def test_synthesize_for_model(debug=False):
all_models1 = [{'x': False},
{'x': True}]
all_models2 = [{'p': False, 'q': False},
{'p': False, 'q': True},
{'p': True, 'q': False},
{'p': True, 'q': True}]
all_models3 = [{'r1': False, 'r12': False, 'p37': False},
{'r1': False, 'r12': False, 'p37': True},
{'r1': False, 'r12': True, 'p37': False},
{'r1': False, 'r12': True, 'p37': True},
{'r1': True, 'r12': False, 'p37': False},
{'r1': True, 'r12': False, 'p37': True},
{'r1': True, 'r12': True, 'p37': False},
{'r1': True, 'r12': True, 'p37': True}]
for all_models in [all_models1, all_models2, all_models3]:
for idx in range(len(all_models)):
if debug:
print('Testing synthesis of formula for model', all_models[idx])
f = synthesize_for_model(frozendict(all_models[idx]))
assert type(f) is Formula, 'Expected a formula, got ' + str(f)
assert is_clause(f), str(f) + ' should be a clause'
all_values = [False] * len(all_models)
all_values[idx] = True
for model,value in zip(all_models, all_values):
assert evaluate(f, frozendict(model)) == value
def test_evaluate_inference(debug=False):
from propositions.proofs import InferenceRule
# Test 1
rule1 = InferenceRule([Formula.parse('p'), Formula.parse('q')],
Formula.parse('r'))
for model in all_models(['p', 'q', 'r']):
if debug:
print('Testing evaluation of inference rule', rule1, 'in model',
model)
assert evaluate_inference(rule1, frozendict(model)) == \
(not model['p']) or (not model['q']) or model['r']
# Test 2
rule2 = InferenceRule([Formula.parse('(x|y)')],
Formula.parse('x'))
for model in all_models(['x', 'y']):
if debug:
print('Testing evaluation of inference rule', rule2, 'in model',
model)
assert evaluate_inference(rule2, frozendict(model)) == \
(not model['y']) or model['x']
# Test 3
rule3 = InferenceRule([Formula.parse(s) for s in ['(p->q)', '(q->r)']],
Formula.parse('r'))
for model in all_models(['p', 'q', 'r']):
if debug:
print('Testing evaluation of inference rule', rule3, 'in model',
model)
assert evaluate_inference(rule3, frozendict(model)) == \
(model['p'] and not model['q']) or \
(model['q'] and not model['r']) or model['r']
def test_merge_specialization_maps(debug=False):
for d1, d2, d in [
({}, {}, {}),
({}, None, None),
(None, {}, None),
(None, None, None),
({'p':'q'}, {'r':'s'}, {'p':'q', 'r':'s'}),
({'p':'q'}, {}, {'p':'q'}),
({}, {'p':'q'}, {'p':'q'}),
({'p':'q'}, {'p':'r'}, None),
({'p':'q'}, None, None),
(None, {'p':'q'}, None),
({'x':'p1', 'y':'p2'}, {'x':'p1', 'z':'p3'},
{'x':'p1', 'y':'p2', 'z':'p3'}),
({'x':'p1', 'y':'p2'}, {'x':'p1', 'y':'p3'}, None),
({'x':'p1', 'y':'p2'}, {'x':'p1', 'y':'p2', 'z':'p3'},
{'x':'p1', 'y':'p2', 'z':'p3'}),
({'x':'p1', 'y':'p2', 'z':'p3'}, {'x':'p1', 'y':'p2'},
{'x':'p1', 'y':'p2', 'z':'p3'})]:
if debug:
print('Testing merging of dictionaries', d1, d2)
dd = InferenceRule.merge_specialization_maps(
frozendict({v: Formula.parse(d1[v]) for v in d1}) if d1 is not None
else None,
frozendict({v: Formula.parse(d2[v]) for v in d2}) if d2 is not None
else None)
assert dd == ({v: Formula.parse(d[v]) for v in d}
