.. Copyright (C) 2001-2019 NLTK Project
.. For license information, see LICENSE.TXT

======================
Nonmonotonic Reasoning
======================

    >>> from nltk import *
    >>> from nltk.inference.nonmonotonic import *
    >>> from nltk.sem import logic
    >>> logic._counter._value = 0
    >>> read_expr = logic.Expression.fromstring

------------------------
Closed Domain Assumption
------------------------

The only entities in the domain are those found in the assumptions or goal.
If the domain only contains "A" and "B", then the expression "exists x.P(x)" can
be replaced with "P(A) | P(B)" and an expression "all x.P(x)" can be replaced
with "P(A) & P(B)".

    >>> p1 = read_expr(r'all x.(man(x) -> mortal(x))')
    >>> p2 = read_expr(r'man(Socrates)')
    >>> c = read_expr(r'mortal(Socrates)')
    >>> prover = Prover9Command(c, [p1,p2])
    >>> prover.prove()
    True
    >>> cdp = ClosedDomainProver(prover)
    >>> for a in cdp.assumptions(): print(a) # doctest: +SKIP
    (man(Socrates) -> mortal(Socrates))
    man(Socrates)
    >>> cdp.prove()
    True

    >>> p1 = read_expr(r'exists x.walk(x)')
    >>> p2 = read_expr(r'man(Socrates)')
    >>> c = read_expr(r'walk(Socrates)')
    >>> prover = Prover9Command(c, [p1,p2])
    >>> prover.prove()
    False
    >>> cdp = ClosedDomainProver(prover)
    >>> for a in cdp.assumptions(): print(a) # doctest: +SKIP
    walk(Socrates)
    man(Socrates)
    >>> cdp.prove()
    True

    >>> p1 = read_expr(r'exists x.walk(x)')
    >>> p2 = read_expr(r'man(Socrates)')
    >>> p3 = read_expr(r'-walk(Bill)')
    >>> c = read_expr(r'walk(Socrates)')
    >>> prover = Prover9Command(c, [p1,p2,p3])
    >>> prover.prove()
    False
    >>> cdp = ClosedDomainProver(prover)
    >>> for a in cdp.assumptions(): print(a) # doctest: +SKIP
    (walk(Socrates) | walk(Bill))
    man(Socrates)
    -walk(Bill)
    >>> cdp.prove()
    True

    >>> p1 = read_expr(r'walk(Socrates)')
    >>> p2 = read_expr(r'walk(Bill)')
    >>> c = read_expr(r'all x.walk(x)')
    >>> prover = Prover9Command(c, [p1,p2])
    >>> prover.prove()
    False
    >>> cdp = ClosedDomainProver(prover)
    >>> for a in cdp.assumptions(): print(a) # doctest: +SKIP
    walk(Socrates)
    walk(Bill)
    >>> print(cdp.goal()) # doctest: +SKIP
    (walk(Socrates) & walk(Bill))
    >>> cdp.prove()
    True

    >>> p1 = read_expr(r'girl(mary)')
    >>> p2 = read_expr(r'dog(rover)')
    >>> p3 = read_expr(r'all x.(girl(x) -> -dog(x))')
    >>> p4 = read_expr(r'all x.(dog(x) -> -girl(x))')
    >>> p5 = read_expr(r'chase(mary, rover)')
    >>> c = read_expr(r'exists y.(dog(y) & all x.(girl(x) -> chase(x,y)))')
    >>> prover = Prover9Command(c, [p1,p2,p3,p4,p5])
    >>> print(prover.prove())
    False
    >>> cdp = ClosedDomainProver(prover)
    >>> for a in cdp.assumptions(): print(a) # doctest: +SKIP
    girl(mary)
    dog(rover)
    ((girl(rover) -> -dog(rover)) & (girl(mary) -> -dog(mary)))
    ((dog(rover) -> -girl(rover)) & (dog(mary) -> -girl(mary)))
    chase(mary,rover)
    >>> print(cdp.goal()) # doctest: +SKIP
    ((dog(rover) & (girl(rover) -> chase(rover,rover)) & (girl(mary) -> chase(mary,rover))) | (dog(mary) & (girl(rover) -> chase(rover,mary)) & (girl(mary) -> chase(mary,mary))))
    >>> print(cdp.prove())
    True

-----------------------
Unique Names Assumption
-----------------------

No two entities in the domain represent the same entity unless it can be
explicitly proven that they do.  Therefore, if the domain contains "A" and "B",
then add the assumption "-(A = B)" if it is not the case that
"<assumptions> \|- (A = B)".

