Boolean Polynomials

Elements of the quotient ring

\[\GF{2}[x_1,...,x_n]/<x_1^2+x_1,...,x_n^2+x_n>.\]

are called boolean polynomials. Boolean polynomials arise naturally in cryptography, coding theory, formal logic, chip design and other areas. This implementation is a thin wrapper around the PolyBoRi library by Michael Brickenstein and Alexander Dreyer.

“Boolean polynomials can be modelled in a rather simple way, with both coefficients and degree per variable lying in {0, 1}. The ring of Boolean polynomials is, however, not a polynomial ring, but rather the quotient ring of the polynomial ring over the field with two elements modulo the field equations \(x^2=x\) for each variable \(x\). Therefore, the usual polynomial data structures seem not to be appropriate for fast Groebner basis computations. We introduce a specialised data structure for Boolean polynomials based on zero-suppressed binary decision diagrams (ZDDs), which is capable of handling these polynomials more efficiently with respect to memory consumption and also computational speed. Furthermore, we concentrate on high-level algorithmic aspects, taking into account the new data structures as well as structural properties of Boolean polynomials.” - [BD2007]

For details on the internal representation of polynomials see

http://polybori.sourceforge.net/zdd.html

AUTHORS:

• Michael Brickenstein: PolyBoRi author

• Alexander Dreyer: PolyBoRi author

• Burcin Erocal <burcin@erocal.org>: main Sage wrapper author

• Martin Albrecht <malb@informatik.uni-bremen.de>: some contributions to the Sage wrapper

• Simon King <simon.king@uni-jena.de>: Adopt the new coercion model. Fix conversion from univariate polynomial rings. Pickling of BooleanMonomialMonoid (via UniqueRepresentation) and BooleanMonomial.

• Charles Bouillaguet <charles.bouillaguet@gmail.com>: minor changes to improve compatibility with MPolynomial and make the variety() function work on ideals of BooleanPolynomial’s.

EXAMPLES:

Consider the ideal

\[<ab + cd + 1, ace + de, abe + ce, bc + cde + 1>.\]

First, we compute the lexicographical Groebner basis in the polynomial ring

\[R = \GF{2}[a,b,c,d,e].\]

sage: P.<a,b,c,d,e> = PolynomialRing(GF(2), 5, order='lex')
sage: I1 = ideal([a*b + c*d + 1, a*c*e + d*e, a*b*e + c*e, b*c + c*d*e + 1])
sage: for f in I1.groebner_basis():
....:   f
a + c^2*d + c + d^2*e
b*c + d^3*e^2 + d^3*e + d^2*e^2 + d*e + e + 1
b*e + d*e^2 + d*e + e
c*e + d^3*e^2 + d^3*e + d^2*e^2 + d*e
d^4*e^2 + d^4*e + d^3*e + d^2*e^2 + d^2*e + d*e + e

If one wants to solve this system over the algebraic closure of \(\GF{2}\) then this Groebner basis was the one to consider. If one wants solutions over \(\GF{2}\) only then one adds the field polynomials to the ideal to force the solutions in \(\GF{2}\).

sage: J = I1 + sage.rings.ideal.FieldIdeal(P)
sage: for f in J.groebner_basis():
....:   f
a + d + 1
b + 1
c + 1
d^2 + d
e

So the solutions over \(\GF{2}\) are \(\{e=0, d=1, c=1, b=1, a=0\}\) and \(\{e=0, d=0, c=1, b=1, a=1\}\).

We can express the restriction to \(\GF{2}\) by considering the quotient ring. If \(I\) is an ideal in \(\Bold{F}[x_1, ..., x_n]\) then the ideals in the quotient ring \(\Bold{F}[x_1, ..., x_n]/I\) are in one-to-one correspondence with the ideals of \(\Bold{F}[x_0, ..., x_n]\) containing \(I\) (that is, the ideals \(J\) satisfying \(I \subset J \subset P\)).

sage: Q = P.quotient( sage.rings.ideal.FieldIdeal(P) )
sage: I2 = ideal([Q(f) for f in I1.gens()])
sage: for f in I2.groebner_basis():
....:     f
abar + dbar + 1
bbar + 1
cbar + 1
ebar

This quotient ring is exactly what PolyBoRi handles well:

sage: B.<a,b,c,d,e> = BooleanPolynomialRing(5, order='lex')
sage: I2 = ideal([B(f) for f in I1.gens()])
sage: for f in I2.groebner_basis():
....:   f
a + d + 1
b + 1
c + 1
e

Note that d^2 + d is not representable in B == Q. Also note, that PolyBoRi cannot play out its strength in such small examples, i.e. working in the polynomial ring might be faster for small examples like this.

Implementation specific notes

PolyBoRi comes with a Python wrapper. However this wrapper does not match Sage’s style and is written using Boost. Thus Sage’s wrapper is a reimplementation of Python bindings to PolyBoRi’s C++ library. This interface is written in Cython like all of Sage’s C/C++ library interfaces. An interface in PolyBoRi style is also provided which is effectively a reimplementation of the official Boost wrapper in Cython. This means that some functionality of the official wrapper might be missing from this wrapper and this wrapper might have bugs not present in the official Python interface.

Access to the original PolyBoRi interface

The re-implementation PolyBoRi’s native wrapper is available to the user too:

sage: from sage.rings.polynomial.pbori import *
sage: declare_ring([Block('x',2),Block('y',3)],globals())
Boolean PolynomialRing in x0, x1, y0, y1, y2
sage: r
Boolean PolynomialRing in x0, x1, y0, y1, y2

sage: [Variable(i, r) for i in range(r.ngens())]
[x(0), x(1), y(0), y(1), y(2)]

For details on this interface see:

http://polybori.sourceforge.net/doc/tutorial/tutorial.html.

Also, the interface provides functions for compatibility with Sage accepting convenient Sage data types which are slower than their native PolyBoRi counterparts. For instance, sets of points can be represented as tuples of tuples (Sage) or as BooleSet (PolyBoRi) and naturally the second option is faster.

class sage.rings.polynomial.pbori.pbori.BooleConstant

Bases: object

Construct a boolean constant (modulo 2) from integer value:

INPUT:

• i - an integer

EXAMPLES:

sage: from sage.rings.polynomial.pbori.pbori import BooleConstant
sage: [BooleConstant(i) for i in range(5)]
[0, 1, 0, 1, 0]

deg()

Get degree of boolean constant.

EXAMPLES:

sage: from sage.rings.polynomial.pbori.pbori import BooleConstant
sage: BooleConstant(0).deg()
-1
sage: BooleConstant(1).deg()
0

has_constant_part()

This is true for \(BooleConstant(1)\).

EXAMPLES:

sage: from sage.rings.polynomial.pbori.pbori import BooleConstant
sage: BooleConstant(1).has_constant_part()
True
sage: BooleConstant(0).has_constant_part()
False

is_constant()

This is always true for in this case.

EXAMPLES:

sage: from sage.rings.polynomial.pbori.pbori import BooleConstant
sage: BooleConstant(1).is_constant()
True
sage: BooleConstant(0).is_constant()
True

is_one()

Check whether boolean constant is one.

EXAMPLES:

sage: from sage.rings.polynomial.pbori.pbori import BooleConstant
sage: BooleConstant(0).is_one()
False
sage: BooleConstant(1).is_one()
True

is_zero()

Check whether boolean constant is zero.

EXAMPLES:

sage: from sage.rings.polynomial.pbori.pbori import BooleConstant
sage: BooleConstant(1).is_zero()
False
sage: BooleConstant(0).is_zero()
True

variables()

Get variables (return always and empty tuple).

