Implement Polyhedron.fromstring

[linpy.git] / pypol / linear.py

diff --git a/pypol/linear.py b/pypol/linear.py
index a562a3f..0712e1e 100644 (file)
--- a/pypol/linear.py
+++ b/pypol/linear.py
@@ -1,19 +1,19 @@
-import ctypes, ctypes.util
+import ast

 import functools
 import numbers
 import functools
 import numbers

+import re

 
 from fractions import Fraction, gcd
 
 
 from fractions import Fraction, gcd
 

-from . import isl
-from .isl import libisl
+from pypol import isl
+from pypol.isl import libisl

 
 
 __all__ = [
 
 
 __all__ = [

-    'Expression',
-    'constant', 'symbol', 'symbols',
+    'Expression', 'Constant', 'Symbol', 'symbols',

     'eq', 'le', 'lt', 'ge', 'gt',
     'Polyhedron',
     'eq', 'le', 'lt', 'ge', 'gt',
     'Polyhedron',

-    'empty', 'universe'
+    'Empty', 'Universe'

 ]
 
 
 ]
 
 


@@ -23,7 +23,7 @@ def _polymorphic_method(func):
         if isinstance(b, Expression):
             return func(a, b)
         if isinstance(b, numbers.Rational):
         if isinstance(b, Expression):
             return func(a, b)
         if isinstance(b, numbers.Rational):

-            b = constant(b)
+            b = Constant(b)

             return func(a, b)
         return NotImplemented
     return wrapper
             return func(a, b)
         return NotImplemented
     return wrapper


@@ -34,7 +34,7 @@ def _polymorphic_operator(func):
     @functools.wraps(func)
     def wrapper(a, b):
         if isinstance(a, numbers.Rational):
     @functools.wraps(func)
     def wrapper(a, b):
         if isinstance(a, numbers.Rational):

-            a = constant(a)
+            a = Constant(a)

             return func(a, b)
         elif isinstance(a, Expression):
             return func(a, b)
             return func(a, b)
         elif isinstance(a, Expression):
             return func(a, b)


@@ -55,27 +55,72 @@ class Expression:
             if constant:
                 raise TypeError('too many arguments')
             return cls.fromstring(coefficients)
             if constant:
                 raise TypeError('too many arguments')
             return cls.fromstring(coefficients)

-        self = super().__new__(cls)
-        self._coefficients = {}

         if isinstance(coefficients, dict):
             coefficients = coefficients.items()
         if isinstance(coefficients, dict):
             coefficients = coefficients.items()

-        if coefficients is not None:
-            for symbol, coefficient in coefficients:
-                if isinstance(symbol, Expression) and symbol.issymbol():
-                    symbol = str(symbol)
-                elif not isinstance(symbol, str):
-                    raise TypeError('symbols must be strings')
-                if not isinstance(coefficient, numbers.Rational):
-                    raise TypeError('coefficients must be rational numbers')
-                if coefficient != 0:
-                    self._coefficients[symbol] = coefficient
+        if coefficients is None:
+            return Constant(constant)
+        coefficients = [(symbol, coefficient)
+                for symbol, coefficient in coefficients if coefficient != 0]
+        if len(coefficients) == 0:
+            return Constant(constant)
+        elif len(coefficients) == 1 and constant == 0:
+            symbol, coefficient = coefficients[0]
+            if coefficient == 1:
+                return Symbol(symbol)
+        self = object().__new__(cls)
+        self._coefficients = {}
+        for symbol, coefficient in coefficients:
+            if isinstance(symbol, Symbol):
+                symbol = symbol.name
+            elif not isinstance(symbol, str):
+                raise TypeError('symbols must be strings or Symbol instances')
+            if isinstance(coefficient, Constant):
+                coefficient = coefficient.constant
+            if not isinstance(coefficient, numbers.Rational):
+                raise TypeError('coefficients must be rational numbers or Constant instances')
+            self._coefficients[symbol] = coefficient
+        if isinstance(constant, Constant):
+            constant = constant.constant

         if not isinstance(constant, numbers.Rational):
         if not isinstance(constant, numbers.Rational):

-            raise TypeError('constant must be a rational number')
+            raise TypeError('constant must be a rational number or a Constant instance')

         self._constant = constant
         self._symbols = tuple(sorted(self._coefficients))
         self._dimension = len(self._symbols)
         return self
 
         self._constant = constant
         self._symbols = tuple(sorted(self._coefficients))
         self._dimension = len(self._symbols)
         return self
 

