# Natural Language Toolkit: Logic # # Author: Dan Garrette # # Copyright (C) 2001-2023 NLTK Project # URL: # For license information, see LICENSE.TXT """ A version of first order predicate logic, built on top of the typed lambda calculus. """ import operator import re from collections import defaultdict from functools import reduce, total_ordering from nltk.internals import Counter from nltk.util import Trie APP = "APP" _counter = Counter() class Tokens: LAMBDA = "\\" LAMBDA_LIST = ["\\"] # Quantifiers EXISTS = "exists" EXISTS_LIST = ["some", "exists", "exist"] ALL = "all" ALL_LIST = ["all", "forall"] IOTA = "iota" IOTA_LIST = ["iota"] # Punctuation DOT = "." OPEN = "(" CLOSE = ")" COMMA = "," # Operations NOT = "-" NOT_LIST = ["not", "-", "!"] AND = "&" AND_LIST = ["and", "&", "^"] OR = "|" OR_LIST = ["or", "|"] IMP = "->" IMP_LIST = ["implies", "->", "=>"] IFF = "<->" IFF_LIST = ["iff", "<->", "<=>"] EQ = "=" EQ_LIST = ["=", "=="] NEQ = "!=" NEQ_LIST = ["!="] # Collections of tokens BINOPS = AND_LIST + OR_LIST + IMP_LIST + IFF_LIST QUANTS = EXISTS_LIST + ALL_LIST + IOTA_LIST PUNCT = [DOT, OPEN, CLOSE, COMMA] TOKENS = BINOPS + EQ_LIST + NEQ_LIST + QUANTS + LAMBDA_LIST + PUNCT + NOT_LIST # Special SYMBOLS = [x for x in TOKENS if re.match(r"^[-\\.(),!&^|>=<]*$", x)] def boolean_ops(): """ Boolean operators """ names = ["negation", "conjunction", "disjunction", "implication", "equivalence"] for pair in zip(names, [Tokens.NOT, Tokens.AND, Tokens.OR, Tokens.IMP, Tokens.IFF]): print("%-15s\t%s" % pair) def equality_preds(): """ Equality predicates """ names = ["equality", "inequality"] for pair in zip(names, [Tokens.EQ, Tokens.NEQ]): print("%-15s\t%s" % pair) def binding_ops(): """ Binding operators """ names = ["existential", "universal", "lambda"] for pair in zip(names, [Tokens.EXISTS, Tokens.ALL, Tokens.LAMBDA, Tokens.IOTA]): print("%-15s\t%s" % pair) class LogicParser: """A lambda calculus expression parser.""" def __init__(self, type_check=False): """ :param type_check: should type checking be performed to their types? :type type_check: bool """ assert isinstance(type_check, bool) self._currentIndex = 0 self._buffer = [] self.type_check = type_check """A list of tuples of quote characters. The 4-tuple is comprised of the start character, the end character, the escape character, and a boolean indicating whether the quotes should be included in the result. Quotes are used to signify that a token should be treated as atomic, ignoring any special characters within the token. The escape character allows the quote end character to be used within the quote. If True, the boolean indicates that the final token should contain the quote and escape characters. This method exists to be overridden""" self.quote_chars = [] self.operator_precedence = dict( [(x, 1) for x in Tokens.LAMBDA_LIST] + [(x, 2) for x in Tokens.NOT_LIST] + [(APP, 3)] + [(x, 4) for x in Tokens.EQ_LIST + Tokens.NEQ_LIST] + [(x, 5) for x in Tokens.QUANTS] + [(x, 6) for x in Tokens.AND_LIST] + [(x, 7) for x in Tokens.OR_LIST] + [(x, 8) for x in Tokens.IMP_LIST] + [(x, 9) for x in Tokens.IFF_LIST] + [(None, 10)] ) self.right_associated_operations = [APP] def parse(self, data, signature=None): """ Parse the expression. :param data: str for the input to be parsed :param signature: ``dict`` that maps variable names to type strings :returns: a parsed Expression """ data = data.rstrip() self._currentIndex = 0 self._buffer, mapping = self.process(data) try: result = self.process_next_expression(None) if self.inRange(0): raise UnexpectedTokenException(self._currentIndex + 1, self.token(0)) except LogicalExpressionException as e: msg = "{}\n{}\n{}^".format(e, data, " " * mapping[e.index - 1]) raise LogicalExpressionException(None, msg) from e if self.type_check: result.typecheck(signature) return result def process(self, data): """Split the data into tokens""" out = [] mapping = {} tokenTrie = Trie(self.get_all_symbols()) token = "" data_idx = 0 token_start_idx = data_idx while data_idx < len(data): cur_data_idx = data_idx quoted_token, data_idx = self.process_quoted_token(data_idx, data) if quoted_token: if not token: token_start_idx = cur_data_idx token += quoted_token continue st = tokenTrie c = data[data_idx] symbol = "" while c in st: symbol += c st = st[c] if len(data) - data_idx > len(symbol): c = data[data_idx + len(symbol)] else: break if Trie.LEAF in st: # token is a complete symbol if token: mapping[len(out)] = token_start_idx out.append(token) token = "" mapping[len(out)] = data_idx out.append(symbol) data_idx += len(symbol) else: if data[data_idx] in " \t\n": # any whitespace if token: mapping[len(out)] = token_start_idx out.append(token) token = "" else: if not token: token_start_idx = data_idx token += data[data_idx] data_idx += 1 if token: mapping[len(out)] = token_start_idx out.append(token) mapping[len(out)] = len(data) mapping[len(out) + 1] = len(data) + 1 return out, mapping def process_quoted_token(self, data_idx, data): token = "" c = data[data_idx] i = data_idx for start, end, escape, incl_quotes in self.quote_chars: if c == start: if incl_quotes: token += c i += 1 while data[i] != end: if data[i] == escape: if incl_quotes: token += data[i] i += 1 if len(data) == i: # if there are no more chars raise LogicalExpressionException( None, "End of input reached. " "Escape character [%s] found at end." % escape, ) token += data[i] else: token += data[i] i += 1 if len(data) == i: raise LogicalExpressionException( None, "End of input reached. " "Expected: [%s]" % end ) if incl_quotes: token += data[i] i += 1 if not token: raise LogicalExpressionException(None, "Empty quoted token found") break return token, i def get_all_symbols(self): """This method exists to be overridden""" return Tokens.SYMBOLS def inRange(self, location): """Return TRUE if the given location is within the buffer""" return self._currentIndex + location < len(self._buffer) def token(self, location=None): """Get the next waiting token. If a location is given, then return the token at currentIndex+location without advancing currentIndex; setting it gives lookahead/lookback capability.""" try: if location is None: tok = self._buffer[self._currentIndex] self._currentIndex += 1 else: tok = self._buffer[self._currentIndex + location] return tok except IndexError as e: raise ExpectedMoreTokensException(self._currentIndex + 1) from e def isvariable(self, tok): return tok not in Tokens.TOKENS def process_next_expression(self, context): """Parse the next complete expression from the stream and return it.""" try: tok = self.token() except ExpectedMoreTokensException as e: raise ExpectedMoreTokensException( self._currentIndex + 1, message="Expression expected." ) from e accum = self.handle(tok, context) if not accum: raise UnexpectedTokenException( self._currentIndex, tok, message="Expression expected." ) return self.attempt_adjuncts(accum, context) def handle(self, tok, context): """This method is intended to be overridden for logics that use different operators or expressions""" if self.isvariable(tok): return self.handle_variable(tok, context) elif tok in Tokens.NOT_LIST: return self.handle_negation(tok, context) elif tok in Tokens.LAMBDA_LIST: return self.handle_lambda(tok, context) elif tok in Tokens.QUANTS: return self.handle_quant(tok, context) elif tok == Tokens.OPEN: return self.handle_open(tok, context) def attempt_adjuncts(self, expression, context): cur_idx = None while cur_idx != self._currentIndex: # while adjuncts are added cur_idx = self._currentIndex expression = self.attempt_EqualityExpression(expression, context) expression = self.attempt_ApplicationExpression(expression, context) expression = self.attempt_BooleanExpression(expression, context) return expression def handle_negation(self, tok, context): return self.make_NegatedExpression(self.process_next_expression(Tokens.NOT)) def make_NegatedExpression(self, expression): return NegatedExpression(expression) def handle_variable(self, tok, context): # It's either: 1) a predicate expression: sees(x,y) # 2) an application expression: P(x) # 3) a solo variable: john OR x accum = self.make_VariableExpression(tok) if self.inRange(0) and self.token(0) == Tokens.OPEN: # The predicate has arguments if not isinstance(accum, FunctionVariableExpression) and not isinstance( accum, ConstantExpression ): raise LogicalExpressionException( self._currentIndex, "'%s' is an illegal predicate name. " "Individual variables may not be used as " "predicates." % tok, ) self.token() # swallow the Open Paren # curry the arguments accum = self.make_ApplicationExpression( accum, self.process_next_expression(APP) ) while self.inRange(0) and self.token(0) == Tokens.COMMA: self.token() # swallow the comma accum = self.make_ApplicationExpression( accum, self.process_next_expression(APP) ) self.assertNextToken(Tokens.CLOSE) return accum def get_next_token_variable(self, description): try: tok = self.token() except ExpectedMoreTokensException as e: raise ExpectedMoreTokensException(e.index, "Variable expected.") from e if isinstance(self.make_VariableExpression(tok), ConstantExpression): raise LogicalExpressionException( self._currentIndex, "'%s' is an illegal variable name. " "Constants may not be %s." % (tok, description), ) return Variable(tok) def handle_lambda(self, tok, context): # Expression is a lambda expression if not self.inRange(0): raise ExpectedMoreTokensException( self._currentIndex + 2, message="Variable and Expression expected following lambda operator.", ) vars = [self.get_next_token_variable("abstracted")] while True: if not self.inRange(0) or ( self.token(0) == Tokens.DOT and not self.inRange(1) ): raise ExpectedMoreTokensException( self._currentIndex + 2, message="Expression expected." ) if not self.isvariable(self.token(0)): break # Support expressions like: \x y.M == \x.\y.M vars.append(self.get_next_token_variable("abstracted")) if self.inRange(0) and self.token(0) == Tokens.DOT: self.token() # swallow the dot accum = self.process_next_expression(tok) while vars: accum = self.make_LambdaExpression(vars.pop(), accum) return accum def handle_quant(self, tok, context): # Expression is a quantified expression: some x.M factory = self.get_QuantifiedExpression_factory(tok) if not self.inRange(0): raise ExpectedMoreTokensException( self._currentIndex + 2, message="Variable and Expression expected following quantifier '%s'." % tok, ) vars = [self.get_next_token_variable("quantified")] while True: if not self.inRange(0) or ( self.token(0) == Tokens.DOT and not self.inRange(1) ): raise ExpectedMoreTokensException( self._currentIndex + 2, message="Expression expected." ) if not self.isvariable(self.token(0)): break # Support expressions like: some x y.M == some x.some y.M vars.append(self.get_next_token_variable("quantified")) if self.inRange(0) and self.token(0) == Tokens.DOT: self.token() # swallow the dot accum = self.process_next_expression(tok) while vars: accum = self.make_QuanifiedExpression(factory, vars.pop(), accum) return accum def get_QuantifiedExpression_factory(self, tok): """This method serves as a hook for other logic parsers that have different quantifiers""" if tok in Tokens.EXISTS_LIST: return ExistsExpression elif tok in Tokens.ALL_LIST: return AllExpression elif tok in Tokens.IOTA_LIST: return IotaExpression else: self.assertToken(tok, Tokens.QUANTS) def make_QuanifiedExpression(self, factory, variable, term): return factory(variable, term) def handle_open(self, tok, context): # Expression is in parens accum = self.process_next_expression(None) self.assertNextToken(Tokens.CLOSE) return accum def attempt_EqualityExpression(self, expression, context): """Attempt to make an equality expression. If the next token is an equality operator, then an EqualityExpression will be returned. Otherwise, the parameter will be returned.""" if self.inRange(0): tok = self.token(0) if tok in Tokens.EQ_LIST + Tokens.NEQ_LIST and self.has_priority( tok, context ): self.token() # swallow the "=" or "!=" expression = self.make_EqualityExpression( expression, self.process_next_expression(tok) ) if tok in Tokens.NEQ_LIST: expression = self.make_NegatedExpression(expression) return expression def make_EqualityExpression(self, first, second): """This method serves as a hook for other logic parsers that have different equality expression classes""" return EqualityExpression(first, second) def attempt_BooleanExpression(self, expression, context): """Attempt to make a boolean expression. If the next token is a boolean operator, then a BooleanExpression will be returned. Otherwise, the parameter will be returned.""" while self.inRange(0): tok = self.token(0) factory = self.get_BooleanExpression_factory(tok) if factory and self.has_priority(tok, context): self.token() # swallow the operator expression = self.make_BooleanExpression( factory, expression, self.process_next_expression(tok) ) else: break return expression def get_BooleanExpression_factory(self, tok): """This method serves as a hook for other logic parsers that have different boolean operators""" if tok in Tokens.AND_LIST: return AndExpression elif tok in Tokens.OR_LIST: return OrExpression elif tok in Tokens.IMP_LIST: return ImpExpression elif tok in Tokens.IFF_LIST: return IffExpression else: return None def make_BooleanExpression(self, factory, first, second): return