Some optimizations

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Brandon Rozek 2024-04-15 00:08:00 -04:00
parent 9f985740e0
commit 20ccacc166
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5 changed files with 132 additions and 57 deletions

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@ -23,38 +23,38 @@ x = PropositionalVariable("x")
y = PropositionalVariable("y")
z = PropositionalVariable("z")
implication_rules = {
Rule({}, Implication(x, x)),
Rule(set(), Implication(x, x)),
Rule({Implication(x, y), Implication(y, z)}, Implication(x, z)),
Rule({}, Implication(Implication(x, Implication(x, y)), Implication(x, y))),
Rule({}, Implication(Implication(x, Implication(y, z)), Implication(y, Implication(x, z)))),
Rule({}, Implication(Implication(x, y), Implication(Implication(z, x), Implication(z, y)))),
Rule({}, Implication(Implication(x, y), Implication(Implication(y, z), Implication(x, z)))),
Rule({Implication(x, Implication(x, y)),}, Implication(x, y)),
Rule({Implication(x, Implication(y, z)),}, Implication(y, Implication(x, z))),
Rule({Implication(x, y),}, Implication(Implication(z, x), Implication(z, y))),
Rule({Implication(x, y),}, Implication(Implication(y, z), Implication(x, z))),
Rule({Implication(x, y), x}, y)
negation_rules = {
Rule({}, Implication(Negation(Negation(x)), x)),
Rule({}, Implication(x, Negation(Negation(x)))),
Rule({Negation(Negation(x)),}, x),
Rule({x,}, Negation(Negation(x))),
Rule({Implication(x, y)}, Implication(Negation(y), Negation(x))),
Rule({}, Implication(Implication(x, y), Implication(Negation(y), Negation(x))))
Rule({Implication(x, y),}, Implication(Negation(y), Negation(x)))
conjunction_rules = {
Rule({y, z}, Conjunction(y, z)),
Rule({}, Implication(Conjunction(x, y), x)),
Rule({}, Implication(Conjunction(x, y), y)),
Rule({}, Implication(Conjunction(Implication(x, y), Implication(x, z)), Implication(x, Conjunction(y, z))))
Rule({Conjunction(x, y),}, x),
Rule({Conjunction(x, y),}, y),
Rule({Conjunction(Implication(x, y), Implication(x, z)),}, Implication(x, Conjunction(y, z)))
disjunction_rules = {
Rule({}, Implication(x, Disjunction(x, y))),
Rule({}, Implication(y, Disjunction(x, y))),
Rule({}, Implication(Conjunction(Implication(x, z), Implication(y, z)), Implication(Disjunction(x, y), z))),
Rule({}, Implication(Conjunction(x, Disjunction(y, z)), Disjunction(Conjunction(x, y), Conjunction(x, z))))
Rule({x,}, Disjunction(x, y)),
Rule({y,}, Disjunction(x, y)),
Rule({Conjunction(Implication(x, z), Implication(y, z)),}, Implication(Disjunction(x, y), z)),
Rule({Conjunction(x, Disjunction(y, z)),}, Disjunction(Conjunction(x, y), Conjunction(x, z)))
logic_rules = implication_rules | negation_rules | conjunction_rules | disjunction_rules
operations = {Negation, Conjunction, Disjunction, Implication}
@ -118,7 +118,7 @@ interpretation = {
# Generate models of R of a given size
model_size = 2
satisfiable_models = generate_model(R_logic, model_size)
satisfiable_models = generate_model(R_logic, model_size, print_model=True)
print(f"There are {len(satisfiable_models)} satisfiable models of element length {model_size}")

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@ -0,0 +1,4 @@
from typing import Set
def set_to_str(x: Set) -> str:
return "{" + ", ".join((str(xi) for xi in x)) + "}"