if d is not None else None), "got " + dd
def __test_synthesize_clause(clause_synthesizer, for_model, debug):
all_models1 = [{'x': False},
{'x': True}]
all_models2 = [{'p': False, 'q': False},
{'p': False, 'q': True},
{'p': True, 'q': False},
{'p': True, 'q': True}]
all_models3 = [{'r1': False, 'r12': False, 'p37': False},
{'r1': False, 'r12': False, 'p37': True},
{'r1': False, 'r12': True, 'p37': False},
{'r1': False, 'r12': True, 'p37': True},
{'r1': True, 'r12': False, 'p37': False},
{'r1': True, 'r12': False, 'p37': True},
{'r1': True, 'r12': True, 'p37': False},
{'r1': True, 'r12': True, 'p37': True}]
for all_models in [all_models1, all_models2, all_models3]:
for idx in range(len(all_models)):
if debug:
print('Testing', clause_synthesizer.__qualname__, 'for model',
all_models[idx])
f = clause_synthesizer(frozendict(all_models[idx]))
assert type(f) is Formula, 'Expected a formula, got ' + str(f)
if for_model:
assert is_conjunctive_clause(f), \
str(f) + ' should be a conjunctive clause'
all_values = [False] * len(all_models)
all_values[idx] = True
else:
assert is_disjunctive_clause(f), \
str(f) + ' should be a disjunctive clause'
all_values = [True] * len(all_models)
all_values[idx] = False
for model, value in zip(all_models, all_values):
assert evaluate(f, frozendict(model)) == value
def evaluate_formula(
self, formula: Formula, assignment: Mapping[str,
T] = frozendict()) -> bool:
"""Calculates the truth value of the given formula in the current model,
for the given assignment of values to free occurrences of variables
names.
Parameters:
formula: formula to calculate the truth value of, for the constants,
functions, and relations of which the current model has
meanings.
assignment: mapping from each variable name that has a free
occurrence in the given formula to a universe element to which
it is to be evaluated.
Returns:
The truth value of the given formula in the current model, for the
given assignment of values to free occurrences of variable names.
"""
assert formula.constants().issubset(self.constant_meanings.keys())
assert formula.free_variables().issubset(assignment.keys())
for function, arity in formula.functions():
assert function in self.function_meanings and \
self.function_arities[function] == arity
for relation, arity in formula.relations():
assert relation in self.relation_meanings and \
self.relation_arities[relation] in {-1, arity}
# Task 7.8
return self.evaluate_helper(assignment, formula)
def evaluate_term(
self, term: Term, assignment: Mapping[str, T] = frozendict()) -> T:
"""Calculates the value of the given term in the current model, for the
given assignment of values to variables names.
Parameters:
term: term to calculate the value of, for the constants and
functions of which the current model has meanings.
assignment: mapping from each variable name in the given term to a
universe element to which it is to be evaluated.
Returns:
The value (in the universe of the current model) of the given
term in the current model, for the given assignment of values to
variable names.