EXAMPLES:

sage: from sage.rings.polynomial.pbori.pbori import BooleConstant
sage: BooleConstant(0).variables()
()
sage: BooleConstant(1).variables()
()

class sage.rings.polynomial.pbori.pbori.BooleSet

Bases: object

Return a new set of boolean monomials. This data type is also implemented on the top of ZDDs and allows to see polynomials from a different angle. Also, it makes high-level set operations possible, which are in most cases faster than operations handling individual terms, because the complexity of the algorithms depends only on the structure of the diagrams.

Objects of type BooleanPolynomial can easily be converted to the type BooleSet by using the member function BooleanPolynomial.set().

INPUT:

• param - either a CCuddNavigator, a BooleSet or None.

• ring - a boolean polynomial ring.

EXAMPLES:

sage: from sage.rings.polynomial.pbori.pbori import BooleSet
sage: B.<a,b,c,d> = BooleanPolynomialRing(4)
sage: BS = BooleSet(a.set())
sage: BS
{{a}}

sage: BS = BooleSet((a*b + c + 1).set())
sage: BS
{{a,b}, {c}, {}}

sage: from sage.rings.polynomial.pbori.pbori import *
sage: from sage.rings.polynomial.pbori.PyPolyBoRi import Monomial
sage: BooleSet([Monomial(B)])
{{}}

Note

BooleSet prints as {} but are not Python dictionaries.

cartesian_product(rhs)

Return the Cartesian product of this set and the set rhs.

The Cartesian product of two sets X and Y is the set of all possible ordered pairs whose first component is a member of X and whose second component is a member of Y.

\[X\times Y = \{(x,y) | x\in X\;\mathrm{and}\;y\in Y\}.\]

EXAMPLES:

sage: B = BooleanPolynomialRing(5,'x')
sage: x0,x1,x2,x3,x4 = B.gens()
sage: f = x1*x2+x2*x3
sage: s = f.set(); s
{{x1,x2}, {x2,x3}}
sage: g = x4 + 1
sage: t = g.set(); t
{{x4}, {}}
sage: s.cartesian_product(t)
{{x1,x2,x4}, {x1,x2}, {x2,x3,x4}, {x2,x3}}

change(ind)

Swaps the presence of x_i in each entry of the set.

EXAMPLES:

sage: P.<a,b,c> = BooleanPolynomialRing()
sage: f = a+b
sage: s = f.set(); s
{{a}, {b}}
sage: s.change(0)
{{a,b}, {}}
sage: s.change(1)
{{a,b}, {}}
sage: s.change(2)
{{a,c}, {b,c}}

diff(rhs)

Return the set theoretic difference of this set and the set rhs.

The difference of two sets \(X\) and \(Y\) is defined as:

\[X \ Y = \{x | x\in X\;\mathrm{and}\;x\not\in Y\}.\]

EXAMPLES:

sage: B = BooleanPolynomialRing(5,'x')
sage: x0,x1,x2,x3,x4 = B.gens()
sage: f = x1*x2+x2*x3
sage: s = f.set(); s
{{x1,x2}, {x2,x3}}
sage: g = x2*x3 + 1
sage: t = g.set(); t
{{x2,x3}, {}}
sage: s.diff(t)
{{x1,x2}}

divide(rhs)

Divide each element of this set by the monomial rhs and return a new set containing the result.

EXAMPLES:

sage: B.<a,b,c,d,e,f> = BooleanPolynomialRing(order='lex')
sage: f = b*e + b*c*d + b
sage: s = f.set(); s
{{b,c,d}, {b,e}, {b}}
sage: s.divide(b.lm())
{{c,d}, {e}, {}}

sage: f = b*e + b*c*d + b + c
sage: s = f.set()
sage: s.divide(b.lm())
{{c,d}, {e}, {}}

divisors_of(m)

Return those members which are divisors of m.

INPUT:

• m - a boolean monomial

EXAMPLES:

sage: B = BooleanPolynomialRing(5,'x')
sage: x0,x1,x2,x3,x4 = B.gens()
sage: f = x1*x2+x2*x3
sage: s = f.set()
sage: s.divisors_of((x1*x2*x4).lead())
{{x1,x2}}

empty()

Return True if this set is empty.

EXAMPLES:

sage: B.<a,b,c,d> = BooleanPolynomialRing(4)
sage: BS = (a*b + c).set()
sage: BS.empty()
False

sage: BS = B(0).set()
sage: BS.empty()
True

include_divisors()

Extend this set to include all divisors of the elements already in this set and return the result as a new set.

EXAMPLES:

sage: B.<a,b,c,d,e,f> = BooleanPolynomialRing()
sage: f = a*d*e + a*f + b*d*e + c*d*e + 1
sage: s = f.set(); s
{{a,d,e}, {a,f}, {b,d,e}, {c,d,e}, {}}

sage: s.include_divisors()
{{a,d,e}, {a,d}, {a,e}, {a,f}, {a}, {b,d,e}, {b,d}, {b,e},
 {b}, {c,d,e}, {c,d}, {c,e}, {c}, {d,e}, {d}, {e}, {f}, {}}

intersect(other)

Return the set theoretic intersection of this set and the set rhs.

The union of two sets \(X\) and \(Y\) is defined as:

\[X \cap Y = \{x | x\in X\;\mathrm{and}\;x\in Y\}.\]

EXAMPLES:

sage: B = BooleanPolynomialRing(5,'x')
sage: x0,x1,x2,x3,x4 = B.gens()
sage: f = x1*x2+x2*x3
sage: s = f.set(); s
{{x1,x2}, {x2,x3}}
sage: g = x2*x3 + 1
sage: t = g.set(); t
{{x2,x3}, {}}
sage: s.intersect(t)
{{x2,x3}}

minimal_elements()

Return a new set containing a divisor of all elements of this set.

EXAMPLES:

sage: B.<a,b,c,d,e,f> = BooleanPolynomialRing()
sage: f = a*d*e + a*f + a*b*d*e + a*c*d*e + a
sage: s = f.set(); s
{{a,b,d,e}, {a,c,d,e}, {a,d,e}, {a,f}, {a}}
sage: s.minimal_elements()
{{a}}

multiples_of(m)

Return those members which are multiples of m.

INPUT:

• m - a boolean monomial

EXAMPLES:

sage: B = BooleanPolynomialRing(5,'x')
sage: x0,x1,x2,x3,x4 = B.gens()
sage: f = x1*x2+x2*x3
sage: s = f.set()
sage: s.multiples_of(x1.lm())
{{x1,x2}}

n_nodes()

Return the number of nodes in the ZDD.

EXAMPLES:

sage: B = BooleanPolynomialRing(5,'x')
sage: x0,x1,x2,x3,x4 = B.gens()
sage: f = x1*x2+x2*x3
sage: s = f.set(); s
{{x1,x2}, {x2,x3}}
sage: s.n_nodes()
4

navigation()

Navigators provide an interface to diagram nodes, accessing their index as well as the corresponding then- and else-branches.

You should be very careful and always keep a reference to the original object, when dealing with navigators, as navigators contain only a raw pointer as data. For the same reason, it is necessary to supply the ring as argument, when constructing a set out of a navigator.

