def if_A3(F):
# (!B->!A) -> ((!B->A)->B)
# ________ ____________
# F1 F2
try:
F1 = F.left
F2 = F.right
B = F2.right
A = F2.left.right
nB = notFormula(B)
nA = notFormula(A)
if F1.left == nB and F1.right == nA and \
F2 == Node(Node(nB, A), B):
return True
a3 = A3(A, B)
if a3 == F:
return True
except:
# F is variable or does not have left or right Node
pass
return False
def build_deduction(hypoth, F, G, proof=[]):
"""
hypoth, F |- G => hypoth |- F->G
proof = [F1, ..., Fn-1] + [G]
F = Fn
"""
if len(proof) == 0 or proof[-1] != G:
proof.append(G)
res = []
P_index = -1
for p in proof:
P_index += 1
if p == F:
res.extend(build_TL(F))
elif axiom.if_Axiom(p) or p in hypoth:
# A1 for F, G
F1 = axiom.A1(p, F)
# MP for F and F1
F2 = axiom.MP(p, F1)
res.extend([F1, F2])
else:
# smth weird happens here
_str = p.msg
if not _str.startswith("MP for"):
continue
_str = _str.replace("MP for", "")
sFr, sFs = _str.split(',')
Fr = formula.buildFormula(formula.prepareString(sFr))
Fs = formula.buildFormula(formula.prepareString(sFs))
# A2 for G, Fr, F
F1 = axiom.A2(F, Fr, p)
# MP for F1, G->Fs
F2 = axiom.MP(Node(F, Fs), F1)
# MP for G->Fr, F2
F3 = axiom.MP(Node(F, Fr), F2)
res.extend([F1, F2, F3])
return res
def A1(F, G):
"""
F -> (G -> F) axiom (1)
"""
if F is None or G is None:
return None
tree = Node(G, F)
tree = Node(F, tree)
tree.msg = "A1 for " + str(F) + ", " + str(G)
return tree
def build_T7(F, G):
# (F->G)->((!F->G)->G)
nF = notFormula(F)
nG = notFormula(G)
F1 = Node(F, G)
F1.msg = "t7 hypoth"
F2 = Node(nF, G)
F2.msg = "t7 hypoth"
f3 = build_T5(F, G)
F3 = f3[-1]
F4 = axiom.A3(F, G)
f5 = syl_1(F3, F4)
F5 = f5[-1]
F6 = MP(F1, F5)
f7 = build_T4(nG, F)
F7 = f7[-1]
# (!F->G)->(!F->!!G)
F8 = nF
F9 = Node(nF, G)
F10 = MP(F8, F9)
f11 = build_T2(G)
F11 = f11[-1]
F12 = MP(F10, F11)
f13 = build_deduction([F1, F2, F9], F8, F12, [F8, F9, F10] + f11 + [F12])
F13 = f13[-1]
f14 = build_deduction([], F9, F13, f13)
F14 = f14[-1]
#...
f15 = syl_1(F14, F7)
F15 = f15[-1]
f16 = syl_1(F15, F6)
F16 = f16[-1]
proof = [F1, F2] + f14 + [F4] + f5
f17 = build_deduction([], F1, F16, proof)
proof = f17
return proof
def T3(F, G):
"""
!F -> (F->G)
"""
if type(F) is Node:
nf = Node(F.left, F.right, _not=not F._not)
else:
nf = Variable(F.symbol, _not=not F._not)
fg = Node(F, G)
msg = "T3 for " + str(F) + ", " + str(G)
return Node(nf, fg, msg=msg)
def build_TL(F):
"""
Make L theorem with F
:param F: Variable or Node
:return: list of formal conclusions to get F->F
"""
ff = Node(F, F)
# A2 for F, F->F, F
F1 = axiom.A2(F, ff, F)
# A1 for F, F->F
F2 = axiom.A1(F, ff)
# MP for F1 and F2
F3 = axiom.MP(F2, F1)
# A1 for F and F
F4 = axiom.A1(F, F)
# MP for F3, F4
F5 = axiom.MP(F3, F4)
return [F1, F2, F3, F4, F5]
def T6(F, G):
"""
F -> (!G -> !(F->G) )
"""
if type(G) is Node:
ng = Node(G.left, G.right, _not=not G._not)
else:
ng = Variable(G.symbol, _not=not G._not)
nFG = Node(F, G, _not=True)
f2 = Node(ng, nFG)
msg = "T6 for " + str(F) + ", " + str(G)
return Node(F, f2, msg=msg)
def T7(F, G):
"""
(F->G) -> ( (!F->G) ->G )
"""
if type(F) is Node:
nf = Node(F.left, F.right, _not=not F._not)
else:
nf = Variable(F.symbol, _not=not F._not)
fg = Node(F, G)
nf_g = Node(nf, G)
nfg_G = Node(nf_g, G) # (!F->G)->G