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@ -1,9 +1,11 @@
File which generates all the models
from logic import Logic
from common import set_to_str
from logic import Logic, Operation, Rule, get_operations_from_term, PropositionalVariable
from model import ModelValue, Model, satisfiable, ModelFunction
from itertools import combinations, chain, product
from typing import Set
def possible_designations(iterable):
"""Powerset without the empty and complete set"""
@ -23,13 +25,59 @@ def possible_functions(operation, carrier_set):
yield ModelFunction(new_function, operation.symbol)
def possible_interpretations(logic, carrier_set):
def only_rules_with(rules: Set[Rule], operation: Operation) -> Set[Rule]:
result_rules = []
for rule in rules:
is_valid = True
for t in (rule.premises | {rule.conclusion,}):
t_operations = get_operations_from_term(t)
if len(t_operations) > 1:
is_valid = False
if len(t_operations) == 0:
t_operation = next(iter(t_operations))
if t_operation != operation:
is_valid = False
if is_valid:
return result_rules
def possible_interpretations(
logic: Logic, carrier_set: Set[ModelValue],
designated_values: Set[ModelValue]):
operations = []
model_functions = []
for operation in logic.operations:
model_functions.append(possible_functions(operation, carrier_set))
candidate_functions = list(possible_functions(operation, carrier_set))
passed_functions = []
Only consider functions that at least pass
in the rules with the operation by itself.
restricted_rules = only_rules_with(logic.rules, operation)
if len(restricted_rules) > 0:
small_logic = Logic({operation,}, restricted_rules)
for f in candidate_functions:
small_model = Model(carrier_set, {f,}, designated_values)
interp = {operation: f}
if satisfiable(small_logic, small_model, interp):
passed_functions = candidate_functions
if len(passed_functions) == 0:
raise Exception("No interpretation satisfies the axioms for the operation " + str(operation))
f"Operation {operation.symbol} has {len(passed_functions)} candidate functions"
functions_choice = product(*model_functions)
for functions in functions_choice:
@ -39,25 +87,36 @@ def possible_interpretations(logic, carrier_set):
interpretation[operation] = function
yield interpretation
def generate_model(logic: Logic, number_elements: int):
def generate_model(logic: Logic, number_elements: int, num_solutions: int = -1, print_model=False):
carrier_set = {
ModelValue("a" + str(i)) for i in range(number_elements)
possible_designated_values = possible_designations(carrier_set)
possible_interps = possible_interpretations(logic, carrier_set)
satisfied_models = []
checked = 0
for designated_values, interpretation in product(possible_designated_values, possible_interps):
checked += 1
for designated_values in possible_designated_values:
designated_values = set(designated_values)
model = Model(carrier_set, set(interpretation.values()), designated_values)
if satisfiable(logic, model, interpretation):
print("Checked", checked)
print("Considering models for designated values", set_to_str(designated_values))
possible_interps = possible_interpretations(logic, carrier_set, designated_values)
for interpretation in possible_interps:
is_valid = True
model = Model(carrier_set, set(interpretation.values()), designated_values)
# Iteratively test possible interpretations
# by adding one axiom at a time
for rule in logic.rules:
small_logic = Logic(logic.operations, {rule,})
if not satisfiable(small_logic, model, interpretation):
is_valid = False
if is_valid:
if print_model:
print(model, flush=True)
if num_solutions >= 0 and len(satisfied_models) >= num_solutions:
return satisfied_models
return satisfied_models

