transformations

Interfaces for transforming representations of expressions.

transformations.bexpr module

Transformation functions for expressions.

tt.transformations.bexpr.apply_de_morgans(expr)

Convert an expression to a form with De Morgan’s Law applied.

Returns:A new expression object, transformed so that De Morgan’s Law has been applied to negated ANDs and ORs. Return type:BooleanExpression Raises:InvalidArgumentTypeError – If expr is not a valid type.

Here’s a couple of simple examples showing De Morgan’s Law being applied to a negated AND and a negated OR:

>>> from tt import apply_de_morgans
>>> apply_de_morgans(r'~(A /\ B)')
<BooleanExpression "~A \/ ~B">
>>> apply_de_morgans(r'~(A \/ B)')
<BooleanExpression "~A /\ ~B">

tt.transformations.bexpr.apply_idempotent_law(expr)

Convert an expression to a form with the Idempotent Law applied.

Returns:A new expression object, transformed so that the Idempotent Law has been applied to applicable clauses. Return type:BooleanExpression Raises:InvalidArgumentTypeError – If expr is not a valid data type.

This transformation will apply the Idempotent Law to clauses of AND and OR operators containing redundant operands. Here are a couple of simple examples:

>>> from tt import apply_idempotent_law
>>> apply_idempotent_law('A and A')
<BooleanExpression "A">
>>> apply_idempotent_law('B or B')
<BooleanExpression "B">

This transformation will consider similarly-negated operands to be redundant; for example:

>>> from tt import apply_idempotent_law
>>> apply_idempotent_law('~A and ~~~A')
<BooleanExpression "~A">
>>> apply_idempotent_law('B or ~B or ~~B or ~~~B or ~~~~B or ~~~~~B')
<BooleanExpression "B or ~B">

Let’s also take a quick look at this transformation’s ability to prune redundant operands from CNF and DNF clauses:

>>> from tt import apply_idempotent_law
>>> apply_idempotent_law('(A and B and C and C and B) or (A and A)')
<BooleanExpression "(A and B and C) or A">

Of important note is that this transformation will not recursively apply the Idempotent Law to operands that bubble up. Here’s an example illustrating this case:

>>> from tt import apply_idempotent_law
>>> apply_idempotent_law('(A or A) and (A or A)')
<BooleanExpression "A and A">

tt.transformations.bexpr.apply_identity_law(expr)

Convert an expression to a form with the Identity Law applied.

It should be noted that this transformation will also annihilate terms when possible. One such case where this would be applicable is the expression A and 0, which would be transformed to the constant value 0.

Returns:A new expression object, transformed so that the Identity Law has been applied to applicable ANDs and ORs. Return type:BooleanExpression Raises:InvalidArgumentTypeError – If expr is not a valid type.

Here are a few simple examples showing the behavior of this transformation across all two-operand scenarios:

>>> from tt import apply_identity_law
>>> apply_identity_law('A and 1')
<BooleanExpression "A">
>>> apply_identity_law('A and 0')
<BooleanExpression "0">
>>> apply_identity_law('A or 0')
<BooleanExpression "A">
>>> apply_identity_law('A or 1')
<BooleanExpression "1">

tt.transformations.bexpr.apply_inverse_law(expr)

Convert an expression to a form with the Inverse Law applied.

Returns:A new expression object, transformed so that the Inverse Law has been applied to applicable ANDs and ORs. Return type:BooleanExpression Raises:InvalidArgumentTypeError – If expr is not a valid type.

This transformation will apply the Identity Law to simple binary expressions consisting of negated and non-negated forms of the same operand. Let’s take a look:

>>> from tt.transformations import apply_inverse_law
>>> apply_inverse_law('A and ~A')
<BooleanExpression "0">
>>> apply_inverse_law('A or B or ~B or C')
<BooleanExpression "1">

This transformation will also apply the behavior expected of the Inverse Law when negated and non-negated forms of the same operand appear in the same CNF or DNF clause in an expression. If you don’t believe me, take a look for yourself:

>>> from tt.transformations import apply_inverse_law
>>> apply_inverse_law('(A or B or ~A) -> (C and ~C)')
<BooleanExpression "1 -> 0">
>>> apply_inverse_law('(A or !!!A) xor (not C or not not C)')
<BooleanExpression "1 xor 1">

tt.transformations.bexpr.coalesce_negations(expr)

Convert an expression to a form with all negations condensed.

