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"""Representations and Inference for Logic (Chapters 7-9, 12)

Covers both Propositional and First-Order Logic. First we have four

important data types:

KB Abstract class holds a knowledge base of logical expressions

KB_Agent Abstract class subclasses agents.Agent

Expr A logical expression, imported from utils.py

substitution Implemented as a dictionary of var:value pairs, {x:1, y:x}

Be careful: some functions take an Expr as argument, and some take a KB.

Logical expressions can be created with Expr or expr, imported from utils, TODO

or with expr, which adds the capability to write a string that uses

the connectives ==>, <==, <=>, or <=/=>. But be careful: these have the

operator precedence of commas; you may need to add parens to make precedence work.

See logic.ipynb for examples.

Then we implement various functions for doing logical inference:

pl_true Evaluate a propositional logical sentence in a model

tt_entails Say if a statement is entailed by a KB

pl_resolution Do resolution on propositional sentences

dpll_satisfiable See if a propositional sentence is satisfiable

WalkSAT Try to find a solution for a set of clauses

And a few other functions:

to_cnf Convert to conjunctive normal form

unify Do unification of two FOL sentences

diff, simp Symbolic differentiation and simplification

"""

from utils import (

removeall, unique, first, argmax, probability,

isnumber, issequence, Expr, expr, subexpressions

)

import agents

import itertools

import random

from collections import defaultdict

# ______________________________________________________________________________

class KB:

"""A knowledge base to which you can tell and ask sentences.

To create a KB, first subclass this class and implement

tell, ask_generator, and retract. Why ask_generator instead of ask?

The book is a bit vague on what ask means --

For a Propositional Logic KB, ask(P & Q) returns True or False, but for an

FOL KB, something like ask(Brother(x, y)) might return many substitutions

such as {x: Cain, y: Abel}, {x: Abel, y: Cain}, {x: George, y: Jeb}, etc.

So ask_generator generates these one at a time, and ask either returns the

first one or returns False."""

def __init__(self, sentence=None):

raise NotImplementedError

def tell(self, sentence):

"""Add the sentence to the KB."""

raise NotImplementedError

def ask(self, query):

"""Return a substitution that makes the query true, or, failing that, return False."""

return first(self.ask_generator(query), default=False)

def ask_generator(self, query):

"""Yield all the substitutions that make query true."""

raise NotImplementedError

def retract(self, sentence):

"""Remove sentence from the KB."""

raise NotImplementedError

class PropKB(KB):

"""A KB for propositional logic. Inefficient, with no indexing."""

def __init__(self, sentence=None):

self.clauses = []

if sentence:

self.tell(sentence)

def tell(self, sentence):

"""Add the sentence's clauses to the KB."""

self.clauses.extend(conjuncts(to_cnf(sentence)))

def ask_generator(self, query):

"""Yield the empty substitution {} if KB entails query; else no results."""

if tt_entails(Expr('&', *self.clauses), query):

yield {}

def ask_if_true(self, query):

"""Return True if the KB entails query, else return False."""

for _ in self.ask_generator(query):

return True

return False

def retract(self, sentence):

"""Remove the sentence's clauses from the KB."""

for c in conjuncts(to_cnf(sentence)):

if c in self.clauses:

self.clauses.remove(c)

# ______________________________________________________________________________

def KB_AgentProgram(KB):

"""A generic logical knowledge-based agent program. [Figure 7.1]"""

steps = itertools.count()

def program(percept):

t = next(steps)

KB.tell(make_percept_sentence(percept, t))

action = KB.ask(make_action_query(t))

KB.tell(make_action_sentence(action, t))

return action

def make_percept_sentence(percept, t):

return Expr("Percept")(percept, t)

def make_action_query(t):

return expr("ShouldDo(action, {})".format(t))

def make_action_sentence(action, t):

return Expr("Did")(action[expr('action')], t)

return program

def is_symbol(s):

"""A string s is a symbol if it starts with an alphabetic char."""

return isinstance(s, str) and s[:1].isalpha()

def is_var_symbol(s):

"""A logic variable symbol is an initial-lowercase string."""

return is_symbol(s) and s[0].islower()

def is_prop_symbol(s):

"""A proposition logic symbol is an initial-uppercase string."""

return is_symbol(s) and s[0].isupper()

def variables(s):

"""Return a set of the variables in expression s.

>>> variables(expr('F(x, x) & G(x, y) & H(y, z) & R(A, z, 2)')) == {x, y, z}

True

"""

return {x for x in subexpressions(s) if is_variable(x)}

def is_definite_clause(s):

"""Returns True for exprs s of the form A & B & ... & C ==> D,

where all literals are positive. In clause form, this is

~A | ~B | ... | ~C | D, where exactly one clause is positive.

>>> is_definite_clause(expr('Farmer(Mac)'))

True

"""

if is_symbol(s.op):

return True

elif s.op == '==>':

antecedent, consequent = s.args

return (is_symbol(consequent.op) and

all(is_symbol(arg.op) for arg in conjuncts(antecedent)))

else:

return False

def parse_definite_clause(s):

"""Return the antecedents and the consequent of a definite clause."""

assert is_definite_clause(s)

if is_symbol(s.op):

return [], s

else:

antecedent, consequent = s.args

return conjuncts(antecedent), consequent

# Useful constant Exprs used in examples and code:

A, B, C, D, E, F, G, P, Q, x, y, z = map(Expr, 'ABCDEFGPQxyz')

# ______________________________________________________________________________

def tt_entails(kb, alpha):

"""Does kb entail the sentence alpha? Use truth tables. For propositional

kb's and sentences. [Figure 7.10]. Note that the 'kb' should be an

Expr which is a conjunction of clauses.

>>> tt_entails(expr('P & Q'), expr('Q'))

True

"""

assert not variables(alpha)

symbols = list(prop_symbols(kb & alpha))

return tt_check_all(kb, alpha, symbols, {})

def tt_check_all(kb, alpha, symbols, model):

"""Auxiliary routine to implement tt_entails."""

if not symbols:

if pl_true(kb, model):

result = pl_true(alpha, model)

assert result in (True, False)

return result

else:

return True

else:

P, rest = symbols[0], symbols[1:]

return (tt_check_all(kb, alpha, rest, extend(model, P, True)) and

tt_check_all(kb, alpha, rest, extend(model, P, False)))

def prop_symbols(x):

"""Return the set of all propositional symbols in x."""

if not isinstance(x, Expr):

return set()

elif is_prop_symbol(x.op):

return {x}

else:

return {symbol for arg in x.args for symbol in prop_symbols(arg)}

def constant_symbols(x):

"""Return the set of all constant symbols in x."""

if not isinstance(x, Expr):

return set()

elif is_prop_symbol(x.op) and not x.args:

return {x}

else:

return {symbol for arg in x.args for symbol in constant_symbols(arg)}

def predicate_symbols(x):

"""Return a set of (symbol_name, arity) in x.

All symbols (even functional) with arity > 0 are considered."""

if not isinstance(x, Expr) or not x.args:

return set()

pred_set = {(x.op, len(x.args))} if is_prop_symbol(x.op) else set()

pred_set.update({symbol for arg in x.args for symbol in predicate_symbols(arg)})

return pred_set

def tt_true(s):

"""Is a propositional sentence a tautology?

>>> tt_true('P | ~P')

True

"""

s = expr(s)

return tt_entails(True, s)

def pl_true(exp, model={}):

"""Return True if the propositional logic expression is true in the model,

and False if it is false. If the model does not specify the value for

every proposition, this may return None to indicate 'not obvious';

this may happen even when the expression is tautological."""

if exp in (True, False):

return exp

op, args = exp.op, exp.args

if is_prop_symbol(op):

return model.get(exp)

elif op == '~':

p = pl_true(args[0], model)

if p is None:

return None

else:

return not p

elif op == '|':

result = False

for arg in args:

p = pl_true(arg, model)

if p is True:

return True

if p is None:

result = None

return result

elif op == '&':

result = True

for arg in args:

p = pl_true(arg, model)

if p is False:

return False

if p is None:

result = None

return result

p, q = args

if op == '==>':

return pl_true(~p | q, model)

elif op == '<==':

return pl_true(p | ~q, model)

pt = pl_true(p, model)

if pt is None:

return None

qt = pl_true(q, model)

if qt is None:

return None

if op == '<=>':

return pt == qt

elif op == '^': # xor or 'not equivalent'

return pt != qt

else:

raise ValueError("illegal operator in logic expression" + str(exp))

# ______________________________________________________________________________

# Convert to Conjunctive Normal Form (CNF)

def to_cnf(s):

"""Convert a propositional logical sentence to conjunctive normal form.

