X-Git-Url: https://scm.cri.ensmp.fr/git/linpy.git/blobdiff_plain/7fff25bf40e4db570565586fb49165d1675002c2..916d092d8a3bc9b0f2097c4f6c953f3a69b668e7:/linpy/linexprs.py

diff --git a/linpy/linexprs.py b/linpy/linexprs.py
index bc36fda..3c00f22 100644
--- a/linpy/linexprs.py
+++ b/linpy/linexprs.py
@@ -25,7 +25,7 @@ from fractions import Fraction, gcd
 
 
 __all__ = [
-    'Expression',
+    'LinExpr',
     'Symbol', 'Dummy', 'symbols',
     'Rational',
 ]
@@ -34,7 +34,7 @@ __all__ = [
 def _polymorphic(func):
     @functools.wraps(func)
     def wrapper(left, right):
-        if isinstance(right, Expression):
+        if isinstance(right, LinExpr):
             return func(left, right)
         elif isinstance(right, numbers.Rational):
             right = Rational(right)
@@ -43,19 +43,50 @@ def _polymorphic(func):
     return wrapper
 
 
-class Expression:
+class LinExpr:
     """
-    This class implements linear expressions.
+    A linear expression consists of a list of coefficient-variable pairs
+    that capture the linear terms, plus a constant term. Linear expressions
+    are used to build constraints. They are temporary objects that typically
+    have short lifespans.
+
+    Linear expressions are generally built using overloaded operators. For
+    example, if x is a Symbol, then x + 1 is an instance of LinExpr.
+
+    LinExpr instances are hashable, and should be treated as immutable.
     """
 
     def __new__(cls, coefficients=None, constant=0):
         """
-        Create a new expression.
+        Return a linear expression from a dictionary or a sequence, that maps
+        symbols to their coefficients, and a constant term. The coefficients and
+        the constant term must be rational numbers.
+
+        For example, the linear expression x + 2*y + 1 can be constructed using
+        one of the following instructions:
+
+        >>> x, y = symbols('x y')
+        >>> LinExpr({x: 1, y: 2}, 1)
+        >>> LinExpr([(x, 1), (y, 2)], 1)
+
+        However, it may be easier to use overloaded operators:
+
+        >>> x, y = symbols('x y')
+        >>> x + 2*y + 1
+
+        Alternatively, linear expressions can be constructed from a string:
+
+        >>> LinExpr('x + 2y + 1')
+
+        A linear expression with a single symbol of coefficient 1 and no
+        constant term is automatically subclassed as a Symbol instance. A linear
+        expression with no symbol, only a constant term, is automatically
+        subclassed as a Rational instance.
         """
         if isinstance(coefficients, str):
             if constant != 0:
                 raise TypeError('too many arguments')
-            return Expression.fromstring(coefficients)
+            return LinExpr.fromstring(coefficients)
         if coefficients is None:
             return Rational(constant)
         if isinstance(coefficients, Mapping):
@@ -86,39 +117,42 @@ class Expression:
 
     def coefficient(self, symbol):
         """
-        Return the coefficient value of the given symbol.
+        Return the coefficient value of the given symbol, or 0 if the symbol
+        does not appear in the expression.
         """
         if not isinstance(symbol, Symbol):
             raise TypeError('symbol must be a Symbol instance')
-        return Rational(self._coefficients.get(symbol, 0))
+        return self._coefficients.get(symbol, Fraction(0))
 
     __getitem__ = coefficient
 
     def coefficients(self):
         """
-        Return a list of the coefficients of an expression
+        Iterate over the pairs (symbol, value) of linear terms in the
+        expression. The constant term is ignored.
         """
-        for symbol, coefficient in self._coefficients.items():
-            yield symbol, Rational(coefficient)
+        yield from self._coefficients.items()
 
     @property
     def constant(self):
         """
-        Return the constant value of an expression.
+        The constant term of the expression.
         """
-        return Rational(self._constant)
+        return self._constant
 
     @property
     def symbols(self):
         """
-        Return a list of symbols in an expression.
+        The tuple of symbols present in the expression, sorted according to
+        Symbol.sortkey().
         """
         return self._symbols
 
     @property
     def dimension(self):
         """
-        Create and return a new linear expression from a string or a list of coefficients and a constant.
+        The dimension of the expression, i.e. the number of symbols present in
+        it.
         """
         return self._dimension
 