+    @classmethod
+    def _fromast(cls, node):
+        if isinstance(node, ast.Module):
+            assert 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)
+        elif isinstance(node, ast.Num):
+            return Constant(node.n)
+        elif isinstance(node, ast.UnaryOp):
+            if isinstance(node.op, ast.USub):
+                return -cls._fromast(node.operand)
+        elif isinstance(node, ast.BinOp):
+            left = cls._fromast(node.left)
+            right = cls._fromast(node.right)
+            if isinstance(node.op, ast.Add):
+                return left + right
+            elif isinstance(node.op, ast.Sub):
+                return left - right
+            elif isinstance(node.op, ast.Mult):
+                return left * right
+            elif isinstance(node.op, ast.Div):
+                return left / right
+        raise SyntaxError('invalid syntax')
+
+    @classmethod
+    def fromstring(cls, string):
+        string = re.sub(r'(\d+|\))\s*([^\W\d_]\w*|\()', r'\1*\2', string)
+        tree = ast.parse(string, 'eval')
+        return cls._fromast(tree)
+

     @property
     def symbols(self):
         return self._symbols
     @property
     def symbols(self):
         return self._symbols


@@ -85,10 +130,10 @@ class Expression:
         return self._dimension
 
     def coefficient(self, symbol):
         return self._dimension
 
     def coefficient(self, symbol):

-        if isinstance(symbol, Expression) and symbol.issymbol():
+        if isinstance(symbol, Symbol):

             symbol = str(symbol)
         elif not isinstance(symbol, str):
             symbol = str(symbol)
         elif not isinstance(symbol, str):

-            raise TypeError('symbol must be a string')
+            raise TypeError('symbol must be a string or a Symbol instance')

         try:
             return self._coefficients[symbol]
         except KeyError:
         try:
             return self._coefficients[symbol]
         except KeyError:


@@ -105,30 +150,18 @@ class Expression:
         return self._constant
 
     def isconstant(self):
         return self._constant
 
     def isconstant(self):

-        return len(self._coefficients) == 0
+        return False

 
     def values(self):
         for symbol in self.symbols:
             yield self.coefficient(symbol)
         yield self.constant
 
 
     def values(self):
         for symbol in self.symbols:
             yield self.coefficient(symbol)
         yield self.constant
 

-    def values_int(self):
-        for symbol in self.symbols:
-            return self.coefficient(symbol)
-        return int(self.constant)
-
-    @property
-    def symbol(self):
-        if not self.issymbol():
-            raise ValueError('not a symbol: {}'.format(self))
-        for symbol in self.symbols:
-            return symbol
-

     def issymbol(self):
     def issymbol(self):

-        return len(self._coefficients) == 1 and self._constant == 0
+        return False

 
     def __bool__(self):
 
     def __bool__(self):

-        return (not self.isconstant()) or bool(self.constant)
+        return True

 
     def __pos__(self):
         return self
 
     def __pos__(self):
         return self


@@ -139,7 +172,7 @@ class Expression:
     @_polymorphic_method
     def __add__(self, other):
         coefficients = dict(self.coefficients())
     @_polymorphic_method
     def __add__(self, other):
         coefficients = dict(self.coefficients())

-        for symbol, coefficient in other.coefficients:
+        for symbol, coefficient in other.coefficients():

             if symbol in coefficients:
                 coefficients[symbol] += coefficient
             else:
             if symbol in coefficients:
                 coefficients[symbol] += coefficient
             else:


@@ -152,7 +185,7 @@ class Expression:
     @_polymorphic_method
     def __sub__(self, other):
         coefficients = dict(self.coefficients())
     @_polymorphic_method
     def __sub__(self, other):
         coefficients = dict(self.coefficients())

-        for symbol, coefficient in other.coefficients:
+        for symbol, coefficient in other.coefficients():

             if symbol in coefficients:
                 coefficients[symbol] -= coefficient
             else:
             if symbol in coefficients:
                 coefficients[symbol] -= coefficient
             else:


@@ -205,8 +238,8 @@ class Expression:
     def __str__(self):
         string = ''
         i = 0
     def __str__(self):
         string = ''
         i = 0