factory(first, second) def attempt_ApplicationExpression(self, expression, context): """Attempt to make an application expression. The next tokens are a list of arguments in parens, then the argument expression is a function being applied to the arguments. Otherwise, return the argument expression.""" if self.has_priority(APP, context): if self.inRange(0) and self.token(0) == Tokens.OPEN: if ( not isinstance(expression, LambdaExpression) and not isinstance(expression, ApplicationExpression) and not isinstance(expression, FunctionVariableExpression) and not isinstance(expression, ConstantExpression) ): raise LogicalExpressionException( self._currentIndex, ("The function '%s" % expression) + "' is not a Lambda Expression, an " "Application Expression, or a " "functional predicate, so it may " "not take arguments.", ) self.token() # swallow then open paren # curry the arguments accum = self.make_ApplicationExpression( expression, self.process_next_expression(APP) ) while self.inRange(0) and self.token(0) == Tokens.COMMA: self.token() # swallow the comma accum = self.make_ApplicationExpression( accum, self.process_next_expression(APP) ) self.assertNextToken(Tokens.CLOSE) return accum return expression def make_ApplicationExpression(self, function, argument): return ApplicationExpression(function, argument) def make_VariableExpression(self, name): return VariableExpression(Variable(name)) def make_LambdaExpression(self, variable, term): return LambdaExpression(variable, term) def has_priority(self, operation, context): return self.operator_precedence[operation] < self.operator_precedence[ context ] or ( operation in self.right_associated_operations and self.operator_precedence[operation] == self.operator_precedence[context] ) def assertNextToken(self, expected): try: tok = self.token() except ExpectedMoreTokensException as e: raise ExpectedMoreTokensException( e.index, message="Expected token '%s'." % expected ) from e if isinstance(expected, list): if tok not in expected: raise UnexpectedTokenException(self._currentIndex, tok, expected) else: if tok != expected: raise UnexpectedTokenException(self._currentIndex, tok, expected) def assertToken(self, tok, expected): if isinstance(expected, list): if tok not in expected: raise UnexpectedTokenException(self._currentIndex, tok, expected) else: if tok != expected: raise UnexpectedTokenException(self._currentIndex, tok, expected) def __repr__(self): if self.inRange(0): msg = "Next token: " + self.token(0) else: msg = "No more tokens" return "<" + self.__class__.__name__ + ": " + msg + ">" def read_logic(s, logic_parser=None, encoding=None): """ Convert a file of First Order Formulas into a list of {Expression}s. :param s: the contents of the file :type s: str :param logic_parser: The parser to be used to parse the logical expression :type logic_parser: LogicParser :param encoding: the encoding of the input string, if it is binary :type encoding: str :return: a list of parsed formulas. :rtype: list(Expression) """ if encoding is not None: s = s.decode(encoding) if logic_parser is None: logic_parser = LogicParser() statements = [] for linenum, line in enumerate(s.splitlines()): line = line.strip() if line.startswith("#") or line == "": continue try: statements.append(logic_parser.parse(line)) except LogicalExpressionException as e: raise ValueError(f"Unable to parse line {linenum}: {line}") from e return statements @total_ordering class Variable: def __init__(self, name): """ :param name: the name of the variable """ assert isinstance(name, str), "%s is not a string" % name self.name = name def __eq__(self, other): return isinstance(other, Variable) and self.name == other.name def __ne__(self, other): return not self == other def __lt__(self, other): if not isinstance(other, Variable): raise TypeError return self.name < other.name def substitute_bindings(self, bindings): return bindings.get(self, self) def __hash__(self): return hash(self.name) def __str__(self): return self.name def __repr__(self): return "Variable('%s')" % self.name def unique_variable(pattern=None, ignore=None): """ Return a new, unique variable. :param pattern: ``Variable`` that is being replaced. The new variable must be the same type. :param term: a set of ``Variable`` objects that should not be returned from this function. :rtype: Variable """ if pattern is not None: if is_indvar(pattern.name): prefix = "z" elif is_funcvar(pattern.name): prefix = "F" elif is_eventvar(pattern.name): prefix = "e0" else: assert False, "Cannot generate a unique constant" else: prefix = "z" v = Variable(f"{prefix}{_counter.get()}") while ignore is not None and v in ignore: v = Variable(f"{prefix}{_counter.get()}") return v def skolem_function(univ_scope=None): """ Return a skolem function over the variables in univ_scope param univ_scope """ skolem = VariableExpression(Variable("F%s" % _counter.get())) if univ_scope: for v in list(univ_scope): skolem = skolem(VariableExpression(v)) return skolem class Type: def __repr__(self): return "%s" % self def __hash__(self): return hash("%s" % self) @classmethod def fromstring(cls, s): return read_type(s) class ComplexType(Type): def __init__(self, first, second): assert isinstance(first, Type), "%s is not a Type" % first assert isinstance(second, Type), "%s is not a Type" % second self.first = first self.second = second def __eq__(self, other): return ( isinstance(other, ComplexType) and self.first == other.first and self.second == other.second ) def __ne__(self, other): return not self == other __hash__ = Type.__hash__ def matches(self, other): if isinstance(other, ComplexType): return self.first.matches(other.first) and self.second.matches(other.second) else: return self == ANY_TYPE def resolve(self, other): if other == ANY_TYPE: return self elif isinstance(other, ComplexType): f = self.first.resolve(other.first) s = self.second.resolve(other.second) if f and s: return ComplexType(f, s) else: return None elif self == ANY_TYPE: return other else: return None def __str__(self): if self == ANY_TYPE: return "%s" % ANY_TYPE else: return f"<{self.first},{self.second}>" def str(self): if self == ANY_TYPE: return ANY_TYPE.str() else: return f"({self.first.str()} -> {self.second.str()})" class BasicType(Type): def __eq__(self, other): return isinstance(other, BasicType) and ("%s" % self) == ("%s" % other) def __ne__(self, other): return not self == other __hash__ = Type.