msg = "T7 for " + str(F) + ", " + str(G)
return Node(fg, nfg_G, msg=msg)
def deduction(hypothesis, tree):
"""
Gamma, F |- G => Gamma |- F->G
hypothesis is a node
"""
msg = "Deduction theorem for " + str(hypothesis) + ", " + str(tree)
new_tree = Node(hypothesis, tree, msg=msg)
return new_tree
def _create_varfunc(self, quant, name):
assert quant != None
if not self.debug and self.rename:
name = 'VARFUNC'
else:
name = name.split(':')[0]
node = Node(NodeType.VARFUNC, name)
node.quant = quant
quant.vfunc.append(node)
self.graph.append(node)
return node
def A3(F, G):
"""
(!G -> !F) -> ( (!G -> F) -> G ) axiom (3)
"""
try:
notG = notFormula(G)
notF = notFormula(F)
ngnf = Node(notG, notF)
ngf = Node(notG, F)
ngf_g = Node(ngf, G)
msg = "A3 for " + str(F) + ", " + str(G)
return Node(ngnf, ngf_g, msg=msg)
except:
return None
def T5(F, G):
"""
(F->G) -> (!G -> !F)
"""
if type(F) is Node:
nf = Node(F.left, F.right, _not=not F._not)
else:
nf = Variable(F.symbol, _not=not F._not)
if type(G) is Node:
ng = Node(G.left, G.right, _not=not G._not)
else:
ng = Variable(G.symbol, _not=not G._not)
f1 = Node(F, G)
f2 = Node(ng, nf)
msg = "T5 for " + str(F) + ", " + str(G)
return Node(f1, f2, msg=msg)
def _create_node(self, name):
'''Create and return a node
Parameters
----------
name : str
Name of the node
Returns
-------
Node
A newly created node
'''
nname, nodetype = self._name_type(name)
node = Node(nodetype, nname)
self.graph.append(node)
return node
def _create_const(self, name):
'''Create and return a Const node
Parameters
----------
name : str
Name of the constant.
Returns
-------
Node
A newly created constant node
'''
node = Node(NodeType.CONST, name)
self.graph.append(node)
self.scope.declare_const(name, node)
return node
def convert(self, formula):
'''Convert tupled formula parse tree into graph
Parameters
----------
formula : tuple
Tupled tree representation of a formula
Returns
-------
Graph
DAG representation of a formula
'''
self.scope = Scope()
self.graph = []
self.root = None
self.tail = None
Node.reset_id()
append_infer_node = False
if isinstance(formula,
tuple) and len(formula) == 2 and formula[0] == '|-:c':
formula = formula[1]
append_infer_node = True
while isinstance(formula, tuple) and formula[0] == '!!':
formula = formula[2]
node = self._formula_to_graph(formula)
if self.tail is not None:
self.tail.outgoing.append(node)
node.incoming.append(self.tail)
else:
self.root = node
if append_infer_node:
node = Node(NodeType.CONSTFUNC, '|-:c')
self.graph.append(node)
node.outgoing.append(self.root)
self.root.incoming.append(node)
self.root = node
if not self.debug:
self._finalize_graph()
return self.graph
def S1(F1, F2):
"""
F->G, G->H |- F->H
F1 = F->G
F2 = G->H
"""
try:
if F1.right == F2.left:
msg = "S1 for " + str(F1) + ", " + str(F2)
node = Node(F1.right, F2.left, msg=msg)
return node
except:
pass
return None
def S2(F1, F2):
"""
F->(G->H), G |- F->H
F1 = F->(G->H)
F2 = G
"""
try:
msg = "S2 for " + str(F1) + ", " + str(F2)
if F1.right.left == F2:
node = Node(F1.left, F1.right.right, msg=msg)
except:
pass
return None
def _create_quant(self, name, free_var=False):
'''Create and return a Quant node
Parameters
----------
name : str
Name of the quantifier.
