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287 lines
9.8 KiB
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287 lines
9.8 KiB
Plaintext
.. Copyright (C) 2001-2019 NLTK Project
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.. For license information, see LICENSE.TXT
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======================
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Nonmonotonic Reasoning
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======================
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>>> from nltk import *
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>>> from nltk.inference.nonmonotonic import *
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>>> from nltk.sem import logic
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>>> logic._counter._value = 0
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>>> read_expr = logic.Expression.fromstring
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------------------------
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Closed Domain Assumption
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------------------------
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The only entities in the domain are those found in the assumptions or goal.
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If the domain only contains "A" and "B", then the expression "exists x.P(x)" can
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be replaced with "P(A) | P(B)" and an expression "all x.P(x)" can be replaced
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with "P(A) & P(B)".
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>>> p1 = read_expr(r'all x.(man(x) -> mortal(x))')
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>>> p2 = read_expr(r'man(Socrates)')
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>>> c = read_expr(r'mortal(Socrates)')
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>>> prover = Prover9Command(c, [p1,p2])
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>>> prover.prove()
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True
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>>> cdp = ClosedDomainProver(prover)
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>>> for a in cdp.assumptions(): print(a) # doctest: +SKIP
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(man(Socrates) -> mortal(Socrates))
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man(Socrates)
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>>> cdp.prove()
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True
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>>> p1 = read_expr(r'exists x.walk(x)')
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>>> p2 = read_expr(r'man(Socrates)')
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>>> c = read_expr(r'walk(Socrates)')
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>>> prover = Prover9Command(c, [p1,p2])
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>>> prover.prove()
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False
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>>> cdp = ClosedDomainProver(prover)
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>>> for a in cdp.assumptions(): print(a) # doctest: +SKIP
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walk(Socrates)
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man(Socrates)
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>>> cdp.prove()
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True
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>>> p1 = read_expr(r'exists x.walk(x)')
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>>> p2 = read_expr(r'man(Socrates)')
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>>> p3 = read_expr(r'-walk(Bill)')
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>>> c = read_expr(r'walk(Socrates)')
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>>> prover = Prover9Command(c, [p1,p2,p3])
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>>> prover.prove()
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False
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>>> cdp = ClosedDomainProver(prover)
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>>> for a in cdp.assumptions(): print(a) # doctest: +SKIP
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(walk(Socrates) | walk(Bill))
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man(Socrates)
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-walk(Bill)
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>>> cdp.prove()
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True
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>>> p1 = read_expr(r'walk(Socrates)')
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>>> p2 = read_expr(r'walk(Bill)')
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>>> c = read_expr(r'all x.walk(x)')
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>>> prover = Prover9Command(c, [p1,p2])
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>>> prover.prove()
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False
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>>> cdp = ClosedDomainProver(prover)
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>>> for a in cdp.assumptions(): print(a) # doctest: +SKIP
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walk(Socrates)
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walk(Bill)
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>>> print(cdp.goal()) # doctest: +SKIP
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(walk(Socrates) & walk(Bill))
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>>> cdp.prove()
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True
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>>> p1 = read_expr(r'girl(mary)')
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>>> p2 = read_expr(r'dog(rover)')
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>>> p3 = read_expr(r'all x.(girl(x) -> -dog(x))')
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>>> p4 = read_expr(r'all x.(dog(x) -> -girl(x))')
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>>> p5 = read_expr(r'chase(mary, rover)')
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>>> c = read_expr(r'exists y.(dog(y) & all x.(girl(x) -> chase(x,y)))')
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>>> prover = Prover9Command(c, [p1,p2,p3,p4,p5])
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>>> print(prover.prove())
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False
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>>> cdp = ClosedDomainProver(prover)
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>>> for a in cdp.assumptions(): print(a) # doctest: +SKIP
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girl(mary)
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dog(rover)
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((girl(rover) -> -dog(rover)) & (girl(mary) -> -dog(mary)))
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((dog(rover) -> -girl(rover)) & (dog(mary) -> -girl(mary)))
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chase(mary,rover)
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>>> print(cdp.goal()) # doctest: +SKIP
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((dog(rover) & (girl(rover) -> chase(rover,rover)) & (girl(mary) -> chase(mary,rover))) | (dog(mary) & (girl(rover) -> chase(rover,mary)) & (girl(mary) -> chase(mary,mary))))
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>>> print(cdp.prove())
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True
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-----------------------
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Unique Names Assumption
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-----------------------
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No two entities in the domain represent the same entity unless it can be
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explicitly proven that they do. Therefore, if the domain contains "A" and "B",
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then add the assumption "-(A = B)" if it is not the case that
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"<assumptions> \|- (A = B)".