"""
assert term.constants().issubset(self.constant_meanings.keys())
assert term.variables().issubset(assignment.keys())
for function, arity in term.functions():
assert function in self.function_meanings and \
self.function_arities[function] == arity
# Decide based on the term type
term_type = term.get_type()
if term_type == Term.TermType.T_CONST:
return self.constant_meanings[term.root]
elif term_type == Term.TermType.T_VARIABLE:
return assignment[term.root]
elif term_type == Term.TermType.T_FUNCTION:
# Evaluate the arguments
args = [self.evaluate_term(t, assignment) for t in term.arguments]
return self.function_meanings[term.root][tuple(args)]
def test_evaluate_all_operators(debug=False):
infix1 = '(p+q7)'
models_values1 = [
({'p': True, 'q7': False}, True),
({'p': False, 'q7': False}, False),
({'p': True, 'q7': True}, False)
]
infix2 = '~(p<->q7)'
models_values2 = [
({'p': True, 'q7': False}, True),
({'p': False, 'q7': False}, False),
({'p': True, 'q7': True}, False)
]
infix3 = '~((x-&x)-|(y-&y))'
models_values3 = [
({'x': True, 'y': False}, True),
({'x': False, 'y': False}, True),
({'x': True, 'y': True}, False)
]
for infix,models_values in [[infix1, models_values1],
[infix2, models_values2],
[infix3, models_values3]]:
formula = Formula.parse(infix)
for model,value in models_values:
if debug:
print('Testing evaluation of formula', formula, 'in model',
model)
assert evaluate(formula, frozendict(model)) == value
def test_reduce_assumption(debug=False):
for f, m, v in [ ('(y->x)', {'x':True}, 'y'),
('(p->p)', {}, 'p'),
('(p->(r->q))', {'p':True, 'q':True}, 'r')]:
f = Formula.parse(f)
m[v]=True
pt = prove_in_model(f, frozendict(m))
m[v]=False
pf = prove_in_model(f, frozendict(m))
if debug:
print("testing reduce assumption on", pt.statement, "and",
pf.statement)
p = reduce_assumption(pt,pf)
assert p.statement.conclusion == pf.statement.conclusion
assert p.statement.assumptions[:] == pt.statement.assumptions[:-1]
assert p.rules == AXIOMATIC_SYSTEM
assert p.is_valid(), offending_line(p)
def test_formulas_capturing_model(debug=False):
for q,a in [({'p':True},['p']),
({'q7':False}, ['~q7']),
({'x1':True, 'x2':False, 'x3':True}, ['x1', '~x2', 'x3']),
({'q3':False, 'p13':False, 'r':True}, ['~p13', '~q3', 'r'])]:
if debug:
print("Testing formulas_capturing_model on", q)
aa = [Formula.parse(f) for f in a]
assert formulas_capturing_model(frozendict(q)) == aa
def test_prove_in_model_full(debug=False):
for (f, m, a, cp) in [ ('x', {'x':True}, ['x'], ''),
('x',{'x':False}, ['~x'], '~'),
('~x', {'x':False}, ['~x'], ''),
('~x', {'x':True}, ['x'], '~'),
('x', {'x':True, 'z5':False}, ['x', '~z5'], ''),
('(p->~p)', {'p':True}, ['p'], '~'),
('(p->~p)', {'p':False}, ['~p'], ''),
('(p->q)', {'p':True, 'q':True}, ['p','q'], ''),
('(p->q)', {'p':True, 'q':False}, ['p','~q'], '~'),
('(p->q)', {'p':False, 'q':True}, ['~p','q'], ''),
('(p->q)', {'p':False, 'q':False}, ['~p', '~q'], ''),
('~~~~y7', {'y7':True}, ['y7'], ''),
('~~~~y7', {'y7':False}, ['~y7'], '~'),
('~(~p->~q)', {'p':True, 'q':True}, ['p','q'], '~'),
('~(~p->~q)', {'p':False, 'q':True}, ['~p','q'], ''),
('((p1->p2)->(p3->p4))',
{'p1':True, 'p2':True, 'p3':True, 'p4':True},
['p1','p2','p3','p4'], ''),
('((p1->p2)->(p3->p4))',
{'p1':True, 'p2':True, 'p3':True, 'p4':False},
['p1','p2','p3','~p4'], '~'),
('((p1->p2)->(p3->p4))',
{'p1':True, 'p2':False, 'p3':True, 'p4':False},
['p1','~p2','p3','~p4'], ''),
('(~(~x->~~y)->(z->(~x->~~~z)))',
{'z':True, 'x':False, 'y':False},
['~x','~y','z'], '~'),