That is, to the form ((A | ~B | ...) & (B | C | ...) & ...) [p. 253]

>>> to_cnf('~(B | C)')

(~B & ~C)

"""

s = expr(s)

if isinstance(s, str):

s = expr(s)

s = eliminate_implications(s) # Steps 1, 2 from p. 253

s = move_not_inwards(s) # Step 3

return distribute_and_over_or(s) # Step 4

def eliminate_implications(s):

"""Change implications into equivalent form with only &, |, and ~ as logical operators."""

s = expr(s)

if not s.args or is_symbol(s.op):

return s # Atoms are unchanged.

args = list(map(eliminate_implications, s.args))

a, b = args[0], args[-1]

if s.op == '==>':

return b | ~a

elif s.op == '<==':

return a | ~b

elif s.op == '<=>':

return (a | ~b) & (b | ~a)

elif s.op == '^':

assert len(args) == 2 # TODO: relax this restriction

return (a & ~b) | (~a & b)

else:

assert s.op in ('&', '|', '~')

return Expr(s.op, *args)

def move_not_inwards(s):

"""Rewrite sentence s by moving negation sign inward.

>>> move_not_inwards(~(A | B))

(~A & ~B)"""

s = expr(s)

if s.op == '~':

def NOT(b):

return move_not_inwards(~b)

a = s.args[0]

if a.op == '~':

return move_not_inwards(a.args[0]) # ~~A ==> A

if a.op == '&':

return associate('|', list(map(NOT, a.args)))

if a.op == '|':

return associate('&', list(map(NOT, a.args)))

return s

elif is_symbol(s.op) or not s.args:

return s

else:

return Expr(s.op, *list(map(move_not_inwards, s.args)))

def distribute_and_over_or(s):

"""Given a sentence s consisting of conjunctions and disjunctions

of literals, return an equivalent sentence in CNF.

>>> distribute_and_over_or((A & B) | C)

((A | C) & (B | C))

"""

s = expr(s)

if s.op == '|':

s = associate('|', s.args)

if s.op != '|':

return distribute_and_over_or(s)

if len(s.args) == 0:

return False

if len(s.args) == 1:

return distribute_and_over_or(s.args[0])

conj = first(arg for arg in s.args if arg.op == '&')

if not conj:

return s

others = [a for a in s.args if a is not conj]

rest = associate('|', others)

return associate('&', [distribute_and_over_or(c | rest)

for c in conj.args])

elif s.op == '&':

return associate('&', list(map(distribute_and_over_or, s.args)))

else:

return s

def associate(op, args):

"""Given an associative op, return an expression with the same

meaning as Expr(op, *args), but flattened -- that is, with nested

instances of the same op promoted to the top level.

>>> associate('&', [(A&B),(B|C),(B&C)])

(A & B & (B | C) & B & C)

>>> associate('|', [A|(B|(C|(A&B)))])

(A | B | C | (A & B))

"""

args = dissociate(op, args)

if len(args) == 0:

return _op_identity[op]

elif len(args) == 1:

return args[0]

else:

return Expr(op, *args)

_op_identity = {'&': True, '|': False, '+': 0, '*': 1}

def dissociate(op, args):

"""Given an associative op, return a flattened list result such

that Expr(op, *result) means the same as Expr(op, *args)."""

result = []

def collect(subargs):

for arg in subargs:

if arg.op == op:

collect(arg.args)

else:

result.append(arg)

collect(args)

return result

def conjuncts(s):

"""Return a list of the conjuncts in the sentence s.

>>> conjuncts(A & B)

[A, B]

>>> conjuncts(A | B)

[(A | B)]

"""

return dissociate('&', [s])

def disjuncts(s):

"""Return a list of the disjuncts in the sentence s.