EXAMPLES:

sage: from sage.rings.polynomial.pbori.pbori import BooleSet
sage: B = BooleanPolynomialRing(5,'x')
sage: x0,x1,x2,x3,x4 = B.gens()
sage: f = x1*x2+x2*x3*x4+x2*x4+x3+x4+1
sage: s = f.set(); s
{{x1,x2}, {x2,x3,x4}, {x2,x4}, {x3}, {x4}, {}}

sage: nav = s.navigation()
sage: BooleSet(nav, s.ring())
{{x1,x2}, {x2,x3,x4}, {x2,x4}, {x3}, {x4}, {}}

sage: nav.value()
1

sage: nav_else = nav.else_branch()

sage: BooleSet(nav_else, s.ring())
{{x2,x3,x4}, {x2,x4}, {x3}, {x4}, {}}

sage: nav_else.value()
2

ring()

Return the parent ring.

EXAMPLES:

sage: B = BooleanPolynomialRing(5,'x')
sage: x0,x1,x2,x3,x4 = B.gens()
sage: f = x1*x2+x2*x3*x4+x2*x4+x3+x4+1
sage: f.set().ring() is B
True

set()

Return self.

EXAMPLES:

sage: B.<a,b,c,d> = BooleanPolynomialRing(4)
sage: BS = (a*b + c).set()
sage: BS.set() is BS
True

size_double()

Return the size of this set as a floating point number.

EXAMPLES:

sage: B = BooleanPolynomialRing(5,'x')
sage: x0,x1,x2,x3,x4 = B.gens()
sage: f = x1*x2+x2*x3
sage: s = f.set()
sage: s.size_double()
2.0

stable_hash()

A hash value which is stable across processes.

EXAMPLES:

sage: B.<x,y> = BooleanPolynomialRing()
sage: x.set() is x.set()
False
sage: x.set().stable_hash() == x.set().stable_hash()
True

Note

This function is part of the upstream PolyBoRi interface. In Sage all hashes are stable.

subset0(i)

Return a set of those elements in this set which do not contain the variable indexed by i.

INPUT:

• i - an index

EXAMPLES:

sage: BooleanPolynomialRing(5,'x')
Boolean PolynomialRing in x0, x1, x2, x3, x4
sage: B = BooleanPolynomialRing(5,'x')
sage: B.inject_variables()
Defining x0, x1, x2, x3, x4
sage: f = x1*x2+x2*x3
sage: s = f.set(); s
{{x1,x2}, {x2,x3}}
sage: s.subset0(1)
{{x2,x3}}

subset1(i)

Return a set of those elements in this set which do contain the variable indexed by i and evaluate the variable indexed by i to 1.

INPUT:

• i - an index

EXAMPLES:

sage: BooleanPolynomialRing(5,'x')
Boolean PolynomialRing in x0, x1, x2, x3, x4
sage: B = BooleanPolynomialRing(5,'x')
sage: B.inject_variables()
Defining x0, x1, x2, x3, x4
sage: f = x1*x2+x2*x3
sage: s = f.set(); s
{{x1,x2}, {x2,x3}}
sage: s.subset1(1)
{{x2}}

union(rhs)

Return the set theoretic union of this set and the set rhs.

The union of two sets \(X\) and \(Y\) is defined as:

\[X \cup Y = \{x | x\in X\;\mathrm{or}\;x\in Y\}.\]

EXAMPLES:

sage: B = BooleanPolynomialRing(5,'x')
sage: x0,x1,x2,x3,x4 = B.gens()
sage: f = x1*x2+x2*x3
sage: s = f.set(); s
{{x1,x2}, {x2,x3}}
sage: g = x2*x3 + 1
sage: t = g.set(); t
{{x2,x3}, {}}
sage: s.union(t)
{{x1,x2}, {x2,x3}, {}}

vars()

Return the variables in this set as a monomial.

EXAMPLES:

sage: B.<a,b,c,d,e,f> = BooleanPolynomialRing(order='lex')
sage: f = a + b*e + d*f + e + 1
sage: s = f.set()
sage: s
{{a}, {b,e}, {d,f}, {e}, {}}
sage: s.vars()
a*b*d*e*f

class sage.rings.polynomial.pbori.pbori.BooleSetIterator

Bases: object

Helper class to iterate over boolean sets.

class sage.rings.polynomial.pbori.pbori.BooleanMonomial

Bases: sage.structure.element.MonoidElement

Construct a boolean monomial.

INPUT:

• parent - parent monoid this element lives in

EXAMPLES:

sage: from sage.rings.polynomial.pbori.pbori import BooleanMonomialMonoid, BooleanMonomial
sage: P.<x,y,z> = BooleanPolynomialRing(3)
sage: M = BooleanMonomialMonoid(P)
sage: BooleanMonomial(M)
1

Note

Use the BooleanMonomialMonoid__call__() method and not this constructor to construct these objects.

deg()

Return degree of this monomial.

EXAMPLES:

sage: from sage.rings.polynomial.pbori.pbori import BooleanMonomialMonoid
sage: P.<x,y,z> = BooleanPolynomialRing(3)
sage: M = BooleanMonomialMonoid(P)
sage: M(x*y).deg()
2

sage: M(x*x*y*z).deg()
3

Note

This function is part of the upstream PolyBoRi interface.

degree(x=None)

Return the degree of this monomial in x, where x must be one of the generators of the polynomial ring.

INPUT:

• x - boolean multivariate polynomial (a generator of the polynomial ring). If x is not specified (or is None), return the total degree of this monomial.

EXAMPLES:

sage: from sage.rings.polynomial.pbori.pbori import BooleanMonomialMonoid
sage: P.<x,y,z> = BooleanPolynomialRing(3)
sage: M = BooleanMonomialMonoid(P)
sage: M(x*y).degree()
2
sage: M(x*y).degree(x)
1
sage: M(x*y).degree(z)
0

divisors()

Return a set of boolean monomials with all divisors of this monomial.

EXAMPLES:

sage: B.<x,y,z> = BooleanPolynomialRing(3)
sage: f = x*y
sage: m = f.lm()
sage: m.divisors()
{{x,y}, {x}, {y}, {}}

gcd(rhs)

Return the greatest common divisor of this boolean monomial and rhs.

INPUT:

• rhs - a boolean monomial

EXAMPLES:

sage: B.<a,b,c,d> = BooleanPolynomialRing()
sage: a,b,c,d = a.lm(), b.lm(), c.lm(), d.lm()
sage: (a*b).gcd(b*c)
b
sage: (a*b*c).gcd(d)
1

index()

Return the variable index of the first variable in this monomial.

EXAMPLES:

sage: B.<x,y,z> = BooleanPolynomialRing(3)
sage: f = x*y
sage: m = f.lm()
sage: m.index()
0

Note

This function is part of the upstream PolyBoRi interface.

iterindex()

Return an iterator over the indices of the variables in self.

EXAMPLES:

sage: from sage.rings.polynomial.pbori.pbori import BooleanMonomialMonoid
sage: P.<x,y,z> = BooleanPolynomialRing(3)
sage: M = BooleanMonomialMonoid(P)
sage: list(M(x*z).iterindex())
[0, 2]

multiples(rhs)

Return a set of boolean monomials with all multiples of this monomial up to the bound rhs.

INPUT:

• rhs - a boolean monomial

EXAMPLES:

sage: B.<x,y,z> = BooleanPolynomialRing(3)
sage: f = x
sage: m = f.lm()
sage: g = x*y*z
sage: n = g.lm()
sage: m.multiples(n)
{{x,y,z}, {x,y}, {x,z}, {x}}
sage: n.multiples(m)
{{x,y,z}}

Note

The returned set always contains self even if the bound rhs is smaller than self.

navigation()

Navigators provide an interface to diagram nodes, accessing their index as well as the corresponding then- and else-branches.

