LogicInAMPL < OpsRes

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---+ <a name="top"></a> Logic in AMPL

   1 <a href="#relational">Relational Operators</a> 
   1 <a href="#operators">Logical Operators</a> 
   1 <a href="#logical">Logical Expressions</a> 
   1 <a href="#conditional">Conditional Expressions</a> 
   1 <a href="#structure">Conditional Structures</a> 
   1 <a href="#binary">Binary Parameters</a> 

---++ <a name="relational"></a> Relational Operators

Relational operators are used to compare two expressions. They are most commonly used in [[ConstraintsInAMPL][constraints]], but not exclusively. The relational operators are:
| *Expression* | *Meaning* |
| < | Less than |
| <= | Less than or equal to |
| > | Greater than |
| >= | Greater than or equal to |
| == | Equal to |
| <> | Not equal to |

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---++ <a name="operators"></a> Logical Operators

Logical operators are used to combine [[#logical][logical expressions]]. They are most commonly used in [[#conditional][conditional statements]], [[#structure][conditional structures]] and [[LoopingInAMPL#condition][conditional loops]]. The logical operators are =not=, =and= and =or=. They are used as shown in the table below:

| *Expression* | *Result* |
| =not <expression>= | True if <span><expression></span> is false, false if <span><expression></span> is true |
| =<e1> and <e2>= | True if <span><e1></span> and <span><e2></span> are both true, otherwise false |
| =<e1> or <e2>= | True if either <span><e1></span> or <span><e2></span> are true, otherwise false |

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---++ <a name="logical"></a> Logical Expressions

Logical expressions are expressions that will evaluate to either true or false. Logical expressions are usually defined in terms of the [[#relational][relational operators]]:
<pre>Lower[r] <= sum {i in INGREDIENTS} Contributes[r, i] * Amount[i] </pre>

<pre>sum {s in SURFBOARDS} Recipe[m, s] * Production[s] <= Supply[m]; </pre>

However, there are some special logical expressions in AMPL for use with [[SetsInAMPL][sets]]:
   1 <span class="WYSIWYG_TT"><e> in <SET></span> is _true_ if <span class="WYSIWYG_TT"><e></span> is a member of <span class="WYSIWYG_TT"><SET></span>; 
   1 <span class="WYSIWYG_TT"><e> not in <SET></span> is _false_ if <span class="WYSIWYG_TT"><e></span> is a member of <span class="WYSIWYG_TT"><SET></span>; 
   1 <span class="WYSIWYG_TT">exists {<e> in <SET>} <expression></span> is _true_ if some <span class="WYSIWYG_TT"><e></span> in <span class="WYSIWYG_TT"><SET></span> has <span class="WYSIWYG_TT"><expression></span> being true; 
   1 <span class="WYSIWYG_TT">forall {<e> in <SET>} <expression></span> is _true_ if all <span class="WYSIWYG_TT"><e></span> in <span class="WYSIWYG_TT"><SET></span> have <span class="WYSIWYG_TT"><expression></span> being true; 
   1 <span class="WYSIWYG_TT"><SUBSET> within <SET></span> is _true_ if all the elements in <span class="WYSIWYG_TT"><SUBSET></span> are in <span class="WYSIWYG_TT"><SET></span>; 
   1 <span class="WYSIWYG_TT"><SUBSET> not within <SET></span> is _true_ if some element in <span class="WYSIWYG_TT"><SUBSET></span> is _not_ in <span class="WYSIWYG_TT"><SET></span>. 

Logical expressions can be built up from other logical expressions, [[#binary][binary parameters]] and [[#operators][logical operators]].

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---++ <a name="conditional"></a> Conditional Expressions

A conditional expression is very much like the =IF= function in Microsoft Excel:
<pre>param ifvalue := if <some logical expression> then                    <a value>                 [else                    <another value>]; </pre>

If the logical expression is true then =ifvalue= will be set to <span><a value></span>, otherwise it is set to 0 (by default) or, if the else part of the expression is present, <span><another value></span>. Note that if the =else= keyword is present, then no =;= needs to be included after <span><a value></span>.

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---++ <a name="structure"></a> Conditional Structures

A conditional structure is the same as the classical _if_-_then_-_else_ statement in programming languages like MATLAB, Fortran, Visual Basic and C++:
<pre>if <logical expression> then   <a statement>; [else   <another statement>;] </pre>

Note here that even if the =else= keyword is present you need to end <span><a statement></span> with =;=. If you want to include more than one statement within the conditional structures you can use ={= and =}= to enclose your statements:
<pre>if <logical expression> then {   <some statements> } [else {   <some other statements> }] </pre>

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---++ <a name="binary"></a> Binary Parameters

In AMPL we can create _binary_ parameters by using the =binary= keyword in the parameter declaration:
<pre>param stillSearching binary; </pre>

Binary parameters are used in a similar way to boolean variables (in Matlab, C, etc) and logical variables (Fortran). If a binary parameter has the value 0 this is equivalent to false, and 1 is equivalent to true. Binary parameters can be used with [[#conditional][conditional expressions]] to hold a true/false result from a [[#logical][logical expression]]:
<pre>param isGreater binary;  let isGreater := if 4 > 5 then 1 else 0; # isGreater = 0 (false) let isGreater := if 6 > 5 then 1; # else 0 is the default, isGreater = 1 (true) </pre> so the syntax is <pre>let <binary parameter> := if <expression> then 1; </pre>

You can also set binary parameters within [[#structure][conditional structures]]
<pre>binary <binary parameter>;  if <expression> then   let <binary parameter> := 1; else   let <binary parameter> := 0; </pre>

Binary parameters may be used in [[#logical][logical expressions]] or as the condition in a [[#condition][conditional statement]] or [[#structure][conditional structure]]. They are very useful for building complex conditional statements or structures:
<pre>Some example from depth first searching or column generation </pre> %RED%Coming soon!%ENDCOLOR% and controlling [[LoopingInAMPL#condition][conditional loops]]: <pre>Some example from depth first searching or column generation </pre> %RED%Coming soon!%ENDCOLOR%

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-- Main.TWikiAdminGroup - 18 Mar 2008
Topic revision: r7 - 2009-09-18 - MichaelOSullivan