reserved and cannot change their meaning
during program execution.
Euclid performs its computation
by returning answers (answer substitutions) together with
list of constraints to queries.
A query or goal is a rule without a head.
An answer substitution is an assignment of terms
to variables that when substituted to the original query
the statement becomes true in the logic. The set of constraints
returned with the answer is a set of (possibly) satisfiable
constraints that need to hold so that the original query
could be satisfied. Logically this answer processes can
be viewed as partial computation where constraints are
carried over until the end and a trusted method simplifies
them to a presentable set.
Examples:
- gravity_force_field(M,G,F).
will return the constrains between the real variables
M,G and F.
- gravity_force_field(10,9.81,F).
will return the value of the force acting on a mass
of 10 mass units if the gravitational acceleration is
9.81 acceleration units.