You should be very careful and always keep a reference to the original object, when dealing with navigators, as navigators contain only a raw pointer as data. For the same reason, it is necessary to supply the ring as argument, when constructing a set out of a navigator.

EXAMPLES:

sage: from sage.rings.polynomial.pbori.pbori import BooleSet
sage: B = BooleanPolynomialRing(5,'x')
sage: x0,x1,x2,x3,x4 = B.gens()
sage: f = x1*x2+x2*x3*x4+x2*x4+x3+x4+1
sage: m = f.lm(); m
x1*x2

sage: nav = m.navigation()
sage: BooleSet(nav, B)
{{x1,x2}}

sage: nav.value()
1

reducible_by(rhs)

Return True if self is reducible by rhs.

INPUT:

• rhs - a boolean monomial

EXAMPLES:

sage: B.<x,y,z> = BooleanPolynomialRing(3)
sage: f = x*y
sage: m = f.lm()
sage: m.reducible_by((x*y).lm())
True
sage: m.reducible_by((x*z).lm())
False

ring()

Return the corresponding boolean ring.

EXAMPLES:

sage: B.<a,b,c,d> = BooleanPolynomialRing(4)
sage: a.lm().ring() is B
True

set()

Return a boolean set of variables in this monomials.

EXAMPLES:

sage: B.<x,y,z> = BooleanPolynomialRing(3)
sage: f = x*y
sage: m = f.lm()
sage: m.set()
{{x,y}}

stable_hash()

A hash value which is stable across processes.

EXAMPLES:

sage: B.<x,y> = BooleanPolynomialRing()
sage: x.lm() is x.lm()
False
sage: x.lm().stable_hash() == x.lm().stable_hash()
True

Note

This function is part of the upstream PolyBoRi interface. In Sage all hashes are stable.

variables()

Return a tuple of the variables in this monomial.

EXAMPLES:

sage: from sage.rings.polynomial.pbori.pbori import BooleanMonomialMonoid
sage: P.<x,y,z> = BooleanPolynomialRing(3)
sage: M = BooleanMonomialMonoid(P)
sage: M(x*z).variables() # indirect doctest
(x, z)

class sage.rings.polynomial.pbori.pbori.BooleanMonomialIterator

Bases: object

An iterator over the variable indices of a monomial.

class sage.rings.polynomial.pbori.pbori.BooleanMonomialMonoid(polring)

Bases: sage.structure.unique_representation.UniqueRepresentation, sage.monoids.monoid.Monoid_class

Construct a boolean monomial monoid given a boolean polynomial ring.

This object provides a parent for boolean monomials.

INPUT:

• polring - the polynomial ring our monomials lie in

EXAMPLES:

sage: from sage.rings.polynomial.pbori.pbori import BooleanMonomialMonoid
sage: P.<x,y> = BooleanPolynomialRing(2)
sage: M = BooleanMonomialMonoid(P)
sage: M
MonomialMonoid of Boolean PolynomialRing in x, y

sage: M.gens()
(x, y)
sage: type(M.gen(0))
<type 'sage.rings.polynomial.pbori.pbori.BooleanMonomial'>

Since trac ticket #9138, boolean monomial monoids are unique parents and are fit into the category framework:

sage: loads(dumps(M)) is M
True
sage: TestSuite(M).run()

gen(i=0)

Return the i-th generator of self.

INPUT:

• i - an integer

EXAMPLES:

sage: from sage.rings.polynomial.pbori.pbori import BooleanMonomialMonoid
sage: P.<x,y,z> = BooleanPolynomialRing(3)
sage: M = BooleanMonomialMonoid(P)
sage: M.gen(0)
x
sage: M.gen(2)
z

sage: P = BooleanPolynomialRing(1000, 'x')
sage: M = BooleanMonomialMonoid(P)
sage: M.gen(50)
x50

gens()

Return the tuple of generators of this monoid.

EXAMPLES:

sage: from sage.rings.polynomial.pbori.pbori import BooleanMonomialMonoid
sage: P.<x,y,z> = BooleanPolynomialRing(3)
sage: M = BooleanMonomialMonoid(P)
sage: M.gens()
(x, y, z)

ngens()

Return the number of variables in this monoid.

EXAMPLES:

sage: from sage.rings.polynomial.pbori.pbori import BooleanMonomialMonoid
sage: P = BooleanPolynomialRing(100, 'x')
sage: M = BooleanMonomialMonoid(P)
sage: M.ngens()
100

class sage.rings.polynomial.pbori.pbori.BooleanMonomialVariableIterator

Bases: object

class sage.rings.polynomial.pbori.pbori.BooleanMulAction

Bases: sage.categories.action.Action

class sage.rings.polynomial.pbori.pbori.BooleanPolynomial

Bases: sage.rings.polynomial.multi_polynomial.MPolynomial

Construct a boolean polynomial object in the given boolean polynomial ring.

INPUT:

• parent - a boolean polynomial ring

Note

Do not use this method to construct boolean polynomials, but use the appropriate __call__ method in the parent.

constant()

Return True if this element is constant.

EXAMPLES:

sage: B.<x,y,z> = BooleanPolynomialRing(3)
sage: x.constant()
False

sage: B(1).constant()
True

Note

This function is part of the upstream PolyBoRi interface.

constant_coefficient()

Return the constant coefficient of this boolean polynomial.

EXAMPLES:

sage: B.<a,b> = BooleanPolynomialRing()
sage: a.constant_coefficient()
0
sage: (a+1).constant_coefficient()
1

deg()

Return the degree of self. This is usually equivalent to the total degree except for weighted term orderings which are not implemented yet.

EXAMPLES:

sage: P.<x,y> = BooleanPolynomialRing(2)
sage: (x+y).degree()
1

sage: P(1).degree()
0

sage: (x*y + x + y + 1).degree()
2

Note

This function is part of the upstream PolyBoRi interface.

degree(x=None)

Return the maximal degree of this polynomial in x, where x must be one of the generators for the parent of this polynomial.

If x is not specified (or is None), return the total degree, which is the maximum degree of any monomial.

EXAMPLES:

sage: P.<x,y> = BooleanPolynomialRing(2)
sage: (x+y).degree()
1

sage: P(1).degree()
0

sage: (x*y + x + y + 1).degree()
2

sage: (x*y + x + y + 1).degree(x)
1

elength()

Return elimination length as used in the SlimGB algorithm.

EXAMPLES:

sage: P.<x,y> = BooleanPolynomialRing(2)
sage: x.elength()
1
sage: f = x*y + 1
sage: f.elength()
2

REFERENCES:

• Michael Brickenstein; SlimGB: Groebner Bases with Slim Polynomials http://www.mathematik.uni-kl.de/~zca/Reports_on_ca/35/paper_35_full.ps.gz

Note

This function is part of the upstream PolyBoRi interface.

first_term()

Return the first term with respect to the lexicographical term ordering.

EXAMPLES:

sage: B.<a,b,z> = BooleanPolynomialRing(3,order='lex')
sage: f = b*z + a + 1
sage: f.first_term()
a

Note

This function is part of the upstream PolyBoRi interface.

graded_part(deg)

Return graded part of this boolean polynomial of degree deg.

INPUT:

• deg - a degree

EXAMPLES:

sage: B.<a,b,c,d> = BooleanPolynomialRing(4)
sage: f = a*b*c + c*d + a*b + 1
sage: f.graded_part(2)
a*b + c*d

sage: f.graded_part(0)
1

has_constant_part()

Return True if this boolean polynomial has a constant part, i.e. if 1 is a term.