Returns:A new expression object, transformed so that all “runs” of logical NOTs are condensed into the minimal equivalent number. Return type:BooleanExpression Raises:InvalidArgumentTypeError – If expr is not a valid type.

Here’s a simple example showing the basic premise of this transformation:

>>> from tt import coalesce_negations
>>> coalesce_negations('~~A or ~B or ~~~C or ~~~~D')
<BooleanExpression "A or ~B or ~C or D">

This transformation works on more complex expressions, too:

>>> coalesce_negations('!!(A -> not not B) or ~(~(A xor B))')
<BooleanExpression "(A -> B) or (A xor B)">

It should be noted that this transformation will also apply negations to constant operands, as well. The behavior for this functionality is as follows:

>>> coalesce_negations('~0')
<BooleanExpression "1">
>>> coalesce_negations('~1')
<BooleanExpression "0">
>>> coalesce_negations('~~~0 -> ~1 -> not 1')
<BooleanExpression "1 -> 0 -> 0">

tt.transformations.bexpr.distribute_ands(expr)

Convert an expression to distribute ANDs over ORed clauses.

Parameters:expr (str or BooleanExpression) – The expression to transform. Returns:A new expression object, transformed to distribute ANDs over ORed clauses. Return type:BooleanExpression Raises:InvalidArgumentTypeError – If expr is not a valid type.

Here’s a couple of simple examples:

>>> from tt import distribute_ands
>>> distribute_ands('A and (B or C or D)')
<BooleanExpression "(A and B) or (A and C) or (A and D)">
>>> distribute_ands('(A or B) and C')
<BooleanExpression "(A and C) or (B and C)">

And an example involving distributing a sub-expression:

>>> distribute_ands('(A and B) and (C or D or E)')
<BooleanExpression "(A and B and C) or (A and B and D) or (A and B and E)">

tt.transformations.bexpr.distribute_ors(expr)

Convert an expression to distribute ORs over ANDed clauses.

Parameters:expr (str or BooleanExpression) – The expression to transform. Returns:A new expression object, transformed to distribute ORs over ANDed clauses. Return type:BooleanExpression Raises:InvalidArgumentTypeError – If expr is not a valid type.

Here’s a couple of simple examples:

>>> from tt import distribute_ors
>>> distribute_ors('A or (B and C and D and E)')
<BooleanExpression "(A or B) and (A or C) and (A or D) and (A or E)">
>>> distribute_ors('(A and B) or C')
<BooleanExpression "(A or C) and (B or C)">

And an example involving distributing a sub-expression:

>>> distribute_ors('(A or B) or (C and D)')
<BooleanExpression "(A or B or C) and (A or B or D)">

tt.transformations.bexpr.to_cnf(expr)

Convert an expression to conjunctive normal form (CNF).

This transformation only guarantees to produce an equivalent form of the passed expression in conjunctive normal form; the transformed expression may be an inefficent representation of the passed expression.

Parameters:expr (str or BooleanExpression) – The expression to transform. Returns:A new expression object, transformed to be in CNF. Return type:BooleanExpression Raises:InvalidArgumentTypeError – If expr is not a valid type.

Here are a few examples:

>>> from tt import to_cnf
>>> b = to_cnf('(A nor B) impl C')
>>> b
<BooleanExpression "A or B or C">
>>> b.is_cnf
True
>>> b = to_cnf(r'~(~(A /\ B) /\ C /\ D)')
>>> b
<BooleanExpression "(A \/ ~C \/ ~D) /\ (B \/ ~C \/ ~D)">
>>> b.is_cnf
True

tt.transformations.bexpr.to_primitives(expr)

Convert an expression to a form with only primitive operators.

All operators will be transformed equivalent form composed only of the logical AND, OR,and NOT operators. Symbolic operators in the passed expression will remain symbolic in the transformed expression and the same applies for plain English operators.

Parameters:expr (str or BooleanExpression) – The expression to transform. Returns:A new expression object, transformed to contain only primitive operators. Return type:BooleanExpression Raises:InvalidArgumentTypeError – If expr is not a valid type.

Here’s a simple transformation of exclusive-or:

>>> from tt import to_primitives
>>> to_primitives('A xor B')
<BooleanExpression "(A and not B) or (not A and B)">

And another example of if-and-only-if (using symbolic operators):

>>> to_primitives('A <-> B')
<BooleanExpression "(A /\ B) \/ (~A /\ ~B)">