"""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 csp import parse_neighbors, UniversalDict from utils import ( removeall, unique, first, argmax, probability, isnumber, issequence, Expr, expr, subexpressions ) #from agents import Agent, Glitter, Bump, Stench, Breeze, Scream from search import astar_search, PlanRoute 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. >>> is_symbol('R2D2') True """ return isinstance(s, str) and s[:1].isalpha() def is_var_symbol(s): """A logic variable symbol is an initial-lowercase string. >>> is_var_symbol('EXE') False """ return is_symbol(s) and s[0].islower() def is_prop_symbol(s): """A proposition logic symbol is an initial-uppercase string. >>> is_prop_symbol('exe') False """ 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, a, x, y, z, u = map(Expr, 'ABCDEFGPQaxyzu') # ______________________________________________________________________________ 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. >>> pl_true(P, {}) is None True """ 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). >>> dissociate('&', [A & B]) [A, B] """ 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] >>> pl_resolution(horn_clauses_KB, A) True """ 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)) """ Definite clauses KB example """ definite_clauses_KB = PropDefiniteKB() for clause in ['(B & F)==>E', '(A & E & F)==>G', '(B & C)==>F', '(A & B)==>D', '(E & F)==>H', '(H & I)==>J', 'A', 'B', 'C']: definite_clauses_KB.tell(expr(clause)) # ______________________________________________________________________________ # 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. >>> dpll_satisfiable(A |'<=>'| B) == {A: True, B: True} True """ 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 >>> WalkSAT([A & ~A], 0.5, 100) is None True """ # 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 # ______________________________________________________________________________ # Map Coloring Problems def MapColoringSAT(colors, neighbors): """Make a SAT for the problem of coloring a map with different colors for any two adjacent regions. Arguments are a list of colors, and a dict of {region: [neighbor,...]} entries. This dict may also be specified as a string of the form defined by parse_neighbors.""" if isinstance(neighbors, str): neighbors = parse_neighbors(neighbors) colors = UniversalDict(colors) clauses = [] for state in neighbors.keys(): clause = [expr(state + '_' + c) for c in colors[state]] clauses.append(clause) for t in itertools.combinations(clause, 2): clauses.append([~t[0], ~t[1]]) visited = set() adj = set(neighbors[state]) - visited visited.add(state) for n_state in adj: for col in colors[n_state]: clauses.append([expr('~' + state + '_' + col), expr('~' + n_state + '_' + col)]) return associate('&', map(lambda c: associate('|', c), clauses)) australia_sat = MapColoringSAT(list('RGB'), """SA: WA NT Q NSW V; NT: WA Q; NSW: Q V; T: """) france_sat = MapColoringSAT(list('RGBY'), """AL: LO FC; AQ: MP LI PC; AU: LI CE BO RA LR MP; BO: CE IF CA FC RA AU; BR: NB PL; CA: IF PI LO FC BO; CE: PL NB NH IF BO AU LI PC; FC: BO CA LO AL RA; IF: NH PI CA BO CE; LI: PC CE AU MP AQ; LO: CA AL FC; LR: MP AU RA PA; MP: AQ LI AU LR; NB: NH CE PL BR; NH: PI IF CE NB; NO: PI; PA: LR RA; PC: PL CE LI AQ; PI: NH NO CA IF; PL: BR NB CE PC; RA: AU BO FC PA LR""") usa_sat = MapColoringSAT(list('RGBY'), """WA: OR ID; OR: ID NV CA; CA: NV AZ; NV: ID UT AZ; ID: MT WY UT; UT: WY CO AZ; MT: ND SD WY; WY: SD NE CO; CO: NE KA OK NM; NM: OK TX AZ; ND: MN SD; SD: MN IA NE; NE: IA MO KA; KA: MO OK; OK: MO AR TX; TX: AR LA; MN: WI IA; IA: WI IL MO; MO: IL KY TN AR; AR: MS TN LA; LA: MS; WI: MI IL; IL: IN KY; IN: OH KY; MS: TN AL; AL: TN GA FL; MI: OH IN; OH: PA WV KY; KY: WV VA TN; TN: VA NC GA; GA: NC SC FL; PA: NY NJ DE MD WV; WV: MD VA; VA: MD DC NC; NC: SC; NY: VT MA CT NJ; NJ: DE; DE: MD; MD: DC; VT: NH MA; MA: NH RI CT; CT: RI; ME: NH; HI: ; AK: """) # ______________________________________________________________________________ # Expr functions for WumpusKB and HybridWumpusAgent def facing_east(time): return Expr('FacingEast', time) def facing_west(time): return Expr('FacingWest', time) def facing_north(time): return Expr('FacingNorth', time) def facing_south(time): return Expr('FacingSouth', time) def wumpus(x, y): return Expr('W', x, y) def pit(x, y): return Expr('P', x, y) def breeze(x, y): return Expr('B', x, y) def stench(x, y): return Expr('S', x, y) def wumpus_alive(time): return Expr('WumpusAlive', time) def have_arrow(time): return Expr('HaveArrow', time) def percept_stench(time): return Expr('Stench', time) def percept_breeze(time): return Expr('Breeze', time) def percept_glitter(time): return Expr('Glitter', time) def percept_bump(time): return Expr('Bump', time) def percept_scream(time): return Expr('Scream', time) def move_forward(time): return Expr('Forward', time) def shoot(time): return Expr('Shoot', time) def turn_left(time): return Expr('TurnLeft', time) def turn_right(time): return Expr('TurnRight', time) def ok_to_move(x, y, time): return Expr('OK', x, y, time) def location(x, y, time=None): if time is None: return Expr('L', x, y) else: return Expr('L', x, y, time) # Symbols def implies(lhs, rhs): return Expr('==>', lhs, rhs) def equiv(lhs, rhs): return Expr('<=>', lhs, rhs) # Helper Function def new_disjunction(sentences): t = sentences[0] for i in range(1, len(sentences)): t |= sentences[i] return t # ______________________________________________________________________________ class WumpusKB(PropKB): """ Create a Knowledge Base that contains the a temporal "Wumpus physics" and temporal rules with time zero. """ def __init__(self, dimrow): super().__init__() self.dimrow = dimrow self.tell(~wumpus(1, 1)) self.tell(~pit(1, 1)) for y in range(1, dimrow + 1): for x in range(1, dimrow + 1): pits_in = list() wumpus_in = list() if x > 1: # West room exists pits_in.append(pit(x - 1, y)) wumpus_in.append(wumpus(x - 1, y)) if y < dimrow: # North room exists pits_in.append(pit(x, y + 1)) wumpus_in.append(wumpus(x, y + 1)) if x < dimrow: # East room exists pits_in.append(pit(x + 1, y)) wumpus_in.append(wumpus(x + 1, y)) if y > 1: # South room exists pits_in.append(pit(x, y - 1)) wumpus_in.append(wumpus(x, y - 1)) self.tell(equiv(breeze(x, y), new_disjunction(pits_in))) self.tell(equiv(stench(x, y), new_disjunction(wumpus_in))) # Rule that describes existence of at least one Wumpus wumpus_at_least = list() for x in range(1, dimrow + 1): for y in range(1, dimrow + 1): wumpus_at_least.append(wumpus(x, y)) self.tell(new_disjunction(wumpus_at_least)) # Rule that describes existence of at most one Wumpus for i in range(1, dimrow + 1): for j in range(1, dimrow + 1): for u in range(1, dimrow + 1): for v in range(1, dimrow + 1): if i != u or j != v: self.tell(~wumpus(i, j) | ~wumpus(u, v)) # Temporal rules at time zero self.tell(location(1, 1, 0)) for i in range(1, dimrow + 1): for j in range(1, dimrow + 1): self.tell(implies(location(i, j, 0), equiv(percept_breeze(0), breeze(i, j)))) self.tell(implies(location(i, j, 0), equiv(percept_stench(0), stench(i, j)))) if i != 1 or j != 1: self.tell(~location(i, j, 0)) self.tell(wumpus_alive(0)) self.tell(have_arrow(0)) self.tell(facing_east(0)) self.tell(~facing_north(0)) self.tell(~facing_south(0)) self.tell(~facing_west(0)) def make_action_sentence(self, action, time): actions = [move_forward(time), shoot(time), turn_left(time), turn_right(time)] for a in actions: if action is a: self.tell(action) else: self.tell(~a) def make_percept_sentence(self, percept, time): # Glitter, Bump, Stench, Breeze, Scream flags = [0, 0, 0, 0, 0] # Things perceived if isinstance(percept, Glitter): flags[0] = 1 self.tell(percept_glitter(time)) elif