-        for symbol in symbols:
-            coefficient = self[symbol]
+        for symbol in self.symbols:
+            coefficient = self.coefficient(symbol)

             if coefficient == 1:
                 if i == 0:
                     string += symbol
             if coefficient == 1:
                 if i == 0:
                     string += symbol


@@ -255,10 +288,6 @@ class Expression:
         string += '}}, {!r})'.format(self.constant)
         return string
 
         string += '}}, {!r})'.format(self.constant)
         return string
 

-    @classmethod
-    def fromstring(cls, string):
-        raise NotImplementedError
-

     @_polymorphic_method
     def __eq__(self, other):
         # "normal" equality
     @_polymorphic_method
     def __eq__(self, other):
         # "normal" equality


@@ -268,7 +297,7 @@ class Expression:
                 self.constant == other.constant
 
     def __hash__(self):
                 self.constant == other.constant
 
     def __hash__(self):

-        return hash((self._coefficients, self._constant))
+        return hash((tuple(sorted(self._coefficients.items())), self._constant))

 
     def _toint(self):
         lcm = functools.reduce(lambda a, b: a*b // gcd(a, b),
 
     def _toint(self):
         lcm = functools.reduce(lambda a, b: a*b // gcd(a, b),


@@ -296,42 +325,84 @@ class Expression:
         return Polyhedron(inequalities=[(self - other)._toint() - 1])
 
 
         return Polyhedron(inequalities=[(self - other)._toint() - 1])
 
 

-def constant(numerator=0, denominator=None):
-    if denominator is None and isinstance(numerator, numbers.Rational):
-        return Expression(constant=numerator)
-    else:
-        return Expression(constant=Fraction(numerator, denominator))
+class Constant(Expression):
+
+    def __new__(cls, numerator=0, denominator=None):
+        self = object().__new__(cls)
+        if denominator is None:
+            if isinstance(numerator, numbers.Rational):
+                self._constant = numerator
+            elif isinstance(numerator, Constant):
+                self._constant = numerator.constant
+            else:
+                raise TypeError('constant must be a rational number or a Constant instance')
+        else:
+            self._constant = Fraction(numerator, denominator)
+        self._coefficients = {}
+        self._symbols = ()
+        self._dimension = 0
+        return self
+
+    def isconstant(self):
+        return True
+
+    def __bool__(self):
+        return bool(self.constant)
+
+    def __repr__(self):
+        return '{}({!r})'.format(self.__class__.__name__, self._constant)
+
+
+class Symbol(Expression):
+
+    def __new__(cls, name):
+        if isinstance(name, Symbol):
+            name = name.name
+        elif not isinstance(name, str):
+            raise TypeError('name must be a string or a Symbol instance')
+        self = object().__new__(cls)
+        self._coefficients = {name: 1}
+        self._constant = 0
+        self._symbols = tuple(name)
+        self._name = name
+        self._dimension = 1
+        return self
+
+    @property
+    def name(self):
+        return self._name
+
+    def issymbol(self):
+        return True

 
 

-def symbol(name):
-    if not isinstance(name, str):
-        raise TypeError('name must be a string')
-    return Expression(coefficients={name: 1})
+    def __repr__(self):
+        return '{}({!r})'.format(self.__class__.__name__, self._name)

 
 def symbols(names):
     if isinstance(names, str):
         names = names.replace(',', ' ').split()
 
 def symbols(names):
     if isinstance(names, str):
         names = names.replace(',', ' ').split()

-    return (symbol(name) for name in names)
+    return (Symbol(name) for name in names)

 
 
 @_polymorphic_operator
 def eq(a, b):
 
 
 @_polymorphic_operator
 def eq(a, b):

-    return a._eq(b)
+    return a.__eq__(b)

 
 @_polymorphic_operator
 def le(a, b):
 
 @_polymorphic_operator
 def le(a, b):

-    return a <= b
+    return a.__le__(b)

 
 @_polymorphic_operator
 def lt(a, b):
 
 @_polymorphic_operator
 def lt(a, b):

-    return a < b
+    return a.__lt__(b)

 
 @_polymorphic_operator
 def ge(a, b):
 
 @_polymorphic_operator
 def ge(a, b):

-    return a >= b
+    return a.__ge__(b)

 
 @_polymorphic_operator
 def gt(a, b):
 
