# Natural Language Toolkit: First-order Resolution-based Theorem Prover 
#
# Author: Dan Garrette <dhgarrette@gmail.com>
#
# Copyright (C) 2001-2009 NLTK Project
# URL: <http://www.nltk.org>
# For license information, see LICENSE.TXT

from nltk import defaultdict
from nltk.sem.logic import *
from nltk.sem.util import skolemize
from api import Prover, BaseProverCommand

"""
Module for a resolution-based First Order theorem prover.
"""

class ProverParseError(Exception): pass

class ResolutionProver(Prover):
    ANSWER_KEY = 'ANSWER'
    _assume_false=True
    
    def _prove(self, goal=None, assumptions=None, verbose=False):
        """
        @param goal: Input expression to prove
        @type goal: L{logic.Expression}
        @param assumptions: Input expressions to use as assumptions in the proof
        @type assumptions: L{list} of logic.Expression objects
        """
        if not assumptions:
            assumptions = []
            
        result = None
        try:
            clauses = []
            if goal:
                clauses.extend(clausify(-goal))
            for a in assumptions:
                clauses.extend(clausify(a))
            result, clauses = self._attempt_proof(clauses)
            if verbose: 
                print ResolutionProverCommand._decorate_clauses(clauses)
        except RuntimeError, e:
            if self._assume_false and str(e).startswith('maximum recursion depth exceeded'):
                result = False
                clauses = []
            else:
                if verbose:
                    print e
                else:
                    raise e
        return (result, clauses)

    def _attempt_proof(self, clauses):
        #map indices to lists of indices, to store attempted unifications
        tried = defaultdict(list)
        
        i = 0
        while i < len(clauses):
            if not clauses[i].is_tautology():
                #since we try clauses in order, we should start after the last
                #index tried
                if tried[i]: 
                    j = tried[i][-1] + 1
                else: 
                    j = i + 1 #nothing tried yet for 'i', so start with the next
                    
                while j < len(clauses):
                    #don't: 1) unify a clause with itself, 
                    #       2) use tautologies
                    if i != j and j and not clauses[j].is_tautology():
                        tried[i].append(j) 
                        newclauses = clauses[i].unify(clauses[j])
                        if newclauses:
                            for newclause in newclauses:
                                newclause._parents = (i+1, j+1)
                                clauses.append(newclause)
                                if not len(newclause): #if there's an empty clause
                                    return (True, clauses) 
                            i=-1 #since we added a new clause, restart from the top 
                            break
                    j += 1
            i += 1
        return (False, clauses)
        
class ResolutionProverCommand(BaseProverCommand):
    def __init__(self, goal=None, assumptions=None, prover=None):
        """
        @param goal: Input expression to prove
        @type goal: L{logic.Expression}
        @param assumptions: Input expressions to use as assumptions in
            the proof.
        @type assumptions: C{list} of L{logic.Expression}
        """
        if prover is not None:
            assert isinstance(prover, ResolutionProver)
        else:
            prover = ResolutionProver()

        BaseProverCommand.__init__(self, prover, goal, assumptions)
        self._clauses = None
    
    def prove(self, verbose=False):
        """
        Perform the actual proof.  Store the result to prevent unnecessary
        re-proving.
        """
        if self._result is None:
            self._result, clauses = self._prover._prove(self.goal(), 
                                                        self.assumptions(),
                                                        verbose)
            self._clauses = clauses
            self._proof = ResolutionProverCommand._decorate_clauses(clauses)
        return self._result

    def find_answers(self, verbose=False):
        self.prove(verbose)
        
        answers = set()
        answer_ex = VariableExpression(Variable(ResolutionProver.ANSWER_KEY))
        for clause in self._clauses:
            for term in clause:
                if isinstance(term, ApplicationExpression) and\
                   term.function == answer_ex and\
                   not isinstance(term.argument, IndividualVariableExpression):
                    answers.add(term.argument)
        return answers
    
    @staticmethod
    def _decorate_clauses(clauses):
        """
        Decorate the proof output.
        """
        out = ''
        max_clause_len = max([len(str(clause)) for clause in clauses])
        max_seq_len = len(str(len(clauses)))
        for i in range(len(clauses)):
            parents = 'A'
            taut = ''
            if clauses[i].is_tautology():
                taut = 'Tautology'
            if clauses[i]._parents:
                parents = str(clauses[i]._parents)
            parents = ' '*(max_clause_len-len(str(clauses[i]))+1) + parents
            seq = ' '*(max_seq_len-len(str(i+1))) + str(i+1)
            out += '[%s] %s %s %s\n' % (seq, clauses[i], parents, taut)
        return out 
    
class Clause(list):
    def __init__(self, data):
        list.__init__(self, data)
        self._is_tautology = None
        self._parents = None
    
    def unify(self, other, bindings=None, used=None, skipped=None, debug=False):
        """
        Attempt to unify this Clause with the other, returning a list of 
        resulting, unified, Clauses.
        