__hash__ def matches(self, other): return other == ANY_TYPE or self == other def resolve(self, other): if self.matches(other): return self else: return None class EntityType(BasicType): def __str__(self): return "e" def str(self): return "IND" class TruthValueType(BasicType): def __str__(self): return "t" def str(self): return "BOOL" class EventType(BasicType): def __str__(self): return "v" def str(self): return "EVENT" class AnyType(BasicType, ComplexType): def __init__(self): pass @property def first(self): return self @property def second(self): return self def __eq__(self, other): return isinstance(other, AnyType) or other.__eq__(self) def __ne__(self, other): return not self == other __hash__ = Type.__hash__ def matches(self, other): return True def resolve(self, other): return other def __str__(self): return "?" def str(self): return "ANY" TRUTH_TYPE = TruthValueType() ENTITY_TYPE = EntityType() EVENT_TYPE = EventType() ANY_TYPE = AnyType() def read_type(type_string): assert isinstance(type_string, str) type_string = type_string.replace(" ", "") # remove spaces if type_string[0] == "<": assert type_string[-1] == ">" paren_count = 0 for i, char in enumerate(type_string): if char == "<": paren_count += 1 elif char == ">": paren_count -= 1 assert paren_count > 0 elif char == ",": if paren_count == 1: break return ComplexType( read_type(type_string[1:i]), read_type(type_string[i + 1 : -1]) ) elif type_string[0] == "%s" % ENTITY_TYPE: return ENTITY_TYPE elif type_string[0] == "%s" % TRUTH_TYPE: return TRUTH_TYPE elif type_string[0] == "%s" % ANY_TYPE: return ANY_TYPE else: raise LogicalExpressionException( None, "Unexpected character: '%s'." % type_string[0] ) class TypeException(Exception): def __init__(self, msg): super().__init__(msg) class InconsistentTypeHierarchyException(TypeException): def __init__(self, variable, expression=None): if expression: msg = ( "The variable '%s' was found in multiple places with different" " types in '%s'." % (variable, expression) ) else: msg = ( "The variable '%s' was found in multiple places with different" " types." % (variable) ) super().__init__(msg) class TypeResolutionException(TypeException): def __init__(self, expression, other_type): super().__init__( "The type of '%s', '%s', cannot be resolved with type '%s'" % (expression, expression.type, other_type) ) class IllegalTypeException(TypeException): def __init__(self, expression, other_type, allowed_type): super().__init__( "Cannot set type of %s '%s' to '%s'; must match type '%s'." % (expression.__class__.__name__, expression, other_type, allowed_type) ) def typecheck(expressions, signature=None): """ Ensure correct typing across a collection of ``Expression`` objects. :param expressions: a collection of expressions :param signature: dict that maps variable names to types (or string representations of types) """ # typecheck and create master signature for expression in expressions: signature = expression.typecheck(signature) # apply master signature to all expressions for expression in expressions[:-1]: expression.typecheck(signature) return signature class SubstituteBindingsI: """ An interface for classes that can perform substitutions for variables. """ def substitute_bindings(self, bindings): """ :return: The object that is obtained by replacing each variable bound by ``bindings`` with its values. Aliases are already resolved. (maybe?) :rtype: (any) """ raise NotImplementedError() def variables(self): """ :return: A list of all variables in this object. """ raise NotImplementedError() class Expression(SubstituteBindingsI): """This is the base abstract object for all logical expressions""" _logic_parser = LogicParser() _type_checking_logic_parser = LogicParser(type_check=True) @classmethod def fromstring(cls, s, type_check=False, signature=None): if type_check: return cls._type_checking_logic_parser.parse(s, signature) else: return cls._logic_parser.parse(s, signature) def __call__(self, other, *additional): accum = self.applyto(other) for a in additional: accum = accum(a) return accum def applyto(self, other): assert isinstance(other, Expression), "%s is not an Expression" % other return ApplicationExpression(self, other) def __neg__(self): return NegatedExpression(self) def negate(self): """If this is a negated expression, remove the negation. Otherwise add a negation.""" return -self def __and__(self, other): if not isinstance(other, Expression): raise TypeError("%s is not an Expression" % other) return AndExpression(self, other) def __or__(self, other): if not isinstance(other, Expression): raise TypeError("%s is not an Expression" % other) return OrExpression(self, other) def __gt__(self, other): if not isinstance(other, Expression): raise TypeError("%s is not an Expression" % other) return ImpExpression(self, other) def __lt__(self, other): if not isinstance(other, Expression): raise TypeError("%s is not an Expression" % other) return IffExpression(self, other) def __eq__(self, other): return NotImplemented def __ne__(self, other): return not self == other def equiv(self, other, prover=None): """ Check for logical equivalence. Pass the expression (self <-> other) to the theorem prover. If the prover says it is valid, then the self and other are equal. :param other: an ``Expression`` to check equality against :param prover: a ``nltk.inference.api.Prover`` """ assert isinstance(other, Expression), "%s is not an Expression" % other if prover is None: from nltk.inference import Prover9 prover = Prover9() bicond = IffExpression(self.simplify(), other.simplify()) return prover.prove(bicond) def __hash__(self): return hash(repr(self)) def substitute_bindings(self, bindings): expr = self for var in expr.variables(): if var in bindings: val = bindings[var] if isinstance(val, Variable): val = self.make_VariableExpression(val) elif not isinstance(val, Expression): raise ValueError( "Can not substitute a non-expression " "value into an expression: %r" % (val,) ) # Substitute bindings in the target value. val = val.substitute_bindings(bindings) # Replace var w/ the target value. expr = expr.replace(var, val) return expr.simplify() def typecheck(self, signature=None): """ Infer and check types. Raise exceptions if necessary. :param signature: dict that maps variable names to types (or string representations of types) :return: the signature, plus any additional type mappings """ sig = defaultdict(list) if signature: for key in signature: val = signature[key] varEx = VariableExpression(Variable(key)) if isinstance(val, Type): varEx.type = val else: varEx.type = read_type(val) sig[key].append(varEx) self._set_type(signature=sig) return {key: sig[key][0].type for key in sig} def findtype(self, variable): """ Find the type of the given variable as it is used in this expression. For example, finding the type of "P" in "P(x) & Q(x,y)" yields "" :param