 @_polymorphic_operator
 def gt(a, b):

-    return a > b
+    return a.__gt__(b)

 
 
 class Polyhedron:
 
 
 class Polyhedron:


@@ -370,6 +441,43 @@ class Polyhedron:
         self._symbols = tuple(sorted(self._symbols))
         return self
 
         self._symbols = tuple(sorted(self._symbols))
         return self
 

+    @classmethod
+    def fromstring(cls, string):
+        string = string.strip()
+        string = re.sub(r'^\{\s*|\s*\}$', '', string)
+        string = re.sub(r'([^<=>])=([^<=>])', r'\1==\2', string)
+        string = re.sub(r'(\d+|\))\s*([^\W\d_]\w*|\()', r'\1*\2', string)
+        equalities = []
+        inequalities = []
+        for cstr in re.split(r',|;|and|&&|/\\|∧', string, flags=re.I):
+            tree = ast.parse(cstr.strip(), 'eval')
+            if not isinstance(tree, ast.Module) or len(tree.body) != 1:
+                raise SyntaxError('invalid syntax')
+            node = tree.body[0]
+            if not isinstance(node, ast.Expr):
+                raise SyntaxError('invalid syntax')
+            node = node.value
+            if not isinstance(node, ast.Compare):
+                raise SyntaxError('invalid syntax')
+            left = Expression._fromast(node.left)
+            for i in range(len(node.ops)):
+                op = node.ops[i]
+                right = Expression._fromast(node.comparators[i])
+                if isinstance(op, ast.Lt):
+                    inequalities.append(right - left - 1)
+                elif isinstance(op, ast.LtE):
+                    inequalities.append(right - left)
+                elif isinstance(op, ast.Eq):
+                    equalities.append(left - right)
+                elif isinstance(op, ast.GtE):
+                    inequalities.append(left - right)
+                elif isinstance(op, ast.Gt):
+                    inequalities.append(left - right - 1)
+                else:
+                    raise SyntaxError('invalid syntax')
+                left = right
+        return cls(equalities, inequalities)
+

     @property
     def equalities(self):
         return self._equalities
     @property
     def equalities(self):
         return self._equalities


@@ -378,9 +486,6 @@ class Polyhedron:
     def inequalities(self):
         return self._inequalities
 
     def inequalities(self):
         return self._inequalities
 

-    def isempty(self):
-        return bool(libisl.isl_basic_set_is_empty(self._bset))
-

     @property
     def constraints(self):
         return self._constraints
     @property
     def constraints(self):
         return self._constraints


@@ -401,26 +506,42 @@ class Polyhedron:
         raise NotImplementedError
 
     def __eq__(self, other):
         raise NotImplementedError
 
     def __eq__(self, other):

-        raise NotImplementedError
+        # works correctly when symbols is not passed
+        # should be equal if values are the same even if symbols are different
+        bset = self._toisl()
+        other = other._toisl()
+        return bool(libisl.isl_basic_set_plain_is_equal(bset, other))

 
     def isempty(self):
 
     def isempty(self):

-        return self == empty
+        bset = self._toisl()
+        return bool(libisl.isl_basic_set_is_empty(bset))

 
     def isuniverse(self):
 
     def isuniverse(self):

-        return self == universe
+        bset = self._toisl()
+        return bool(libisl.isl_basic_set_is_universe(bset))

 
     def isdisjoint(self, other):
         # return true if the polyhedron has no elements in common with other
 
     def isdisjoint(self, other):
         # return true if the polyhedron has no elements in common with other

-        raise NotImplementedError
+        #symbols = self._symbolunion(other)
+        bset = self._toisl()
+        other = other._toisl()
+        return bool(libisl.isl_set_is_disjoint(bset, other))

 
     def issubset(self, other):
 
     def issubset(self, other):

-        raise NotImplementedError
+        # check if self(bset) is a subset of other
+        symbols = self._symbolunion(other)
+        bset = self._toisl(symbols)
+        other = other._toisl(symbols)
+        return bool(libisl.isl_set_is_strict_subset(other, bset))

 
     def __le__(self, other):
         return self.issubset(other)
 
     def __lt__(self, other):
 
     def __le__(self, other):
         return self.issubset(other)
 
     def __lt__(self, other):