        @param other: C{Clause} with which to unify
        @param bindings: C{BindingDict} containing bindings that should be used
        during the unification
        @param used: C{tuple} of two C{list}s of atoms.  The first lists the 
        atoms from 'self' that were successfully unified with atoms from 
        'other'.  The second lists the atoms from 'other' that were successfully
        unified with atoms from 'self'.
        @param skipped: C{tuple} of two C{Clause}s.  The first is a list of all
        the atoms from the 'self' Clause that have not been unified with 
        anything on the path.  The second is same thing for the 'other' Clause.
        @param debug: C{bool} indicating whether debug statements should print
        @return: C{list} containing all the resulting C{Clause}s that could be
        obtained by unification
        """
        if bindings is None: bindings = BindingDict()
        if used is None: used = ([],[])
        if skipped is None: skipped = ([],[])
        if isinstance(debug, bool): debug = DebugObject(debug)

        newclauses = _iterate_first(self, other, bindings, used, skipped, _complete_unify_path, debug)

        #remove subsumed clauses.  make a list of all indices of subsumed
        #clauses, and then remove them from the list
        subsumed = []
        for i, c1 in enumerate(newclauses):
            if i not in subsumed:
                for j, c2 in enumerate(newclauses):
                    if i!=j and j not in subsumed and c1.subsumes(c2):
                        subsumed.append(j)
        result = []            
        for i in range(len(newclauses)):
            if i not in subsumed:
                result.append(newclauses[i])

        return result

    def isSubsetOf(self, other):
        """
        Return True iff every term in 'self' is a term in 'other'.
        
        @param other: C{Clause}
        @return: C{bool}
        """
        for a in self:
            if a not in other:
                return False
        return True

    def subsumes(self, other):
        """
        Return True iff 'self' subsumes 'other', this is, if there is a 
        substitution such that every term in 'self' can be unified with a term
        in 'other'.
        
        @param other: C{Clause}
        @return: C{bool}
        """
        negatedother = []
        for atom in other:
            if isinstance(atom, NegatedExpression):
                negatedother.append(atom.term)
            else:
                negatedother.append(-atom)
        
        negatedotherClause = Clause(negatedother)

        bindings = BindingDict()
        used = ([],[])
        skipped = ([],[])
        debug = DebugObject(False)

        return len(_iterate_first(self, negatedotherClause, bindings, used, 
                                      skipped, _subsumes_finalize, 
                                      debug)) > 0
        
    def __getslice__(self, start, end):
        return Clause(list.__getslice__(self, start, end))
    
    def __sub__(self, other):
        return Clause([a for a in self if a not in other])
    
    def __add__(self, other):
        return Clause(list.__add__(self, other))
    
    def is_tautology(self):
        """
        Self is a tautology if it contains ground terms P and -P.  The ground 
        term, P, must be an exact match, ie, not using unification. 
        """
        if self._is_tautology is not None:
            return self._is_tautology
        for i,a in enumerate(self):
            if not isinstance(a, EqualityExpression):
                j = len(self)-1
                while j > i:
                    b = self[j]
                    if isinstance(a, NegatedExpression):
                        if a.term == b:  
                            self._is_tautology = True
                            return True
                    elif isinstance(b, NegatedExpression):
                        if a == b.term:
                            self._is_tautology = True
                            return True
                    j -= 1
        self._is_tautology = False
        return False
    
    def free(self):
        s = set()
        for atom in self:
            s |= atom.free(False)
        return s
    
    def replace(self, variable, expression):
        """
        Replace every instance of variable with expression across every atom
        in the clause
        
        @param variable: C{Variable}
        @param expression: C{Expression}
        """
        return Clause([atom.replace(variable, expression) for atom in self])
    
    def substitute_bindings(self, bindings):
        """
        Replace every binding 
        
        @param bindings: A C{list} of tuples mapping Variable Expressions to the
        Expressions to which they are bound
        @return: C{Clause}
        """
        return Clause([atom.substitute_bindings(bindings) for atom in self])
    
    def __str__(self):
        return '{' + ', '.join([str(item) for item in self]) + '}'

    def __repr__(self):
        return str(self)

def _iterate_first(first, second, bindings, used, skipped, finalize_method, debug):
    """
    This method facilitates movement through the terms of 'self'
    """
    debug.line('unify(%s,%s) %s'%(first, second, bindings))

    if not len(first) or not len(second): #if no more recursions can be performed
        return finalize_method(first, second, bindings, used, skipped, debug)
    else: 
        #explore this 'self' atom
        result = _iterate_second(first, second, bindings, used, skipped, finalize_method, debug+1)
        
        #skip this possible 'self' atom
        newskipped = (skipped[0]+[first[0]], skipped[1])
        result += _iterate_first(first[1:], second, bindings, used, newskipped, finalize_method, debug+1)
                 
        try:
            newbindings, newused, unused = _unify_terms(first[0], second[0], bindings, used)
            #Unification found, so progress with this line of unification
            #put skipped and unused terms back into play for later unification.
            newfirst = first[1:] + skipped[0] + unused[0]
            newsecond = second[1:] + skipped[1] + unused[1]
            result += _iterate_first(newfirst, newsecond, newbindings, newused, ([],[]), finalize_method, debug+1)
        except BindingException:
            #the atoms could not be unified,
            pass 
            
        return result            

def _iterate_second(first, second, bindings, used, skipped, finalize_method, debug):
    """
    This method facilitates movement through the terms of 'other'
    """
    debug.line('unify(%s,%s) %s'%(first, second, bindings))

    if not len(first) or not len(second): #if no more recursions can be performed
        return finalize_method(first, second, bindings, used, skipped, debug)
    else:
        #skip this possible pairing and move to the next
        newskipped = (skipped[0], skipped[1]+[second[0]])
        result = _iterate_second(first, second[1:], bindings, used, newskipped, finalize_method, debug+1)

        try:
            newbindings, newused, unused = _unify_terms(first[0], second[0], bindings, used)
            #Unification found, so progress with this line of unification
            #put skipped and unused terms back into play for later unification.
            newfirst = first[1:] + skipped[0] + unused[0]
            newsecond = second[1:] + skipped[1] + unused[1]
            result += _iterate_second(newfirst, newsecond, newbindings, newused, ([],[]), finalize_method, debug+1)
        except BindingException:
            #the atoms could not be unified,
            pass 
            
        return result
    
def _unify_terms(a, b, bindings=None, used=None):
    """
    This method attempts to unify two terms.  Two expressions are unifiable 
    if there exists a substitution function S such that S(a) == S(-b).

    @param a: C{Expression} 
    @param b: C{Expression} 
    @param bindings: C{BindingDict} a starting set of bindings with which 
    the unification must be consistent
    @return: C{BindingDict} A dictionary of the bindings required to unify
    @raise C{BindingException}: If the terms cannot be unified
    """
    assert isinstance(a, Expression)
    assert isinstance(b, Expression)

    if bindings is None: bindings = BindingDict()
    if used is None: used = ([],[])

    # Use resolution
    if isinstance(a, NegatedExpression) and isinstance(b, ApplicationExpression):
        newbindings = most_general_unification(a.term, b, bindings)
        newused = (used[0]+[a], used[1]+[b])
        unused = ([],[])
    elif isinstance(a, ApplicationExpression) and isinstance(b, NegatedExpression):
        newbindings = most_general_unification(a, b.term, bindings)
        newused = (used[0]+[a], used[1]+[b])
        unused = ([],[])

    # Use demodulation 
    elif isinstance(a, EqualityExpression):
        newbindings = BindingDict([(a.first.variable, a.second)])
        newused = (used[0]+[a], used[1])
        unused = ([],[b])
    elif isinstance(b, EqualityExpression):
        newbindings = BindingDict([(b.first.variable, b.second)])
        newused = (used[0], used[1]+[b])
        unused = ([a],[])

    else:
        raise BindingException((a, b))
            
    return newbindings, newused, unused

def _complete_unify_path(first, second, bindings, used, skipped, debug):
    if used[0] or used[1]: #if bindings were made along the path
        newclause = Clause(skipped[0] + skipped[1] + first + second)
        debug.line('  -> New Clause: %s' % newclause)
        return [newclause.substitute_bindings(bindings)]
    else: #no bindings made means no unification occurred.  so no result
        debug.line('  -> End')
        return []

def _subsumes_finalize(first, second, bindings, used, skipped, debug):
    if not len(skipped[0]) and not len(first):
        #If there are no skipped terms and no terms left in 'first', then 
        #all of the terms in the original 'self' were unified with terms
        #in 'other'.  Therefore, there exists a binding (this one) such that
        #every term in self can be unified with a term in other, which
        #is the definition of subsumption. 
        return [True]
    else:
        return []

