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_is_sound_inference(debug=False):
from propositions.proofs import InferenceRule
for assumptions,conclusion,tautological in [
[[], '(~p|p)', True],
[[], '(p|p)', False],
[[], '(~p|q)', False],
[['(~p|q)', 'p'], 'q', True],
[['(p|q)', 'p'], 'q', False],
[['(p|q)', '(~p|r)'], '(q|r)', True],
[['(p->q)', '(q->r)'], 'r', False],
[['(p->q)', '(q->r)'], '(p->r)', True],
[['(x|y)'], '(y|x)', True],
[['x'], '(x|y)', True],
[['(x&y)'], 'x', True],
[['x'], '(x&y)', False]]:
rule = InferenceRule(
[Formula.parse(assumption) for assumption in assumptions],
Formula.parse(conclusion))
if debug:
print('Testing whether', rule, 'is sound')
assert is_sound_inference(rule) == tautological
for rule in [MP, I0, I1, D, I2, N, NI, NN, R,
A, NA1, NA2, O1, O2, NO, T, NF,
N_ALTERNATIVE, AE1, AE2, OE]:
if debug:
print('Testing that', rule, 'is sound')
assert is_sound_inference(rule)
def propositional_prove(formula: str, assumptions: List[str]):
from propositions.syntax import Formula
from propositions.tautology import proof_or_counterexample, prove_in_model, evaluate, all_models
from propositions.tautology import prove_tautology, encode_as_formula
from propositions.proofs import InferenceRule
f = Formula.parse(formula)
if f is None:
print_error('Could not parse the given formula')
return None
collected_assumptions = []
for assumption in assumptions:
a = Formula.parse(assumption)
if a is None:
print_error('Could not parse the given assumption')
return None
collected_assumptions.append(a)
f = encode_as_formula(InferenceRule(collected_assumptions, f))
valid_proofs = []
for model in all_models(list(f.variables())):
if not evaluate(f, model):
valid_proofs.append(prove_in_model(f, model))
if len(valid_proofs) > 0:
print_result('Found valid proof:\n' + str(valid_proofs))
return True
print_result('No valid proof.\n' + str(proof_or_counterexample(f)))
return False
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)
evaluation_result = evaluate_inference(rule1, frozendict(model)) == \
(not model['p']) or (not model['q']) or model['r']
if not evaluation_result:
print("FAILED.")
print("*" * 25)
print("EXPECTED:", (not model['p']) or (not model['q'])
or model['r'])
print(
"REASON: ",
"(not {}) or (not {}) or {}".format(model['p'], model['q'],
model['r']))
print("*" * 25)
assert evaluation_result
# 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 prove_from_skeleton_proof(formula: Formula,
skeleton_proof: PropositionalProof,
substitution_map: Mapping[str, Formula]) -> \
Proof:
"""Converts the given proof of a propositional skeleton of the given
predicate-logic formula into a proof of that predicate-logic formula.
Parameters:
formula: predicate-logic formula to prove.
skeleton_proof: valid propositional-logic proof of a propositional
skeleton of the given formula, from no assumptions and via
`~propositions.axiomatic_systems.AXIOMATIC_SYSTEM`.
substitution_map: map from each atomic propositional subformula of the
skeleton of the given formula that is proven in the given proof to
the respective predicate-logic subformula of the given formula.
Returns:
A valid predicate-logic proof of the given formula from the axioms
`PROPOSITIONAL_AXIOMATIC_SYSTEM_SCHEMAS` via only assumption lines and
MP lines.
"""
assert len(skeleton_proof.statement.assumptions) == 0 and \
skeleton_proof.rules.issubset(PROPOSITIONAL_AXIOMATIC_SYSTEM) and \
skeleton_proof.is_valid()
assert Formula.from_propositional_skeleton(
skeleton_proof.statement.conclusion, substitution_map) == formula
# Task 9.11.2
# create the proof, translate lines, assumption and conclusion
# creating lines:
predicate_line_formulas = [
Formula.from_propositional_skeleton(line.formula, substitution_map)
for line in skeleton_proof.lines
]
real_predicate_lines = []
for predicate_formula, skeleton_line in zip(predicate_line_formulas,
skeleton_proof.lines):
# if line was proven using MP
if skeleton_line.rule == MP:
# if the line is MP create a predicate MP line
real_predicate_lines.append(
Proof.MPLine(predicate_formula, skeleton_line.assumptions[0],
skeleton_line.assumptions[1]))
else: # line must be from an axiom
line_to_specialization_map = PropositionalInferenceRule.formula_specialization_map(
skeleton_line.rule.conclusion, skeleton_line.formula)
instantiation_map = axiom_specialization_map_to_schema_instantiation_map(
line_to_specialization_map, substitution_map)
real_predicate_lines.append(
Proof.AssumptionLine(
predicate_formula,
PROPOSITIONAL_AXIOM_TO_SCHEMA[skeleton_line.rule],
instantiation_map))
# create the final proof
return Proof(PROPOSITIONAL_AXIOMATIC_SYSTEM_SCHEMAS, formula,
real_predicate_lines)
def _prove_from_skeleton_proof(formula: Formula,
skeleton_proof: PropositionalProof,
substitution_map: Mapping[str, Formula]) -> \
Proof:
"""Converts the given proof of a propositional skeleton of the given
predicate-logic formula to a proof of that predicate-logic formula.
