def test_contraposition_theorem():
kb = KnowledgeBase()
kb.add_axiom(Proposition('''p -> q'''))
kb.add_goal(Proposition('''(~q -> ~p)'''))
assert kb.prove() == True
def test_implication_distribution():
kb = KnowledgeBase()
kb.add_axiom(Proposition('''p -> (q -> r)'''))
kb.add_goal(Proposition('''(p -> q) -> (p -> r)'''))
assert kb.prove() == True
def test_biconditional_elimination_v4():
kb = KnowledgeBase()
axioms = ['p = q', '¬q']
[kb.add_axiom(Proposition(statement)) for statement in axioms]
kb.add_goal(Proposition('''!p'''))
assert kb.prove() == True
def test_biconditional_elimination_unsat_v5():
kb = KnowledgeBase()
axioms = ['j <-> r', 'r v j']
[kb.add_axiom(Proposition(statement)) for statement in axioms]
kb.add_goal(Proposition('''!(r & j)'''))
assert kb.prove() == False
def test_biconditional_elimination_unsat_v3():
kb = KnowledgeBase()
axioms = ['p = q', '!p']
[kb.add_axiom(Proposition(statement)) for statement in axioms]
kb.add_goal(Proposition('''q'''))
assert kb.prove() == False
def test_hypothetical_syllogism_fails_when_conclusion_negated():
kb = KnowledgeBase()
axioms = ['a -> b', 'b -> c', 'c -> d', 'd -> e', 'e -> f', 'f -> g', 'a']
[kb.add_axiom(Proposition(statement)) for statement in axioms]
kb.add_goal(Proposition('''~g'''))
assert kb.prove() == False
def test_constructive_dilemma():
kb = KnowledgeBase()
axioms = ['a -> b', 'c -> d', 'a v c']
[kb.add_axiom(Proposition(statement)) for statement in axioms]
kb.add_goal(Proposition('''(b v d)'''))
assert kb.prove() == True
def test_case_analysis():
kb = KnowledgeBase()
axioms = ['a -> b', 'c -> b', 'a v c']
[kb.add_axiom(Proposition(statement)) for statement in axioms]
kb.add_goal(Proposition('''b'''))
assert kb.prove() == True
def test_disjunctive_syllogism_v2():
kb = KnowledgeBase()
axioms = ['p v q', '~q']
[kb.add_axiom(Proposition(statement)) for statement in axioms]
kb.add_goal(Proposition('''p'''))
assert kb.prove() == True
def test_absorption():
kb = KnowledgeBase()
kb.add_axiom(Proposition('''(p -> q)'''))
kb.add_axiom(Proposition('''(p -> (p & q))'''))
kb.add_goal(Proposition('''(p -> q)'''))
assert kb.prove() == True
def test_modus_tollens():
kb = KnowledgeBase()
axioms = ['p -> q', '~q']
[kb.add_axiom(Proposition(statement)) for statement in axioms]
kb.add_goal(Proposition('''~p'''))
assert kb.prove() == True
def test_biconditional_introduction():
kb = KnowledgeBase()
axioms = ['e -> f', 'f -> e']
[kb.add_axiom(Proposition(statement)) for statement in axioms]
kb.add_goal(Proposition('''e <-> f'''))
assert kb.prove() == True
def test_hypothetical_syllogism():
kb = KnowledgeBase()
axioms = ['a -> b', 'b -> c', 'c -> d', 'd -> e', 'e -> f', 'f -> g', 'a']
[kb.add_axiom(Proposition(statement)) for statement in axioms]
kb.add_goal(Proposition('''g'''))
assert kb.prove() == True
def test_multiple_goal_clauses():
kb = KnowledgeBase()
kb.add_axiom(Proposition('''p -> q'''))
kb.add_axiom(Proposition('''r -> s'''))
kb.add_goal(Proposition('''(p v r) -> (q v s)'''))
assert kb.prove() == True
def test_biconditional_elimination_v6():
kb = KnowledgeBase()
axioms = ['j <-> r', '~r v ~j']
[kb.add_axiom(Proposition(statement)) for statement in axioms]
kb.add_goal(Proposition('''(~r & ~j)'''))
assert kb.prove() == True
def test_affirming_the_consequent():
''' Affirming the consequent should return False '''
kb = KnowledgeBase()
axioms = ['p -> q', 'q']
[kb.add_axiom(Proposition(statement)) for statement in axioms]
kb.add_goal(Proposition('''p'''))
assert kb.prove() == False
def test_another_multiple_goal_clauses():
kb = KnowledgeBase()
kb.add_axiom(Proposition('''(p -> q) -> q'''))
kb.add_axiom(Proposition('''(p -> p) -> r'''))
kb.add_axiom(Proposition('''(r -> s) -> ~(s -> q)'''))
kb.add_goal(Proposition('''r'''))
assert kb.prove() == True
def test_exercise_5_4():
''' http://intrologic.stanford.edu/exercises/exercise_05_04.html '''
kb = KnowledgeBase()
kb.add_axiom(Proposition('''p -> q'''))
kb.add_axiom(Proposition('''r -> s'''))
kb.add_goal(Proposition('''(p v r) -> (q v s)'''))
assert kb.prove() == True
def test_lem_and_lnc_imply_one_another():
kb = KnowledgeBase()
lem = Proposition('''p v ~p''')
lnc = Proposition('''~(p & ~p)''')
kb.add_axiom(lem)
kb.add_goal(lnc)
assert kb.prove() == True
def test_lnc_conflicts_with_negated_lnc():
""" You should *not* be able to derive a negation of a theorem (from itself)! """
kb = KnowledgeBase()
lnc = Proposition('''~(p & ~p)''')
negated_lnc = Proposition('''~(~(p & ~p))''')
kb.add_axiom(lnc)
kb.add_goal(negated_lnc)
assert kb.prove() == False
def test_constructive_dilemma():
'''
It is the inference that, if P implies Q and R implies S and either P or R is true, then Q or S has to be true.