isinstance(percept, Bump): flags[1] = 1 self.tell(percept_bump(time)) elif isinstance(percept, Stench): flags[2] = 1 self.tell(percept_stench(time)) elif isinstance(percept, Breeze): flags[3] = 1 self.tell(percept_breeze(time)) elif isinstance(percept, Scream): flags[4] = 1 self.tell(percept_scream(time)) # Things not perceived for i in range(len(flags)): if flags[i] == 0: if i == 0: self.tell(~percept_glitter(time)) elif i == 1: self.tell(~percept_bump(time)) elif i == 2: self.tell(~percept_stench(time)) elif i == 3: self.tell(~percept_breeze(time)) elif i == 4: self.tell(~percept_scream(time)) def add_temporal_sentences(self, time): if time == 0: return t = time - 1 # current location rules for i in range(1, self.dimrow + 1): for j in range(1, self.dimrow + 1): self.tell(implies(location(i, j, time), equiv(percept_breeze(time), breeze(i, j)))) self.tell(implies(location(i, j, time), equiv(percept_stench(time), stench(i, j)))) s = list() s.append( equiv( location(i, j, time), location(i, j, time) & ~move_forward(time) | percept_bump(time))) if i != 1: s.append(location(i - 1, j, t) & facing_east(t) & move_forward(t)) if i != self.dimrow: s.append(location(i + 1, j, t) & facing_west(t) & move_forward(t)) if j != 1: s.append(location(i, j - 1, t) & facing_north(t) & move_forward(t)) if j != self.dimrow: s.append(location(i, j + 1, t) & facing_south(t) & move_forward(t)) # add sentence about location i,j self.tell(new_disjunction(s)) # add sentence about safety of location i,j self.tell( equiv(ok_to_move(i, j, time), ~pit(i, j) & ~wumpus(i, j) & wumpus_alive(time)) ) # Rules about current orientation a = facing_north(t) & turn_right(t) b = facing_south(t) & turn_left(t) c = facing_east(t) & ~turn_left(t) & ~turn_right(t) s = equiv(facing_east(time), a | b | c) self.tell(s) a = facing_north(t) & turn_left(t) b = facing_south(t) & turn_right(t) c = facing_west(t) & ~turn_left(t) & ~turn_right(t) s = equiv(facing_west(time), a | b | c) self.tell(s) a = facing_east(t) & turn_left(t) b = facing_west(t) & turn_right(t) c = facing_north(t) & ~turn_left(t) & ~turn_right(t) s = equiv(facing_north(time), a | b | c) self.tell(s) a = facing_west(t) & turn_left(t) b = facing_east(t) & turn_right(t) c = facing_south(t) & ~turn_left(t) & ~turn_right(t) s = equiv(facing_south(time), a | b | c) self.tell(s) # Rules about last action self.tell(equiv(move_forward(t), ~turn_right(t) & ~turn_left(t))) # Rule about the arrow self.tell(equiv(have_arrow(time), have_arrow(t) & ~shoot(t))) # Rule about Wumpus (dead or alive) self.tell(equiv(wumpus_alive(time), wumpus_alive(t) & ~percept_scream(time))) def ask_if_true(self, query): return pl_resolution(self, query) # ______________________________________________________________________________ class WumpusPosition: def __init__(self, x, y, orientation): self.X = x self.Y = y self.orientation = orientation def get_location(self): return self.X, self.Y def set_location(self, x, y): self.X = x self.Y = y def get_orientation(self): return self.orientation def set_orientation(self, orientation): self.orientation = orientation def __eq__(self, other): if other.get_location() == self.get_location() and \ other.get_orientation() == self.get_orientation(): return True else: return False # ______________________________________________________________________________ """ class HybridWumpusAgent(Agent): # An agent for the wumpus world that does logical inference. [Figure 7.20] def __init__(self, dimentions): self.dimrow = dimentions self.kb = WumpusKB(self.dimrow) self.t = 0 self.plan = list() self.current_position = WumpusPosition(1, 1, 'UP') super().