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>>> p1 = read_expr(r'man(Socrates)')
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>>> p2 = read_expr(r'man(Bill)')
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>>> c = read_expr(r'exists x.exists y.-(x = y)')
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>>> prover = Prover9Command(c, [p1,p2])
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>>> prover.prove()
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False
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>>> unp = UniqueNamesProver(prover)
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>>> for a in unp.assumptions(): print(a) # doctest: +SKIP
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man(Socrates)
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man(Bill)
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-(Socrates = Bill)
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>>> unp.prove()
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True
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>>> p1 = read_expr(r'all x.(walk(x) -> (x = Socrates))')
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>>> p2 = read_expr(r'Bill = William')
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>>> p3 = read_expr(r'Bill = Billy')
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>>> c = read_expr(r'-walk(William)')
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>>> prover = Prover9Command(c, [p1,p2,p3])
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>>> prover.prove()
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False
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>>> unp = UniqueNamesProver(prover)
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>>> for a in unp.assumptions(): print(a) # doctest: +SKIP
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all x.(walk(x) -> (x = Socrates))
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(Bill = William)
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(Bill = Billy)
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-(William = Socrates)
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-(Billy = Socrates)
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-(Socrates = Bill)
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>>> unp.prove()
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True
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-----------------------
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Closed World Assumption
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-----------------------
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The only entities that have certain properties are those that is it stated
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have the properties. We accomplish this assumption by "completing" predicates.
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If the assumptions contain "P(A)", then "all x.(P(x) -> (x=A))" is the completion
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of "P". If the assumptions contain "all x.(ostrich(x) -> bird(x))", then
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"all x.(bird(x) -> ostrich(x))" is the completion of "bird". If the
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assumptions don't contain anything that are "P", then "all x.-P(x)" is the
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completion of "P".
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>>> p1 = read_expr(r'walk(Socrates)')
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>>> p2 = read_expr(r'-(Socrates = Bill)')
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>>> c = read_expr(r'-walk(Bill)')
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>>> prover = Prover9Command(c, [p1,p2])
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>>> prover.prove()
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False
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>>> cwp = ClosedWorldProver(prover)
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>>> for a in cwp.assumptions(): print(a) # doctest: +SKIP
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walk(Socrates)
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-(Socrates = Bill)
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all z1.(walk(z1) -> (z1 = Socrates))
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>>> cwp.prove()
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True
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>>> p1 = read_expr(r'see(Socrates, John)')
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>>> p2 = read_expr(r'see(John, Mary)')
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>>> p3 = read_expr(r'-(Socrates = John)')
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>>> p4 = read_expr(r'-(John = Mary)')
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>>> c = read_expr(r'-see(Socrates, Mary)')
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>>> prover = Prover9Command(c, [p1,p2,p3,p4])
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>>> prover.prove()
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False
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>>> cwp = ClosedWorldProver(prover)
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>>> for a in cwp.assumptions(): print(a) # doctest: +SKIP
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see(Socrates,John)
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see(John,Mary)
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-(Socrates = John)
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-(John = Mary)
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all z3 z4.(see(z3,z4) -> (((z3 = Socrates) & (z4 = John)) | ((z3 = John) & (z4 = Mary))))
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>>> cwp.prove()
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True
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>>> p1 = read_expr(r'all x.(ostrich(x) -> bird(x))')
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>>> p2 = read_expr(r'bird(Tweety)')
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>>> p3 = read_expr(r'-ostrich(Sam)')
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>>> p4 = read_expr(r'Sam != Tweety')
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>>> c = read_expr(r'-bird(Sam)')
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>>> prover = Prover9Command(c, [p1,p2,p3,p4])
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>>> prover.prove()
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False
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>>> cwp = ClosedWorldProver(prover)
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>>> for a in cwp.assumptions(): print(a) # doctest: +SKIP
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all x.(ostrich(x) -> bird(x))
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bird(Tweety)
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-ostrich(Sam)
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-(Sam = Tweety)
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all z7.-ostrich(z7)
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all z8.(bird(z8) -> ((z8 = Tweety) | ostrich(z8)))
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>>> print(cwp.prove())
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True
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-----------------------
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Multi-Decorator Example
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-----------------------
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Decorators can be nested to utilize multiple assumptions.
