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Introducing integers and arithmetic operations into ASP, expanding our logical toolset beyond simple atoms to handle numerical computations and comparisons.

Up to this point we have used only string-based atoms. We're going to venture in to the realm of integers and math! Please note that only 

 numbers in ASP.  This limitation exists because ASP is designed for exact reasoning and finite domains. Floating point numbers, with their inherent imprecision and infinite possible values between any two numbers, would make it impossible to guarantee the solver finds all stable models in finite time.

Terms in predicates can include integers:

Integers can be used with pooling as well:

We have seen examples before that include multiple instances of pooling. This example describes a 2x2 grid

Don't forget the parenthesis otherwise you won't get the desired results!

 (which also shows that you can create predicates with different arity within a single atom).

. Note that intervals can only be applied to integers. 

Intervals don't suffer from the same precedence problem and can be used without parenthesis:

. (Please note that atoms within an answer set are ordered 

The following arithmetic operators are available with their expected precedence:

(Bitwise operators are also available but are rarely used.)

addition(1 + 1) .
subtraction(3 - 7) .
multiplication(2 * 6) .
exponentiation(3 ** 2) .
division(7 / 3) .
remainder(7 \ 3) .
absolute_value(|-15|) .
combination(4 * 3 + 3 / 5) .

These arithmetic operators can be applied in any term such as within intervals:

The following comparison operators are available

equal ← 1 = 1 .
not_equal1 ← 2 = 1 .

not_equal2 ← 2 != 1 .
not_equal3 ← 2 != 2 .

less_than1 ← 3 < 1 + 1 .
less_than2 ← 4 + 5 < 5 * 4 .

greater_than1 ← 3 > 1 + 1 .
greater_than2 ← 4 + 5 > 5 * 4 .

(Remember that rules where the body is replaced with a truth value of 

 for all (variable-free and without arithmetic operations) atoms so that it is possible (though not common) to compare atoms and numbers:

greater_than ← a > 1 .
   less_than ← a < 1 .

 a variable; it is an atom (and it doesn't need to exist in the head of a rule). It is more common to compare atoms:

greater_than ← b > a .
   less_than ← b < a .

Logic Programming Rules! | Rules whose Body Evaluates to 

" section. Look at the following cases and think about what you would expect the answer set to contain:

example(1) ← 2 = 2 .
example(2) ← 2 = (1 ; 2 ; 3) .
example(3) ← 2 = 1..3 .

Let's expand the pooling and intervals for clarity and note any rules whose body evaluates to 

example(1) ← 2 = 2 .

example(2) ← 2 = 1 . % FALSE
example(2) ← 2 = 2 .
example(2) ← 2 = 3 . % FALSE

example(3) ← 2 = 1 . % FALSE
example(3) ← 2 = 2 .
example(3) ← 2 = 3 . % FALSE

example(4) ← 3 < 1..4 .
example(5) ← 3 > 1..4 .

(Expand the intervals and cross out any rules whose body evaluates to 

example(6) ← 1..3 < 1..4 .
example(7) ← 1..3 = 1..4 .

Combining variables and rules increases the expressivity of rules.

". Expanding the interval makes it clear what the answer set contains:

p(X, X*X, X+1) ← X = 1 .
p(X, X*X, X+1) ← X = 2 .
p(X, X*X, X+1) ← X = 3 .

Arithmetic operators can be used with intervals as well on both the left- and right-hand sides of the "

p(X) ← X + 3 = 1..3 .
q(X) ←     X = (1..3) - 3 .

Notice that the parenthesis are necessary in the second rule since "

Intervals can be combined with other intervals:

Multiple variables with intervals can be used:

And intervals can be used in the head of the rule as well:

ASP-04 | Numbers and Math

Now things start to get exciting!

Facts

Intervals

Operators

Arithmetic Operators

Comparison Operators

Equality vs Belongs To

Rules and Variables

Solver Limitations

More from Building Blocks in Surface Tension

Variables

Rules Part II

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