__init__(self.execute) def execute(self, percept): self.kb.make_percept_sentence(percept, self.t) self.kb.add_temporal_sentences(self.t) temp = list() for i in range(1, self.dimrow + 1): for j in range(1, self.dimrow + 1): if self.kb.ask_if_true(location(i, j, self.t)): temp.append(i) temp.append(j) if self.kb.ask_if_true(facing_north(self.t)): self.current_position = WumpusPosition(temp[0], temp[1], 'UP') elif self.kb.ask_if_true(facing_south(self.t)): self.current_position = WumpusPosition(temp[0], temp[1], 'DOWN') elif self.kb.ask_if_true(facing_west(self.t)): self.current_position = WumpusPosition(temp[0], temp[1], 'LEFT') elif self.kb.ask_if_true(facing_east(self.t)): self.current_position = WumpusPosition(temp[0], temp[1], 'RIGHT') safe_points = list() for i in range(1, self.dimrow + 1): for j in range(1, self.dimrow + 1): if self.kb.ask_if_true(ok_to_move(i, j, self.t)): safe_points.append([i, j]) if self.kb.ask_if_true(percept_glitter(self.t)): goals = list() goals.append([1, 1]) self.plan.append('Grab') actions = self.plan_route(self.current_position, goals, safe_points) self.plan.extend(actions) self.plan.append('Climb') if len(self.plan) == 0: unvisited = list() for i in range(1, self.dimrow + 1): for j in range(1, self.dimrow + 1): for k in range(self.t): if self.kb.ask_if_true(location(i, j, k)): unvisited.append([i, j]) unvisited_and_safe = list() for u in unvisited: for s in safe_points: if u not in unvisited_and_safe and s == u: unvisited_and_safe.append(u) temp = self.plan_route(self.current_position, unvisited_and_safe, safe_points) self.plan.extend(temp) if len(self.plan) == 0 and self.kb.ask_if_true(have_arrow(self.t)): possible_wumpus = list() for i in range(1, self.dimrow + 1): for j in range(1, self.dimrow + 1): if not self.kb.ask_if_true(wumpus(i, j)): possible_wumpus.append([i, j]) temp = self.plan_shot(self.current_position, possible_wumpus, safe_points) self.plan.extend(temp) if len(self.plan) == 0: not_unsafe = list() for i in range(1, self.dimrow + 1): for j in range(1, self.dimrow + 1): if not self.kb.ask_if_true(ok_to_move(i, j, self.t)): not_unsafe.append([i, j]) temp = self.plan_route(self.current_position, not_unsafe, safe_points) self.plan.extend(temp) if len(self.plan) == 0: start = list() start.append([1, 1]) temp = self.plan_route(self.current_position, start, safe_points) self.plan.extend(temp) self.plan.append('Climb') action = self.plan[0] self.plan = self.plan[1:] self.kb.make_action_sentence(action, self.t) self.t += 1 return action def plan_route(self, current, goals, allowed): problem = PlanRoute(current, goals, allowed, self.dimrow) return astar_search(problem).solution() def plan_shot(self, current, goals, allowed): shooting_positions = set() for loc in goals: x = loc[0] y = loc[1] for i in range(1, self.dimrow + 1): if i < x: shooting_positions.add(WumpusPosition(i, y, 'EAST')) if i > x: shooting_positions.add(WumpusPosition(i, y, 'WEST')) if i < y: shooting_positions.add(WumpusPosition(x, i, 'NORTH')) if i > y: shooting_positions.add(WumpusPosition(x, i, 'SOUTH')) # Can't have a shooting position from any of the rooms the Wumpus could reside orientations = ['EAST', 'WEST', 'NORTH', 'SOUTH'] for loc in goals: for orientation in orientations: shooting_positions.remove(WumpusPosition(loc[0], loc[1], orientation)) actions = list() actions.extend(self.plan_route(current, shooting_positions, allowed)) actions.append('Shoot') return actions """ # ______________________________________________________________________________ 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] >>> transition = {'A': {'Left': 'A', 'Right': 'B'}, 'B': {'Left': 'A', 'Right': 'C'}, 'C': {'Left': 'B', 'Right': 'C'}} >>> SAT_plan('A', transition, 'C', 2) is None True """ # 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] >>> unify(x, 3, {}) {x: 3} """ 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: new_s = extend(s, var, x) cascade_substitution(new_s) return new_s 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. >>> extend({x: 1}, y, 2) == {x: 1, y: 2} True """ 