@@ -127,23 +161,25 @@ class Expression:
 
     def isconstant(self):
         """
-        Return true if an expression is a constant.
+        Return True if the expression only consists of a constant term. In this
+        case, it is a Rational instance.
         """
         return False
 
     def issymbol(self):
         """
-        Return true if an expression is a symbol.
+        Return True if an expression only consists of a symbol with coefficient
+        1. In this case, it is a Symbol instance.
         """
         return False
 
     def values(self):
         """
-        Return the coefficient and constant values of an expression.
+        Iterate over the coefficient values in the expression, and the constant
+        term.
         """
-        for coefficient in self._coefficients.values():
-            yield Rational(coefficient)
-        yield Rational(self._constant)
+        yield from self._coefficients.values()
+        yield self._constant
 
     def __bool__(self):
         return True
@@ -157,26 +193,26 @@ class Expression:
     @_polymorphic
     def __add__(self, other):
         """
-        Return the sum of two expressions.
+        Return the sum of two linear expressions.
         """
         coefficients = defaultdict(Fraction, self._coefficients)
         for symbol, coefficient in other._coefficients.items():
             coefficients[symbol] += coefficient
         constant = self._constant + other._constant
-        return Expression(coefficients, constant)
+        return LinExpr(coefficients, constant)
 
     __radd__ = __add__
 
     @_polymorphic
     def __sub__(self, other):
         """
-        Return the difference between two expressions.
+        Return the difference between two linear expressions.
         """
         coefficients = defaultdict(Fraction, self._coefficients)
         for symbol, coefficient in other._coefficients.items():
             coefficients[symbol] -= coefficient
         constant = self._constant - other._constant
-        return Expression(coefficients, constant)
+        return LinExpr(coefficients, constant)
 
     @_polymorphic
     def __rsub__(self, other):
@@ -184,79 +220,96 @@ class Expression:
 
     def __mul__(self, other):
         """
-        Return the product of two expressions if other is a rational number.
+        Return the product of the linear expression by a rational.
         """
         if isinstance(other, numbers.Rational):
             coefficients = ((symbol, coefficient * other)
                 for symbol, coefficient in self._coefficients.items())
             constant = self._constant * other
-            return Expression(coefficients, constant)
+            return LinExpr(coefficients, constant)
         return NotImplemented
 
     __rmul__ = __mul__
 
     def __truediv__(self, other):
+        """
+        Return the quotient of the linear expression by a rational.
+        """
         if isinstance(other, numbers.Rational):
             coefficients = ((symbol, coefficient / other)
                 for symbol, coefficient in self._coefficients.items())
             constant = self._constant / other
-            return Expression(coefficients, constant)
+            return LinExpr(coefficients, constant)
         return NotImplemented
 
     @_polymorphic
     def __eq__(self, other):
         """
-        Test whether two expressions are equal
+        Test whether two linear expressions are equal. Unlike methods
+        LinExpr.__lt__(), LinExpr.__le__(), LinExpr.__ge__(), LinExpr.__gt__(),
+        the result is a boolean value, not a polyhedron. To express that two
+        linear expressions are equal or not equal, use functions Eq() and Ne()
+        instead.
         """
-        return isinstance(other, Expression) and \
-            self._coefficients == other._coefficients and \
+        return self._coefficients == other._coefficients and \
             self._constant == other._constant
 
-    def __le__(self, other):
-        from .polyhedra import Le
-        return Le(self, other)
-
+    @_polymorphic
     def __lt__(self, other):
-        from .polyhedra import Lt
-        return Lt(self, other)
+        from .polyhedra import Polyhedron
+        return Polyhedron([], [other - self - 1])
+
+    @_polymorphic
+    def __le__(self, other):
+        from .polyhedra import Polyhedron
+        return Polyhedron([], [other - self])
 
+    @_polymorphic
     def __ge__(self, other):
-        from .polyhedra import Ge
-        return Ge(self, other)
+        from .polyhedra import Polyhedron
+        return Polyhedron([], [self - other])
 