def clausify(expression):
    """
    Skolemize, clausify, and standardize the variables apart.
    """
    clause_list = []
    for clause in _clausify(skolemize(expression)):
        for free in clause.free():
            if is_indvar(free.name):
                newvar = VariableExpression(unique_variable())
                clause = clause.replace(free, newvar)
        clause_list.append(clause)
    return clause_list
    
def _clausify(expression):
    """
    @param expression: a skolemized expression in CNF
    """
    if isinstance(expression, AndExpression):
        return _clausify(expression.first) + _clausify(expression.second)
    elif isinstance(expression, OrExpression):
        first = _clausify(expression.first)
        second = _clausify(expression.second)
        assert len(first) == 1
        assert len(second) == 1
        return [first[0] + second[0]]
    elif isinstance(expression, EqualityExpression):
        return [Clause([expression])]
    elif isinstance(expression, ApplicationExpression):
        return [Clause([expression])]
    elif isinstance(expression, NegatedExpression):
        if isinstance(expression.term, ApplicationExpression):
            return [Clause([expression])]
        elif isinstance(expression.term, EqualityExpression):
            return [Clause([expression])]
    raise ProverParseError()
    
    
class BindingDict(object):
    def __init__(self, binding_list=None):
        """
        @param binding_list: C{list} of (C{AbstractVariableExpression}, C{AtomicExpression}) to initialize the dictionary
        """
        self.d = {}

        if binding_list:
            for (v, b) in binding_list:
                self[v] = b
    
    def __setitem__(self, variable, binding):
        """
        A binding is consistent with the dict if its variable is not already bound, OR if its 
        variable is already bound to its argument.
        
        @param variable: C{Variable} The variable to bind
        @param binding: C{Expression} The atomic to which 'variable' should be bound
        @raise BindingException: If the variable cannot be bound in this dictionary
        """
        assert isinstance(variable, Variable)
        assert isinstance(binding, Expression) 
        
        try:
            existing = self[variable]
        except KeyError:
            existing = None
            
        if not existing or binding == existing:
            self.d[variable] = binding
        elif isinstance(binding, IndividualVariableExpression):
            # Since variable is already bound, try to bind binding to variable
            try:
                existing = self[binding.variable]
            except KeyError:
                existing = None
                
            binding2 = VariableExpression(variable)
                
            if not existing or binding2 == existing:
                self.d[binding.variable] = binding2
            else:
                raise BindingException('Variable %s already bound to another '
                                       'value' % (variable))
        else:
            raise BindingException('Variable %s already bound to another '
                                   'value' % (variable))

    def __getitem__(self, variable):
        """
        Return the expression to which 'variable' is bound
        """
        assert isinstance(variable, Variable)

        intermediate = self.d[variable]
        while intermediate:
            try:
                intermediate = self.d[intermediate]
            except KeyError:
                return intermediate
            
    def __contains__(self, item):
        return item in self.d

    def __add__(self, other):
        """
        @param other: C{BindingDict} The dict with which to combine self
        @return: C{BindingDict} A new dict containing all the elements of both parameters
        @raise BindingException: If the parameter dictionaries are not consistent with each other
        """
        try:
            combined = BindingDict()
            for v in self.d:
                combined[v] = self.d[v]
            for v in other.d:
                combined[v] = other.d[v]
            return combined
        except BindingException:
            raise BindingException("Attempting to add two contradicting "
                                   "BindingDicts: '%s' and '%s'" 
                                   % (self, other))

    def __len__(self):
        return len(self.d)
    
    def __str__(self):
        return '{' + ', '.join(['%s: %s' % (v, self.d[v]) for v in self.d]) + '}'

    def __repr__(self):
        return str(self)


def most_general_unification(a, b, bindings=None):
    """
    Find the most general unification of the two given expressions
    
    @param a: C{Expression}
    @param b: C{Expression}
    @param bindings: C{BindingDict} a starting set of bindings with which the
                     unification must be consistent
    @return: a list of bindings
    @raise BindingException: if the Expressions cannot be unified
    """
    if bindings is None:
        bindings = BindingDict()
    
    if a == b:
        return bindings
    elif isinstance(a, IndividualVariableExpression):
        return _mgu_var(a, b, bindings)
    elif isinstance(b, IndividualVariableExpression):
        return _mgu_var(b, a, bindings)
    elif isinstance(a, ApplicationExpression) and\
         isinstance(b, ApplicationExpression):
        return most_general_unification(a.function, b.function, bindings) +\
               most_general_unification(a.argument, b.argument, bindings)
    raise BindingException((a, b))

def _mgu_var(var, expression, bindings):
    if var.variable in expression.free(False):
        raise BindingException((var, expression))
    else:
        return BindingDict([(var.variable, expression)]) + bindings
    
    
class BindingException(Exception):
    def __init__(self, arg):
        if isinstance(arg, tuple):
            Exception.__init__(self, "'%s' cannot be bound to '%s'" % arg)
        else:
            Exception.__init__(self, arg)
    
class UnificationException(Exception):
    def __init__(self, a, b):
        Exception.__init__(self, "'%s' cannot unify with '%s'" % (a,b))
    
    
class DebugObject(object):
    def __init__(self, enabled=True, indent=0):
        self.enabled = enabled
        self.indent = indent
    
    def __add__(self, i):
        return DebugObject(self.enabled, self.indent+i)

    def line(self, line):
        if self.enabled: 
            print '    '*self.indent + line


def testResolutionProver():
    resolution_test(r'man(x)')
    resolution_test(r'(man(x) -> man(x))')
    resolution_test(r'(man(x) -> --man(x))')
    resolution_test(r'-(man(x) and -man(x))')
    resolution_test(r'(man(x) or -man(x))')
    resolution_test(r'(man(x) -> man(x))')
    resolution_test(r'-(man(x) and -man(x))')
    resolution_test(r'(man(x) or -man(x))')
    resolution_test(r'(man(x) -> man(x))')
    resolution_test(r'(man(x) iff man(x))')
    resolution_test(r'-(man(x) iff -man(x))')
    resolution_test('all x.man(x)')
    resolution_test('-all x.some y.F(x,y) & some x.all y.(-F(x,y))')
    resolution_test('some x.all y.sees(x,y)')

    p1 = LogicParser().parse(r'all x.(man(x) -> mortal(x))')
    p2 = LogicParser().parse(r'man(Socrates)')
    c = LogicParser().parse(r'mortal(Socrates)')
    print '%s, %s |- %s: %s' % (p1, p2, c, ResolutionProver().prove(c, [p1,p2]))
    
    p1 = LogicParser().parse(r'all x.(man(x) -> walks(x))')
    p2 = LogicParser().parse(r'man(John)')
    c = LogicParser().parse(r'some y.walks(y)')
    print '%s, %s |- %s: %s' % (p1, p2, c, ResolutionProver().prove(c, [p1,p2]))
    
    p = LogicParser().parse(r'some e1.some e2.(believe(e1,john,e2) & walk(e2,mary))')
    c = LogicParser().parse(r'some e0.walk(e0,mary)')
    print '%s |- %s: %s' % (p, c, ResolutionProver().prove(c, [p]))
    
def resolution_test(e):
    f = LogicParser().parse(e)
    t = ResolutionProver().prove(f)
    print '|- %s: %s' % (f, t)

def test_clausify():
    lp = LogicParser()
    
    print clausify(lp.parse('P(x) | Q(x)'))
    print clausify(lp.parse('(P(x) & Q(x)) | R(x)'))
    print clausify(lp.parse('P(x) | (Q(x) & R(x))'))
    print clausify(lp.parse('(P(x) & Q(x)) | (R(x) & S(x))'))
        
    print clausify(lp.parse('P(x) | Q(x) | R(x)'))
    print clausify(lp.parse('P(x) | (Q(x) & R(x)) | S(x)'))

    print clausify(lp.parse('exists x.P(x) | Q(x)'))

    print clausify(lp.parse('-(-P(x) & Q(x))'))
    print clausify(lp.parse('P(x) <-> Q(x)'))
    print clausify(lp.parse('-(P(x) <-> Q(x))'))
    print clausify(lp.parse('-(all x.P(x))'))
    print clausify(lp.parse('-(some x.P(x))'))
    
    print clausify(lp.parse('some x.P(x)'))
    print clausify(lp.parse('some x.all y.P(x,y)'))
    print clausify(lp.parse('all y.some x.P(x,y)'))
    print clausify(lp.parse('all z.all y.some x.P(x,y,z)'))
    print clausify(lp.parse('all x.(all y.P(x,y) -> -all y.(Q(x,y) -> R(x,y)))'))

if __name__ == '__main__':
    test_clausify()
    print
    testResolutionProver()
    print
    
    p = LogicParser().parse('man(x)')
    print ResolutionProverCommand(p, [p]).prove()