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Merge bb2024d254 into 51c26dd9fc
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2 changed files with 606 additions and 2 deletions
388
vsp.py
388
vsp.py
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@ -2,8 +2,8 @@
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Check to see if the model has the variable
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sharing property.
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"""
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from itertools import product
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from typing import List, Optional, Set, Tuple
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from itertools import product, chain, combinations
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from typing import List, Generator, Optional, Set, Tuple
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from common import set_to_str
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from model import (
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Model, model_closure, ModelFunction, ModelValue
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@ -125,3 +125,387 @@ def has_vsp(model: Model, impfunction: ModelFunction,
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return VSP_Result(True, model.name, carrier_set_left, carrier_set_right)
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return VSP_Result(False, model.name)
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class SVSP_Result:
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def __init__(
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self, has_svsp: bool, model_name: Optional[str] = None,
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subalgebra1: Optional[Set[ModelValue]] = None,
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subalgebra2: Optional[Set[ModelValue]] = None,
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U: Optional[Set[ModelValue]] = None,
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L: Optional[Set[ModelValue]] = None):
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self.has_svsp = has_svsp
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self.model_name = model_name
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self.subalgebra1 = subalgebra1
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self.subalgebra2 = subalgebra2
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self.U = U
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self.L = L
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def __str__(self):
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if not self.has_svsp:
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return f"Model {self.model_name} does not have the signed variable sharing property."
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return f"""Model {self.model_name} has the signed variable sharing property.
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Subalgebra 1: {set_to_str(self.subalgebra1)}
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Subalgebra 2: {set_to_str(self.subalgebra2)}
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U: {set_to_str(self.U)}
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L: {set_to_str(self.L)}
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"""
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def powerset_minus_empty(s):
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return chain.from_iterable(combinations(s, r) for r in range(1, len(s) + 1))
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def find_k1_k2(model: Model, impfunction: ModelFunction,
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negation_defined: bool) -> Generator[Tuple[Set[ModelValue], Set[ModelValue]], None, None]:
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"""
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Returns a list of possible subalgebra pairs (K1, K2)
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for SVSP. This is less efficient than the VSP version
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due to interaction with the L and U sets in SVSP.
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"""
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assert model.ordering is not None, "Expected ordering table in model"
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top = model.ordering.top()
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bottom = model.ordering.bottom()
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# Compute I the set of tuples (x, y) where
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# x -> y does not take a designiated value
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I: List[Tuple[ModelValue, ModelValue]] = []
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for (x, y) in product(model.carrier_set, model.carrier_set):
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if impfunction(x, y) not in model.designated_values:
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I.append((x, y))
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Is = powerset_minus_empty(I)
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# Find the subalgebras which falsify implication
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for xys in Is:
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xs = {xy[0] for xy in xys}
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# Discard ({⊥} ∪ A', B) subalgebras
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if bottom is not None and bottom in xs:
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continue
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# Discard ({⊤} ∪ A', B) subalgebras when negation is defined
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if top is not None and negation_defined and top in xs:
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continue
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ys = {xy[1] for xy in xys}
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# Discard (A, {⊤} ∪ B') subalgebras
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if top is not None and top in ys:
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continue
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# Discard (A, {⊥} ∪ B') subalgebras when negation is defined
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if bottom is not None and negation_defined and bottom in ys:
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continue
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order_dependent = False
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for (xi, yi) in product(xs, ys):
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# Discard ({a} ∪ A', {b} ∪ B') subalgebras when a <= b
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if model.ordering.is_lt(xi, yi):
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order_dependent = True
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break
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# Discard ({a} ∪ A', {b} ∪ B') subalgebras when b <= a and negation is defined
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if negation_defined and model.ordering.is_lt(yi, xi):
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order_dependent = True
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break
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if order_dependent:
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continue
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# Compute the left closure of the set containing xi under all the operations
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carrier_set_left: Set[ModelValue] = model_closure({xi,}, model.logical_operations, bottom)
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# Discard ({⊥} ∪ A', B) subalgebras
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if bottom is not None and bottom in carrier_set_left:
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continue
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# Discard ({⊤} ∪ A', B) subalgebras when negation is defined
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if top is not None and negation_defined and top in carrier_set_left:
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continue
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# Compute the closure of all operations
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# with just the ys
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carrier_set_right: Set[ModelValue] = model_closure({yi,}, model.logical_operations, top)
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# Discard (A, {⊤} ∪ B') subalgebras
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if top is not None and top in carrier_set_right:
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continue
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# Discard (A, {⊥} ∪ B') subalgebras when negation is defined
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if bottom is not None and negation_defined and bottom in carrier_set_right:
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continue
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# Discard subalgebras that intersect
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if not carrier_set_left.isdisjoint(carrier_set_right):
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continue
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# Check whether for all pairs in the subalgebra,
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# that implication is falsified.