free_var : bool
True if quant is associated with a free variable
Returns
-------
Node
A newly created quantifier node
'''
node = Node(NodeType.QUANT, name)
self.graph.append(node)
if not free_var:
self.scope.declare_var(node)
return node
def build_T4(F, G):
# (!G->!F)->(F->G)
nF = notFormula(F)
nG = notFormula(G)
F1 = Node(nG, nF)
F2 = F
F3 = axiom.A3(F, G)
F4 = MP(F1, F3)
F5 = axiom.A1(F, nG)
f6 = syl_1(F5, F4)
proof = [F3, F4, F5]
proof.extend(f6)
proof = build_deduction([F1, F], F1, f6[-1], proof)
return proof
def A2(F, G, H):
"""
(F -> (G -> H)) -> ( (F -> G) -> (F -> H) ) axiom (2)
"""
gh = Node(G, H)
fgh = Node(F, gh)
fg = Node(F, G)
fh = Node(F, H)
fgfh = Node(fg, fh)
msg = "A2 for " + str(F) + ", " + str(G) + ", " + str(H)
res = Node(fgh, fgfh, msg=msg)
return res
def build_T5(F, G):
"""
(F->G)->(!G->!F)
"""
nF = notFormula(F)
nG = notFormula(G)
nnF = notFormula(nF)
nnG = notFormula(nG)
F1 = Node(F, G)
f2 = build_T2(G)
F2 = f2[-1]
F3 = axiom.A3(nG, nF)
f4 = syl_1(F1, F2) # f->!!g
F4 = f4[-1]
f5 = build_T1(F)
F5 = f5[-1]
f6 = syl_1(F5, F4) # !!f->!!g
F6 = f6[-1]
F7 = MP(F6, F3) # (!!f->!g)->!f
F8 = axiom.A1(nG, nnF)
f9 = syl_1(F8, F7)
F9 = f9[-1]
proof = f2 + [F3] + f4 + f5 + f6 + [F7, F8] + f9
proof = build_deduction([F1, nG], F1, F9, proof)
return proof
def build_T6(F, G):
fg = Node(F, G)
fg.msg = "hypoth t6"
t5 = build_T5(fg, G)
T5 = t5[-1]
ng = notFormula(G)
ng.msg = "hypoth t6"
F1 = F
F1.msg = "hypoth t6"
# we need (F->G)->G
F2 = MP(F1, fg)
ded = build_deduction([F1, ng], fg, G, [F1, F2])
F3 = ded[-1]
F4 = MP(F3, T5)
proof = [F1, ng, F2] + ded + [F4]
proof = build_deduction([], F, F4, proof)
return proof
def _create_var(self, quant, name):
'''Create and return a Var node
This function won't declare the variable!
Parameters
----------
quant : Quant
Quantifier that quantifies this variable.
Returns
-------
Node
A newly created variable node
'''
assert quant != None
if not self.debug and self.rename:
name = 'VAR'
else:
name = name.split(':')[0]
node = Node(NodeType.VAR, name)
node.quant = quant
quant.vvalue = node
self.graph.append(node)
return node
return False
def if_Axiom(F):
"""
Check if F is Axiom
"""
return if_A1(F) or if_A2(F) or if_A3(F)
if __name__ == '__main__':
f1 = Variable('F')
f2 = Node(Variable('F'), Variable('G'))
print(MP(f1, f2))
exit()
f1 = "(A->B)"
f2 = "(A->!(B->!C))"
tests = [
"F->(G->F)", "G->(F->G)", "F->(F->F)", "!F->((A->B)->!F)",
"(A->(B->C))->((A->B)->(A->C))",
"({0}->(B->{1}))->(({0}->B)->({0}->{1}))".format(f1, f2),
"(!G->!F)->((!G->F)->G)", "(!G->!G)->((!G->G)->G)",
"(!(f1)->!(f2))->((!(f1)->(f2))->(f1))".replace('f1',
f1).replace('f2', f2)
]
from formula import prepareString, buildFormula, IncorrectInput
def _create_constfunc(self, name):
node = Node(NodeType.CONSTFUNC, name)
self.graph.append(node)
return node
def Calmar_Theorem(F, values):
'''
Calmar's Theorem implementation
:param F: Propositional calculus formula: Variable or Node
:return:
'''
if type(F) == Variable:
v = Variable(F.symbol, _not=True if values[F.symbol] == 1 else 0)
return [v]
if F._not: # F = !G
G = Node(None, None)
G.left = F.left # Copy G, but F = !G
G.right = F.right
if F.calculate(values) == 1:
G.msg = "Calmar's theorem, 1.a) F^(alpha) = G^(alpha) => hypoth |- G^() = F^()"
return [G]
else:
# !f = !!g
t2 = build_T2(G)
T2 = t2[-1]
nng = MP(G, T2)
res = t2
res.append(nng)
return res
else: # F = G->H
res = []
G = F.left
H = F.right
if G.calculate(values) == 0:
nG = notFormula(G)
t3 = build_T3(G, H)
T = t3[-1]
mp = MP(nG, T)
res = t3
res.append(mp)
elif H.calculate(values) == 1:
# h->(g->h)
h = H
hgh = axiom.A1(H, G)
gh = MP(h, hgh)
return [hgh, gh]
else:
# F = !(G->H)
# have G, !H. proof !(G->H)
nH = notFormula(H)
t6 = build_T6(G, H)
T6 = t6[-1]
f1 = MP(G, T6)
f2 = MP(nH, f1)
res = t6
res.extend([f1, f2])
return res
def TL(F):
msg = "L theorem for " + str(F)
return Node(F, F, msg=msg)
proof.extend(deductC2)
proof.extend(t7)
proof.extend([mp1, mp2])
curLen = len(proof) #for debug
vector += 1
n += 1
return proof
if __name__ == '__main__':
f = Variable('F')
g = Variable('G')
nf = notFormula(f)
nnf = notFormula(nf)
Ad_Theorem(Node(nnf, f))
t4 = build_T4(f, g)
T4 = t4[-1]
t5 = build_T5(f, g)
t6 = build_T6(f, g)
t7 = build_T7(f, g)
T7 = t7[-1]
print('no exception? it is small victory')