# Sample usage for nonmonotonic¶

## Nonmonotonic Reasoning¶

```>>> from nltk.test.setup_fixt import check_binary
>>> check_binary('mace4')
```
```>>> from nltk import *
>>> from nltk.inference.nonmonotonic import *
>>> from nltk.sem import logic
>>> logic._counter._value = 0
```

### 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))')
>>> prover = Prover9Command(c, [p1,p2])
>>> prover.prove()
True
>>> cdp = ClosedDomainProver(prover)
>>> for a in cdp.assumptions(): print(a)
(man(Socrates) -> mortal(Socrates))
man(Socrates)
>>> cdp.prove()
True
```
```>>> p1 = read_expr(r'exists x.walk(x)')
>>> prover = Prover9Command(c, [p1,p2])
>>> prover.prove()
False
>>> cdp = ClosedDomainProver(prover)
>>> for a in cdp.assumptions(): print(a)
walk(Socrates)
man(Socrates)
>>> cdp.prove()
True
```
```>>> p1 = read_expr(r'exists x.walk(x)')
>>> prover = Prover9Command(c, [p1,p2,p3])
>>> prover.prove()
False
>>> cdp = ClosedDomainProver(prover)
>>> for a in cdp.assumptions(): print(a)
(walk(Socrates) | walk(Bill))
man(Socrates)
-walk(Bill)
>>> cdp.prove()
True
```
```>>> p1 = read_expr(r'walk(Socrates)')
>>> prover = Prover9Command(c, [p1,p2])
>>> prover.prove()
False
>>> cdp = ClosedDomainProver(prover)
>>> for a in cdp.assumptions(): print(a)
walk(Socrates)
walk(Bill)
>>> print(cdp.goal())
(walk(Socrates) & walk(Bill))
>>> cdp.prove()
True
```
```>>> p1 = read_expr(r'girl(mary)')
>>> p3 = read_expr(r'all x.(girl(x) -> -dog(x))')
>>> p4 = read_expr(r'all x.(dog(x) -> -girl(x))')
>>> 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)
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())
((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)')
>>> 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)
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')
>>> prover = Prover9Command(c, [p1,p2,p3])
>>> prover.prove()
False
>>> unp = UniqueNamesProver(prover)
>>> for a in unp.assumptions(): print(a)
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)')
>>> prover = Prover9Command(c, [p1,p2])
>>> prover.prove()
False
>>> cwp = ClosedWorldProver(prover)
>>> for a in cwp.assumptions(): print(a)
walk(Socrates)
-(Socrates = Bill)
all z1.(walk(z1) -> (z1 = Socrates))
>>> cwp.prove()
True
```
```>>> p1 = read_expr(r'see(Socrates, John)')
>>> p3 = read_expr(r'-(Socrates = John)')
>>> p4 = read_expr(r'-(John = Mary)')
>>> prover = Prover9Command(c, [p1,p2,p3,p4])
>>> prover.prove()
False
>>> cwp = ClosedWorldProver(prover)
>>> for a in cwp.assumptions(): print(a)
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))')
>>> p4 = read_expr(r'Sam != Tweety')
>>> prover = Prover9Command(c, [p1,p2,p3,p4])
>>> prover.prove()
False
>>> cwp = ClosedWorldProver(prover)
>>> for a in cwp.assumptions(): print(a)
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)')
>>> 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))'))
```

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)'))
```

print the augmented assumptions list

```>>> prover = Prover9Command(None, premises)
>>> command = UniqueNamesProver(ClosedWorldProver(prover))
>>> for a in command.assumptions(): print(a)
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