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falsified = True
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for (x2, y2) in product(carrier_set_left, carrier_set_right):
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if impfunction(x2, y2) in model.designated_values:
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falsified = False
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break
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if falsified:
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yield (carrier_set_left, carrier_set_right)
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def find_candidate_u_l(
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model: Model, impfn: ModelFunction, negfn: Optional[ModelFunction],
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K1: Set[ModelValue], K2: Set[ModelValue]) -> Generator[Tuple[Set[ModelValue], Set[ModelValue]], None, None]:
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# Compute I the set of tuples (x, y) where
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# x -> y does not take a designiated value
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I: List[Tuple[ModelValue, ModelValue]] = []
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if negfn is None:
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# NOTE: K2 ∩ U = ∅ if ∀x(x → x) ∈ T
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# NOTE: K1 ∩ L = ∅ if ∀x(x → x) ∈ T
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for (x, y) in product(model.carrier_set - K2, model.carrier_set - K1):
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if impfn(x, y) not in model.designated_values:
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I.append((x, y))
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else:
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# NOTE: K1, K2, L, and U are pairwise distinct
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CmK1uK2 = model.carrier_set - (K1 | K2)
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for (x, y) in product(CmK1uK2, CmK1uK2):
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if impfn(x, y) not in model.designated_values:
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I.append((x, y))
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Is = powerset_minus_empty(I)
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F = model.carrier_set - model.designated_values
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has_double_negation_eq = False
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if negfn is not None:
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has_double_negation_eq = True
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for x in model.carrier_set:
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if negfn(negfn(x)) != x:
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has_double_negation_eq = False
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break
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for ULs in Is:
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unsat = False
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U = {UL[0] for UL in ULs}
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L = {UL[1] for UL in ULs}
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# U and L are distinct
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if U.intersection(L):
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continue
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if has_double_negation_eq:
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# NOTE: U is the negation image of L, that is, U = {¬x | x ∈ L}, if ∀x(x = ¬¬x).
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U2 = {negfn(x) for x in L}
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if U != U2:
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continue
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yield (U, L)
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LFi = F.intersection(L)
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for (x, y) in product(U, L):
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# Required property: ∀x ∈ U, y ∈ L(x → y ∈ L ∩ F)
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if impfn(x, y) not in LFi:
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unsat = True
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break
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# Required Property: ∀x ∈ L, y ∈ U(x → y ∈ U)
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if impfn(y, x) not in U:
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unsat = True
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break
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if unsat:
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continue
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if negfn is not None:
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for x in L:
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# Required Property: ∀x(x ∈ L ⇒ ¬x ∈ U)
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if negfn(x) not in U:
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unsat = True
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break
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if unsat:
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continue
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for x in U:
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# Required Property: ∀x(x ∈ U ⇒ ¬x ∈ L)
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if negfn(x) not in L:
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unsat = True
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break
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if unsat:
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continue
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# Passed all required properties
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yield (U, L)
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def has_svsp(model: Model, impfn: ModelFunction,
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conjfn: Optional[ModelFunction],
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disjfn: Optional[ModelFunction],
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negfn: Optional[ModelFunction]) -> SVSP_Result:
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"""
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Checks whether a model has the signed
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variable sharing property.