('T', {}, [], ''),
('F', {}, [], '~'),
('(p|q)', {'p': True, 'q': True}, ['p', 'q'], ''),
('(p|q)', {'p': True, 'q': False}, ['p', '~q'], ''),
('(p|q)', {'p': False, 'q': True}, ['~p', 'q'], ''),
('(p|q)',
{'p': False, 'q': False}, ['~p', '~q'], '~'),
('(p&q)', {'p': True, 'q': True}, ['p', 'q'], ''),
('(p&q)', {'p': True, 'q': False}, ['p', '~q'], '~'),
('(p&q)', {'p': False, 'q': True}, ['~p', 'q'], '~'),
('(p&q)',
{'p': False, 'q': False}, ['~p', '~q'], '~'),
('~(~(q|p)&(r->~(s|q)))',
{'p': False, 'q': False, 'r': False, 's': False},
['~p', '~q', '~r', '~s'], '~'),
('~(~(q|p)&(r->~(s|q)))',
{'p': False, 'q': False, 'r': True, 's': True},
['~p', '~q', 'r', 's'], '')
]:
c = Formula.parse(cp+f)
f = Formula.parse(f)
a = (Formula.parse(v) for v in a)
if debug:
print('Testing prove_in_model_full on formula',f, 'in model', m)
p = prove_in_model_full(f, frozendict(m))
assert p.statement == InferenceRule(a, c)
assert p.rules == AXIOMATIC_SYSTEM_FULL
assert p.is_valid(), offending_line(p)
def prove_tautology(tautology: Formula, model: Model = frozendict()) -> Proof:
"""Proves the given tautology from the formulas that capture the given
model.
Parameters:
tautology: tautology that contains no constants or operators beyond
``'->'`` and ``'~'``, to prove.
model: model over a (possibly empty) prefix (with respect to the
alphabetical order) of the variables of `tautology`, from whose
formulas to prove.
Returns:
A valid proof of the given tautology from the formulas that capture the
given model, in the order returned by
`formulas_capturing_model`\ ``(``\ `model`\ ``)``, via
`~propositions.axiomatic_systems.AXIOMATIC_SYSTEM`.
Examples:
>>> proof = prove_tautology(Formula.parse('(~(p->p)->q)'),
... {'p': True, 'q': False})
>>> proof.is_valid()
True
>>> proof.statement.conclusion
(~(p->p)->q)
>>> proof.statement.assumptions
(p, ~q)
>>> proof.rules == AXIOMATIC_SYSTEM
True
>>> proof = prove_tautology(Formula.parse('(~(p->p)->q)'))
>>> proof.is_valid()
True
>>> proof.statement.conclusion
(~(p->p)->q)
>>> proof.statement.assumptions
()
>>> proof.rules == AXIOMATIC_SYSTEM
True
"""
assert is_tautology(tautology)
assert tautology.operators().issubset({'->', '~'})
assert is_model(model)
assert sorted(tautology.variables())[:len(model)] == sorted(model.keys())
if len(model) == len(tautology.variables()):
return prove_in_model(tautology, model)
variables_not_in_model = sorted(
[v for v in tautology.variables() if v not in model])
check_on_var = variables_not_in_model[0]
variables_not_in_model.remove(check_on_var)
new_model = {check_on_var: True}
new_model.update(model)
aff_proof = prove_tautology(tautology, new_model)
new_model[check_on_var] = False
neg_proof = prove_tautology(tautology, new_model)
return reduce_assumption(aff_proof, neg_proof)
def __test_synthesize(synthesizer, dnf, debug):
all_variables1 = ['p']
all_models1 = [{'p': False}, {'p': True}]
value_lists1 = [(False, False), (False, True), (True, False), (True, True)]
all_variables2 = ['p', 'q']
all_models2 = [{'p': False, 'q': False},
{'p': False, 'q': True},
{'p': True, 'q': False},
{'p': True, 'q': True}]
value_lists2 = [(True, False, False, True),
(True, True, True, True),
(False, False, False, False)]
all_variables3 = ['r1', 'r12', 'p37']
all_models3 = [{'r1': False, 'r12': False, 'p37': False},
{'r1': False, 'r12': False, 'p37': True},
{'r1': False, 'r12': True, 'p37': False},
{'r1': False, 'r12': True, 'p37': True},
{'r1': True, 'r12': False, 'p37': False},
{'r1': True, 'r12': False, 'p37': True},
