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