    >>> p1 = read_expr(r'man(Socrates)')
    >>> p2 = read_expr(r'man(Bill)')
    >>> c = read_expr(r'exists x.exists y.-(x = y)')
    >>> prover = Prover9Command(c, [p1,p2])
    >>> prover.prove()
    False
    >>> unp = UniqueNamesProver(prover)
    >>> for a in unp.assumptions(): print(a) # doctest: +SKIP
    man(Socrates)
    man(Bill)
    -(Socrates = Bill)
    >>> unp.prove()
    True

    >>> p1 = read_expr(r'all x.(walk(x) -> (x = Socrates))')
    >>> p2 = read_expr(r'Bill = William')
    >>> p3 = read_expr(r'Bill = Billy')
    >>> c = read_expr(r'-walk(William)')
    >>> prover = Prover9Command(c, [p1,p2,p3])
    >>> prover.prove()
    False
    >>> unp = UniqueNamesProver(prover)
    >>> for a in unp.assumptions(): print(a) # doctest: +SKIP
    all x.(walk(x) -> (x = Socrates))
    (Bill = William)
    (Bill = Billy)
    -(William = Socrates)
    -(Billy = Socrates)
    -(Socrates = Bill)
    >>> unp.prove()
    True

-----------------------
Closed World Assumption
-----------------------

The only entities that have certain properties are those that is it stated
have the properties.  We accomplish this assumption by "completing" predicates.

If the assumptions contain "P(A)", then "all x.(P(x) -> (x=A))" is the completion
of "P".  If the assumptions contain "all x.(ostrich(x) -> bird(x))", then
"all x.(bird(x) -> ostrich(x))" is the completion of "bird".  If the
assumptions don't contain anything that are "P", then "all x.-P(x)" is the
completion of "P".

    >>> p1 = read_expr(r'walk(Socrates)')
    >>> p2 = read_expr(r'-(Socrates = Bill)')
    >>> c = read_expr(r'-walk(Bill)')
    >>> prover = Prover9Command(c, [p1,p2])
    >>> prover.prove()
    False
    >>> cwp = ClosedWorldProver(prover)
    >>> for a in cwp.assumptions(): print(a) # doctest: +SKIP
    walk(Socrates)
    -(Socrates = Bill)
    all z1.(walk(z1) -> (z1 = Socrates))
    >>> cwp.prove()
    True

    >>> p1 = read_expr(r'see(Socrates, John)')
    >>> p2 = read_expr(r'see(John, Mary)')
    >>> p3 = read_expr(r'-(Socrates = John)')
    >>> p4 = read_expr(r'-(John = Mary)')
    >>> c = read_expr(r'-see(Socrates, Mary)')
    >>> prover = Prover9Command(c, [p1,p2,p3,p4])
    >>> prover.prove()
    False
    >>> cwp = ClosedWorldProver(prover)
    >>> for a in cwp.assumptions(): print(a) # doctest: +SKIP
    see(Socrates,John)
    see(John,Mary)
    -(Socrates = John)
    -(John = Mary)
    all z3 z4.(see(z3,z4) -> (((z3 = Socrates) & (z4 = John)) | ((z3 = John) & (z4 = Mary))))
    >>> cwp.prove()
    True

    >>> p1 = read_expr(r'all x.(ostrich(x) -> bird(x))')
    >>> p2 = read_expr(r'bird(Tweety)')
    >>> p3 = read_expr(r'-ostrich(Sam)')
    >>> p4 = read_expr(r'Sam != Tweety')
    >>> c = read_expr(r'-bird(Sam)')
    >>> prover = Prover9Command(c, [p1,p2,p3,p4])
    >>> prover.prove()
    False
    >>> cwp = ClosedWorldProver(prover)
    >>> for a in cwp.assumptions(): print(a) # doctest: +SKIP
    all x.(ostrich(x) -> bird(x))
    bird(Tweety)
    -ostrich(Sam)
    -(Sam = Tweety)
    all z7.-ostrich(z7)
    all z8.(bird(z8) -> ((z8 = Tweety) | ostrich(z8)))
    >>> print(cwp.prove())
    True

-----------------------
Multi-Decorator Example
-----------------------

Decorators can be nested to utilize multiple assumptions.