{'r1': True, 'r12': True, 'p37': False},
{'r1': True, 'r12': True, 'p37': True}]
value_lists3 = [(True, False, True, True, False, True, False, True),
(True, True, True, True, True, True, True, True),
(False, False, False, False, False, False, False, False)]
for all_variables, all_models, value_lists in [
[all_variables1, all_models1, value_lists1],
[all_variables2, all_models2, value_lists2],
[all_variables3, all_models3, value_lists3]]:
for all_values in value_lists:
if debug:
print('Testing', synthesizer.__qualname__, 'for variables',
all_variables, 'and model-values', all_values)
formula = synthesizer(tuple(all_variables), all_values)
assert type(formula) is Formula, \
'Expected a formula, got ' + str(formula)
if dnf:
assert is_dnf(formula), str(formula) + ' should be a DNF'
else:
assert is_cnf(formula), str(formula) + ' should be a CNF'
assert formula.variables().issubset(set(all_variables))
for model, value in zip(all_models, all_values):
assert evaluate(formula, frozendict(model)) == value, \
str(formula) + ' does not evaluate to ' + str(value) + \
' on ' + str(model)
def prove_tautology(tautology: Formula, model: Model = frozendict()) -> Proof:
"""Proves the given tautology from the formulae that capture the given
model.
Parameters:
tautology: tautology that contains no constants or operators beyond
``'->'`` and ``'~'``, to prove.
model: model over a (possibly empty) prefix (with respect to the
alphabetical order) of the variables of `tautology`, from whose
formulae to prove.
Returns:
A valid proof of the given tautology from the formulae that capture the
given model, in the order returned by
`formulae_capturing_model`\ ``(``\ `model`\ ``)``, via
`~propositions.axiomatic_systems.AXIOMATIC_SYSTEM`.
Examples:
If the given model is the empty dictionary, then the returned proof is
of the given tautology from no assumptions.
"""
assert is_tautology(tautology)
assert tautology.operators().issubset({'->', '~'})
assert is_model(model)
assert sorted(tautology.variables())[:len(model)] == sorted(model.keys())
# Task 6.3a
statement = InferenceRule(formulae_capturing_model(model), tautology)
# rules are AXIOMATIC_SYSTEM
list_of_var_in_formula = list(tautology.variables())
list_of_var_in_formula.sort()
for var in list_of_var_in_formula:
if var not in model:
new_model = dict(model)
new_model[var] = True
# proof 1 is with that var with value True
proof1 = prove_tautology(tautology, new_model)
new_model[var] = False
# proof 2 is with that var with value False
proof2 = prove_tautology(tautology, new_model)
# proof without that var
return reduce_assumption(proof1, proof2)
# if the model contains all the var in the tautology formula
return prove_in_model(tautology, model)
def prove_tautology(tautology: Formula, model: Model = frozendict()) -> Proof:
"""Proves the given tautology from the formulae that capture the given
model.
Parameters:
tautology: tautology that contains no constants or operators beyond
``'->'`` and ``'~'``, to prove.
model: model over a (possibly empty) prefix (with respect to the
alphabetical order) of the variables of `tautology`, from whose
formulae to prove.
Returns:
A valid proof of the given tautology from the formulae that capture the
given model, in the order returned by
`formulae_capturing_model`\ ``(``\ `model`\ ``)``, via
`~propositions.axiomatic_systems.AXIOMATIC_SYSTEM`.
Examples:
If the given model is the empty dictionary, then the returned proof is
of the given tautology from no assumptions.