+    @_polymorphic
     def __gt__(self, other):
-        from .polyhedra import Gt
-        return Gt(self, other)
+        from .polyhedra import Polyhedron
+        return Polyhedron([], [self - other - 1])
 
     def scaleint(self):
         """
-        Multiply an expression by a scalar to make all coefficients integer values.
+        Return the expression multiplied by its lowest common denominator to
+        make all values integer.
         """
-        lcm = functools.reduce(lambda a, b: a*b // gcd(a, b),
+        lcd = functools.reduce(lambda a, b: a*b // gcd(a, b),
             [value.denominator for value in self.values()])
-        return self * lcm
+        return self * lcd
 
     def subs(self, symbol, expression=None):
         """
-        Subsitute symbol by expression in equations and return the resulting
-        expression.
+        Substitute the given symbol by an expression and return the resulting
+        expression. Raise TypeError if the resulting expression is not linear.
+
+        >>> x, y = symbols('x y')
+        >>> e = x + 2*y + 1
+        >>> e.subs(y, x - 1)
+        3*x - 1
+
+        To perform multiple substitutions at once, pass a sequence or a
+        dictionary of (old, new) pairs to subs.
+
+        >>> e.subs({x: y, y: x})
+        2*x + y + 1
         """
         if expression is None:
-            if isinstance(symbol, Mapping):
-                symbol = symbol.items()
-            substitutions = symbol
+            substitutions = dict(symbol)
         else:
-            substitutions = [(symbol, expression)]
-        result = self
-        for symbol, expression in substitutions:
+            substitutions = {symbol: expression}
+        for symbol in substitutions:
             if not isinstance(symbol, Symbol):
                 raise TypeError('symbols must be Symbol instances')
-            coefficients = [(othersymbol, coefficient)
-                for othersymbol, coefficient in result._coefficients.items()
-                if othersymbol != symbol]
-            coefficient = result._coefficients.get(symbol, 0)
-            constant = result._constant
-            result = Expression(coefficients, constant) + coefficient*expression
+        result = self._constant
+        for symbol, coefficient in self._coefficients.items():
+            expression = substitutions.get(symbol, symbol)
+            result += coefficient * expression
         return result
 
     @classmethod
@@ -284,17 +337,21 @@ class Expression:
                 return left / right
         raise SyntaxError('invalid syntax')
 
-    _RE_NUM_VAR = re.compile(r'(\d+|\))\s*([^\W\d_]\w*|\()')
+    _RE_NUM_VAR = re.compile(r'(\d+|\))\s*([^\W\d]\w*|\()')
 
     @classmethod
     def fromstring(cls, string):
         """
-        Create an expression from a string.
+        Create an expression from a string. Raise SyntaxError if the string is
+        not properly formatted.
         """
-        # add implicit multiplication operators, e.g. '5x' -> '5*x'
-        string = Expression._RE_NUM_VAR.sub(r'\1*\2', string)
+        # Add implicit multiplication operators, e.g. '5x' -> '5*x'.
+        string = LinExpr._RE_NUM_VAR.sub(r'\1*\2', string)
         tree = ast.parse(string, 'eval')
-        return cls._fromast(tree)
+        expression = cls._fromast(tree)
+        if not isinstance(expression, cls):
+            raise SyntaxError('invalid syntax')
+        return expression
 
     def __repr__(self):
         string = ''
@@ -320,30 +377,6 @@ class Expression:
             string += ' - {}'.format(-constant)
         return string
 
-    def _repr_latex_(self):
-        string = ''
-        for i, (symbol, coefficient) in enumerate(self.coefficients()):
-            if coefficient == 1:
-                if i != 0:
-                    string += ' + '
-            elif coefficient == -1:
-                string += '-' if i == 0 else ' - '
-            elif i == 0:
-                string += '{}'.format(coefficient._repr_latex_().strip('$'))
-            elif coefficient > 0:
-                string += ' + {}'.format(coefficient._repr_latex_().strip('$'))
-            elif coefficient < 0:
-                string += ' - {}'.format((-coefficient)._repr_latex_().strip('$'))
-            string += '{}'.format(symbol._repr_latex_().strip('$'))
-        constant = self.constant
-        if len(string) == 0:
-            string += '{}'.format(constant._repr_latex_().strip('$'))
-        elif constant > 0:
-            string += ' + {}'.format(constant._repr_latex_().strip('$'))
-        elif constant < 0:
-            string += ' - {}'.format((-constant)._repr_latex_().strip('$'))
-        return '$${}$$'.format(string)
-
     def _parenstr(self, always=False):
         string = str(self)
         if not always and (self.isconstant() or self.issymbol()):
@@ -352,123 +385,168 @@ class Expression:
             return '({})'.format(string)
 