    >>> p1 = read_expr(r'see(Socrates, John)')
    >>> p2 = read_expr(r'see(John, Mary)')
    >>> c = read_expr(r'-see(Socrates, Mary)')
    >>> prover = Prover9Command(c, [p1,p2])
    >>> print(prover.prove())
    False
    >>> cmd = ClosedDomainProver(UniqueNamesProver(ClosedWorldProver(prover)))
    >>> print(cmd.prove())
    True

-----------------
Default Reasoning
-----------------
    >>> logic._counter._value = 0
    >>> premises = []

define the taxonomy
    >>> premises.append(read_expr(r'all x.(elephant(x)        -> animal(x))'))
    >>> premises.append(read_expr(r'all x.(bird(x)            -> animal(x))'))
    >>> premises.append(read_expr(r'all x.(dove(x)            -> bird(x))'))
    >>> premises.append(read_expr(r'all x.(ostrich(x)         -> bird(x))'))
    >>> premises.append(read_expr(r'all x.(flying_ostrich(x)  -> ostrich(x))'))

default the properties using abnormalities
    >>> premises.append(read_expr(r'all x.((animal(x)  & -Ab1(x)) -> -fly(x))')) #normal animals don't fly
    >>> premises.append(read_expr(r'all x.((bird(x)    & -Ab2(x)) -> fly(x))'))  #normal birds fly
    >>> premises.append(read_expr(r'all x.((ostrich(x) & -Ab3(x)) -> -fly(x))')) #normal ostriches don't fly

specify abnormal entities
    >>> premises.append(read_expr(r'all x.(bird(x)           -> Ab1(x))')) #flight
    >>> premises.append(read_expr(r'all x.(ostrich(x)        -> Ab2(x))')) #non-flying bird
    >>> premises.append(read_expr(r'all x.(flying_ostrich(x) -> Ab3(x))')) #flying ostrich

define entities
    >>> premises.append(read_expr(r'elephant(el)'))
    >>> premises.append(read_expr(r'dove(do)'))
    >>> premises.append(read_expr(r'ostrich(os)'))

print the augmented assumptions list
    >>> prover = Prover9Command(None, premises)
    >>> command = UniqueNamesProver(ClosedWorldProver(prover))
    >>> for a in command.assumptions(): print(a) # doctest: +SKIP
    all x.(elephant(x) -> animal(x))
    all x.(bird(x) -> animal(x))
    all x.(dove(x) -> bird(x))
    all x.(ostrich(x) -> bird(x))
    all x.(flying_ostrich(x) -> ostrich(x))
    all x.((animal(x) & -Ab1(x)) -> -fly(x))
    all x.((bird(x) & -Ab2(x)) -> fly(x))
    all x.((ostrich(x) & -Ab3(x)) -> -fly(x))
    all x.(bird(x) -> Ab1(x))
    all x.(ostrich(x) -> Ab2(x))
    all x.(flying_ostrich(x) -> Ab3(x))
    elephant(el)
    dove(do)
    ostrich(os)
    all z1.(animal(z1) -> (elephant(z1) | bird(z1)))
    all z2.(Ab1(z2) -> bird(z2))
    all z3.(bird(z3) -> (dove(z3) | ostrich(z3)))
    all z4.(dove(z4) -> (z4 = do))
    all z5.(Ab2(z5) -> ostrich(z5))
    all z6.(Ab3(z6) -> flying_ostrich(z6))
    all z7.(ostrich(z7) -> ((z7 = os) | flying_ostrich(z7)))
    all z8.-flying_ostrich(z8)
    all z9.(elephant(z9) -> (z9 = el))
    -(el = os)
    -(el = do)
    -(os = do)

    >>> UniqueNamesProver(ClosedWorldProver(Prover9Command(read_expr('-fly(el)'), premises))).prove()
    True
    >>> UniqueNamesProver(ClosedWorldProver(Prover9Command(read_expr('fly(do)'), premises))).prove()
    True
    >>> UniqueNamesProver(ClosedWorldProver(Prover9Command(read_expr('-fly(os)'), premises))).prove()
    True