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"""
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# NOTE: No models with only one designated
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# value satisfies SVSP
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if len(model.designated_values) == 1:
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return SVSP_Result(False, model.name)
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F = model.carrier_set - model.designated_values
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starops = [conjfn, disjfn]
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K1K2s = find_k1_k2(model, impfn, negfn is not None)
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for K1, K2 in K1K2s:
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ULs = find_candidate_u_l(model, impfn, negfn, K1, K2)
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for U, L in ULs:
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unsat = False
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K1Uu = K1 | U
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K1Lu = K1 | L
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K1LuFi = K1Lu.intersection(F) # (K1 ∪ L) ∩ F
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K2Uu = K2 | U
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K2Lu = K2 | L
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K2LuFi = K2Lu.intersection(F) # (K2 ∪ L) ∩ F
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# (6)
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for x, y in product(K1, U):
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# b) x → y ∈ K1 ∪ U
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if impfn(x, y) not in K1Uu:
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unsat = True
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break
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# c) y → x ∈ K1 ∪ L
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if impfn(y, x) not in K1Lu:
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unsat = True
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break
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# a) x ∗ y, y ∗ x, y ∗ z ∈ K1 ∪ U
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for z in U:
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for op in starops:
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if op is not None:
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if op(x, y) not in K1Uu:
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unsat = True
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break
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if op(y, x) not in K1Uu:
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unsat = True
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break
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if op(y, z) not in K1Uu:
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unsat = True
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break
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if unsat:
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break
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if unsat:
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# Verification for these set of matrices failed
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break
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if unsat:
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# Move onto the next candidates K1, K2, U, L
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continue
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# (7)
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for x, y in product(K1, L):
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# b) x → y ∈ (K1 ∪ L) ∩ F
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if impfn(x, y) not in K1LuFi:
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unsat = True
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break
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# c) y → x ∈ K1 ∪ U
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if impfn(y, x) not in K1Uu:
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unsat = True
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break
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# a) x ∗ y, y ∗ x, y ∗ z ∈ K1 ∪ L
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for z in L:
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for op in starops:
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if op is not None:
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if op(x, y) not in K1Lu:
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unsat = True
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break
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if op(y, x) not in K1Lu:
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unsat = True
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break
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if op(y, z) not in K1Lu:
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unsat = True
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break
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if unsat:
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break
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if unsat:
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break
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if unsat:
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continue
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# (8)
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for x, y in product(K2, U):
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# b) x → y ∈ K2 ∪ U
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if impfn(x, y) not in K2Uu:
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unsat = True
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break
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# c) y → x ∈ (K2 ∪ L) ∩ F
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if impfn(y, x) not in K2LuFi:
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unsat = True
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break
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# a) x ∗ y, y ∗ x, y ∗ z ∈ K2 ∪ U
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for z in U:
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for op in starops:
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if op is not None:
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if op(x, y) not in K2Uu:
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unsat = True
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break
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if op(y, x) not in K2Uu:
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unsat = True
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break
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if op(y, z) not in K2Uu:
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unsat = True
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break
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if unsat:
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break
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if unsat:
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break
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if unsat:
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continue
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# (9)
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for x, y in product(K2, L):
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# b) x → y ∈ K2 ∪ L
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if impfn(x, y) not in K2Lu:
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unsat = True
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break
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# c) y → x ∈ K2 ∪ U
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if impfn(y, x) not in K2Uu:
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unsat = True
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break
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# a) x ∗ y, y ∗ x, y ∗ z ∈ K2 ∪ L
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for z in L:
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for op in starops:
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if op is not None:
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if op(x, y) not in K2Lu:
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unsat = True
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break
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if op(y, x) not in K2Lu:
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unsat = True
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break
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if op(y, z) not in K2Lu:
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unsat = True
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break
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if unsat:
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break
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if unsat:
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break
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if not unsat:
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return SVSP_Result(True, model.name, K1, K2, U, L)
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return SVSP_Result(False, model.name)
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