The Fortress Language Specification - CiteSeerX

extends { BinaryOperatorT, ⊙, UnaryOperatorT, ∼, EquivalenceRelationT, = }
where { T coerces ComplementBound⊙ }
property ∀(a:T) (a ⊙ (∼ a)) :=: ComplementBound⊙
end
A set of values with a binary operator ⊙ has complements if and only if there is a specific value e such that for every
value a there is another value a ′ such that the result of applying ⊙ to a and a ′ (in either order) equals the specified
value e . This value may be obtained by coercing the object named ComplementBound⊙ to type T . The unary
operator ∼ returns the complement of its argument with respect to the operator ⊙ .
Note that the trait HasComplements is similar to the trait HasInverses, but HasComplements does not require that
that specified element be an identity of the binary operator.
trait DeMorganT extends DeMorganT, , , ∼,opr ,opr ,opr ∼
extends { BinaryOperatorT, , BinaryOperatorT, ,
UnaryOperatorT, ∼, EquivalenceRelationT, = }
property ∀(a:T, b: T) (∼ (a b)) :=: ((∼ a) (∼ b))
end
This trait expresses De Morgan’s law for two binary operators and and a unary operator ∼ : the expressions
∼ (a b) and (∼ a) (∼ b) produce equal results.
trait BooleanAlgebraT extends BooleanAlgebraT, , , ∼, ∨,opr ,opr ,opr ∼,opr ∨
extends { CommutativeT, , AssociativeT, ,
CommutativeT, , AssociativeT, ,
IdempotentBinaryOperatorT, ,
IdempotentBinaryOperatorT, ,
HasIdentityT, , HasIdentityT, ,
HasComplementsT, , ∼, HasComplementsT, , ∼,
DistributiveT, , , DistributiveT, , ,
DeMorganT, , , ∼, DeMorganT, , , ∼,
RingT, ∨,IDENTITY, }
property ∀(a:T) (∼ (∼ a)) :=: a
opr ∨(self,other: T):T = (self (∼ other)) ((∼ self) other)
property castT(Identity∨) = castT(Identity)
property castT(ComplementBound) = castT(Identity)
property castT(ComplementBound) = castT(Identity)
end
A boolean algebra is an algebraic structure consisting of a set of values, three binary operators , , and ∨ , and a
unary operator ∼ , such that the operators obey a number of specific properties:
• is commutative, associative, and idempotent, and has a unique identity
• is commutative, associative, and idempotent, and has a unique identity
• has complements with respect to ∼
• has complements with respect to ∼
• and distribute over each other
• De Morgan’s law applies to , , and ∼ , and also to , , and ∼
• The values form a ring with ∨ as the addition operator, IDENTITY as the additive inverse operator, and as
the multiplication operator
260