Logical Expressions

Logical expressions

Logical expressions are expressions that evaluate to a logical value-0.0 for false and 1.0 for true. AIMMS supports several types of logical expressions.

logical-expression:

Numerical expressions as logical

As AIMMS permits numerical expressions as logical expressions it is important to discuss how numerical expressions are interpreted logically, and how logical expressions are interpreted numerically. Numerical expressions that evaluate to zero (0.0) are false, while all others (including ZERO, NA and UNDF) are true. A false logical expression evaluates to zero (0.0), while a true logical expression evaluates to one (1.0). If one or more of the operands of a logical operator is UNDF or NA, the numerical value is also UNDF or NA. Note that AIMMS will not accept expressions that evaluate to UNDF or NA in the condition in control flow statements, where it must be known whether the result of that condition is equal to 0.0 or not (see also Flow Control Statements).

Example

this table illustrates the different interpretation of a number of numerical and logical expressions as either a numerical or a logical expression. See also this table for the results associated with the AND operator.

Table 15 Numerical and logical values

Expression	Numerical value	Logical value
3*(2 > 1)	3.0	true
3*(1 > 2)	0.0	false
(1 < 2) + (2 < 3)	2.0	true
max((1 < 2),(2 < 3))	1.0	true
2 AND 0.0	0.0	false
2 AND ZERO	1.0	true
2 AND NA	NA	true
UNDF < 0	UNDF	true

Logical Operator Expressions

Unary and binary logical operators

AIMMS supports the unary logical operator NOT and the binary logical operators AND, OR, and XOR. this table gives the logical results of these operators for zero and nonzero operands.

Table 16 Logical operators

Operands		Result
a	b	a AND b	a OR b	a XOR b	NOT a
0	0	0	0	0	1
0	nonzero	0	1	1	1
nonzero	0	0	1	1	0
nonzero	nonzero	1	1	0	0

Precedence order

The precedence order of these operators from highest to lowest is given by NOT, AND, OR, and XOR respectively. Whenever the precedence order is not immediately clear, it is advisable to use parentheses. Besides preventing unwanted mistakes, it also make your model easier to understand and maintain.

Example

The expression

NOT a AND b XOR c OR d

is parsed by AIMMS as if it were written

((NOT a) AND b) XOR (c OR d).

Execution order

Due to the sparse execution system underlying AIMMS it is not guaranteed that logical expressions containing binary logical operators are executed in a strict left-to-right order. If you are a C/C++ programmer (where logical conditions are executed in a strict left-to-right order), you should take extra care to ensure that your logical conditions do not depend on this assumption.

Numerical Comparison

Numerical comparison

Numerical relationships compare two numerical expressions, using one of the relational operators =, <>, >, >=, <, or <=. Numerical inclusions are equivalent to two numerical relationships, and indicate whether a given expression lies within two bounds.

Syntax

expression-relationship:

expression-inclusion:

Numerical tolerances

For two real numbers \(x\) and \(y\) the result of the comparison \(x \gtrless y\), where \(\gtrless\) denotes any relational operator, depends on two tolerances

• Equality_Absolute_Tolerance(denoted as \(\varepsilon_a\)), and

• Equality_Relative_Tolerance(denoted as \(\varepsilon_r\)).

You can set these tolerances through the options dialog box. Their default values are \(0\) and \(10^{-13}\), respectively. If the number \(\varepsilon_{x,y}\) is given by the formula

\[\varepsilon_{x,y} = \max(\varepsilon_a,\varepsilon_r\cdot x,\varepsilon_r \cdot y),\]

then the relational operators evaluate as shown in the this table.

Table 17 Interpretation of numerical tolerances

AIMMS expression	Evaluates as
\(x \mathtt{=} y\)	\(|x-y| \leq \varepsilon_{x,y}\)
\(x \mathtt{<>} y\)	\(|x-y| > \varepsilon_{x,y}\)
\(x \mathtt{<=} y\)	\(x-y \leq \varepsilon_{x,y}\)
\(x \mathtt{<} y\)	\(x-y < -\varepsilon_{x,y}\)

Comparison for extended arithmetic

For any combination of an ordinary real number with one of the special symbols ZERO, INF, and -INF, the relational operators behave as expected. If any of the operands is either NA or UNDF, relationships other than = and <> also evaluate to NA or UNDF and hence, as a logical expression, to true. In addition, the logical expressions INF = INF and -INF = -INF evaluate to true.

Testing for zero value

One can formulate numerous logical expressions to test for a zero value, and one should be clear on the desired result. The following example makes the point.

p_inv(i)             := 1 / p(i);
p_inv(i | p(i))      := 1 / p(i);
p_inv(i | p(i) <> 0) := 1 / p(i);

The first assignment will produce a runtime error when p(i) assumes a value of 0 or ZERO. The second assignment will filter out the 0’s, but not the ZERO values because ZERO evaluates to the logical value “true”. The last assignment will never produce runtime errors, because of the numerical comparison to 0.

