#author("2017-03-02T16:49:19+09:00","default:LMNtal","LMNtal")
#author("2023-02-14T00:42:32+09:00","default:LMNtal","LMNtal")
//[[Documentation]]

#mathjax

*Guards [#v398323a]

**Rules with a Guard [#x7373323]

Guards specify applicability conditions of rewrite rules.  The syntax of a rule with a guard is:
>'''Head''' :- '''Guard''' | '''Body'''
>\( \textit{Head}\ \texttt{:-}\ \textit{Guard}\ \texttt{|}\ \textit{Body} \)

where '''Guard''' is a multiset of ''type constraints'' of the form
'''c'''($'''p'''&size(10){1};, ..., $'''p'''&size(10){'''n'''};).
where \( \textit{Guard} \) is a multiset of ''type constraints'' of the form
\( c(\texttt{\$}p_1, \dots, \texttt{\$}p_n) \).

Type constraints constrains the shapes of processes
(or the names of unary atoms)
received by the process contexts $'''p'''&size(10){1};, ..., $'''p'''&size(10){'''n'''};.
The ''type constraint name'' '''c''' is drawn from a built-in set and specifies which kind of constraints is imposed.
received by the process contexts \( \texttt{\$}p_1, \dots, \texttt{\$}p_n \).
The ''type constraint name'' \( c \) is drawn from a built-in set and specifies which kind of constraints is imposed.

A constraint of the form ''uniq''($'''p'''&size(10){1};, ..., $'''p'''&size(10){'''n'''};) is
A constraint of the form ''uniq''(\( \texttt{\$}p_1, \dots, \texttt{\$}p_n \)) is
also allowed.  This is a control structure rather than a type constraint
and used to avoid infinite rule application (see below).

***Examples [#oc28c486]

Here is an example rule with guard:

 waitint(X), $p[X] :- int($p) | ok.

This can be abbreviated to

 waitint($p) :- int($p) | ok.

and can be thought of representing the following infinite number of rules:

 waitint(0)  :- ok.
 waitint(1)  :- ok.
 waitint(-1) :- ok.
 waitint(2)  :- ok.
 waitint(-2) :- ok.
 ...
 waitint(0)  :- ok.   waitint(1)  :- ok.    waitint(-1) :- ok.
 waitint(2)  :- ok.   waitint(-2) :- ok.   ...

The following are examples of some type constraints that can be written in '''Guard''':

-&color(#8B4513){int($p)}; ... specifies that $p must be an integer atom.
-&color(#8B4513){4($p)}; ... specifies that $p must be a unary integer atom of value 4 (i.e., 4(X)).
-&color(#8B4513){$p < $q}; ... specifies that $p and $q are integer atoms
such that the value of $p is less than that of $q.
-&color(#8B4513){$r = $p +. $q}; ... specifies that $p, $q, and $r are floating
point number atoms such that the sum of the values of $p and $q is equal to the value of $r. 

***Notes [#z6515d94]

Each type constraint name (such as int or <)
has its own mode of usage
that specifies which of its arguments are input arguments.
The effect of the constraint specified by a type constraint
is enabled only after the shapes (or values)
of its input arguments are all determined.
For example, $r = $p + $q proceeds only when $p and $q are determined.

The same abbreviation scheme as defined for atoms
applies to type constraints when a process context name
$'''p'''&size(10){'''k'''};
occurs exactly twice in the rule.
For example, p($n) :- $n>$z, 0($z) | ok can be abbreviated
to p($n) :- $n>0 | ok.
For example, 
 p($n) :- $n > $z, 0($z) | ok
can be abbreviated to:
 p($n) :- $n > 0 | ok

**Typed Process Contexts [#z5d7f69f]
// **Typed Process Contexts [#z5d7f69f]

A process context constrained in '''Guard'''
is said to be a ''typed process context''.
// As a syntactic sugar,
// typed process context names can be written as link names.
// For inscance, the above example can be written as:
//  waitint(X) :- int(X) | ok.
// // ( Res = gen(N) :- N > 0 | Res = [N|gen(N-1)] ), p(gen(10))
// However, the original form, waitint($p) :- int($p) | ok, is preferred
// because, unlike link names, typed process context names has no constraints
// on the number of their occurrences.

**Avoiding Infinite Rule Application [#w149d3ab]

A constraint of the form
''uniq''($'''p'''&size(10){1};, ..., $'''p'''&size(10){'''n'''};)
succeeds if each $'''p'''&size(10){'''k'''}; is a '''ground''' structure
(connected graph with exactly one free link; see below) 
and the rule has not
been applied to the tuple
$'''p'''&size(10){1};, ..., $'''p'''&size(10){'''n'''};
before.

As a special case of '''n'''=0,
''uniq'' succeeds if the rule in question has not been used
before.  The ''uniq''() test is a general tool for avoiding 
infinite application of rules whose right-hand side is a 
super(multi)set of the left-hand side.


**Guard Library [#udf848a7]

The following type constraints can be used in guards.
The + (input) sign preceding a process context name means that the name should
appear in the head, while the - (output) sign means that the name should not
appear in the head.