Parameters:
formula: predicate-logic formula to prove.
skeleton_proof: valid propositional-logic proof of a propositional
skeleton of the given formula, from no assumptions and via
`~propositions.axiomatic_systems.AXIOMATIC_SYSTEM`, and containing
no constants or operators beyond ``'~'``, ``'->'``, ``'|'``, and
``'&'``.
substitution_map: mapping from each atomic propositional subformula of
the skeleton of the given formula that is proven in the given proof
to the respective predicate-logic subformula of the given formula.
Returns:
A valid predicate-logic proof of the given formula from the axioms
`PROPOSITIONAL_AXIOMATIC_SYSTEM_SCHEMAS` via only assumption lines and
MP lines.
"""
assert len(skeleton_proof.statement.assumptions) == 0 and \
skeleton_proof.rules.issubset(PROPOSITIONAL_AXIOMATIC_SYSTEM) and \
skeleton_proof.is_valid()
assert Formula.from_propositional_skeleton(
skeleton_proof.statement.conclusion, substitution_map) == formula
for line in skeleton_proof.lines:
for operator in line.formula.operators():
assert is_unary(operator) or is_binary(operator)
# Task 9.11.2
lines = []
for line in skeleton_proof.lines:
form = Formula.from_propositional_skeleton(line.formula,
substitution_map)
if line.rule == MP:
mp_line = Proof.MPLine(form, line.assumptions[0],
line.assumptions[1])
lines.append(mp_line)
else: # rule is axiom
rule = PROPOSITIONAL_AXIOM_TO_SCHEMA[line.rule]
# map of format {'p': ~z33}
propositional_specialization_map = PropositionalInferenceRule.specialization_map(
line.rule,
PropositionalInferenceRule(line.rule.assumptions,
line.formula))
# map of format {'P': ~Ey[~x=y]}
scheme_map = _axiom_specialization_map_to_schema_instantiation_map(
propositional_specialization_map, substitution_map)
lines.append(Proof.AssumptionLine(form, rule, scheme_map))
return Proof(PROPOSITIONAL_AXIOMATIC_SYSTEM_SCHEMAS, formula, lines)
def prove_from_skeleton_proof(formula: Formula,
skeleton_proof: PropositionalProof,
substitution_map: Mapping[str, Formula]) -> \
Proof:
"""Converts the given proof of a propositional skeleton of the given
predicate-logic formula into a proof of that predicate-logic formula.
Parameters:
formula: predicate-logic formula to prove.
skeleton_proof: valid propositional-logic proof of a propositional
skeleton of the given formula, from no assumptions and via
`~propositions.axiomatic_systems.AXIOMATIC_SYSTEM`.
substitution_map: map from each atomic propositional subformula of the
skeleton of the given formula that is proven in the given proof to
the respective predicate-logic subformula of the given formula.
Returns:
A valid predicate-logic proof of the given formula from the axioms
`PROPOSITIONAL_AXIOMATIC_SYSTEM_SCHEMAS` via only assumption lines and
MP lines.
"""
assert len(skeleton_proof.statement.assumptions) == 0 and \
skeleton_proof.rules.issubset(PROPOSITIONAL_AXIOMATIC_SYSTEM) and \
skeleton_proof.is_valid()
assert Formula.from_propositional_skeleton(
skeleton_proof.statement.conclusion, substitution_map) == formula
# Task 9.11.2
assumptions = PROPOSITIONAL_AXIOMATIC_SYSTEM_SCHEMAS
conclusion = formula
lines = []
for skeleton_line in skeleton_proof.lines:
line_formula = Formula.from_propositional_skeleton(
skeleton_line.formula, substitution_map)
if PropositionalProof.Line.is_assumption(skeleton_line):
assumption_schema = Schema(line_formula, set())
line = Proof.AssumptionLine(line_formula, assumption_schema,
dict())
else:
if skeleton_line.rule == MP:
ante_line_num, cond_line_num = [
skeleton_line.assumptions[i] for i in [0, 1]
]
line = Proof.MPLine(line_formula, ante_line_num, cond_line_num)
else:
schema_rule = PROPOSITIONAL_AXIOM_TO_SCHEMA[skeleton_line.rule]
prop_spec_map = PropositionalInferenceRule.formula_specialization_map(
skeleton_line.rule.conclusion, skeleton_line.formula)
inst_map = axiom_specialization_map_to_schema_instantiation_map(
prop_spec_map, substitution_map)
line = Proof.AssumptionLine(line_formula, schema_rule,
inst_map)
lines.append(line)
return Proof(assumptions, conclusion, lines)
def prove_from_skeleton_proof(formula: Formula,
skeleton_proof: PropositionalProof,
substitution_map: Mapping[str, Formula]) -> \
Proof:
"""Converts the given proof of a propositional skeleton of the given
predicate-logic formula into a proof of that predicate-logic formula.