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@ -1,4 +1,4 @@
from typing import Any, Set
from typing import Any, Set, Tuple
from functools import lru_cache
class Operation:
@ -38,14 +38,10 @@ class PropositionalVariable(Term):
def __hash__(self):
return self.hashed_value
# def __setattr__(self, name: str, value: Any):
# raise Exception("Propositional variables are immutable")
def __str__(self):
# def PropTerm(Term):
# def __init__(self, v: PropositionalVariable):
# self.v = v
class OpTerm(Term):
def __init__(self, operation: Operation, arguments):
@ -65,12 +61,12 @@ class OpTerm(Term):
arg_strs = [str(a) for a in self.arguments]
return self.operation.symbol + "(" + ",".join(arg_strs) + ")"
# Standard operators
Negation = Operation("¬", 1)
Conjunction = Operation("", 2)
Disjunction = Operation("", 2)
Implication = Operation("", 2)
class Inequation:
def __init__(self, antecedant : Term, consequent: Term):
self.antecedant = antecedant
@ -104,18 +100,19 @@ class Logic:
self.rules = rules
def get_prop_var_from_term(t: Term):
def get_prop_var_from_term(t: Term) -> Set[PropositionalVariable]:
if isinstance(t, PropositionalVariable):
return {t,}
result = set()
result: Set[PropositionalVariable] = set()
for arg in t.arguments:
result |= get_prop_var_from_term(arg)
return result
def get_propostional_variables(rules):
vars = set()
def get_propostional_variables(rules: Tuple[Rule]) -> Set[PropositionalVariable]:
vars: Set[PropositionalVariable] = set()
for rule in rules:
# Get all vars in premises
@ -125,4 +122,14 @@ def get_propostional_variables(rules):
# Get vars in conclusion
vars |= get_prop_var_from_term(rule.conclusion)
return vars
return vars
def get_operations_from_term(t: Term) -> Set[Operation]:
if isinstance(t, PropositionalVariable):
return set()
result: Set[Operation] = {t.operation,}
for arg in t.arguments:
result |= get_operations_from_term(arg)
return result

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@ -1,18 +1,18 @@
Defining what it means to be a model
from common import set_to_str
from logic import (
PropositionalVariable, get_propostional_variables, Logic, Term,
from typing import Set, List, Dict
from typing import Set, List, Dict, Tuple
from itertools import product
from functools import lru_cache
__all__ = ['ModelValue', 'ModelFunction', 'Model']
def set_to_str(x):
return "{" + ", ".join((str(xi) for xi in x)) + "}"
class ModelValue:
def __init__(self, name):
@ -80,7 +80,7 @@ Designated Values: {set_to_str(self.designated_values)}
return result
def evaluate_term(t: Term, f: Dict[PropositionalVariable, ModelValue], interpretation: Dict[Operation, ModelFunction]):
def evaluate_term(t: Term, f: Dict[PropositionalVariable, ModelValue], interpretation: Dict[Operation, ModelFunction]) -> ModelValue:
if isinstance(t, PropositionalVariable):
return f[t]
@ -93,24 +93,28 @@ def evaluate_term(t: Term, f: Dict[PropositionalVariable, ModelValue], interpret
return model_function(*model_arguments)
def all_model_valuations(
pvars: Set[PropositionalVariable],
mvalues: Set[ModelValue]):
pvars: Tuple[PropositionalVariable],
mvalues: Tuple[ModelValue]):
pvars = list(pvars)
possible_valuations = [mvalues for _ in pvars]
all_possible_values = product(*possible_valuations)
for valuation in all_possible_values:
mapping = dict()
mapping: Dict[PropositionalVariable, ModelValue] = dict()
assert len(pvars) == len(valuation)
for pvar, value in zip(pvars, valuation):
mapping[pvar] = value
yield mapping
def satisfiable(logic: Logic, model: Model, interpretation: Dict[Operation, ModelFunction]):
pvars = get_propostional_variables(logic.rules)
mappings = all_model_valuations(pvars, model.carrier_set)
def all_model_valuations_cached(
pvars: Tuple[PropositionalVariable],
mvalues: Tuple[ModelValue]):
return list(all_model_valuations(pvars, mvalues))
def satisfiable(logic: Logic, model: Model, interpretation: Dict[Operation, ModelFunction]) -> bool:
pvars = tuple(get_propostional_variables(tuple(logic.rules)))
mappings = all_model_valuations_cached(pvars, tuple(model.carrier_set))
for mapping in mappings:
for rule in logic.rules:
@ -120,6 +124,7 @@ def satisfiable(logic: Logic, model: Model, interpretation: Dict[Operation, Mode
if t not in model.designated_values:
premise_met = False
if not premise_met: