logic.py

"""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
"""
import heapq
import itertools
import random
from collections import defaultdict, Counter

import networkx as nx

from agents import Agent, Glitter, Bump, Stench, Breeze, Scream
from csp import parse_neighbors, UniversalDict
from search import astar_search, PlanRoute
from utils import (remove_all, unique, first, argmax, probability, isnumber,
                   issequence, Expr, expr, subexpressions, extend)


# ______________________________________________________________________________


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:
                clauses.append(associate('|', unique(remove_all(di, disjuncts(ci)) + remove_all(dj, disjuncts(cj)))))
    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 no_branching_heuristic(symbols, clauses):
    return first(symbols), True


def min_clauses(clauses):
    min_len = min(map(lambda c: len(c.args), clauses), default=2)
    return filter(lambda c: len(c.args) == (min_len if min_len > 1 else 2), clauses)


def moms(symbols, clauses):
    """
    MOMS (Maximum Occurrence in clauses of Minimum Size) heuristic
    Returns the literal with the most occurrences in all clauses of minimum size
    """
    scores = Counter(l for c in min_clauses(clauses) for l in prop_symbols(c))
    return max(symbols, key=lambda symbol: scores[symbol]), True


def momsf(symbols, clauses, k=0):
    """
    MOMS alternative heuristic
    If f(x) the number of occurrences of the variable x in clauses with minimum size,
    we choose the variable maximizing [f(x) + f(-x)] * 2^k + f(x) * f(-x)
    Returns x if f(x) >= f(-x) otherwise -x
    """
    scores = Counter(l for c in min_clauses(clauses) for l in disjuncts(c))
    P = max(symbols,
            key=lambda symbol: (scores[symbol] + scores[~symbol]) * pow(2, k) + scores[symbol] * scores[~symbol])
    return P, True if scores[P] >= scores[~P] else False


def posit(symbols, clauses):
    """
    Freeman's POSIT version of MOMs
    Counts the positive x and negative x for each variable x in clauses with minimum size
    Returns x if f(x) >= f(-x) otherwise -x
    """
    scores = Counter(l for c in min_clauses(clauses) for l in disjuncts(c))
    P = max(symbols, key=lambda symbol: scores[symbol] + scores[~symbol])
    return P, True if scores[P] >= scores[~P] else False


def zm(symbols, clauses):
    """
    Zabih and McAllester's version of MOMs
    Counts the negative occurrences only of each variable x in clauses with minimum size
    """
    scores = Counter(l for c in min_clauses(clauses) for l in disjuncts(c) if l.op == '~')
    return max(symbols, key=lambda symbol: scores[~symbol]), True


def dlis(symbols, clauses):
    """
    DLIS (Dynamic Largest Individual Sum) heuristic
    Choose the variable and value that satisfies the maximum number of unsatisfied clauses
    Like DLCS but we only consider the literal (thus Cp and Cn are individual)
    """
    scores = Counter(l for c in clauses for l in disjuncts(c))
    P = max(symbols, key=lambda symbol: scores[symbol])
    return P, True if scores[P] >= scores[~P] else False


def dlcs(symbols, clauses):
    """
    DLCS (Dynamic Largest Combined Sum) heuristic
    Cp the number of clauses containing literal x
    Cn the number of clauses containing literal -x
    Here we select the variable maximizing Cp + Cn
    Returns x if Cp >= Cn otherwise -x
    """
    scores = Counter(l for c in clauses for l in disjuncts(c))
    P = max(symbols, key=lambda symbol: scores[symbol] + scores[~symbol])
    return P, True if scores[P] >= scores[~P] else False


def jw(symbols, clauses):
    """
    Jeroslow-Wang heuristic
    For each literal compute J(l) = \sum{l in clause c} 2^{-|c|}
    Return the literal maximizing J
    """
    scores = Counter()
    for c in clauses:
        for l in prop_symbols(c):
            scores[l] += pow(2, -len(c.args))
    return max(symbols, key=lambda symbol: scores[symbol]), True


def jw2(symbols, clauses):
    """
    Two Sided Jeroslow-Wang heuristic
    Compute J(l) also counts the negation of l = J(x) + J(-x)
    Returns x if J(x) >= J(-x) otherwise -x
    """
    scores = Counter()
    for c in clauses:
        for l in disjuncts(c):
            scores[l] += pow(2, -len(c.args))
    P = max(symbols, key=lambda symbol: scores[symbol] + scores[~symbol])
    return P, True if scores[P] >= scores[~P] else False


