def get_initial_concept_domain(sig):
"""
sig is an ivy_logic.Sig object
"""
concepts = OrderedDict()
concepts['nodes'] = []
concepts['node_labels'] = []
concepts['edges'] = []
# add sort concepts
for s in sorted(sig.sorts.values()):
X = Var('X', s)
concepts[s.name] = Concept([X], Eq(X,X))
concepts['nodes'].append(s.name)
# add equality concept
X = Var('X', TopSort())
Y = Var('Y', TopSort())
concepts['='] = Concept([X, Y], Eq(X, Y))
# add concepts from relations
for c in sorted(sig.symbols.values()):
assert type(c) is Const
if first_order_sort(c.sort):
# first order constant, add unary equality concept
X = Var('X', c.sort)
name = '={}'.format(c.name)
concepts[name] = Concept([X], Eq(X,c))
elif type(c.sort) is FunctionSort and c.sort.arity == 1:
# add unary concept and label
X = Var('X', c.sort.domain[0])
name = '{}'.format(c.name)
concepts[name] = Concept([X], c(X))
elif type(c.sort) is FunctionSort and c.sort.arity == 2:
# add binary concept and edge
X = Var('X', c.sort.domain[0])
Y = Var('Y', c.sort.domain[1])
name = '{}'.format(c.name)
concepts[name] = Concept([X, Y], c(X, Y))
elif type(c.sort) is FunctionSort and c.sort.arity == 3:
# add ternary concept
X = Var('X', c.sort.domain[0])
Y = Var('Y', c.sort.domain[1])
Z = Var('Z', c.sort.domain[2])
name = '{}'.format(c.name)
concepts[name] = Concept([X, Y, Z], c(X, Y, Z))
else:
# skip other symbols
pass
return ConceptDomain(concepts, get_standard_combiners(), get_standard_combinations())
def get_diagram_concept_domain(sig, diagram):
"""
sig is an ivy_logic.Sig object
diagram is a formula
"""
concepts = OrderedDict()
concepts['nodes'] = []
concepts['node_labels'] = []
concepts['edges'] = []
# add equality concept
X = Var('X', TopSort())
Y = Var('Y', TopSort())
concepts['='] = Concept([X, Y], Eq(X, Y))
# add concepts from relations and constants in the signature and
# in the diagram
if sig is not None:
sig_symbols = frozenset(sig.symbols.values())
else:
sig_symbols = frozenset()
for c in sorted(sig_symbols | used_constants(diagram)):
assert type(c) is Const
if first_order_sort(c.sort):
# first order constant, add unary equality concept
X = Var('X', c.sort)
name = '{}:{}'.format(c.name, c.sort)
concepts[name] = Concept([X], Eq(X,c))
concepts['nodes'].append(name)
elif type(c.sort) is FunctionSort and c.sort.arity == 1:
# add unary concept and label
X = Var('X', c.sort.domain[0])
name = '{}'.format(c.name)
concepts[name] = Concept([X], c(X))
elif type(c.sort) is FunctionSort and c.sort.arity == 2:
# add binary concept and edge
X = Var('X', c.sort.domain[0])
Y = Var('Y', c.sort.domain[1])
name = '{}'.format(c.name)
concepts[name] = Concept([X, Y], c(X, Y))
elif type(c.sort) is FunctionSort and c.sort.arity == 3:
# add ternary concept
X = Var('X', c.sort.domain[0])
Y = Var('Y', c.sort.domain[1])
Z = Var('Z', c.sort.domain[2])
name = '{}'.format(c.name)
concepts[name] = Concept([X, Y, Z], c(X, Y, Z))
else:
# skip other symbols
pass
return ConceptDomain(concepts, get_standard_combiners(), get_standard_combinations())
def generate(self):
"""
generate variable with new name
Returns
-------
generated_var : .logic.Var
generated variable
"""
generated_var = Var(self.base_name + str(self.counter))
self.counter += 1
return generated_var
def _materialize_node(self, concept_name):
"""
Materialize a node, returns the witness constant
"""
concept = self.domain.concepts[concept_name]
assert concept.arity == 1
sort = concept.variables[0].sort
assert sort != TopSort()
witnesses = self._get_witnesses(concept_name)
if len(witnesses) > 0:
c = witnesses[0]
else:
c = Const(self._fresh_const_name(), sort)
# TODO: maybe we shouldn't split here, and create the concepts explicitly
X = Var('X', c.sort)
name = '={}'.format(c.name)
self.domain.concepts[name] = Concept(name, [X], Eq(X, c))
self.domain.split(concept_name, name)
self.suppose(concept(c))
return c
_implies_cache[key] = False