>>> disjuncts(A | B)

[A, B]

>>> disjuncts(A & B)

[(A & B)]

"""

return dissociate('|', [s])

# ______________________________________________________________________________

def pl_resolution(KB, alpha):

"""Propositional-logic resolution: say if alpha follows from KB. [Figure 7.12]"""

clauses = KB.clauses + conjuncts(to_cnf(~alpha))

new = set()

while True:

n = len(clauses)

pairs = [(clauses[i], clauses[j])

for i in range(n) for j in range(i+1, n)]

for (ci, cj) in pairs:

resolvents = pl_resolve(ci, cj)

if False in resolvents:

return True

new = new.union(set(resolvents))

if new.issubset(set(clauses)):

return False

for c in new:

if c not in clauses:

clauses.append(c)

def pl_resolve(ci, cj):

"""Return all clauses that can be obtained by resolving clauses ci and cj."""

clauses = []

for di in disjuncts(ci):

for dj in disjuncts(cj):

if di == ~dj or ~di == dj:

dnew = unique(removeall(di, disjuncts(ci)) +

removeall(dj, disjuncts(cj)))

clauses.append(associate('|', dnew))

return clauses

# ______________________________________________________________________________

class PropDefiniteKB(PropKB):

"""A KB of propositional definite clauses."""

def tell(self, sentence):

"""Add a definite clause to this KB."""

assert is_definite_clause(sentence), "Must be definite clause"

self.clauses.append(sentence)

def ask_generator(self, query):

"""Yield the empty substitution if KB implies query; else nothing."""

if pl_fc_entails(self.clauses, query):

yield {}

def retract(self, sentence):

self.clauses.remove(sentence)

def clauses_with_premise(self, p):

"""Return a list of the clauses in KB that have p in their premise.

This could be cached away for O(1) speed, but we'll recompute it."""

return [c for c in self.clauses

if c.op == '==>' and p in conjuncts(c.args[0])]

def pl_fc_entails(KB, q):

"""Use forward chaining to see if a PropDefiniteKB entails symbol q.

[Figure 7.15]

>>> pl_fc_entails(horn_clauses_KB, expr('Q'))

True

"""

count = {c: len(conjuncts(c.args[0]))

for c in KB.clauses

if c.op == '==>'}

inferred = defaultdict(bool)

agenda = [s for s in KB.clauses if is_prop_symbol(s.op)]

while agenda:

p = agenda.pop()

if p == q:

return True

if not inferred[p]:

inferred[p] = True

for c in KB.clauses_with_premise(p):

count[c] -= 1

if count[c] == 0:

agenda.append(c.args[1])

return False

""" [Figure 7.13]

Simple inference in a wumpus world example

"""

wumpus_world_inference = expr("(B11 <=> (P12 | P21)) & ~B11")

""" [Figure 7.16]

Propositional Logic Forward Chaining example

"""

horn_clauses_KB = PropDefiniteKB()

for s in "P==>Q; (L&M)==>P; (B&L)==>M; (A&P)==>L; (A&B)==>L; A;B".split(';'):

horn_clauses_KB.tell(expr(s))

# ______________________________________________________________________________

# DPLL-Satisfiable [Figure 7.17]

def dpll_satisfiable(s):

"""Check satisfiability of a propositional sentence.

This differs from the book code in two ways: (1) it returns a model

rather than True when it succeeds; this is more useful. (2) The

function find_pure_symbol is passed a list of unknown clauses, rather

than a list of all clauses and the model; this is more efficient."""

clauses = conjuncts(to_cnf(s))

symbols = list(prop_symbols(s))

return dpll(clauses, symbols, {})

def dpll(clauses, symbols, model):

"""See if the clauses are true in a partial model."""

unknown_clauses = [] # clauses with an unknown truth value

for c in clauses:

val = pl_true(c, model)

if val is False:

return False

if val is not True:

unknown_clauses.append(c)

if not unknown_clauses:

return model

P, value = find_pure_symbol(symbols, unknown_clauses)

if P:

return dpll(clauses, removeall(P, symbols), extend(model, P, value))

P, value = find_unit_clause(clauses, model)

if P:

return dpll(clauses, removeall(P, symbols), extend(model, P, value))

if not symbols:

raise TypeError("Argument should be of the type Expr.")