-        raise NotImplementedError
+        symbols = self._symbolunion(other)
+        bset = self._toisl(symbols)
+        other = other._toisl(symbols)
+        return bool(libisl.isl_set_is_strict_subset(other, bset))

 
     def issuperset(self, other):
         # test whether every element in other is in the polyhedron
 
     def issuperset(self, other):
         # test whether every element in other is in the polyhedron


@@ -430,6 +551,10 @@ class Polyhedron:
         return self.issuperset(other)
 
     def __gt__(self, other):
         return self.issuperset(other)
 
     def __gt__(self, other):

+        symbols = self._symbolunion(other)
+        bset = self._toisl(symbols)
+        other = other._toisl(symbols)
+        bool(libisl.isl_set_is_strict_subset(other, bset))

         raise NotImplementedError
 
     def union(self, *others):
         raise NotImplementedError
 
     def union(self, *others):


@@ -455,10 +580,13 @@ class Polyhedron:
     def __and__(self, other):
         return self.intersection(other)
 
     def __and__(self, other):
         return self.intersection(other)
 

-    def difference(self, *others):
-        # return a new polyhedron with elements in the polyhedron that are not
-        # in the others
-        raise NotImplementedError
+    def difference(self, other):
+        # return a new polyhedron with elements in the polyhedron that are not in the other
+        symbols = self._symbolunion(other)
+        bset = self._toisl(symbols)
+        other = other._toisl(symbols)
+        difference = libisl.isl_set_subtract(bset, other)
+        return difference

 
     def __sub__(self, other):
         return self.difference(other)
 
     def __sub__(self, other):
         return self.difference(other)


@@ -472,14 +600,15 @@ class Polyhedron:
         return '{{{}}}'.format(', '.join(constraints))
 
     def __repr__(self):
         return '{{{}}}'.format(', '.join(constraints))
 
     def __repr__(self):

-        equalities = list(self.equalities)
-        inequalities = list(self.inequalities)
-        return '{}(equalities={!r}, inequalities={!r})' \
-                ''.format(self.__class__.__name__, equalities, inequalities)
-
-    @classmethod
-    def fromstring(cls, string):
-        raise NotImplementedError
+        if self.isempty():
+            return 'Empty'
+        elif self.isuniverse():
+            return 'Universe'
+        else:
+            equalities = list(self.equalities)
+            inequalities = list(self.inequalities)
+            return '{}(equalities={!r}, inequalities={!r})' \
+                    ''.format(self.__class__.__name__, equalities, inequalities)

 
     def _symbolunion(self, *others):
         symbols = set(self.symbols)
 
     def _symbolunion(self, *others):
         symbols = set(self.symbols)


@@ -487,67 +616,57 @@ class Polyhedron:
             symbols.update(other.symbols)
         return sorted(symbols)
 
             symbols.update(other.symbols)
         return sorted(symbols)
 

-    def _to_isl(self, symbols=None):
+    def _toisl(self, symbols=None):

         if symbols is None:
             symbols = self.symbols
         if symbols is None:
             symbols = self.symbols

-        num_coefficients = len(symbols)
-        space = libisl.isl_space_set_alloc(_main_ctx, 0, num_coefficients)
+        dimension = len(symbols)
+        space = libisl.isl_space_set_alloc(_main_ctx, 0, dimension)

         bset = libisl.isl_basic_set_universe(libisl.isl_space_copy(space))
         ls = libisl.isl_local_space_from_space(space)
         bset = libisl.isl_basic_set_universe(libisl.isl_space_copy(space))
         ls = libisl.isl_local_space_from_space(space)

-        ceq = libisl.isl_equality_alloc(libisl.isl_local_space_copy(ls))
-        cin = libisl.isl_inequality_alloc(libisl.isl_local_space_copy(ls))
-        '''if there are equalities/inequalities, take each constant and coefficient and add as a constraint to the basic set'''
-        if list(self.equalities): #check if any equalities exist
-            for eq in self.equalities:
-                coeff_eq = dict(eq.coefficients())
-                if eq.constant:
-                    value = eq.constant
-                    ceq = libisl.isl_constraint_set_constant_si(ceq, value)
-                for eq in coeff_eq:
-                    num = coeff_eq.get(eq)
-                    iden = symbols.index(eq)
-                    ceq = libisl.isl_constraint_set_coefficient_si(ceq, libisl.isl_dim_set, iden, num)  #use 3 for type isl_dim_set
+        for equality in self.equalities:
+            ceq = libisl.isl_equality_alloc(libisl.isl_local_space_copy(ls))
+            for symbol, coefficient in equality.coefficients():
+                val = str(coefficient).encode()
+                val = libisl.isl_val_read_from_str(_main_ctx, val)
+                dim = symbols.index(symbol)
+                ceq = libisl.isl_constraint_set_coefficient_val(ceq, libisl.isl_dim_set, dim, val)
+            if equality.constant != 0:
+                val = str(equality.constant).encode()
+                val = libisl.isl_val_read_from_str(_main_ctx, val)
+                ceq = libisl.isl_constraint_set_constant_val(ceq, val)