Set and Element Comparison

Set relationships

AIMMS features very powerful logical set comparison operators. Not only can sets and their elements be compared using relational operators, but you can also check for set membership with the IN operator.

Syntax

set-relationship:

Element relationship and inclusion

Set elements that lie in the same set can be compared according to their relative position inside that set. You can also compare the positions of arbitrary set element expressions, as long as AIMMS is able to determine a unique domain set in which the comparison has to take place. The allowed relational operators are =, <>, <, <=, >, and >=. As with numerical expression, AIMMS also allows you to specify an inclusion relationship as a form of repeated comparison to verify whether an element lies within two boundary elements.

Element comparison

The relational operators for element relationships are conveniently defined in terms of the Ord function. Let S be a simple set, i and j indices or element parameters in S, \(\pm\) any of the lag or lead operators +, ++, - or -, \(m\) and \(n\) integer expressions, and \(\gtrless\) one of the operators =, <>, <, <=, >, or >=. The relational operators \(\gtrless\) have the following definition for set elements, provided that the set elements on both sides of the relational operator exist.

\[\begin{split}\mathtt{i}\pm m\gtrless \mathtt{j}\pm n\quad\Leftrightarrow \quad \begin{cases} \mbox{$\mathtt{i}\pm m$ and $\mathtt{j}\pm n$ are both defined, and} \\ \mbox{$\mathtt{Ord(i\pm m,S)}\gtrless \mathtt{Ord(j\pm n,S)}$} \end{cases}\end{split}\]

Note that this type of relational expression evaluates to “false” if one or both of the operands do not refer to existing set elements.

Compare within the same set

Only elements that lie in the same set are comparable using the <, <=, >, and >= operators. The = and <> operators can also be used when the operands merely share the same root set.

Example

The following set assignments demonstrate the correct use of element comparisons.

FuturePeriods := { t in Periods | CurrentPeriod <= t <= PlanningHorizon } ;

BandMatrix := { (i,j) | i - BandWidth <= j <= i + BandWidth } ;

Set membership

Set membership can be tested using the IN operator. This operator checks whether a set element or an element tuple on the left-hand side is a member of the set expression on the right-hand side. Both operands must have the same root set.

Example

Assume that all one-dimensional sets in the following two assignments share the same root set Cities. Then these statements illustrate the correct use of the logical IN operator.

NeighborhoodRoutes := { (i,j) in Routes | j in NeighborhoodCities(i) } ;
ExcludedCities     := { i in ( SmallCities + ForeignCities ) } ;

Set comparisons

Sets can be logically compared using any of the relational operators =, <>, <, <=, > and >=. The inequality operators denote the usual subset relationships. They replace the standard “contained in” operators \(\subsetneq\), \(\subseteq\), \(\supsetneq\) and \(\supseteq\) which are not available on many keyboards.

Example

The following statement illustrates a logical set comparison operator.

IF ( RoutesWithTransport <= NeighborhoodRoutes ) THEN
    DialogMessage( "Solution only contains neighborhood transports" );
ENDIF;

String Comparison

String comparison

Besides their use for comparison of numerical, element- and set-valued expressions, the relational operators =, <>, <, <=, >, and >= can also be used for string comparison. When used for string comparison, AIMMS employs the usual lexicographical ordering. String comparison in AIMMS is case sensitive by default, i.e. strings that only differ in case are considered to be unequal. You can modify this behavior through the option Case_Sensitive_String_Comparison.

Examples

All the following string comparisons evaluate to true.

"The city of Amsterdam" <> "the city of amsterdam"     ! Note case
"The city of Amsterdam" <> "The city of Amsterdam "    ! Note last space
"The city of Amsterdam" <  "The city of Rotterdam"

Logical Iterative Expressions

Logical iterative operators

Logical iterative operators verify whether some or all elements in a domain satisfy a certain logical condition. this table lists all logical iterative operators supported by AIMMS. The second column in this table refers to the required number of expression arguments following the binding domain argument.

Table 18 Logical iterative operators

Name	# Expr.	Meaning
Exists	0	true if the domain is not empty
Atleast	1	true if the domain contains at least \(n\) elements
Atmost	1	true if the domain contains at most \(n\) elements
Exactly	1	true if the domain contains at exactly \(n\) elements
ForAll	1	true if the expression is true for all elements in the domain

Example

The following statements illustrate the use of some of the logical iterative operators listed in this table.

MultipleSupplyCities := { i | Atleast( j | Transport(i,j), 2 ) } ;

IF ( ForAll( i, Exists( j | Transport(i,j) ) ) ) THEN
    DialogMessage( "There are no cities without a transport" );
ENDIF ;