P, symbols = symbols[0], symbols[1:]

return (dpll(clauses, symbols, extend(model, P, True)) or

dpll(clauses, symbols, extend(model, P, False)))

def find_pure_symbol(symbols, clauses):

"""Find a symbol and its value if it appears only as a positive literal

(or only as a negative) in clauses.

>>> find_pure_symbol([A, B, C], [A|~B,~B|~C,C|A])

(A, True)

"""

for s in symbols:

found_pos, found_neg = False, False

for c in clauses:

if not found_pos and s in disjuncts(c):

found_pos = True

if not found_neg and ~s in disjuncts(c):

found_neg = True

if found_pos != found_neg:

return s, found_pos

return None, None

def find_unit_clause(clauses, model):

"""Find a forced assignment if possible from a clause with only 1

variable not bound in the model.

>>> find_unit_clause([A|B|C, B|~C, ~A|~B], {A:True})

(B, False)

"""

for clause in clauses:

P, value = unit_clause_assign(clause, model)

if P:

return P, value

return None, None

def unit_clause_assign(clause, model):

"""Return a single variable/value pair that makes clause true in

the model, if possible.

>>> unit_clause_assign(A|B|C, {A:True})

(None, None)

>>> unit_clause_assign(B|~C, {A:True})

(None, None)

>>> unit_clause_assign(~A|~B, {A:True})

(B, False)

"""

P, value = None, None

for literal in disjuncts(clause):

sym, positive = inspect_literal(literal)

if sym in model:

if model[sym] == positive:

return None, None # clause already True

elif P:

return None, None # more than 1 unbound variable

else:

P, value = sym, positive

return P, value

def inspect_literal(literal):

"""The symbol in this literal, and the value it should take to

make the literal true.

>>> inspect_literal(P)

(P, True)

>>> inspect_literal(~P)

(P, False)

"""

if literal.op == '~':

return literal.args[0], False

else:

return literal, True

# ______________________________________________________________________________

# Walk-SAT [Figure 7.18]

def WalkSAT(clauses, p=0.5, max_flips=10000):

"""Checks for satisfiability of all clauses by randomly flipping values of variables

"""

# Set of all symbols in all clauses

symbols = {sym for clause in clauses for sym in prop_symbols(clause)}

# model is a random assignment of true/false to the symbols in clauses

model = {s: random.choice([True, False]) for s in symbols}

for i in range(max_flips):

satisfied, unsatisfied = [], []

for clause in clauses:

(satisfied if pl_true(clause, model) else unsatisfied).append(clause)

if not unsatisfied: # if model satisfies all the clauses

return model

clause = random.choice(unsatisfied)

if probability(p):

sym = random.choice(list(prop_symbols(clause)))

else:

# Flip the symbol in clause that maximizes number of sat. clauses

def sat_count(sym):

# Return the the number of clauses satisfied after flipping the symbol.

model[sym] = not model[sym]

count = len([clause for clause in clauses if pl_true(clause, model)])

model[sym] = not model[sym]

return count

sym = argmax(prop_symbols(clause), key=sat_count)

model[sym] = not model[sym]

# If no solution is found within the flip limit, we return failure

return None

# ______________________________________________________________________________

class HybridWumpusAgent(agents.Agent):

"""An agent for the wumpus world that does logical inference. [Figure 7.20]"""

def __init__(self):

raise NotImplementedError

def plan_route(current, goals, allowed):

raise NotImplementedError

# ______________________________________________________________________________

def SAT_plan(init, transition, goal, t_max, SAT_solver=dpll_satisfiable):

"""Converts a planning problem to Satisfaction problem by translating it to a cnf sentence.