Source: https://en.wikipedia.org/wiki/Constructive_dilemma
'''
kb = KnowledgeBase()
axioms = ['p -> q', 'r -> s', 'p v r']
[kb.add_axiom(Proposition(statement)) for statement in axioms]
kb.add_goal(Proposition('''q v s'''))
assert kb.prove() == True
def test_medium_sized_kb():
''' https://www.ics.uci.edu/~welling/teaching/271fall09/HW6_sol.pdf '''
kb = KnowledgeBase()
axioms = [
'a', 'b', 'c', '(a & b) -> d', '(b & d) -> f', 'f -> g',
'(a & e) -> h', '(a & c) -> e'
]
[kb.add_axiom(Proposition(statement)) for statement in axioms]
kb.add_goal(Proposition('''h'''))
assert kb.prove() == True
def test_anything_follows_from_a_contradiction():
kb = KnowledgeBase()
negated_lnc = Proposition('''~(~(p & ~p))''')
kb.add_axiom(negated_lnc)
kb.add_goal(Proposition('''z'''))
kb2 = KnowledgeBase()
kb2.add_axiom(negated_lnc)
kb2.add_goal(Proposition('''~r -> s'''))
assert (kb.prove() == True and kb2.prove() == True)
def test_minimal_kb_proof():
kb = KnowledgeBase()
pq = Proposition('''p v q''')
pr = Proposition('''p -> r''')
qr = Proposition('''q -> r''')
kb.add_axiom(pq)
kb.add_axiom(pr)
kb.add_axiom(qr)
r = Proposition('''r''')
kb.add_goal(r)
conclusion = kb.prove()
assert conclusion == True
def test_cc_for_biconditional_introduction():
kb = KnowledgeBase()
axioms = ['e -> f', 'f -> e']
[kb.add_axiom(Proposition(statement)) for statement in axioms]
kb.add_goal(Proposition('''e <-> f'''))
p = kb.prove(return_proof=True)
cc = {
frozenset({'e', 'f'}),
frozenset({'e', '~f'}),
frozenset({'~e', '~f'}),
frozenset({'f', '~e'})
}
assert p.clause_collection == cc
def test_unsatisfiable_clause_collection():
''' http://intrologic.stanford.edu/exercises/exercise_05_03.html '''
goal_as_prop = Proposition('''p v q''')
goal = group_cnf(cnf(expressify(goal_as_prop)))
negated_goal_as_prop = Proposition('''!(p v q)''')
negated_goal = group_cnf(cnf(expressify(negated_goal_as_prop)))
c1, c2, c3, c4 = goal[0], {'~p', 'r'}, {'~p', '~r'}, {'p', '~q'}
goal = c1
unsatisfiable = Proof(goal,
negated_goal,
clause_collection=[c1, c2, c3, c4])
# If a contradiction (empty clause) is found, the set of clauses is unsatisfiable
assert unsatisfiable.find() == True
def test_lem_is_theorem():
kb = KnowledgeBase()
lem = Proposition('''p v ~p''')
kb.add_axiom(lem)
kb.add_goal(lem)
assert kb.prove() == True
def test_lnc_is_theorem():
kb = KnowledgeBase()
lnc = Proposition('''~(p & ~p)''')
kb.add_axiom(lnc)
kb.add_goal(lnc)
assert kb.prove() == True
def test_negated_lnc():
prop = Proposition('''~(~(p and ~p))''')
assert prop.is_contradiction() == True
def test_negated_lem():
prop = Proposition('''~(p or ~p)''')
assert prop.is_contradiction() == True