result.append(False)
elif res == z3.unsat:
_implies_cache[key] = True
result.append(True)
else:
# no caching of unknown results
print "z3 returned: {}".format(res)
assert False
result.append(None)
return result
if __name__ == '__main__':
S = UninterpretedSort('S')
X, Y, Z = (Var(n, S) for n in ['X', 'Y', 'Z'])
BinRel = FunctionSort(S, S, Boolean)
leq = Const('leq', BinRel)
transitive1 = ForAll((X, Y, Z),
Implies(And(leq(X, Y), leq(Y, Z)), leq(X, Z)))
transitive2 = ForAll((X, Y, Z),
Or(Not(leq(X, Y)), Not(leq(Y, Z)), leq(X, Z)))
transitive3 = Not(
Exists((X, Y, Z), And(leq(X, Y), leq(Y, Z), Not(leq(X, Z)))))
antisymmetric = ForAll((X, Y),
Implies(And(leq(X, Y), leq(Y, X), true), Eq(Y, X)))
print z3_implies(transitive1, transitive2)
print z3_implies(transitive2, transitive3)
print z3_implies(transitive3, transitive1)
print z3_implies(transitive3, antisymmetric)
def get_clauses(self):
clause1 = Clause(Atom(self.preds[0], [Var('X'), Var('Y')]), [])
self.clauses = [clause1]
elif ((type(t) is ForAll and type(t.body) is And)
or (type(t) is Exists and type(t.body) is Or)):
return normalize_quantifiers(
type(t.body)(*(type(t)(t.variables, x) for x in t.body)))
elif type(t) in (ForAll, Exists):
return type(t)(free_variables(t.body) & frozenset(t.variables),
normalize_quantifiers(t.body))
else:
assert False, type(t)
def is_tautology_equality(t):
"""Returns True if t is Eq(x,x) for some x"""
return type(t) is Eq and t.t1 == t.t2
if __name__ == '__main__':
from logic import *
s1 = UninterpretedSort('s1')
s2 = UninterpretedSort('s2')
X1 = Var('X', s1)
X2 = Var('X', s2)
f = ForAll([X1], Eq(X2, X2))
assert free_variables(f) == {X2}
assert free_variables(f, by_name=True) == set()
assert free_variables(f.body) == {X2}
assert free_variables(f.body, by_name=True) == {'X'}
def get_structure_concept_domain(state, sig=None):
"""
state is an ivy_interp.State with a .universe
sig is an ivy_logic.Sig object
"""
concepts = OrderedDict()
concepts['nodes'] = []
concepts['node_labels'] = []
# add equality concept
X = Var('X', TopSort())
Y = Var('Y', TopSort())
concepts['='] = Concept([X, Y], Eq(X, Y))
# add nodes for universe elements
elements = [uc for s in state.universe for uc in state.universe[s]]
for uc in sorted(elements):
# add unary equality concept
X = Var('X', uc.sort)
name = uc.name
if str(uc.sort) not in name:
name += ':{}'.format(uc.sort)
concepts[name] = Concept([X], Eq(X,uc))
concepts['nodes'].append(name)
# # find which symbols are equal to which universe constant
# equals = dict(
# (uc, [c for c in symbols
# if c != uc and
# c.sort == s and
# z3_implies(state_formula, Eq(c, uc))])
# for s in state.universe
# for uc in state.universe[s]
# )
# add concepts for relations and constants
state_formula = state.clauses.to_formula()
symbols = used_constants(state_formula)
if sig is not None:
symbols = symbols | frozenset(sig.symbols.values())
symbols = symbols - frozenset(elements)
symbols = sorted(symbols)
for c in symbols:
assert type(c) is Const
if first_order_sort(c.sort):
# first order constant, add unary equality concept
X = Var('X', c.sort)
name = '={}'.format(c.name)
concepts[name] = Concept([X], Eq(X,c))
elif type(c.sort) is FunctionSort and c.sort.arity == 1:
# add unary concept and label
X = Var('X', c.sort.domain[0])
name = '{}'.format(c.name)
concepts[name] = Concept([X], c(X))
elif type(c.sort) is FunctionSort and c.sort.arity == 2:
# add binary concept and edge
X = Var('X', c.sort.domain[0])
Y = Var('Y', c.sort.domain[1])
name = '{}'.format(c.name)
concepts[name] = Concept([X, Y], c(X, Y))
elif type(c.sort) is FunctionSort and c.sort.arity == 3:
# add ternary concept
X = Var('X', c.sort.domain[0])
Y = Var('Y', c.sort.domain[1])
Z = Var('Z', c.sort.domain[2])
name = '{}'.format(c.name)
concepts[name] = Concept([X, Y, Z], c(X, Y, Z))
else:
# skip other symbols
pass