"""
assert is_tautology(tautology)
assert tautology.operators().issubset({'->', '~'})
assert is_model(model)
assert sorted(tautology.variables())[:len(model)] == sorted(model.keys())
# Task 6.3a
if model is not None and len(tautology.variables()) == len(model):
return prove_in_model(tautology, model)
else:
variables = sorted(tautology.variables())
new_model_1 = dict()
if model is not None:
for var in model.keys():
new_model_1[var] = model[var]
new_model_2 = deepcopy(new_model_1)
for var in variables:
if var not in model.keys():
new_model_1[var] = True
new_model_2[var] = False
antecedent_proof_1 = prove_tautology(tautology, new_model_1)
antecedent_proof_2 = prove_tautology(tautology, new_model_2)
return reduce_assumption(antecedent_proof_1, antecedent_proof_2)
def test_specialize(debug=False):
for t in substitutions:
d = t[0]
if debug:
print('Testing substitition dictionary', d)
d = frozendict({k: Formula.parse(d[k]) for k in d})
cases = [ [Formula.parse(c[0]), Formula.parse(c[1])] for c in t[1:]]
for case in cases:
if debug:
print('...checking that', case[0], 'specializes to', case[1])
general = InferenceRule([],case[0])
special = InferenceRule([],case[1])
assert general.specialize(d) == special, \
"got " + str(general.specialize(d).conclusion)
if debug:
print('...now checking all together in a single rule')
general = InferenceRule([case[0] for case in cases[1:]],cases[0][0])
special = InferenceRule([case[1] for case in cases[1:]],cases[0][1])
assert general.specialize(d) == special, \
"got " + str(general.specialize(d))
def test_substitute_variables(debug=False):
# f d result
tests = [('v', {}, 'v'),
('v', {'v': 'p'}, 'p'),
('(F->v12)', {'v12': 'v11'}, '(F->v11)'),
('v', {'q': 'r', 'z': 'w'}, 'v'),
('p', {'p': '(q|q)'}, '(q|q)'),
('~v', {'v': '(q|q)'}, '~(q|q)'),
('(~v|v)', {'v': '(q|q)'}, '(~(q|q)|(q|q))'),
('(q12->w)', {'q12': 'T', 'w': 'x'}, '(T->x)'),
('(v->w)', {'v': 'T', 'w': 'v'}, '(T->v)'),
('((~v&w)|(v->u))', {'v': '(~p->q)', 'u': '~~F'}, '((~(~p->q)&w)|((~p->q)->~~F))'),
('v2', {'v': 'p'}, 'v2')]
for f, d, r in tests:
if debug:
print("Testing substituting variables according to", d, "in formula", f)
f = Formula.parse(f)
d = {k: Formula.parse(d[k]) for k in d}
a = str(f.substitute_variables(frozendict(d)))
assert a == r, "Incorrect answer:" + a
def prove_tautology(tautology: Formula, model: Model = frozendict()) -> Proof:
"""Proves the given tautology from the formulae that capture the given
model.
Parameters:
tautology: tautology that contains no constants or operators beyond
``'->'`` and ``'~'``, to prove.
model: model over a (possibly empty) prefix (with respect to the
alphabetical order) of the variables of `tautology`, from whose
formulae to prove.
Returns:
A valid proof of the given tautology from the formulae that capture the
given model, in the order returned by
`formulae_capturing_model`\ ``(``\ `model`\ ``)``, via
`~propositions.axiomatic_systems.AXIOMATIC_SYSTEM`.
Examples:
If the given model is the empty dictionary, then the returned proof is
of the given tautology from no assumptions.
"""
assert is_tautology(tautology)
assert tautology.operators().issubset({'->', '~'})
assert is_model(model)
assert sorted(tautology.variables())[:len(model)] == sorted(model.keys())
tautology_variables = list(sorted(tautology.variables()))
if len(model) == len(tautology_variables):
return prove_in_model(tautology, model)
else:
#craete a model with added assumption, assigned to be True, and another model
#with the same added assumption assigned to be False
model_with_negation = dict(model)
model_with_negation[tautology_variables[len(model)]] = False
model_with_affirmation = dict(model)
model_with_affirmation[tautology_variables[len(model)]] = True
proof_from_affirmation = prove_tautology(tautology, model_with_affirmation)
proof_from_negation = prove_tautology(tautology, model_with_negation)
return reduce_assumption(proof_from_affirmation, proof_from_negation)