def dpll_satisfiable(s, branching_heuristic=no_branching_heuristic):
    """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
    """
    return dpll(conjuncts(to_cnf(s)), prop_symbols(s), {}, branching_heuristic)


def dpll(clauses, symbols, model, branching_heuristic=no_branching_heuristic):
    """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 None:
            unknown_clauses.append(c)
    if not unknown_clauses:
        return model
    P, value = find_pure_symbol(symbols, unknown_clauses)
    if P:
        return dpll(clauses, remove_all(P, symbols), extend(model, P, value), branching_heuristic)
    P, value = find_unit_clause(clauses, model)
    if P:
        return dpll(clauses, remove_all(P, symbols), extend(model, P, value), branching_heuristic)
    P, value = branching_heuristic(symbols, unknown_clauses)
    return (dpll(clauses, remove_all(P, symbols), extend(model, P, value), branching_heuristic) or
            dpll(clauses, remove_all(P, symbols), extend(model, P, not value), branching_heuristic))


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


# ______________________________________________________________________________
# CDCL - Conflict-Driven Clause Learning with 1UIP Learning Scheme,
# 2WL Lazy Data Structure, VSIDS Branching Heuristic & Restarts


def no_restart(conflicts, restarts, queue_lbd, sum_lbd):
    return False


def luby(conflicts, restarts, queue_lbd, sum_lbd, unit=512):
    # in the state-of-art tested with unit value 1, 2, 4, 6, 8, 12, 16, 32, 64, 128, 256 and 512
    def _luby(i):
        k = 1
        while True:
            if i == (1 << k) - 1:
                return 1 << (k - 1)
            elif (1 << (k - 1)) <= i < (1 << k) - 1:
                return _luby(i - (1 << (k - 1)) + 1)
            k += 1

    return unit * _luby(restarts) == len(queue_lbd)


def glucose(conflicts, restarts, queue_lbd, sum_lbd, x=100, k=0.7):
    # in the state-of-art tested with (x, k) as (50, 0.8) and (100, 0.7)
    # if there were at least x conflicts since the last restart, and then the average LBD of the last
    # x learnt clauses was at least k times higher than the average LBD of all learnt clauses
    return len(queue_lbd) >= x and sum(queue_lbd) / len(queue_lbd) * k > sum_lbd / conflicts


def cdcl_satisfiable(s, vsids_decay=0.95, restart_strategy=no_restart):
    """
    >>> cdcl_satisfiable(A |'<=>'| B) == {A: True, B: True}
    True
    """
    clauses = TwoWLClauseDatabase(conjuncts(to_cnf(s)))
    symbols = prop_symbols(s)
    scores = Counter()
    G = nx.DiGraph()
    model = {}
    dl = 0
    conflicts = 0
    restarts = 1
    sum_lbd = 0
    queue_lbd = []
    while True:
        conflict = unit_propagation(clauses, symbols, model, G, dl)
        if conflict:
            if dl == 0:
                return False
            conflicts += 1
            dl, learn, lbd = conflict_analysis(G, dl)
            queue_lbd.append(lbd)
            sum_lbd += lbd
            backjump(symbols, model, G, dl)
            clauses.add(learn, model)
            scores.update(l for l in disjuncts(learn))
            for symbol in scores:
                scores[symbol] *= vsids_decay
            if restart_strategy(conflicts, restarts, queue_lbd, sum_lbd):
                backjump(symbols, model, G)
                queue_lbd.clear()
                restarts += 1
        else:
            if not symbols:
                return model
            dl += 1
            assign_decision_literal(symbols, model, scores, G, dl)


def assign_decision_literal(symbols, model, scores, G, dl):
    P = max(symbols, key=lambda symbol: scores[symbol] + scores[~symbol])
    value = True if scores[P] >= scores[~P] else False
    symbols.remove(P)
    model[P] = value
    G.add_node(P, val=value, dl=dl)


def unit_propagation(clauses, symbols, model, G, dl):
    def check(c):
        if not model or clauses.get_first_watched(c) == clauses.get_second_watched(c):
            return True
        w1, _ = inspect_literal(clauses.get_first_watched(c))
        if w1 in model:
            return c in (clauses.get_neg_watched(w1) if model[w1] else clauses.get_pos_watched(w1))
        w2, _ = inspect_literal(clauses.get_second_watched(c))
        if w2 in model:
            return c in (clauses.get_neg_watched(w2) if model[w2] else clauses.get_pos_watched(w2))