return ConceptDomain(concepts, get_standard_combiners(), get_standard_combinations())
def get_standard_combiners():
T = TopSort()
UnaryRelation = FunctionSort(T, Boolean)
BinaryRelation = FunctionSort(T, T, Boolean)
X, Y, Z = (Var(n, T) for n in ['X', 'Y', 'Z'])
U = Var('U', UnaryRelation)
U1 = Var('U1', UnaryRelation)
U2 = Var('U2', UnaryRelation)
B = Var('B', BinaryRelation)
B1 = Var('B1', BinaryRelation)
B2 = Var('B2', BinaryRelation)
result = OrderedDict()
result['none'] = ConceptCombiner([U], Not(Exists([X], U(X))))
result['at_least_one'] = ConceptCombiner([U], Exists([X], U(X)))
result['at_most_one'] = ConceptCombiner([U], ForAll([X,Y], Implies(And(U(X), U(Y)), Eq(X,Y))))
result['node_necessarily'] = ConceptCombiner(
[U1, U2],
ForAll([X], Implies(U1(X), U2(X))),
)
result['node_necessarily_not'] = ConceptCombiner(
[U1, U2],
ForAll([X], Implies(U1(X), Not(U2(X)))),
)
result['mutually_exclusive'] = ConceptCombiner(
[U1, U2],
ForAll([X, Y], Not(And(U1(X), U2(Y))))
)
result['all_to_all'] = ConceptCombiner(
[B, U1, U2],
ForAll([X,Y], Implies(And(U1(X), U2(Y)), B(X,Y)))
)
result['none_to_none'] = ConceptCombiner(
[B, U1, U2],
ForAll([X,Y], Implies(And(U1(X), U2(Y)), Not(B(X,Y))))
)
result['total'] = ConceptCombiner(
[B, U1, U2],
ForAll([X], Implies(U1(X), Exists([Y], And(U2(Y), B(X,Y)))))
)
result['functional'] = ConceptCombiner(
[B, U1, U2],
ForAll([X, Y, Z], Implies(And(U1(X), U2(Y), U2(Z), B(X,Y), B(X,Z)), Eq(Y,Z)))
)
result['surjective'] = ConceptCombiner(
[B, U1, U2],
ForAll([Y], Implies(U2(Y), Exists([X], And(U1(X), B(X,Y)))))
)
result['injective'] = ConceptCombiner(
[B, U1, U2],
ForAll([X, Y, Z], Implies(And(U1(X), U1(Y), U2(Z), B(X,Z), B(Y,Z)), Eq(X,Y)))
)
result['node_info'] = ['none', 'at_least_one', 'at_most_one']
if False:
# this just slows us down, and it's not clear it's needed
# later this should be made customizable by the user
result['edge_info'] = ['all_to_all', 'none_to_none', 'total',
'functional', 'surjective', 'injective']
else:
result['edge_info'] = ['all_to_all', 'none_to_none']
result['node_label'] = ['node_necessarily', 'node_necessarily_not']
return result
if sorts is not None:
for (s, tt), sort in zip(pairs, sorts):
unify(s, sort)
return [tt() for s, tt in pairs]
if __name__ == '__main__':
S = UninterpretedSort('S')
T = TopSort()
UnaryRelationS = FunctionSort(S, Boolean)
BinaryRelationS = FunctionSort(S, S, Boolean)
UnaryRelationT = FunctionSort(T, Boolean)
BinaryRelationT = FunctionSort(T, T, Boolean)
XS, YS = (Var(n, S) for n in ['X', 'Y'])
XT, YT = (Var(n, T) for n in ['X', 'Y'])
xs, ys = (Const(n, S) for n in ['x', 'y'])
xt, yt = (Const(n, T) for n in ['x', 'y'])
rs = Const('r', BinaryRelationS)
ps = Const('p', UnaryRelationS)
rt = Const('r', BinaryRelationT)
pt = Const('p', UnaryRelationT)
tt = Const('tt', T)
TT = Var('TT', T)
f1 = And(ps(XS), ps(xs))
cf1 = concretize_sorts(f1)
print repr(f1)
if __name__ == '__main__':
from collections import ConceptDict
from logic import (Var, Const, Apply, Eq, Not, And, Or, Implies, ForAll,
Exists)
from logic import UninterpretedSort, FunctionSort, first_order_sort, Boolean
from logic_util import free_variables, substitute, substitute_apply
from concept import Concept, ConceptCombiner, ConceptDomain
def test(st):
print st, "=", eval(st)
S = UninterpretedSort('S')
UnaryRelation = FunctionSort(S, Boolean)
BinaryRelation = FunctionSort(S, S, Boolean)
X, Y, Z = (Var(n, S) for n in ['X', 'Y', 'Z'])
U = Var('U', UnaryRelation)
U1 = Var('U1', UnaryRelation)
U2 = Var('U2', UnaryRelation)
B = Var('B', BinaryRelation)
B1 = Var('B1', BinaryRelation)
B2 = Var('B2', BinaryRelation)
nstar = Const('nstar', BinaryRelation)
x = Const('x', S)
y = Const('y', S)
c11 = Concept('xy', [X], And(Eq(x, X), Eq(y, X)))
c10 = Concept('x', [X], And(Eq(x, X), Not(Eq(y, X))))
c01 = Concept('y', [X], And(Not(Eq(x, X)), Eq(y, X)))
c00 = Concept('other', [X], And(Not(Eq(x, X)), Not(Eq(y, X))))