EXAMPLES:

sage: B.<a,b,c,d> = BooleanPolynomialRing(4)
sage: f = a*b*c + c*d + a*b + 1
sage: f.has_constant_part()
True

sage: f = a*b*c + c*d + a*b
sage: f.has_constant_part()
False

is_constant()

Check if self is constant.

EXAMPLES:

sage: P.<x,y> = BooleanPolynomialRing(2)
sage: P(1).is_constant()
True

sage: P(0).is_constant()
True

sage: x.is_constant()
False

sage: (x*y).is_constant()
False

is_equal(right)

EXAMPLES:

sage: B.<a,b,z> = BooleanPolynomialRing(3)
sage: f = a*z + b + 1
sage: g = b + z
sage: f.is_equal(g)
False

sage: f.is_equal((f + 1) - 1)
True

Note

This function is part of the upstream PolyBoRi interface.

is_homogeneous()

Return True if this element is a homogeneous polynomial.

EXAMPLES:

sage: P.<x, y> = BooleanPolynomialRing()
sage: (x+y).is_homogeneous()
True
sage: P(0).is_homogeneous()
True
sage: (x+1).is_homogeneous()
False

is_one()

Check if self is 1.

EXAMPLES:

sage: P.<x,y> = BooleanPolynomialRing(2)
sage: P(1).is_one()
True

sage: P.one().is_one()
True

sage: x.is_one()
False

sage: P(0).is_one()
False

is_pair()

Check if self has exactly two terms.

EXAMPLES:

sage: P.<x,y> = BooleanPolynomialRing(2)
sage: P(0).is_pair()
False

sage: x.is_pair()
False

sage: P(1).is_pair()
False

sage: (x*y).is_pair()
False

sage: (x + y).is_pair()
True

sage: (x + 1).is_pair()
True

sage: (x*y + 1).is_pair()
True

sage: (x + y + 1).is_pair()
False

sage: ((x + 1)*(y + 1)).is_pair()
False

is_singleton()

Check if self has at most one term.

EXAMPLES:

sage: P.<x,y> = BooleanPolynomialRing(2)
sage: P(0).is_singleton()
True

sage: x.is_singleton()
True

sage: P(1).is_singleton()
True

sage: (x*y).is_singleton()
True

sage: (x + y).is_singleton()
False

sage: (x + 1).is_singleton()
False

sage: (x*y + 1).is_singleton()
False

sage: (x + y + 1).is_singleton()
False

sage: ((x + 1)*(y + 1)).is_singleton()
False

is_singleton_or_pair()

Check if self has at most two terms.

EXAMPLES:

sage: P.<x,y> = BooleanPolynomialRing(2)
sage: P(0).is_singleton_or_pair()
True

sage: x.is_singleton_or_pair()
True

sage: P(1).is_singleton_or_pair()
True

sage: (x*y).is_singleton_or_pair()
True

sage: (x + y).is_singleton_or_pair()
True

sage: (x + 1).is_singleton_or_pair()
True

sage: (x*y + 1).is_singleton_or_pair()
True

sage: (x + y + 1).is_singleton_or_pair()
False

sage: ((x + 1)*(y + 1)).is_singleton_or_pair()
False

is_unit()

Check if self is invertible in the parent ring.

Note that this condition is equivalent to being 1 for boolean polynomials.

EXAMPLES:

sage: P.<x,y> = BooleanPolynomialRing(2)
sage: P.one().is_unit()
True

sage: x.is_unit()
False

is_univariate()

Return True if self is a univariate polynomial.

This means that self contains at most one variable.

EXAMPLES:

sage: P.<x,y,z> = BooleanPolynomialRing()
sage: f = x + 1
sage: f.is_univariate()
True
sage: f = y*x + 1
sage: f.is_univariate()
False
sage: f = P(0)
sage: f.is_univariate()
True

is_zero()

Check if self is zero.

EXAMPLES:

sage: P.<x,y> = BooleanPolynomialRing(2)
sage: P(0).is_zero()
True

sage: x.is_zero()
False

sage: P(1).is_zero()
False

lead()

Return the leading monomial of boolean polynomial, with respect to to the order of parent ring.

EXAMPLES:

sage: P.<x,y,z> = BooleanPolynomialRing(3)
sage: (x+y+y*z).lead()
x

sage: P.<x,y,z> = BooleanPolynomialRing(3, order='deglex')
sage: (x+y+y*z).lead()
y*z

Note

This function is part of the upstream PolyBoRi interface.

lead_deg()

Return the total degree of the leading monomial of self.

EXAMPLES:

sage: P.<x,y,z> = BooleanPolynomialRing(3)
sage: p = x + y*z
sage: p.lead_deg()
1

sage: P.<x,y,z> = BooleanPolynomialRing(3,order='deglex')
sage: p = x + y*z
sage: p.lead_deg()
2

sage: P(0).lead_deg()
0

Note

This function is part of the upstream PolyBoRi interface.

lead_divisors()

Return a BooleSet of all divisors of the leading monomial.

EXAMPLES:

sage: B.<a,b,z> = BooleanPolynomialRing(3)
sage: f = a*b + z + 1
sage: f.lead_divisors()
{{a,b}, {a}, {b}, {}}

Note

This function is part of the upstream PolyBoRi interface.

lex_lead()

Return the leading monomial of boolean polynomial, with respect to the lexicographical term ordering.

EXAMPLES:

sage: P.<x,y,z> = BooleanPolynomialRing(3)
sage: (x+y+y*z).lex_lead()
x

sage: P.<x,y,z> = BooleanPolynomialRing(3, order='deglex')
sage: (x+y+y*z).lex_lead()
x

sage: P(0).lex_lead()
0

Note

This function is part of the upstream PolyBoRi interface.

lex_lead_deg()

Return degree of leading monomial with respect to the lexicographical ordering.

EXAMPLES:

sage: B.<x,y,z> = BooleanPolynomialRing(3,order='lex')
sage: f = x + y*z
sage: f
x + y*z
sage: f.lex_lead_deg()
1

sage: B.<x,y,z> = BooleanPolynomialRing(3,order='deglex')
sage: f = x + y*z
sage: f
y*z + x
sage: f.lex_lead_deg()
1

Note

This function is part of the upstream PolyBoRi interface.

lm()

Return the leading monomial of this boolean polynomial, with respect to the order of parent ring.

EXAMPLES:

sage: P.<x,y,z> = BooleanPolynomialRing(3)
sage: (x+y+y*z).lm()
x

sage: P.<x,y,z> = BooleanPolynomialRing(3, order='deglex')
sage: (x+y+y*z).lm()
y*z

sage: P(0).lm()
0

lt()

Return the leading term of this boolean polynomial, with respect to the order of the parent ring.

Note that for boolean polynomials this is equivalent to returning leading monomials.

EXAMPLES:

sage: P.<x,y,z> = BooleanPolynomialRing(3)
sage: (x+y+y*z).lt()
x

sage: P.<x,y,z> = BooleanPolynomialRing(3, order='deglex')
sage: (x+y+y*z).lt()
y*z

map_every_x_to_x_plus_one()

Map every variable x_i in this polynomial to x_i + 1.

EXAMPLES:

sage: B.<a,b,z> = BooleanPolynomialRing(3)
sage: f = a*b + z + 1; f
a*b + z + 1
sage: f.map_every_x_to_x_plus_one()
a*b + a + b + z + 1
sage: f(a+1,b+1,z+1)
a*b + a + b + z + 1

monomial_coefficient(mon)

Return the coefficient of the monomial mon in self, where mon must have the same parent as self.