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 cascade_substitution(s): """This method allows to return a correct unifier in normal form and perform a cascade substitution to s. For every mapping in s perform a cascade substitution on s.get(x) and if it is replaced with a function ensure that all the function terms are correct updates by passing over them again. This issue fix: https://github.com/aimacode/aima-python/issues/1053 unify(expr('P(A, x, F(G(y)))'), expr('P(z, F(z), F(u))')) must return {z: A, x: F(A), u: G(y)} and not {z: A, x: F(z), u: G(y)} >>> s = {x: y, y: G(z)} >>> cascade_substitution(s) >>> print(s) {x: G(z), y: G(z)} Parameters ---------- s : Dictionary This contain a substution """ for x in s: s[x] = subst(s, s.get(x)) if isinstance(s.get(x), Expr) and not is_variable(s.get(x)): # Ensure Function Terms are correct updates by passing over them again. s[x] = subst(s, s.get(x)) 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=None): self.clauses = [] # inefficient: no indexing if initial_clauses: 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)', '(Rabbit(r) & Farmer(f)) ==> Hates(f, r)', '(Mother(m, c)) ==> Loves(m, c)', '(Mother(m, r) & Rabbit(r)) ==> Rabbit(m)', '(Farmer(f)) ==> Human(f)', # Note that this order of conjuncts # would result in infinite recursion: # '(Human(h) & Mother(m, h)) ==> Human(m)' '(Mother(m, h) & Human(h)) ==> Human(m)' ])) crime_kb = FolKB( map(expr, ['(American(x) & Weapon(y) & Sells(x, y, z) & Hostile(z)) ==> Criminal(x)', 'Owns(Nono, M1)', 'Missile(M1)', '(Missile(x) & Owns(Nono, x)) ==> Sells(West, x, Nono)', 'Missile(x) ==> Weapon(x)', 'Enemy(x, America) ==> Hostile(x)', 'American(West)', 'Enemy(Nono, America)' ])) # ______________________________________________________________________________ # Example application (not in the book). # You can use the Expr class to do symbolic differentiation. This used to be # a part of AI; now it is considered a separate field, Symbolic Algebra. def diff(y, x): """Return the symbolic derivative, dy/dx, as an Expr. However, you probably want to simplify the results with simp. >>> diff(x * x, x) ((x * 1) + (x * 1)) """ if y == x: return 1 elif not y.args: return 0 else: u, op, v = y.args[0], y.op, y.args[-1] if op == '+': return diff(u, x) + diff(v, x) elif op == '-' and len(y.args) == 1: return -diff(u, x) elif op == '-': return diff(u, x) - diff(v, x) elif op == '*': return u * diff(v, x) + v * diff(u, x) elif op == '/': return (v * diff(u, x) - u * diff(v, x)) / (v * v) elif op == '**' and isnumber(x.op): return v * u ** (v - 1) * diff(u, x) elif op == '**': return (v * u ** (v - 1) * diff(u, x) + u ** v * Expr('log')(u) * diff(v, x)) elif op == 'log': return diff(u, x) / u else: raise ValueError("Unknown op: {} in diff({}, {})".format(op, y, x)) def simp(x): """Simplify the expression x.""" if isnumber(x) or not x.args: return x args = list(map(simp, x.args)) u, op, v = args[0], x.op, args[-1] if op == '+': if v == 0: return u if u == 0: return v if u == v: return 2 * u if u == -v or v == -u: return 0 elif op == '-' and len(args) == 1: if u.op == '-' and len(u.args) == 1: return u.args[0] # --y ==> y elif op == '-': if v == 0: return u if u == 0: return -v if u == v: return 0 if u == -v or v == -u: return 0 elif op == '*': if u == 0 or v == 0: return 0 if u == 1: return v if v == 1: return u if u == v: return u ** 2 elif op == '/': if u == 0: return 0 if v == 0: return Expr('Undefined') if u == v: return 1 if u == -v or v == -u: return 0 elif op == '**': if u == 0: return 0 if v == 0: return 1 if u == 1: return 1 if v == 1: return u elif op == 'log': if u == 1: return 0 else: raise ValueError("Unknown op: " + op) # If we fall through to here, we can not simplify further return Expr(op, *args) def d(y, x): """Differentiate and then simplify. >>> d(x * x - x, x) ((2 * x) - 1) """ return simp(diff(y, x))