     @classmethod
-    def fromsympy(cls, expr):
+    def fromsympy(cls, expression):
         """
-        Convert sympy object to an expression.
+        Create a linear expression from a SymPy expression. Raise TypeError is
+        the sympy expression is not linear.
         """
         import sympy
         coefficients = []
         constant = 0
-        for symbol, coefficient in expr.as_coefficients_dict().items():
+        for symbol, coefficient in expression.as_coefficients_dict().items():
             coefficient = Fraction(coefficient.p, coefficient.q)
             if symbol == sympy.S.One:
                 constant = coefficient
+            elif isinstance(symbol, sympy.Dummy):
+                # We cannot properly convert dummy symbols with respect to
+                # symbol equalities.
+                raise TypeError('cannot convert dummy symbols')
             elif isinstance(symbol, sympy.Symbol):
                 symbol = Symbol(symbol.name)
                 coefficients.append((symbol, coefficient))
             else:
-                raise ValueError('non-linear expression: {!r}'.format(expr))
-        return Expression(coefficients, constant)
+                raise TypeError('non-linear expression: {!r}'.format(expression))
+        expression = LinExpr(coefficients, constant)
+        if not isinstance(expression, cls):
+            raise TypeError('cannot convert to a {} instance'.format(cls.__name__))
+        return expression
 
     def tosympy(self):
         """
-        Return an expression as a sympy object.  
+        Convert the linear expression to a SymPy expression.
         """
         import sympy
-        expr = 0
+        expression = 0
         for symbol, coefficient in self.coefficients():
             term = coefficient * sympy.Symbol(symbol.name)
-            expr += term
-        expr += self.constant
-        return expr
+            expression += term
+        expression += self.constant
+        return expression
+
+
+class Symbol(LinExpr):
+    """
+    Symbols are the basic components to build expressions and constraints.
+    They correspond to mathematical variables. Symbols are instances of
+    class LinExpr and inherit its functionalities.
 
+    Two instances of Symbol are equal if they have the same name.
+    """
 
-class Symbol(Expression):
+    __slots__ = (
+        '_name',
+        '_constant',
+        '_symbols',
+        '_dimension',
+    )
 
     def __new__(cls, name):
         """
-        Create and return a symbol from a string.
+        Return a symbol with the name string given in argument.
         """
         if not isinstance(name, str):
             raise TypeError('name must be a string')
+        node = ast.parse(name)
+        try:
+            name = node.body[0].value.id
+        except (AttributeError, SyntaxError):
+            raise SyntaxError('invalid syntax')
         self = object().__new__(cls)
-        self._name = name.strip()
-        self._coefficients = {self: Fraction(1)}
+        self._name = name
         self._constant = Fraction(0)
         self._symbols = (self,)
         self._dimension = 1
         return self
 
+    @property
+    def _coefficients(self):
+        # This is not implemented as an attribute, because __hash__ is not
+        # callable in __new__ in class Dummy.
+        return {self: Fraction(1)}
+
     @property
     def name(self):
+        """
+        The name of the symbol.
+        """
         return self._name
 
     def __hash__(self):
         return hash(self.sortkey())
 
     def sortkey(self):
+        """
+        Return a sorting key for the symbol. It is useful to sort a list of
+        symbols in a consistent order, as comparison functions are overridden
+        (see the documentation of class LinExpr).
+
+        >>> sort(symbols, key=Symbol.sortkey)
+        """
         return self.name,
 
     def issymbol(self):
         return True
 
     def __eq__(self, other):
-        return self.sortkey() == other.sortkey()
+        if isinstance(other, Symbol):
+            return self.sortkey() == other.sortkey()
+        return NotImplemented
 
     def asdummy(self):
         """
-        Return a symbol as a Dummy Symbol.
+        Return a new Dummy symbol instance with the same name.
         """
         return Dummy(self.name)
 
-    @classmethod
-    def _fromast(cls, node):
-        if isinstance(node, ast.Module) and len(node.body) == 1:
-            return cls._fromast(node.body[0])
-        elif isinstance(node, ast.Expr):
-            return cls._fromast(node.value)
-        elif isinstance(node, ast.Name):
-            return Symbol(node.id)
-        raise SyntaxError('invalid syntax')
-
     def __repr__(self):
         return self.name
 
-    def _repr_latex_(self):
-        return '$${}$$'.format(self.name)
 