Parameters:
formula: predicate-logic formula to prove.
skeleton_proof: valid propositional-logic proof of a propositional
skeleton of the given formula, from no assumptions and via
`~propositions.axiomatic_systems.AXIOMATIC_SYSTEM`.
substitution_map: map from each atomic propositional subformula of the
skeleton of the given formula that is proven in the given proof to
the respective predicate-logic subformula of the given formula.
Returns:
A valid predicate-logic proof of the given formula from the axioms
`PROPOSITIONAL_AXIOMATIC_SYSTEM_SCHEMAS` via only assumption lines and
MP lines.
"""
assert len(skeleton_proof.statement.assumptions) == 0 and \
skeleton_proof.rules.issubset(PROPOSITIONAL_AXIOMATIC_SYSTEM) and \
skeleton_proof.is_valid()
assert Formula.from_propositional_skeleton(
skeleton_proof.statement.conclusion, substitution_map) == formula
# Task 9.11.2
lines = list()
for line in skeleton_proof.lines:
if len(line.assumptions) == 0:
new_line = Proof.AssumptionLine(
Formula.from_propositional_skeleton(line.formula,
substitution_map),
PROPOSITIONAL_AXIOM_TO_SCHEMA[line.rule],
axiom_specialization_map_to_schema_instantiation_map(
PropositionalInferenceRule.formula_specialization_map(
line.rule.conclusion, line.formula), substitution_map))
else:
new_line = Proof.MPLine(
Formula.from_propositional_skeleton(line.formula,
substitution_map),
line.assumptions[0], line.assumptions[1])
lines.append(new_line)
return Proof(PROPOSITIONAL_AXIOMATIC_SYSTEM_SCHEMAS, formula, lines)
def prove_from_skeleton_proof(formula: Formula,
skeleton_proof: PropositionalProof,
substitution_map: Mapping[str, Formula]) -> \
Proof:
"""Converts the given proof of a propositional skeleton of the given
predicate-logic formula into a proof of that predicate-logic formula.
Parameters:
formula: predicate-logic formula to prove.
skeleton_proof: valid propositional-logic proof of a propositional
skeleton of the given formula, from no assumptions and via
`~propositions.axiomatic_systems.AXIOMATIC_SYSTEM`.
substitution_map: map from each atomic propositional subformula of the
skeleton of the given formula that is proven in the given proof to
the respective predicate-logic subformula of the given formula.
Returns:
A valid predicate-logic proof of the given formula from the axioms
`PROPOSITIONAL_AXIOMATIC_SYSTEM_SCHEMAS` via only assumption lines and
MP lines.
"""
assert len(skeleton_proof.statement.assumptions) == 0 and \
skeleton_proof.rules.issubset(PROPOSITIONAL_AXIOMATIC_SYSTEM) and \
skeleton_proof.is_valid()
assert Formula.from_propositional_skeleton(
skeleton_proof.statement.conclusion, substitution_map) == formula
# Setup the proof builder
builder = Proof.ProofBuilder() \
.with_assumptions(PROPOSITIONAL_AXIOMATIC_SYSTEM_SCHEMAS) \
.with_conclusion(Formula.from_propositional_skeleton(skeleton_proof.statement.conclusion, substitution_map))
for line in skeleton_proof.lines:
# Which rule type do we got
if line.rule == MP:
builder.add_mp_line(Formula.from_propositional_skeleton(line.formula, substitution_map),
line.assumptions[0], line.assumptions[1])
else:
builder.add_claim_line(Formula.from_propositional_skeleton(line.formula, substitution_map),
PROPOSITIONAL_AXIOM_TO_SCHEMA[line.rule],
axiom_specialization_map_to_schema_instantiation_map(
PropositionalInferenceRule.formula_specialization_map(line.rule.conclusion,
line.formula),
substitution_map))
return builder.build()
def test_is_tautological_inference(debug=False):
from propositions.proofs import InferenceRule
for assumptions,conclusion,tautological in [
[[], '(~p|p)', True],
[[], '(p|p)', False],
[[], '(~p|q)', False],
[['(~p|q)', 'p'], 'q', True],
[['(p|q)', 'p'], 'q', False],
[['(p|q)', '(~p|r)'], '(q|r)', True],
[['(p->q)', '(q->r)'], 'r', False],
[['(p->q)', '(q->r)'], '(p->r)', True],
[['(x|y)'], '(y|x)', True],
[['x'], '(x|y)', True],
[['(x&y)'], 'x', True],
[['x'], '(x&y)', False]]:
rule = InferenceRule(
[Formula.from_infix(assumption) for assumption in assumptions],
Formula.from_infix(conclusion))
if debug:
print('Testing whether', rule, 'is a tautological inference')
assert is_tautological_inference(rule) == tautological