Terms

Last updated 4 days ago

Purpose

The purpose of an term is to compute pairs that constitute a relation. We use operators to assemble terms from smaller terms, to express in formal language precisely what is meant in the natural language of the business. The smallest term is a single relation.

Description

An term is a combination of operators and relations. Its meaning is a set of pairs, which is in fact a newly created relation.

Examples

owner

r;s~

I /\ goalkeeper;goalkeeper~

destination;"Algarve" |- spoken;"Portugese"

Syntax

Every term is built out of relations, which are combined by operators. An term has one of the following 8 syntactic structures

<Term> <BinaryOperator> <Term>
<UnaryOpPre> <Term>
<Term> <UnaryOpPost>
<RelationRef> <type>?
I <type>?
V <type>?
<atom>
( <Term> )

Operators

The operators come in families. We advise novices to study only the rule operators, boolean operators and relational operators. There is a wealth of things you can express with just these operators. The residual operators seem a lot harder to learn and the Kleene operators are not fully implemented yet. You can click the hyperlink to navigate to the semantics of each family.

Family

binary operators

binding power

unary operators

binding power

rules

== and

1 (weakest)

boolean

\cup, \cap, and -

2

5

relational

;;, ×\times, and \dagger

4

\smallsmile

5

residual

\\backslash, //, and

3

Kleene

and ++

5

Brackets

Operators with different binding power may be used in the same term without brackets, because the binding power tells how it is interpreted. For example rs;tr\cap s;t means r(s;t)r\cap(s;t) because ;; has a higher binding power than \cap.

Operators with the same binding power must be used unambiguously. For example: r(st)r\cap(s-t) means something different than (rs)t(r\cap s)-t. In such cases Ampersand insists on the use of brackets, so readers without knowledge of the binding powers of the operators can read a term unambiguously.

Repeated uses of an associative operator does not require brackets. So rstr\cap s \cap t is allowed because \cap is associative.

Notation on the keyboard

When coding in Ampersand, these operators are typed with characters on the keyboard. The following table shows the operators in math and their equivalent in code:

operator name

code

math

remark

equivalence (equal)

=

==

use only in a rule

inclusion

|-

\subseteq

use only in a rule

intersect

/\

associative

union

\/

associative

difference (minus)

-

-

complement

-

in code: Prefix; in math: Overline

compose

;

;;

associative

converse (flip)

~

\smallsmile

postfix

left residual

/

//

right residual

\

\\backslash

diamond

<>

\Diamond

relational product

!

\dagger

associative

cartesian product

#

×\times

reflexive transitive closure

*

postfix

transitive closure

+

++

postfix