    def unit_clause(watching):
        w, p = inspect_literal(watching)
        G.add_node(w, val=p, dl=dl)
        G.add_edges_from(zip(prop_symbols(c) - {w}, itertools.cycle([w])), antecedent=c)
        symbols.remove(w)
        model[w] = p

    def conflict_clause(c):
        G.add_edges_from(zip(prop_symbols(c), itertools.cycle('K')), antecedent=c)

    while True:
        bcp = False
        for c in filter(check, clauses.get_clauses()):
            # we need only visit each clause when one of its two watched literals is assigned to 0 because, until
            # this happens, we can guarantee that there cannot be more than n-2 literals in the clause assigned to 0
            first_watched = pl_true(clauses.get_first_watched(c), model)
            second_watched = pl_true(clauses.get_second_watched(c), model)
            if first_watched is None and clauses.get_first_watched(c) == clauses.get_second_watched(c):
                unit_clause(clauses.get_first_watched(c))
                bcp = True
                break
            elif first_watched is False and second_watched is not True:
                if clauses.update_second_watched(c, model):
                    bcp = True
                else:
                    # if the only literal with a non-zero value is the other watched literal then
                    if second_watched is None:  # if it is free, then the clause is a unit clause
                        unit_clause(clauses.get_second_watched(c))
                        bcp = True
                        break
                    else:  # else (it is False) the clause is a conflict clause
                        conflict_clause(c)
                        return True
            elif second_watched is False and first_watched is not True:
                if clauses.update_first_watched(c, model):
                    bcp = True
                else:
                    # if the only literal with a non-zero value is the other watched literal then
                    if first_watched is None:  # if it is free, then the clause is a unit clause
                        unit_clause(clauses.get_first_watched(c))
                        bcp = True
                        break
                    else:  # else (it is False) the clause is a conflict clause
                        conflict_clause(c)
                        return True
        if not bcp:
            return False


def conflict_analysis(G, dl):
    conflict_clause = next(G[p]['K']['antecedent'] for p in G.pred['K'])
    P = next(node for node in G.nodes() - 'K' if G.nodes[node]['dl'] == dl and G.in_degree(node) == 0)
    first_uip = nx.immediate_dominators(G, P)['K']
    G.remove_node('K')
    conflict_side = nx.descendants(G, first_uip)
    while True:
        for l in prop_symbols(conflict_clause).intersection(conflict_side):
            antecedent = next(G[p][l]['antecedent'] for p in G.pred[l])
            conflict_clause = pl_binary_resolution(conflict_clause, antecedent)
            # the literal block distance is calculated by taking the decision levels from variables of all
            # literals in the clause, and counting how many different decision levels were in this set
            lbd = [G.nodes[l]['dl'] for l in prop_symbols(conflict_clause)]
            if lbd.count(dl) == 1 and first_uip in prop_symbols(conflict_clause):
                return 0 if len(lbd) == 1 else heapq.nlargest(2, lbd)[-1], conflict_clause, len(set(lbd))


def pl_binary_resolution(ci, cj):
    for di in disjuncts(ci):
        for dj in disjuncts(cj):
            if di == ~dj or ~di == dj:
                return pl_binary_resolution(associate('|', remove_all(di, disjuncts(ci))),
                                            associate('|', remove_all(dj, disjuncts(cj))))
    return associate('|', unique(disjuncts(ci) + disjuncts(cj)))


def backjump(symbols, model, G, dl=0):
    delete = {node for node in G.nodes() if G.nodes[node]['dl'] > dl}
    G.remove_nodes_from(delete)
    for node in delete:
        del model[node]
    symbols |= delete


class TwoWLClauseDatabase:

    def __init__(self, clauses):
        self.__twl = {}
        self.__watch_list = defaultdict(lambda: [set(), set()])
        for c in clauses:
            self.add(c, None)

    def get_clauses(self):
        return self.__twl.keys()

    def set_first_watched(self, clause, new_watching):
        if len(clause.args) > 2:
            self.__twl[clause][0] = new_watching

    def set_second_watched(self, clause, new_watching):
        if len(clause.args) > 2:
            self.__twl[clause][1] = new_watching

    def get_first_watched(self, clause):
        if len(clause.args) == 2:
            return clause.args[0]
        if len(clause.args) > 2:
            return self.__twl[clause][0]
        return clause

    def get_second_watched(self, clause):
        if len(clause.args) == 2:
            return clause.args[-1]
        if len(clause.args) > 2:
            return self.__twl[clause][1]
        return clause

    def get_pos_watched(self, l):
        return self.__watch_list[l][0]

    def get_neg_watched(self, l):