variable: Variable """ raise NotImplementedError() def _set_type(self, other_type=ANY_TYPE, signature=None): """ Set the type of this expression to be the given type. Raise type exceptions where applicable. :param other_type: Type :param signature: dict(str -> list(AbstractVariableExpression)) """ raise NotImplementedError() def replace(self, variable, expression, replace_bound=False, alpha_convert=True): """ Replace every instance of 'variable' with 'expression' :param variable: ``Variable`` The variable to replace :param expression: ``Expression`` The expression with which to replace it :param replace_bound: bool Should bound variables be replaced? :param alpha_convert: bool Alpha convert automatically to avoid name clashes? """ assert isinstance(variable, Variable), "%s is not a Variable" % variable assert isinstance(expression, Expression), ( "%s is not an Expression" % expression ) return self.visit_structured( lambda e: e.replace(variable, expression, replace_bound, alpha_convert), self.__class__, ) def normalize(self, newvars=None): """Rename auto-generated unique variables""" def get_indiv_vars(e): if isinstance(e, IndividualVariableExpression): return {e} elif isinstance(e, AbstractVariableExpression): return set() else: return e.visit( get_indiv_vars, lambda parts: reduce(operator.or_, parts, set()) ) result = self for i, e in enumerate(sorted(get_indiv_vars(self), key=lambda e: e.variable)): if isinstance(e, EventVariableExpression): newVar = e.__class__(Variable("e0%s" % (i + 1))) elif isinstance(e, IndividualVariableExpression): newVar = e.__class__(Variable("z%s" % (i + 1))) else: newVar = e result = result.replace(e.variable, newVar, True) return result def visit(self, function, combinator): """ Recursively visit subexpressions. Apply 'function' to each subexpression and pass the result of each function application to the 'combinator' for aggregation: return combinator(map(function, self.subexpressions)) Bound variables are neither applied upon by the function nor given to the combinator. :param function: ``Function`` to call on each subexpression :param combinator: ``Function,R>`` to combine the results of the function calls :return: result of combination ``R`` """ raise NotImplementedError() def visit_structured(self, function, combinator): """ Recursively visit subexpressions. Apply 'function' to each subexpression and pass the result of each function application to the 'combinator' for aggregation. The combinator must have the same signature as the constructor. The function is not applied to bound variables, but they are passed to the combinator. :param function: ``Function`` to call on each subexpression :param combinator: ``Function`` with the same signature as the constructor, to combine the results of the function calls :return: result of combination """ return self.visit(function, lambda parts: combinator(*parts)) def __repr__(self): return f"<{self.__class__.__name__} {self}>" def __str__(self): return self.str() def variables(self): """ Return a set of all the variables for binding substitution. The variables returned include all free (non-bound) individual variables and any variable starting with '?' or '@'. :return: set of ``Variable`` objects """ return self.free() | { p for p in self.predicates() | self.constants() if re.match("^[?@]", p.name) } def free(self): """ Return a set of all the free (non-bound) variables. This includes both individual and predicate variables, but not constants. :return: set of ``Variable`` objects """ return self.visit( lambda e: e.free(), lambda parts: reduce(operator.or_, parts, set()) ) def constants(self): """ Return a set of individual constants (non-predicates). :return: set of ``Variable`` objects """ return self.visit( lambda e: e.constants(), lambda parts: reduce(operator.or_, parts, set()) ) def predicates(self): """ Return a set of predicates (constants, not variables). :return: set of ``Variable`` objects """ return self.visit( lambda e: e.predicates(), lambda parts: reduce(operator.or_, parts, set()) ) def simplify(self): """ :return: beta-converted version of this expression """ return self.visit_structured(lambda e: e.simplify(), self.__class__) def make_VariableExpression(self, variable): return VariableExpression(variable) class ApplicationExpression(Expression): r""" This class is used to represent two related types of logical expressions. The first is a Predicate Expression, such as "P(x,y)". A predicate expression is comprised of a ``FunctionVariableExpression`` or ``ConstantExpression`` as the predicate and a list of Expressions as the arguments. The second is a an application of one expression to another, such as "(\x.dog(x))(fido)". The reason Predicate Expressions are treated as Application Expressions is that the Variable Expression predicate of the expression may be replaced with another Expression, such as a LambdaExpression, which would mean that the Predicate should be thought of as being applied to the arguments. The logical expression reader will always curry arguments in a application expression. So, "\x y.see(x,y)(john,mary)" will be represented internally as "((\x y.(see(x))(y))(john))(mary)". This simplifies the internals since there will always be exactly one argument in an application. The str() method will usually print the curried forms of application expressions. The one exception is when the the application expression is really a predicate expression (ie, underlying function is an ``AbstractVariableExpression``). This means that the example from above will be returned as "(\x y.see(x,y)(john))(mary)". """ def __init__(self, function, argument): """ :param function: ``Expression``, for the function expression :param argument: ``Expression``, for the argument """ assert isinstance(function, Expression), "%s is not an Expression" % function assert isinstance(argument, Expression), "%s is not an Expression" % argument self.function = function self.argument = argument def simplify(self): function = self.function.simplify() argument = self.argument.simplify() if isinstance(function, LambdaExpression): return function.term.replace(function.variable, argument).simplify() else: return self.