The Fortress Language Specification - CiteSeerX

The trait Commutative requires the operator ⊙ to be commutative; that is, reversing the operands produces an equal
result.
trait ApproximatelyAssociativeT extends ApproximatelyAssociativeT, ⊙, ≈,opr ⊙,opr ≈
extends { BinaryOperatorT, ⊙, ReflexiveT, ≈, SymmetricT, ≈ }
property ∀(a:T, b: T, c:T) ((a ⊙ b) ⊙ c) :≈: (a ⊙ (b ⊙ c))
end
The trait ApproximatelyAssociative requires the operator ⊙ to be approximately associative; that is, the expressions
(a ⊙ b) ⊙ c and a ⊙ (b ⊙ c) always produce results that are “close enough” to each other as determined by the
specified ≈ predicate.
trait AssociativeT extends AssociativeT, ⊙,opr ⊙
extends { ApproximatelyAssociativeT, ⊙, =, EquivalenceRelationT, = }
end
The trait Associative requires the operator ⊙ to be associative; that is, the expressions (a ⊙ b) ⊙ c and a ⊙ (b ⊙ c)
always produce equal results.
trait IdempotentBinaryOperatorT extends IdempotentBinaryOperatorT, ⊙,opr ⊙
extends { BinaryOperatorT, ⊙, EquivalenceRelationT, = }
property ∀(a:T) (a ⊙ a) :=: a
end
An idempotent binary operator has the property that if its two arguments are the same then the result is equal to each
argument. For example, MAX and MIN are idempotent, as are ∧ and ∨ applied to boolean arguments and ∩ and ∪
applied to sets; but + applied to integers is not idempotent because 1 + 1 does not produce 1 , and ⊕ applied to
boolean arguments is not idempotent because true ⊕ true produces false . The property of idempotency is sometimes
of interest when performing reductions such as MAX
i←1:n a i .
trait HasLeftZeroesT extends HasLeftZeroesT, ⊙,isLeftZero,opr ⊙,ident isLeftZero
extends { BinaryOperatorT, ⊙ }
isLeftZero():Boolean
property ∀(a:T, b: T) a.isLeftZero() →: ((a ⊙ b) = a)
end
A value e is a left zero for a binary operator ⊙ if the result of ⊙ always equals e whenever e is the left-hand operand.
For example, −∞ is a left zero for the MIN operator on floating-point values, and 7FFFFFFF 16 is a left zero for the
MAX operator on values of type Z32 . The purpose of this trait is to specify a method that says whether a given element
is a left zero for ⊙ .
trait HasRightZeroesT extends HasRightZeroesT, ⊙,isRightZero,opr ⊙,ident isRightZero
extends { BinaryOperatorT, ⊙ }
isRightZero():Boolean
property ∀(a:T, b: T) b.isRightZero() →: ((a ⊙ b) = b)
end
A value e is a right zero for a binary operator ⊙ if the result of ⊙ always equals e whenever e is the right-hand
operand. For example, −∞ is a right zero (as well as a left zero) for the MIN operator on floating-point values, and
7FFFFFFF 16 is a right zero (as well as a left zero) for the MAX operator on values of type Z32 . By way of contrast, 0
is a left zero for the arithmetic shift operator on integers, but is not a right zero. The purpose of this trait is to specify
a method that says whether a given element is a right zero for ⊙ .
trait ApproximatelyLeftDistributiveT extends ApproximatelyLeftDistributiveT, ⊗, ⊕, ≈,
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