***Type checking [#l78e805c]

:int(+$i)|check if $i[X] is an integer.
:float(+$f)|check if $f[X] is a floating-point number.
:unary(+$u)|check if $u[X] is a unary atom.  Note that ''int'' and ''float'' are subtypes of ''unary''.
:ground(+$g)|check if $g[X1,...,Xn] (n>0) is a connected graph whose free links are exactly X1,...,Xn.  Note that ''unary'' is a subtype of ''ground''.

***Assignment [#ne364170]

:'='(+$u,-$v)|make sure that $u[X] and $v[Y] are unary atoms with the same name.
:'='(-$u,+$v)|same as above.
:int(+$float,-$int)|cast to int.
:float(+$int,-$float)|cast to float.
:345(-$int)|defined for every integer (not only with 345).
:'-3.14'(-$float)|defined for every float.
:'+'(+$int,+$int,-$int)|binary integer operation;  also: ''&color(#8B4513){'-'};'', ''&color(#8B4513){'*'};'', ''&color(#8B4513){'/'};'', ''&color(#8B4513){mod};'', ''&color(#8B4513){logand};'',  ''&color(#8B4513){logor};'',''&color(#8B4513){logxor};'' (the last three being bitwise operations).
:'+'(+$int,+$int)|unary integer operation;  also: ''&color(#8B4513){'-'};'', ''&color(#8B4513){abs};'', ''&color(#8B4513){lognot};''.
:'+.'(+$float,+$float,-$float)|binary float operation;    also: ''&color(#8B4513){'-.'};'', ''&color(#8B4513){'*.'};'', ''&color(#8B4513){'/.'};''.
:'+.'(+$float,+$float)|unary float operation;    also: ''&color(#8B4513){'-.'};''.

The term notation allows you to write
 abs($x,$v), '*'(3,$v,$w), '+'($w,$y,$z)
as
 $v = abs($x), $w = 3 * $v, $z = $w + $y
which can in turn be written as
 $z = 3 * abs($x) + $y . 

***Comparison [#r7825264]

:'='(+$u,+$v)|check if $u[X1,...,Xn] and $v[Y1,...,Yn] are connected graphs with the same structure.
:'\='(+$u,+$v)|check if $u[X1,...,Xn] and $v[Y1,...,Ym] are connected graphs with different structures.
:'=='(+$u,+$v)|check if $u[X] and $v[Y] are unary atoms with the same name.
:'\=='(+$u,+$v)|check if $u[X] and $v[Y] are unary atoms with different names
(if either of them are not unary, the check fails.)
:'<.'(+$float,+$float)|float comparison;   also: ''&color(#8B4513){'>.'};'', ''&color(#8B4513){'=<.'};'', ''&color(#8B4513){'>=.'};'', ''&color(#8B4513){'=:=.'};'', ''&color(#8B4513){'=\=.'};''.
:'<'(+$int,+$int)|integer comparison; also: ''&color(#8B4513){'>'};'', ''&color(#8B4513){'=<'};'', ''&color(#8B4513){'>='};'', ''&color(#8B4513){'=:='};'', ''&color(#8B4513){'=\='};''.
***Assignment [#ne364170]

:'='(+$u,-$v)|make sure that $u[X] and $v[Y] are unary atoms with the same name.
:'='(-$u,+$v)|same as above.
:int(+$float,-$int)|cast to int.
:float(+$int,-$float)|cast to float.
:345(-$int)|defined for every integer (not only with 345).
:'-3.14'(-$float)|defined for every float.
:'+'(+$int,+$int,-$int)|integer operation;  also: ''&color(#8B4513){'-'};'', ''&color(#8B4513){'*'};'', ''&color(#8B4513){'/'};'', ''&color(#8B4513){mod};''.
:'+.'(+$float,+$float,-$float)|float operation;    also: ''&color(#8B4513){'-.'};'', ''&color(#8B4513){'*.'};'', ''&color(#8B4513){'/.'};''.
The term notation allows you to write
 abs($x,$y), $y < 10
as
 abs($x) < 10 .

***Others [#i37c8a7c]
**Avoiding Infinite Rule Application [#w149d3ab]

A constraint of the form
''uniq''($'''p'''&size(10){1};, ..., $'''p'''&size(10){'''n'''};)
succeeds if each $'''p'''&size(10){'''k'''}; is a '''ground''' structure
(connected graph with exactly one free link; see below) 
and the rule has not
been applied to the tuple
$'''p'''&size(10){1};, ..., $'''p'''&size(10){'''n'''};
before.

As a special case of '''n'''=0,
''uniq'' succeeds if the rule in question has not been used
before.  The ''uniq''() test is a general tool for avoiding 
infinite application of rules whose right-hand side is a 
super(multi)set of the left-hand side.

//***Others [#i37c8a7c]
:uniq(+$g1,...,+$gn)|uniqueness constraint; checks if the rule has not been applied to the tuple $g1[X1], ..., $gn[Xn] (n>=0).
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