INPUT:

• mon - a monomial

EXAMPLES:

sage: P.<x,y> = BooleanPolynomialRing(2)
sage: x.monomial_coefficient(x)
1
sage: x.monomial_coefficient(y)
0
sage: R.<x,y,z,a,b,c>=BooleanPolynomialRing(6)
sage: f=(1-x)*(1+y); f
x*y + x + y + 1

sage: f.monomial_coefficient(1)
1

sage: f.monomial_coefficient(0)
0

monomials()

Return a list of monomials appearing in self ordered largest to smallest.

EXAMPLES:

sage: P.<a,b,c> = BooleanPolynomialRing(3,order='lex')
sage: f = a + c*b
sage: f.monomials()
[a, b*c]

sage: P.<a,b,c> = BooleanPolynomialRing(3,order='deglex')
sage: f = a + c*b
sage: f.monomials()
[b*c, a]
sage: P.zero().monomials()
[]

n_nodes()

Return the number of nodes in the ZDD implementing this polynomial.

EXAMPLES:

sage: B = BooleanPolynomialRing(5,'x')
sage: x0,x1,x2,x3,x4 = B.gens()
sage: f = x1*x2 + x2*x3 + 1
sage: f.n_nodes()
4

Note

This function is part of the upstream PolyBoRi interface.

n_vars()

Return the number of variables used to form this boolean polynomial.

EXAMPLES:

sage: B.<a,b,c,d> = BooleanPolynomialRing(4)
sage: f = a*b*c + 1
sage: f.n_vars()
3

Note

This function is part of the upstream PolyBoRi interface.

navigation()

Navigators provide an interface to diagram nodes, accessing their index as well as the corresponding then- and else-branches.

You should be very careful and always keep a reference to the original object, when dealing with navigators, as navigators contain only a raw pointer as data. For the same reason, it is necessary to supply the ring as argument, when constructing a set out of a navigator.

EXAMPLES:

sage: from sage.rings.polynomial.pbori.pbori import BooleSet
sage: B = BooleanPolynomialRing(5,'x')
sage: x0,x1,x2,x3,x4 = B.gens()
sage: f = x1*x2+x2*x3*x4+x2*x4+x3+x4+1

sage: nav = f.navigation()
sage: BooleSet(nav, B)
{{x1,x2}, {x2,x3,x4}, {x2,x4}, {x3}, {x4}, {}}

sage: nav.value()
1

sage: nav_else = nav.else_branch()

sage: BooleSet(nav_else, B)
{{x2,x3,x4}, {x2,x4}, {x3}, {x4}, {}}

sage: nav_else.value()
2

Note

This function is part of the upstream PolyBoRi interface.

nvariables()

Return the number of variables used to form this boolean polynomial.

EXAMPLES:

sage: B.<a,b,c,d> = BooleanPolynomialRing(4)
sage: f = a*b*c + 1
sage: f.nvariables()
3

reduce(I)

Return the normal form of self w.r.t. I, i.e. return the remainder of self with respect to the polynomials in I. If the polynomial set/list I is not a Groebner basis the result is not canonical.

INPUT:

• I - a list/set of polynomials in self.parent(). If I is an ideal, the generators are used.

EXAMPLES:

sage: B.<x0,x1,x2,x3> = BooleanPolynomialRing(4)
sage: I = B.ideal((x0 + x1 + x2 + x3,
....:              x0*x1 + x1*x2 + x0*x3 + x2*x3,
....:              x0*x1*x2 + x0*x1*x3 + x0*x2*x3 + x1*x2*x3,
....:              x0*x1*x2*x3 + 1))
sage: gb = I.groebner_basis()
sage: f,g,h,i = I.gens()
sage: f.reduce(gb)
0
sage: p = f*g + x0*h + x2*i
sage: p.reduce(gb)
0
sage: p.reduce(I)
x1*x2*x3 + x2
sage: p.reduce([])
x0*x1*x2 + x0*x1*x3 + x0*x2*x3 + x2

Note

If this function is called repeatedly with the same I then it is advised to use PolyBoRi’s GroebnerStrategy object directly, since that will be faster. See the source code of this function for details.

reducible_by(rhs)

Return True if this boolean polynomial is reducible by the polynomial rhs.

INPUT:

• rhs - a boolean polynomial

EXAMPLES:

sage: B.<a,b,c,d> = BooleanPolynomialRing(4,order='deglex')
sage: f = (a*b + 1)*(c + 1)
sage: f.reducible_by(d)
False
sage: f.reducible_by(c)
True
sage: f.reducible_by(c + 1)
True

Note

This function is part of the upstream PolyBoRi interface.

ring()

Return the parent of this boolean polynomial.

EXAMPLES:

sage: B.<a,b,c,d> = BooleanPolynomialRing(4)
sage: a.ring() is B
True

set()

Return a BooleSet with all monomials appearing in this polynomial.

EXAMPLES:

sage: B.<a,b,z> = BooleanPolynomialRing(3)
sage: (a*b+z+1).set()
{{a,b}, {z}, {}}

spoly(rhs)

Return the S-Polynomial of this boolean polynomial and the other boolean polynomial rhs.

EXAMPLES:

sage: B.<a,b,c,d> = BooleanPolynomialRing(4)
sage: f = a*b*c + c*d + a*b + 1
sage: g = c*d + b
sage: f.spoly(g)
a*b + a*c*d + c*d + 1

Note

This function is part of the upstream PolyBoRi interface.

stable_hash()

A hash value which is stable across processes.

EXAMPLES:

sage: B.<x,y> = BooleanPolynomialRing()
sage: x is B.gen(0)
False
sage: x.stable_hash() == B.gen(0).stable_hash()
True

Note

This function is part of the upstream PolyBoRi interface. In Sage all hashes are stable.

subs(in_dict=None, **kwds)

Fixes some given variables in a given boolean polynomial and returns the changed boolean polynomials. The polynomial itself is not affected. The variable, value pairs for fixing are to be provided as dictionary of the form {variable:value} or named parameters (see examples below).

INPUT:

• in_dict - (optional) dict with variable:value pairs

• **kwds - names parameters

EXAMPLES:

sage: P.<x,y,z> = BooleanPolynomialRing(3)
sage: f = x*y + z + y*z + 1
sage: f.subs(x=1)
y*z + y + z + 1
sage: f.subs(x=0)
y*z + z + 1

sage: f.subs(x=y)
y*z + y + z + 1

sage: f.subs({x:1},y=1)
0
sage: f.subs(y=1)
x + 1
sage: f.subs(y=1,z=1)
x + 1
sage: f.subs(z=1)
x*y + y
sage: f.subs({'x':1},y=1)
0

This method can work fully symbolic:

sage: f.subs(x=var('a'),y=var('b'),z=var('c'))
a*b + b*c + c + 1
sage: f.subs({'x':var('a'),'y':var('b'),'z':var('c')})
a*b + b*c + c + 1

terms()

Return a list of monomials appearing in self ordered largest to smallest.

EXAMPLES:

sage: P.<a,b,c> = BooleanPolynomialRing(3,order='lex')
sage: f = a + c*b
sage: f.terms()
[a, b*c]

sage: P.<a,b,c> = BooleanPolynomialRing(3,order='deglex')
sage: f = a + c*b
sage: f.terms()
[b*c, a]

total_degree()

Return the total degree of self.

EXAMPLES:

sage: P.<x,y> = BooleanPolynomialRing(2)
sage: (x+y).total_degree()
1

sage: P(1).total_degree()
0

sage: (x*y + x + y + 1).total_degree()
2

univariate_polynomial(R=None)

Return a univariate polynomial associated to this multivariate polynomial.

