#author("2023-03-13T15:40:11+09:00","default:LMNtal","LMNtal")
//[[Documentation]]

*Hyperlinks and hypergraphs
*Hyperlinks and hypergraphs [#se1fd52b]

LMNtal links represent point-to-point connection between atoms and form an ordinary graph.  In some applications, however, we may want '''hyperlinks''', namely links that interconnect more than two atoms.  A graph structure with hyperlinks forms a '''(hierarchical) hypergraph'''.
LMNtal links represent point-to-point connection between atoms and form an ordinary graph.  In some applications, however, natural modeling calls for '''hyperlinks''', namely links that interconnect more than two atoms.  A graph structure with hyperlinks forms a '''(hierarchical) hypergraph'''.

LMNtal provides syntax and operations for hypergraphs, which will be described below.
Accordingly, LMNtal now provides the syntax and operations for hypergraphs.  The language with this extension is also called HyperLMNtal.

Currently, hyperlinks are supported by the '''SLIM''' runtime
running in the '''ordinary''' (as opposed to non-deterministic) execution mode.
**Creating a hyperlink [#hcd901f8]

**Creating a hyperlink

A hyperlink is created by a ''new'' construct specified in a guard.
 hoge.
 hoge :- new($h) | a($h), b($h).
 
 --> a(!1), b(!1).
Each hyperlink is given a fresh private name, which will be printed as a "''!''" followed by a hexadecimal number. 
Since each hyperlink can be regarded as a unary atom with a fresh local name, we use the syntax of process contexts (starting with a '$') to represent it.

Another view of hyperlinks is a generalization of ordinary links, and they can be written also by attaching '!' to link names as
 hoge.
 hoge :- a(!H), b(!H).
or just
 a(!H), b(!H).

**Type checking
The result of all the above programs will be 
 a(!H4), b(!H4).
That is, a hyperlink will be printed as a '!H' followed by a system-generated hexadecimal number.

**Type checking [#i375b878]

The type constraint ''hlink''($x) checks if $x represents a hyperlink.  
 main.
 main :- new($x) | a($x).
 a($x) :- hlink($x) | b($x).
 
 --> b(!1).
 main :- new($x) | a($y).
 a($x) :- hlink($x) | b($x,$x).
or
 a(!Y).
 a(!X) :- b(!X,!X).
Result:
 b(!H4,!H4).

**Equality checking [#gd567c1c]

**Equality checking

An occurrence of a hyperlink is regarded as a unary atom with a special, private name.  Therefore, the constructs ''=='' and ''\='' can be used for comparing two hyperlinks, e.g.,
Since an occurrence of a hyperlink is regarded as a unary atom with a special private name, the constructs ''=='' and ''\=='' can be used for comparing two hyperlinks, e.g.,
 a($x), b($y) :- hlink($x), hlink($y), $x == $y | ok.
The above rule can be written also as
 a($x), b($x) :- ok.
or
 a(!X), b(!X) :- ok.

**Fusing two hyperlinks [#e554f0f3]

**Fusing two hyperlinks

Two hyperlinks can be fused into one using the "''><''" operator.
 main.
 main :- new($x), new($y) | a($x), b($y).
 a($x), a($y) :- $x \= $y | a($x), b($y), $x >< $y.
 main :- new($x), new($y) | a($x), b($z).
 a($x), b($y) :- $x \== $y | ab($x,$y), $x >< $y.
or
 a(!X), b(!Z).
 a(!X), b(!Y) :- !X \== !Y | ab(!X,!Y), !X >< !Y.
If $x and $y represent different hyperlinks (i.e., have different hyperlink names), they will be merged into one; that is, they are made to have the same hyperlink name.

**Number of occurrences
**Number of occurrences [#hc2d2626]

The ''num'' construct in a guard will return the number of occurrences of the specified hyperlink.
 main.
 main :- new($x) | a($x), b($x).
 a($h) :- $n = num($h) | number($n).
 
 --> b(!1), number(2). 
 a($h) :- $n = num($h) | occurrences($n).
or
 a(!X), b(!X).
 a(!H) :- $n = num(!H) | occurrences($n).
Result:
 b(!H4), occurrences(2). 
Another use:
 main.
 main :- new($x) | a($x).
 a($h) :- $n = num($h), $n < 5 | a($h), a($h).
 
 --> a(!1), a(!1), a(!1), a(!1), a(!1).
 a(!X).
 a(!H) :- num(!H) < 5 | a(!H), a(!H).
Result:
 a(!H4), a(!H4), a(!H4), a(!H4), a(!H4).
Yet another use:
 main :- new($x) | flag(!x), a(!x), a(!x), a(!x), a(!x).
 a($h) :- hlink($h) | .
 flag($h) :- $n = num($h), $n =:= 1 | ok.
 
 --> ok. // ok is created when all a(!1)'s are removed
 flag(!X), a(!X), a(!X), a(!X), a(!X), a(!X).
 a(!H) :- .
 flag(!H) :- num(!H) =:= 1 | ok.
Result:
 ok. // ok is created when all a(!H1)'s are removed
In this way, using hyperlinks, one can check the '''absence''' of a particular hyperlink.

*How to use
*How to use [#nade49a7]

**LaViT
**LaViT [#qcec53b9]

When you execute your program under LaViT (recommended), select the Option tab and turn on the following checkboxes:
- Slimcode Compile Option: ''--slimcode'' and ''--hl-opt''
- Compile Option: ''--slimcode'' and ''--hl-opt''
- Slim Option: ''--hl''

Please see below for the details of those options.

**Compiling
**Compiling [#nd78cdd2]

Currently, programs with hyperlinks can be executed only under SLIM.  To compile them, both ''--slimcode'' and ''--hl'' (or ''--hl-opt'') options need be specified, e.g.,
 lmntal --slimcode --hl hoge.lmn > hoge.il
- ''--hl''     : turn on hyperlinks.
- ''--hl-opt'' : turn on hyperlinks and optimize graph matching using hyperlink connection.

**Execution
**Execution [#aff38a8e]

Specify ''--hl'' when running SLIM, e.g.,
 slim --hl --show-hl hoge.il
- ''--hl''      : turn on hyperlinks
-- currently, this cannot be specified with ''-nd''.
- ''--hl''      : turn on hyperlinks.
- ''--show-hl'' : prints the details of hyperlinks.
-- Without this option, different occurrences of a hyperlink are printed with the same name.
-- With this option, each occurrence of a hyperlink is printed with its own ID.  In addition, a table of hyperlink occurrences will be printed.
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