__class__(function, argument) @property def type(self): if isinstance(self.function.type, ComplexType): return self.function.type.second else: return ANY_TYPE def _set_type(self, other_type=ANY_TYPE, signature=None): """:see Expression._set_type()""" assert isinstance(other_type, Type) if signature is None: signature = defaultdict(list) self.argument._set_type(ANY_TYPE, signature) try: self.function._set_type( ComplexType(self.argument.type, other_type), signature ) except TypeResolutionException as e: raise TypeException( "The function '%s' is of type '%s' and cannot be applied " "to '%s' of type '%s'. Its argument must match type '%s'." % ( self.function, self.function.type, self.argument, self.argument.type, self.function.type.first, ) ) from e def findtype(self, variable): """:see Expression.findtype()""" assert isinstance(variable, Variable), "%s is not a Variable" % variable if self.is_atom(): function, args = self.uncurry() else: # It's not a predicate expression ("P(x,y)"), so leave args curried function = self.function args = [self.argument] found = [arg.findtype(variable) for arg in [function] + args] unique = [] for f in found: if f != ANY_TYPE: if unique: for u in unique: if f.matches(u): break else: unique.append(f) if len(unique) == 1: return list(unique)[0] else: return ANY_TYPE def constants(self): """:see: Expression.constants()""" if isinstance(self.function, AbstractVariableExpression): function_constants = set() else: function_constants = self.function.constants() return function_constants | self.argument.constants() def predicates(self): """:see: Expression.predicates()""" if isinstance(self.function, ConstantExpression): function_preds = {self.function.variable} else: function_preds = self.function.predicates() return function_preds | self.argument.predicates() def visit(self, function, combinator): """:see: Expression.visit()""" return combinator([function(self.function), function(self.argument)]) def __eq__(self, other): return ( isinstance(other, ApplicationExpression) and self.function == other.function and self.argument == other.argument ) def __ne__(self, other): return not self == other __hash__ = Expression.__hash__ def __str__(self): # uncurry the arguments and find the base function if self.is_atom(): function, args = self.uncurry() arg_str = ",".join("%s" % arg for arg in args) else: # Leave arguments curried function = self.function arg_str = "%s" % self.argument function_str = "%s" % function parenthesize_function = False if isinstance(function, LambdaExpression): if isinstance(function.term, ApplicationExpression): if not isinstance(function.term.function, AbstractVariableExpression): parenthesize_function = True elif not isinstance(function.term, BooleanExpression): parenthesize_function = True elif isinstance(function, ApplicationExpression): parenthesize_function = True if parenthesize_function: function_str = Tokens.OPEN + function_str + Tokens.CLOSE return function_str + Tokens.OPEN + arg_str + Tokens.CLOSE def uncurry(self): """ Uncurry this application expression return: A tuple (base-function, arg-list) """ function = self.function args = [self.argument] while isinstance(function, ApplicationExpression): # (\x.\y.sees(x,y)(john))(mary) args.insert(0, function.argument) function = function.function return (function, args) @property def pred(self): """ Return uncurried base-function. If this is an atom, then the result will be a variable expression. Otherwise, it will be a lambda expression. """ return self.uncurry()[0] @property def args(self): """ Return uncurried arg-list """ return self.uncurry()[1] def is_atom(self): """ Is this expression an atom (as opposed to a lambda expression applied to a term)? """ return isinstance(self.pred, AbstractVariableExpression) @total_ordering class AbstractVariableExpression(Expression): """This class represents a variable to be used as a predicate or entity""" def __init__(self, variable): """ :param variable: ``Variable``, for the variable """ assert isinstance(variable, Variable), "%s is not a Variable" % variable self.variable = variable def simplify(self): return self def replace(self, variable, expression, replace_bound=False, alpha_convert=True): """:see: Expression.replace()""" assert isinstance(variable, Variable), "%s is not an Variable" % variable assert isinstance(expression, Expression), ( "%s is not an Expression" % expression ) if self.variable == variable: return expression else: return self def _set_type(self, other_type=ANY_TYPE, signature=None): """:see Expression._set_type()""" assert isinstance(other_type, Type) if signature is None: signature = defaultdict(list) resolution = other_type for varEx in signature[self.variable.name]: resolution = varEx.type.resolve(resolution) if not resolution: raise InconsistentTypeHierarchyException(self) signature[self.variable.name].append(self) for varEx in signature[self.variable.name]: varEx.type = resolution def findtype(self, variable): """:see Expression.findtype()""" assert isinstance(variable, Variable), "%s is not a Variable" % variable if self.variable == variable: return self.type else: return ANY_TYPE def predicates(self): """:see: Expression.predicates()""" return set() def __eq__(self, other): """Allow equality between instances of ``AbstractVariableExpression`` subtypes.""" return ( isinstance(other, AbstractVariableExpression) and self.variable == other.variable ) def __ne__(self, other): return not self == other def __lt__(self, other): if not isinstance(other, AbstractVariableExpression): raise TypeError return self.variable < other.variable __hash__ = Expression.__hash__ def __str__(self): return "%s" % self.variable class IndividualVariableExpression(AbstractVariableExpression): """This class represents variables that take the form of a single lowercase character (other than 'e') followed by zero or more digits.""" def _set_type(self, other_type=ANY_TYPE, signature=None): """:see Expression._set_type()""" assert isinstance(other_type, Type) if signature is None: signature = defaultdict(list) if not other_type.matches(ENTITY_TYPE): raise IllegalTypeException(self, other_type, ENTITY_TYPE) signature[self.variable.name].append(self) def _get_type(self): return ENTITY_TYPE type = property(_get_type, _set_type) def free(self): """:see: Expression.free()""" return {self.variable} def constants(self): """:see: Expression.constants()""" return set() class FunctionVariableExpression(AbstractVariableExpression): """This class represents variables that take the form of a single uppercase character followed by zero or more digits.""" type = ANY_TYPE def free(self): """:see: Expression.free()""" return {self.variable} def constants(self): """:see: Expression.constants()""" return set() class EventVariableExpression(IndividualVariableExpression): """This class represents variables that take the form of a single lowercase 'e' character followed by zero or more digits.""" type = EVENT_TYPE class ConstantExpression(AbstractVariableExpression): """This class represents variables that do not take the form of a single character followed by zero