If this polynomial is not in at most one variable, then a ValueError exception is raised. This is checked using the is_univariate() method. The new Polynomial is over GF(2) and in the variable x if no ring R is provided.

sage: R.<x, y> = BooleanPolynomialRing() sage: f = x - y + x*y + 1 sage: f.univariate_polynomial() Traceback (most recent call last): … ValueError: polynomial must involve at most one variable sage: g = f.subs({x:0}); g y + 1 sage: g.univariate_polynomial () y + 1 sage: g.univariate_polynomial(GF(2)[‘foo’]) foo + 1

Here’s an example with a constant multivariate polynomial:

sage: g = R(1)
sage: h = g.univariate_polynomial(); h
1
sage: h.parent()
Univariate Polynomial Ring in x over Finite Field of size 2 (using GF2X)

variable(i=0)

Return the i-th variable occurring in self. The index i is the index in self.variables()

EXAMPLES:

sage: P.<x,y,z> = BooleanPolynomialRing(3)
sage: f = x*z + z + 1
sage: f.variables()
(x, z)
sage: f.variable(1)
z

variables()

Return a tuple of all variables appearing in self.

EXAMPLES:

sage: P.<x,y,z> = BooleanPolynomialRing(3)
sage: (x + y).variables()
(x, y)

sage: (x*y + z).variables()
(x, y, z)

sage: P.zero().variables()
()

sage: P.one().variables()
()

vars_as_monomial()

Return a boolean monomial with all variables appearing in self.

EXAMPLES:

sage: P.<x,y,z> = BooleanPolynomialRing(3)
sage: (x + y).vars_as_monomial()
x*y

sage: (x*y + z).vars_as_monomial()
x*y*z

sage: P.zero().vars_as_monomial()
1

sage: P.one().vars_as_monomial()
1

Note

This function is part of the upstream PolyBoRi interface.

zeros_in(s)

Return a set containing all elements of s where this boolean polynomial evaluates to zero.

If s is given as a BooleSet, then the return type is also a BooleSet. If s is a set/list/tuple of tuple this function returns a tuple of tuples.

INPUT:

• s - candidate points for evaluation to zero

EXAMPLES:

sage: B.<a,b,c,d> = BooleanPolynomialRing(4)
sage: f = a*b + c + d + 1

Now we create a set of points:

sage: s = a*b + a*b*c + c*d + 1
sage: s = s.set(); s
{{a,b,c}, {a,b}, {c,d}, {}}

This encodes the points (1,1,1,0), (1,1,0,0), (0,0,1,1) and (0,0,0,0). But of these only (1,1,0,0) evaluates to zero.

sage: f.zeros_in(s)
{{a,b}}

sage: f.zeros_in([(1,1,1,0), (1,1,0,0), (0,0,1,1), (0,0,0,0)])
((1, 1, 0, 0),)

class sage.rings.polynomial.pbori.pbori.BooleanPolynomialEntry

Bases: object

p

class sage.rings.polynomial.pbori.pbori.BooleanPolynomialIdeal(ring, gens=[], coerce=True)

Bases: sage.rings.polynomial.multi_polynomial_ideal.MPolynomialIdeal

Construct an ideal in the boolean polynomial ring.

INPUT:

• ring - the ring this ideal is defined in

• gens - a list of generators

• coerce - coerce all elements to the ring ring (default: True)

EXAMPLES:

sage: P.<x0, x1, x2, x3> = BooleanPolynomialRing(4)
sage: I = P.ideal(x0*x1*x2*x3 + x0*x1*x3 + x0*x1 + x0*x2 + x0)
sage: I
Ideal (x0*x1*x2*x3 + x0*x1*x3 + x0*x1 + x0*x2 + x0) of Boolean PolynomialRing in x0, x1, x2, x3
sage: loads(dumps(I)) == I
True

dimension()

Return the dimension of self, which is always zero.

groebner_basis(algorithm='polybori', **kwds)

Return a Groebner basis of this ideal.

INPUT:

• algorithm - either "polybori" (built-in default) or "magma" (requires Magma).

• red_tail - tail reductions in intermediate polynomials, this options affects mainly heuristics. The reducedness of the output polynomials can only be guaranteed by the option redsb (default: True)

• minsb - return a minimal Groebner basis (default: True)

• redsb - return a minimal Groebner basis and all tails are reduced (default: True)

• deg_bound - only compute Groebner basis up to a given degree bound (default: False)

• faugere - turn off or on the linear algebra (default: False)

• linear_algebra_in_last_block - this affects the last block of block orderings and degree orderings. If it is set to True linear algebra takes affect in this block. (default: True)

• gauss_on_linear - perform Gaussian elimination on linear

  polynomials (default: True)

• selection_size - maximum number of polynomials for parallel reductions (default: 1000)

• heuristic - Turn off heuristic by setting heuristic=False (default: True)

• lazy - (default: True)

• invert - setting invert=True input and output get a transformation x+1 for each variable x, which shouldn’t effect the calculated GB, but the algorithm.

• other_ordering_first - possible values are False or an ordering code. In practice, many Boolean examples have very few solutions and a very easy Groebner basis. So, a complex walk algorithm (which cannot be implemented using the data structures) seems unnecessary, as such Groebner bases can be converted quite fast by the normal Buchberger algorithm from one ordering into another ordering. (default: False)

• prot - show protocol (default: False)

• full_prot - show full protocol (default: False)

EXAMPLES:

sage: P.<x0, x1, x2, x3> = BooleanPolynomialRing(4)
sage: I = P.ideal(x0*x1*x2*x3 + x0*x1*x3 + x0*x1 + x0*x2 + x0)
sage: I.groebner_basis()
[x0*x1 + x0*x2 + x0, x0*x2*x3 + x0*x3]

Another somewhat bigger example:

sage: sr = mq.SR(2,1,1,4,gf2=True, polybori=True)
sage: F, s = sr.polynomial_system()
sage: I = F.ideal()
sage: I.groebner_basis()
Polynomial Sequence with 36 Polynomials in 36 Variables

We compute the same example with Magma:

sage: sr = mq.SR(2,1,1,4,gf2=True, polybori=True)
sage: F, s = sr.polynomial_system()
sage: I = F.ideal()
sage: I.groebner_basis(algorithm='magma', prot='sage') # optional - magma
Leading term degree:  1. Critical pairs: 148.
Leading term degree:  2. Critical pairs: 144.
Leading term degree:  3. Critical pairs: 462.
Leading term degree:  1. Critical pairs: 167.
Leading term degree:  2. Critical pairs: 147.
Leading term degree:  3. Critical pairs: 101 (all pairs of current degree eliminated by criteria).

Highest degree reached during computation:  3.
Polynomial Sequence with 35 Polynomials in 36 Variables

interreduced_basis()

If this ideal is spanned by (f_1, ..., f_n) this method returns (g_1, ..., g_s) such that:

• <f_1,...,f_n> = <g_1,...,g_s>

• LT(g_i) != LT(g_j) for all i != j`

• LT(g_i) does not divide m for all monomials m of {g_1,...,g_{i-1},g_{i+1},...,g_s}

EXAMPLES:

sage: sr = mq.SR(1, 1, 1, 4, gf2=True, polybori=True)
sage: F, s = sr.polynomial_system()
sage: I = F.ideal()
sage: I.interreduced_basis()
[k100 + 1, k101 + k001 + 1, k102, k103 + 1, x100 + k001 + 1, x101 + k001, x102, x103 + k001, w100 + 1, w101 + k001 + 1, w102 + 1, w103 + 1, s000 + k001, s001 + k001 + 1, s002, s003 + k001 + 1, k000 + 1, k002 + 1, k003 + 1]

reduce(f)

Reduce an element modulo the reduced Groebner basis for this ideal. This returns 0 if and only if the element is in this ideal. In any case, this reduction is unique up to monomial orders.