[Figure 7.22]"""

# Functions used by SAT_plan

def translate_to_SAT(init, transition, goal, time):

clauses = []

states = [state for state in transition]

# Symbol claiming state s at time t

state_counter = itertools.count()

for s in states:

for t in range(time+1):

state_sym[s, t] = Expr("State_{}".format(next(state_counter)))

# Add initial state axiom

clauses.append(state_sym[init, 0])

# Add goal state axiom

clauses.append(state_sym[goal, time])

# All possible transitions

transition_counter = itertools.count()

for s in states:

for action in transition[s]:

s_ = transition[s][action]

for t in range(time):

# Action 'action' taken from state 's' at time 't' to reach 's_'

action_sym[s, action, t] = Expr(

"Transition_{}".format(next(transition_counter)))

# Change the state from s to s_

clauses.append(action_sym[s, action, t] |'==>'| state_sym[s, t])

clauses.append(action_sym[s, action, t] |'==>'| state_sym[s_, t + 1])

# Allow only one state at any time

for t in range(time+1):

# must be a state at any time

clauses.append(associate('|', [state_sym[s, t] for s in states]))

for s in states:

for s_ in states[states.index(s) + 1:]:

# for each pair of states s, s_ only one is possible at time t

clauses.append((~state_sym[s, t]) | (~state_sym[s_, t]))

# Restrict to one transition per timestep

for t in range(time):

# list of possible transitions at time t

transitions_t = [tr for tr in action_sym if tr[2] == t]

# make sure at least one of the transitions happens

clauses.append(associate('|', [action_sym[tr] for tr in transitions_t]))

for tr in transitions_t:

for tr_ in transitions_t[transitions_t.index(tr) + 1:]:

# there cannot be two transitions tr and tr_ at time t

clauses.append(~action_sym[tr] | ~action_sym[tr_])

# Combine the clauses to form the cnf

return associate('&', clauses)

def extract_solution(model):

true_transitions = [t for t in action_sym if model[action_sym[t]]]

# Sort transitions based on time, which is the 3rd element of the tuple

true_transitions.sort(key=lambda x: x[2])

return [action for s, action, time in true_transitions]

# Body of SAT_plan algorithm

for t in range(t_max):

# dictionaries to help extract the solution from model

state_sym = {}

action_sym = {}

cnf = translate_to_SAT(init, transition, goal, t)

model = SAT_solver(cnf)

if model is not False:

return extract_solution(model)

return None

# ______________________________________________________________________________

def unify(x, y, s={}):

"""Unify expressions x,y with substitution s; return a substitution that

would make x,y equal, or None if x,y can not unify. x and y can be

variables (e.g. Expr('x')), constants, lists, or Exprs. [Figure 9.1]"""

if s is None:

return None

elif x == y:

return s

elif is_variable(x):

return unify_var(x, y, s)

elif is_variable(y):

return unify_var(y, x, s)

elif isinstance(x, Expr) and isinstance(y, Expr):

return unify(x.args, y.args, unify(x.op, y.op, s))

elif isinstance(x, str) or isinstance(y, str):

return None

elif issequence(x) and issequence(y) and len(x) == len(y):

if not x:

return s

return unify(x[1:], y[1:], unify(x[0], y[0], s))

else:

return None

def is_variable(x):

"""A variable is an Expr with no args and a lowercase symbol as the op."""

return isinstance(x, Expr) and not x.args and x.op[0].islower()

def unify_var(var, x, s):

if var in s:

return unify(s[var], x, s)

elif x in s:

return unify(var, s[x], s)

elif occur_check(var, x, s):

return None

else:

return extend(s, var, x)

def occur_check(var, x, s):

"""Return true if variable var occurs anywhere in x

(or in subst(s, x), if s has a binding for x)."""

if var == x:

return True

elif is_variable(x) and x in s:

return occur_check(var, s[x], s)

elif isinstance(x, Expr):

return (occur_check(var, x.op, s) or

occur_check(var, x.args, s))

elif isinstance(x, (list, tuple)):

return first(e for e in x if occur_check(var, e, s))

else:

return False

def extend(s, var, val):

"""Copy the substitution s and extend it by setting var to val; return copy."""

s2 = s.copy()

s2[var] = val

return s2

def subst(s, x):

"""Substitute the substitution s into the expression x.