or more digits.""" type = ENTITY_TYPE def _set_type(self, other_type=ANY_TYPE, signature=None): """:see Expression._set_type()""" assert isinstance(other_type, Type) if signature is None: signature = defaultdict(list) if other_type == ANY_TYPE: # entity type by default, for individuals resolution = ENTITY_TYPE else: resolution = other_type if self.type != ENTITY_TYPE: resolution = resolution.resolve(self.type) for varEx in signature[self.variable.name]: resolution = varEx.type.resolve(resolution) if not resolution: raise InconsistentTypeHierarchyException(self) signature[self.variable.name].append(self) for varEx in signature[self.variable.name]: varEx.type = resolution def free(self): """:see: Expression.free()""" return set() def constants(self): """:see: Expression.constants()""" return {self.variable} def VariableExpression(variable): """ This is a factory method that instantiates and returns a subtype of ``AbstractVariableExpression`` appropriate for the given variable. """ assert isinstance(variable, Variable), "%s is not a Variable" % variable if is_indvar(variable.name): return IndividualVariableExpression(variable) elif is_funcvar(variable.name): return FunctionVariableExpression(variable) elif is_eventvar(variable.name): return EventVariableExpression(variable) else: return ConstantExpression(variable) class VariableBinderExpression(Expression): """This an abstract class for any Expression that binds a variable in an Expression. This includes LambdaExpressions and Quantified Expressions""" def __init__(self, variable, term): """ :param variable: ``Variable``, for the variable :param term: ``Expression``, for the term """ assert isinstance(variable, Variable), "%s is not a Variable" % variable assert isinstance(term, Expression), "%s is not an Expression" % term self.variable = variable self.term = term def replace(self, variable, expression, replace_bound=False, alpha_convert=True): """:see: Expression.replace()""" assert isinstance(variable, Variable), "%s is not a Variable" % variable assert isinstance(expression, Expression), ( "%s is not an Expression" % expression ) # if the bound variable is the thing being replaced if self.variable == variable: if replace_bound: assert isinstance(expression, AbstractVariableExpression), ( "%s is not a AbstractVariableExpression" % expression ) return self.__class__( expression.variable, self.term.replace(variable, expression, True, alpha_convert), ) else: return self else: # if the bound variable appears in the expression, then it must # be alpha converted to avoid a conflict if alpha_convert and self.variable in expression.free(): self = self.alpha_convert(unique_variable(pattern=self.variable)) # replace in the term return self.__class__( self.variable, self.term.replace(variable, expression, replace_bound, alpha_convert), ) def alpha_convert(self, newvar): """Rename all occurrences of the variable introduced by this variable binder in the expression to ``newvar``. :param newvar: ``Variable``, for the new variable """ assert isinstance(newvar, Variable), "%s is not a Variable" % newvar return self.__class__( newvar, self.term.replace(self.variable, VariableExpression(newvar), True) ) def free(self): """:see: Expression.free()""" return self.term.free() - {self.variable} def findtype(self, variable): """:see Expression.findtype()""" assert isinstance(variable, Variable), "%s is not a Variable" % variable if variable == self.variable: return ANY_TYPE else: return self.term.findtype(variable) def visit(self, function, combinator): """:see: Expression.visit()""" return combinator([function(self.term)]) def visit_structured(self, function, combinator): """:see: Expression.visit_structured()""" return combinator(self.variable, function(self.term)) def __eq__(self, other): r"""Defines equality modulo alphabetic variance. If we are comparing \x.M and \y.N, then check equality of M and N[x/y].""" if isinstance(self, other.__class__) or isinstance(other, self.__class__): if self.variable == other.variable: return self.term == other.term else: # Comparing \x.M and \y.N. Relabel y in N with x and continue. varex = VariableExpression(self.variable) return self.term == other.term.replace(other.variable, varex) else: return False def __ne__(self, other): return not self == other __hash__ = Expression.__hash__ class LambdaExpression(VariableBinderExpression): @property def type(self): return ComplexType(self.term.findtype(self.variable), self.term.type) def _set_type(self, other_type=ANY_TYPE, signature=None): """:see Expression._set_type()""" assert isinstance(other_type, Type) if signature is None: signature = defaultdict(list) self.term._set_type(other_type.second, signature) if not self.type.resolve(other_type): raise TypeResolutionException(self, other_type) def __str__(self): variables = [self.variable] term = self.term while term.__class__ == self.__class__: variables.append(term.variable) term = term.term return ( Tokens.LAMBDA + " ".join("%s" % v for v in variables) + Tokens.DOT + "%s" % term ) class QuantifiedExpression(VariableBinderExpression): @property def type(self): return TRUTH_TYPE def _set_type(self, other_type=ANY_TYPE, signature=None): """:see Expression._set_type()""" assert isinstance(other_type, Type) if signature is None: signature = defaultdict(list) if not other_type.matches(TRUTH_TYPE): raise IllegalTypeException(self, other_type, TRUTH_TYPE) self.term._set_type(TRUTH_TYPE, signature) def __str__(self): variables = [self.variable] term = self.term while term.__class__ == self.__class__: variables.append(term.variable) term = term.term return ( self.getQuantifier() + " " + " ".join("%s" % v for v in variables) + Tokens.DOT + "%s" % term ) class ExistsExpression(QuantifiedExpression): def getQuantifier(self): return Tokens.EXISTS class AllExpression(QuantifiedExpression): def getQuantifier(self): return Tokens.ALL class IotaExpression(QuantifiedExpression): def getQuantifier(self): return Tokens.IOTA class NegatedExpression(Expression): def __init__(self, term): assert isinstance(term, Expression), "%s is not an Expression" % term self.term = term @property def type(self): return TRUTH_TYPE def _set_type(self, other_type=ANY_TYPE, signature=None): """:see Expression._set_type()""" assert isinstance(other_type, Type) if signature is None: signature = defaultdict(list) if not other_type.matches(TRUTH_TYPE): raise IllegalTypeException(self, other_type, TRUTH_TYPE) self.term._set_type(TRUTH_TYPE, signature) def findtype(self, variable): assert isinstance(variable, Variable), "%s is not a Variable" % variable return self.term.findtype(variable) def visit(self, function, combinator): """:see: Expression.visit()""" return combinator([function(self.term)]) def negate(self): """:see: Expression.negate()""" return self.term def __eq__(self, other): return isinstance(other, NegatedExpression) and self.term == other.term def __ne__(self, other): return not self == other __hash__ = Expression.