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>>> p1 = read_expr(r'see(Socrates, John)')
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>>> p2 = read_expr(r'see(John, Mary)')
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>>> c = read_expr(r'-see(Socrates, Mary)')
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>>> prover = Prover9Command(c, [p1,p2])
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>>> print(prover.prove())
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False
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>>> cmd = ClosedDomainProver(UniqueNamesProver(ClosedWorldProver(prover)))
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>>> print(cmd.prove())
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True
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-----------------
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Default Reasoning
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-----------------
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>>> logic._counter._value = 0
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>>> premises = []
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define the taxonomy
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>>> premises.append(read_expr(r'all x.(elephant(x) -> animal(x))'))
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>>> premises.append(read_expr(r'all x.(bird(x) -> animal(x))'))
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>>> premises.append(read_expr(r'all x.(dove(x) -> bird(x))'))
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>>> premises.append(read_expr(r'all x.(ostrich(x) -> bird(x))'))
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>>> premises.append(read_expr(r'all x.(flying_ostrich(x) -> ostrich(x))'))
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default the properties using abnormalities
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>>> premises.append(read_expr(r'all x.((animal(x) & -Ab1(x)) -> -fly(x))')) #normal animals don't fly
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>>> premises.append(read_expr(r'all x.((bird(x) & -Ab2(x)) -> fly(x))')) #normal birds fly
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>>> premises.append(read_expr(r'all x.((ostrich(x) & -Ab3(x)) -> -fly(x))')) #normal ostriches don't fly
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specify abnormal entities
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>>> premises.append(read_expr(r'all x.(bird(x) -> Ab1(x))')) #flight
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>>> premises.append(read_expr(r'all x.(ostrich(x) -> Ab2(x))')) #non-flying bird
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>>> premises.append(read_expr(r'all x.(flying_ostrich(x) -> Ab3(x))')) #flying ostrich
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define entities
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>>> premises.append(read_expr(r'elephant(el)'))
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>>> premises.append(read_expr(r'dove(do)'))
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>>> premises.append(read_expr(r'ostrich(os)'))
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print the augmented assumptions list
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>>> prover = Prover9Command(None, premises)
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>>> command = UniqueNamesProver(ClosedWorldProver(prover))
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>>> for a in command.assumptions(): print(a) # doctest: +SKIP
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all x.(elephant(x) -> animal(x))
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all x.(bird(x) -> animal(x))
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all x.(dove(x) -> bird(x))
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all x.(ostrich(x) -> bird(x))
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all x.(flying_ostrich(x) -> ostrich(x))
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all x.((animal(x) & -Ab1(x)) -> -fly(x))
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all x.((bird(x) & -Ab2(x)) -> fly(x))
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all x.((ostrich(x) & -Ab3(x)) -> -fly(x))
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all x.(bird(x) -> Ab1(x))
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all x.(ostrich(x) -> Ab2(x))
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all x.(flying_ostrich(x) -> Ab3(x))
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elephant(el)
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dove(do)
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ostrich(os)
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all z1.(animal(z1) -> (elephant(z1) | bird(z1)))
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all z2.(Ab1(z2) -> bird(z2))
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all z3.(bird(z3) -> (dove(z3) | ostrich(z3)))
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all z4.(dove(z4) -> (z4 = do))
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all z5.(Ab2(z5) -> ostrich(z5))
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all z6.(Ab3(z6) -> flying_ostrich(z6))
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all z7.(ostrich(z7) -> ((z7 = os) | flying_ostrich(z7)))
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all z8.-flying_ostrich(z8)
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all z9.(elephant(z9) -> (z9 = el))
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-(el = os)
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-(el = do)
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-(os = do)
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>>> UniqueNamesProver(ClosedWorldProver(Prover9Command(read_expr('-fly(el)'), premises))).prove()
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True
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>>> UniqueNamesProver(ClosedWorldProver(Prover9Command(read_expr('fly(do)'), premises))).prove()
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True
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>>> UniqueNamesProver(ClosedWorldProver(Prover9Command(read_expr('-fly(os)'), premises))).prove()
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True
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