>>> subst({x: 42, y:0}, F(x) + y)

(F(42) + 0)

"""

if isinstance(x, list):

return [subst(s, xi) for xi in x]

elif isinstance(x, tuple):

return tuple([subst(s, xi) for xi in x])

elif not isinstance(x, Expr):

return x

elif is_var_symbol(x.op):

return s.get(x, x)

else:

return Expr(x.op, *[subst(s, arg) for arg in x.args])

def standardize_variables(sentence, dic=None):

"""Replace all the variables in sentence with new variables."""

if dic is None:

dic = {}

if not isinstance(sentence, Expr):

return sentence

elif is_var_symbol(sentence.op):

if sentence in dic:

return dic[sentence]

else:

v = Expr('v_{}'.format(next(standardize_variables.counter)))

dic[sentence] = v

return v

else:

return Expr(sentence.op,

*[standardize_variables(a, dic) for a in sentence.args])

standardize_variables.counter = itertools.count()

# ______________________________________________________________________________

class FolKB(KB):

"""A knowledge base consisting of first-order definite clauses.

>>> kb0 = FolKB([expr('Farmer(Mac)'), expr('Rabbit(Pete)'),

... expr('(Rabbit(r) & Farmer(f)) ==> Hates(f, r)')])

>>> kb0.tell(expr('Rabbit(Flopsie)'))

>>> kb0.retract(expr('Rabbit(Pete)'))

>>> kb0.ask(expr('Hates(Mac, x)'))[x]

Flopsie

>>> kb0.ask(expr('Wife(Pete, x)'))

False

"""

def __init__(self, initial_clauses=[]):

self.clauses = [] # inefficient: no indexing

for clause in initial_clauses:

self.tell(clause)

def tell(self, sentence):

if is_definite_clause(sentence):

self.clauses.append(sentence)

else:

raise Exception("Not a definite clause: {}".format(sentence))

def ask_generator(self, query):

return fol_bc_ask(self, query)

def retract(self, sentence):

self.clauses.remove(sentence)

def fetch_rules_for_goal(self, goal):

return self.clauses

def fol_fc_ask(KB, alpha):

"""A simple forward-chaining algorithm. [Figure 9.3]"""

# TODO: Improve efficiency

kb_consts = list({c for clause in KB.clauses for c in constant_symbols(clause)})

def enum_subst(p):

query_vars = list({v for clause in p for v in variables(clause)})

for assignment_list in itertools.product(kb_consts, repeat=len(query_vars)):

theta = {x: y for x, y in zip(query_vars, assignment_list)}

yield theta

# check if we can answer without new inferences

for q in KB.clauses:

phi = unify(q, alpha, {})

if phi is not None:

yield phi

while True:

new = []

for rule in KB.clauses:

p, q = parse_definite_clause(rule)

for theta in enum_subst(p):

if set(subst(theta, p)).issubset(set(KB.clauses)):

q_ = subst(theta, q)

if all([unify(x, q_, {}) is None for x in KB.clauses + new]):

new.append(q_)

phi = unify(q_, alpha, {})

if phi is not None:

yield phi

if not new:

break

for clause in new:

KB.tell(clause)

return None

def fol_bc_ask(KB, query):

"""A simple backward-chaining algorithm for first-order logic. [Figure 9.6]

KB should be an instance of FolKB, and query an atomic sentence."""

return fol_bc_or(KB, query, {})

def fol_bc_or(KB, goal, theta):

for rule in KB.fetch_rules_for_goal(goal):

lhs, rhs = parse_definite_clause(standardize_variables(rule))

for theta1 in fol_bc_and(KB, lhs, unify(rhs, goal, theta)):

yield theta1

def fol_bc_and(KB, goals, theta):

if theta is None:

pass

elif not goals:

yield theta

else:

first, rest = goals[0], goals[1:]

for theta1 in fol_bc_or(KB, subst(theta, first), theta):

for theta2 in fol_bc_and(KB, rest, theta1):

yield theta2

# A simple KB that defines the relevant conditions of the Wumpus World as in Fig 7.4.

# See Sec. 7.4.3

wumpus_kb = PropKB()

P11, P12, P21, P22, P31, B11, B21 = expr('P11, P12, P21, P22, P31, B11, B21')

wumpus_kb.tell(~P11)

wumpus_kb.tell(B11 | '<=>' | ((P12 | P21)))

wumpus_kb.tell(B21 | '<=>' | ((P11 | P22 | P31)))

wumpus_kb.tell(~B11)

wumpus_kb.tell(B21)

test_kb = FolKB(

map(expr, ['Farmer(Mac)',

'Rabbit(Pete)',

'Mother(MrsMac, Mac)',

'Mother(MrsRabbit, Pete)',

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