__hash__ def __str__(self): return Tokens.NOT + "%s" % self.term class BinaryExpression(Expression): def __init__(self, first, second): assert isinstance(first, Expression), "%s is not an Expression" % first assert isinstance(second, Expression), "%s is not an Expression" % second self.first = first self.second = second @property def type(self): return TRUTH_TYPE def findtype(self, variable): """:see Expression.findtype()""" assert isinstance(variable, Variable), "%s is not a Variable" % variable f = self.first.findtype(variable) s = self.second.findtype(variable) if f == s or s == ANY_TYPE: return f elif f == ANY_TYPE: return s else: return ANY_TYPE def visit(self, function, combinator): """:see: Expression.visit()""" return combinator([function(self.first), function(self.second)]) def __eq__(self, other): return ( (isinstance(self, other.__class__) or isinstance(other, self.__class__)) and self.first == other.first and self.second == other.second ) def __ne__(self, other): return not self == other __hash__ = Expression.__hash__ def __str__(self): first = self._str_subex(self.first) second = self._str_subex(self.second) return Tokens.OPEN + first + " " + self.getOp() + " " + second + Tokens.CLOSE def _str_subex(self, subex): return "%s" % subex class BooleanExpression(BinaryExpression): def _set_type(self, other_type=ANY_TYPE, signature=None): """:see Expression._set_type()""" assert isinstance(other_type, Type) if signature is None: signature = defaultdict(list) if not other_type.matches(TRUTH_TYPE): raise IllegalTypeException(self, other_type, TRUTH_TYPE) self.first._set_type(TRUTH_TYPE, signature) self.second._set_type(TRUTH_TYPE, signature) class AndExpression(BooleanExpression): """This class represents conjunctions""" def getOp(self): return Tokens.AND def _str_subex(self, subex): s = "%s" % subex if isinstance(subex, AndExpression): return s[1:-1] return s class OrExpression(BooleanExpression): """This class represents disjunctions""" def getOp(self): return Tokens.OR def _str_subex(self, subex): s = "%s" % subex if isinstance(subex, OrExpression): return s[1:-1] return s class ImpExpression(BooleanExpression): """This class represents implications""" def getOp(self): return Tokens.IMP class IffExpression(BooleanExpression): """This class represents biconditionals""" def getOp(self): return Tokens.IFF class EqualityExpression(BinaryExpression): """This class represents equality expressions like "(x = y)".""" def _set_type(self, other_type=ANY_TYPE, signature=None): """:see Expression._set_type()""" assert isinstance(other_type, Type) if signature is None: signature = defaultdict(list) if not other_type.matches(TRUTH_TYPE): raise IllegalTypeException(self, other_type, TRUTH_TYPE) self.first._set_type(ENTITY_TYPE, signature) self.second._set_type(ENTITY_TYPE, signature) def getOp(self): return Tokens.EQ ### Utilities class LogicalExpressionException(Exception): def __init__(self, index, message): self.index = index Exception.__init__(self, message) class UnexpectedTokenException(LogicalExpressionException): def __init__(self, index, unexpected=None, expected=None, message=None): if unexpected and expected: msg = "Unexpected token: '%s'. " "Expected token '%s'." % ( unexpected, expected, ) elif unexpected: msg = "Unexpected token: '%s'." % unexpected if message: msg += " " + message else: msg = "Expected token '%s'." % expected LogicalExpressionException.__init__(self, index, msg) class ExpectedMoreTokensException(LogicalExpressionException): def __init__(self, index, message=None): if not message: message = "More tokens expected." LogicalExpressionException.__init__( self, index, "End of input found. " + message ) def is_indvar(expr): """ An individual variable must be a single lowercase character other than 'e', followed by zero or more digits. :param expr: str :return: bool True if expr is of the correct form """ assert isinstance(expr, str), "%s is not a string" % expr return re.match(r"^[a-df-z]\d*$", expr) is not None def is_funcvar(expr): """ A function variable must be a single uppercase character followed by zero or more digits. :param expr: str :return: bool True if expr is of the correct form """ assert isinstance(expr, str), "%s is not a string" % expr return re.match(r"^[A-Z]\d*$", expr) is not None def is_eventvar(expr): """ An event variable must be a single lowercase 'e' character followed by zero or more digits. :param expr: str :return: bool True if expr is of the correct form """ assert isinstance(expr, str), "%s is not a string" % expr return re.match(r"^e\d*$", expr) is not None def demo(): lexpr = Expression.fromstring print("=" * 20 + "Test reader" + "=" * 20) print(lexpr(r"john")) print(lexpr(r"man(x)")) print(lexpr(r"-man(x)")) print(lexpr(r"(man(x) & tall(x) & walks(x))")) print(lexpr(r"exists x.(man(x) & tall(x) & walks(x))")) print(lexpr(r"\x.man(x)")) print(lexpr(r"\x.man(x)(john)")) print(lexpr(r"\x y.sees(x,y)")) print(lexpr(r"\x y.sees(x,y)(a,b)")) print(lexpr(r"(\x.exists y.walks(x,y))(x)")) print(lexpr(r"exists x.x = y")) print(lexpr(r"exists x.(x = y)")) print(lexpr("P(x) & x=y & P(y)")) print(lexpr(r"\P Q.exists x.(P(x) & Q(x))")) print(lexpr(r"man(x) <-> tall(x)")) print("=" * 20 + "Test simplify" + "=" * 20) print(lexpr(r"\x.\y.sees(x,y)(john)(mary)").simplify()) print(lexpr(r"\x.\y.sees(x,y)(john, mary)").simplify()) print(lexpr(r"all x.(man(x) & (\x.exists y.walks(x,y))(x))").simplify()) print(lexpr(r"(\P.\Q.exists x.(P(x) & Q(x)))(\x.dog(x))(\x.bark(x))").simplify()) print("=" * 20 + "Test alpha conversion and binder expression equality" + "=" * 20) e1 = lexpr("exists x.P(x)") print(e1) e2 = e1.alpha_convert(Variable("z")) print(e2) print(e1 == e2) def demo_errors(): print("=" * 20 + "Test reader errors" + "=" * 20) demoException("(P(x) & Q(x)") demoException("((P(x) &) & Q(x))") demoException("P(x) -> ") demoException("P(x") demoException("P(x,") demoException("P(x,)") demoException("exists") demoException("exists x.") demoException("\\") demoException("\\ x y.") demoException("P(x)Q(x)") demoException("(P(x)Q(x)") demoException("exists x -> y") def demoException(s): try: Expression.fromstring(s) except LogicalExpressionException as e: print(f"{e.__class__.__name__}: {e}") def printtype(ex): print(f"{ex.str()} : {ex.type}") if __name__ == "__main__": demo() # demo_errors()