             bset = libisl.isl_basic_set_add_constraint(bset, ceq)
             bset = libisl.isl_basic_set_add_constraint(bset, ceq)

-        if list(self.inequalities): #check if any inequalities exist
-            for ineq in self.inequalities:
-                coeff_in = dict(ineq.coefficients())
-                if ineq.constant:
-                    value = ineq.constant
-                    cin = libisl.isl_constraint_set_constant_si(cin, value)
-                for ineq in coeff_in:
-                    num = coeff_in.get(ineq)
-                    iden = symbols.index(ineq)
-                    cin = libisl.isl_constraint_set_coefficient_si(cin, libisl.isl_dim_set, iden, num)  #use 3 for type isl_dim_set
+        for inequality in self.inequalities:
+            cin = libisl.isl_inequality_alloc(libisl.isl_local_space_copy(ls))
+            for symbol, coefficient in inequality.coefficients():
+                val = str(coefficient).encode()
+                val = libisl.isl_val_read_from_str(_main_ctx, val)
+                dim = symbols.index(symbol)
+                cin = libisl.isl_constraint_set_coefficient_val(cin, libisl.isl_dim_set, dim, val)
+            if inequality.constant != 0:
+                val = str(inequality.constant).encode()
+                val = libisl.isl_val_read_from_str(_main_ctx, val)
+                cin = libisl.isl_constraint_set_constant_val(cin, val)

             bset = libisl.isl_basic_set_add_constraint(bset, cin)
         bset = isl.BasicSet(bset)
         return bset
 
     @classmethod
             bset = libisl.isl_basic_set_add_constraint(bset, cin)
         bset = isl.BasicSet(bset)
         return bset
 
     @classmethod

-    def from_isl(cls, bset):
-        '''takes basic set in isl form and puts back into python version of polyhedron
-        isl example code gives isl form as:
-            "{[i] : exists (a : i = 2a and i >= 10 and i <= 42)}")
-            our printer is giving form as:
-            b'{ [i0] : 1 = 0 }' '''
+    def _fromisl(cls, bset, symbols):

         raise NotImplementedError
         equalities = ...
         inequalities = ...
         return cls(equalities, inequalities)
         raise NotImplementedError
         equalities = ...
         inequalities = ...
         return cls(equalities, inequalities)

-        #bset = self
-        # if self._equalities:
-        #     constraints = libisl.isl_basic_set_equalities_matrix(bset, 3)
-        # elif self._inequalities:
-        #     constraints = libisl.isl_basic_set_inequalities_matrix(bset, 3)
-        # print(constraints)
-        # return constraints
-
-empty = None #eq(0,1)
-universe = None #Polyhedron()
+        '''takes basic set  in isl form and puts back into python version of polyhedron
+        isl example code gives isl form as:
+            "{[i] : exists (a : i = 2a and i >= 10 and i <= 42)}")
+            our printer is giving form as:
+            { [i0, i1] : 2i1 >= -2 - i0 } '''

 
 

+Empty = eq(0,1)
+Universe = Polyhedron()

 
 if __name__ == '__main__':
 
 if __name__ == '__main__':

-    ex1 = Expression(coefficients={'a': 1, 'x': 2}, constant=2)
-    ex2 = Expression(coefficients={'a': 3  , 'b': 2}, constant=3)
-    p = Polyhedron(inequalities=[ex1, ex2])
-    bs = p._to_isl()
-    print(bs)
+    p1 = Polyhedron('2a + 2b + 1 == 0') # empty
+    print(p1._toisl())
+    p2 = Polyhedron('3x + 2y + 3 == 0') # not empty
+    print(p2._toisl())