-    @classmethod
-    def fromsympy(cls, expr):
-        import sympy
-        if isinstance(expr, sympy.Dummy):
-            return Dummy(expr.name)
-        elif isinstance(expr, sympy.Symbol):
-            return Symbol(expr.name)
-        else:
-            raise TypeError('expr must be a sympy.Symbol instance')
+def symbols(names):
+    """
+    This function returns a tuple of symbols whose names are taken from a comma
+    or whitespace delimited string, or a sequence of strings. It is useful to
+    define several symbols at once.
+
+    >>> x, y = symbols('x y')
+    >>> x, y = symbols('x, y')
+    >>> x, y = symbols(['x', 'y'])
+    """
+    if isinstance(names, str):
+        names = names.replace(',', ' ').split()
+    return tuple(Symbol(name) for name in names)
 
 
 class Dummy(Symbol):
     """
-    This class returns a dummy symbol to ensure that no variables are repeated in an expression
+    A variation of Symbol in which all symbols are unique and identified by
+    an internal count index. If a name is not supplied then a string value
+    of the count index will be used. This is useful when a unique, temporary
+    variable is needed and the name of the variable used in the expression
+    is not important.
+
+    Unlike Symbol, Dummy instances with the same name are not equal:
+
+    >>> x = Symbol('x')
+    >>> x1, x2 = Dummy('x'), Dummy('x')
+    >>> x == x1
+    False
+    >>> x1 == x2
+    False
+    >>> x1 == x1
+    True
     """
+
     _count = 0
 
     def __new__(cls, name=None):
         """
-        Create and return a new dummy symbol.
+        Return a fresh dummy symbol with the name string given in argument.
         """
         if name is None:
             name = 'Dummy_{}'.format(Dummy._count)
-        elif not isinstance(name, str):
-            raise TypeError('name must be a string')
-        self = object().__new__(cls)
+        self = super().__new__(cls, name)
         self._index = Dummy._count
-        self._name = name.strip()
-        self._coefficients = {self: Fraction(1)}
-        self._constant = Fraction(0)
-        self._symbols = (self,)
-        self._dimension = 1
         Dummy._count += 1
         return self
 
@@ -481,23 +559,21 @@ class Dummy(Symbol):
     def __repr__(self):
         return '_{}'.format(self.name)
 
-    def _repr_latex_(self):
-        return '$${}_{{{}}}$$'.format(self.name, self._index)
 
-
-def symbols(names):
+class Rational(LinExpr, Fraction):
     """
-    Transform strings into instances of the Symbol class
+    A particular case of linear expressions are rational values, i.e. linear
+    expressions consisting only of a constant term, with no symbol. They are
+    implemented by the Rational class, that inherits from both LinExpr and
+    fractions.Fraction classes.
     """
-    if isinstance(names, str):
-        names = names.replace(',', ' ').split()
-    return tuple(Symbol(name) for name in names)
-
 
-class Rational(Expression, Fraction):
-    """
-    This class represents integers and rational numbers of any size.
-    """
+    __slots__ = (
+        '_coefficients',
+        '_constant',
+        '_symbols',
+        '_dimension',
+    ) + Fraction.__slots__
 
     def __new__(cls, numerator=0, denominator=None):
         self = object().__new__(cls)
@@ -514,15 +590,9 @@ class Rational(Expression, Fraction):
 
     @property
     def constant(self):
-        """
-        Return rational as a constant.
-        """
         return self
 
     def isconstant(self):
-        """
-        Test whether a value is a constant.
-        """
         return True
 
     def __bool__(self):
@@ -533,26 +603,3 @@ class Rational(Expression, Fraction):
             return '{!r}'.format(self.numerator)
         else:
             return '{!r}/{!r}'.format(self.numerator, self.denominator)
-
-    def _repr_latex_(self):
-        if self.denominator == 1:
-            return '$${}$$'.format(self.numerator)
-        elif self.numerator < 0:
-            return '$$-\\frac{{{}}}{{{}}}$$'.format(-self.numerator,
-                self.denominator)
-        else:
-            return '$$\\frac{{{}}}{{{}}}$$'.format(self.numerator,
-                self.denominator)
-
-    @classmethod
-    def fromsympy(cls, expr):
-        """
-        Create a rational object from a sympy expression
-        """
-        import sympy
-        if isinstance(expr, sympy.Rational):
-            return Rational(expr.p, expr.q)
-        elif isinstance(expr, numbers.Rational):
-            return Rational(expr)
-        else:
-            raise TypeError('expr must be a sympy.Rational instance')