EXAMPLES:

sage: P = PolynomialRing(GF(2),10, 'x')
sage: B = BooleanPolynomialRing(10,'x')
sage: I = sage.rings.ideal.Cyclic(P)
sage: I = B.ideal([B(f) for f in I.gens()])
sage: gb = I.groebner_basis()
sage: I.reduce(gb[0])
0
sage: I.reduce(gb[0] + 1)
1
sage: I.reduce(gb[0]*gb[1])
0
sage: I.reduce(gb[0]*B.gen(1))
0

variety(**kwds)

Return the variety associated to this boolean ideal.

EXAMPLES:

A simple example:

sage: from sage.doctest.fixtures import reproducible_repr
sage: R.<x,y,z> = BooleanPolynomialRing()
sage: I = ideal( [ x*y*z + x*z + y + 1, x+y+z+1 ] )
sage: print(reproducible_repr(I.variety()))
[{x: 0, y: 1, z: 0}, {x: 1, y: 1, z: 1}]

class sage.rings.polynomial.pbori.pbori.BooleanPolynomialIterator

Bases: object

Iterator over the monomials of a boolean polynomial.

class sage.rings.polynomial.pbori.pbori.BooleanPolynomialRing

Bases: sage.rings.polynomial.multi_polynomial_ring_base.MPolynomialRing_base

Construct a boolean polynomial ring with the following parameters:

INPUT:

• n - number of variables (an integer > 1)

• names - names of ring variables, may be a string or list/tuple

• order - term order (default: lex)

EXAMPLES:

sage: R.<x, y, z> = BooleanPolynomialRing()
sage: R
Boolean PolynomialRing in x, y, z

sage: p = x*y + x*z + y*z
sage: x*p
x*y*z + x*y + x*z

sage: R.term_order()
Lexicographic term order

sage: R = BooleanPolynomialRing(5,'x',order='deglex(3),deglex(2)')
sage: R.term_order()
Block term order with blocks:
(Degree lexicographic term order of length 3,
 Degree lexicographic term order of length 2)

sage: R = BooleanPolynomialRing(3,'x',order='deglex')
sage: R.term_order()
Degree lexicographic term order

change_ring(base_ring=None, names=None, order=None)

Return a new multivariate polynomial ring with base ring base_ring, variable names set to names, and term ordering given by order.

When base_ring is not specified, this function returns a BooleanPolynomialRing isomorphic to self. Otherwise, this returns a MPolynomialRing. Each argument above is optional.

INPUT:

• base_ring – a base ring

• names – variable names

• order – a term order

EXAMPLES:

sage: P.<x, y, z> = BooleanPolynomialRing()
sage: P.term_order()
Lexicographic term order
sage: R = P.change_ring(names=('a', 'b', 'c'), order="deglex")
sage: R
Boolean PolynomialRing in a, b, c
sage: R.term_order()
Degree lexicographic term order
sage: T = P.change_ring(base_ring=GF(3))
sage: T
Multivariate Polynomial Ring in x, y, z over Finite Field of size 3
sage: T.term_order()
Lexicographic term order

clone(ordering=None, names=[], blocks=[])

Shallow copy this boolean polynomial ring, but with different ordering, names or blocks if given.

ring.clone(ordering=…, names=…, block=…) generates a shallow copy of ring, but with different ordering, names or blocks if given.

EXAMPLES:

sage: B.<a,b,c> = BooleanPolynomialRing()
sage: B.clone()
Boolean PolynomialRing in a, b, c

sage: B.<x,y,z> = BooleanPolynomialRing(3,order='deglex')
sage: y*z > x
True

Now we call the clone method and generate a compatible, but ‘lex’ ordered, ring:

sage: C = B.clone(ordering=0)
sage: C(y*z) > C(x)
False

Now we change variable names:

sage: P.<x0,x1> = BooleanPolynomialRing(2)
sage: P
Boolean PolynomialRing in x0, x1

sage: Q = P.clone(names=['t'])
sage: Q
Boolean PolynomialRing in t, x1

We can also append blocks to block orderings this way:

sage: R.<x1,x2,x3,x4> = BooleanPolynomialRing(order='deglex(1),deglex(3)')
sage: x2 > x3*x4
False

Now we call the internal method and change the blocks:

sage: S = R.clone(blocks=[3])
sage: S(x2) > S(x3*x4)
True

Note

This is part of PolyBoRi’s native interface.

construction()

A boolean polynomial ring is the quotient of a polynomial ring, in a special implementation.

Before trac ticket #15223, the boolean polynomial rings returned the construction of a polynomial ring, which was of course wrong.

Now, a QuotientFunctor is returned that knows about the \("pbori"\) implementation.

EXAMPLES:

sage: P.<x0, x1, x2, x3> = BooleanPolynomialRing(4,order='degneglex(2),degneglex(2)')
sage: F,O = P.construction()
sage: O
Multivariate Polynomial Ring in x0, x1, x2, x3 over Finite Field of size 2
sage: F
QuotientFunctor
sage: F(O) is P
True

cover_ring()

Return \(R = \GF{2}[x_1,x_2,...,x_n]\) if x_1,x_2,...,x_n is the ordered list of variable names of this ring. R also has the same term ordering as this ring.

EXAMPLES:

sage: B.<x,y> = BooleanPolynomialRing(2)
sage: R = B.cover_ring(); R
Multivariate Polynomial Ring in x, y over Finite Field of size 2

sage: B.term_order() == R.term_order()
True

The cover ring is cached:

sage: B.cover_ring() is B.cover_ring()
True

defining_ideal()

Return \(I = <x_i^2 + x_i> \subset R\) where R = self.cover_ring(), and \(x_i\) any element in the set of variables of this ring.

EXAMPLES:

sage: B.<x,y> = BooleanPolynomialRing(2)
sage: I = B.defining_ideal(); I
Ideal (x^2 + x, y^2 + y) of Multivariate Polynomial Ring
in x, y over Finite Field of size 2

gen(i=0)

Return the i-th generator of this boolean polynomial ring.

INPUT:

• i - an integer or a boolean monomial in one variable

EXAMPLES:

sage: P.<x,y,z> = BooleanPolynomialRing(3)
sage: P.gen()
x
sage: P.gen(2)
z
sage: m = x.monomials()[0]
sage: P.gen(m)
x

gens()

Return the tuple of variables in this ring.

EXAMPLES:

sage: P.<x,y,z> = BooleanPolynomialRing(3)
sage: P.gens()
(x, y, z)

sage: P = BooleanPolynomialRing(10,'x')
sage: P.gens()
(x0, x1, x2, x3, x4, x5, x6, x7, x8, x9)

get_base_order_code()

EXAMPLES:

sage: B.<a,b,c,d,e,f> = BooleanPolynomialRing()
sage: B.get_base_order_code()
0

sage: B.<a,b,c,d,e,f> = BooleanPolynomialRing(order='deglex')
sage: B.get_base_order_code()
1
sage: T = TermOrder('deglex',2) + TermOrder('deglex',2)
sage: B.<a,b,c,d> = BooleanPolynomialRing(4, order=T)
sage: B.get_base_order_code()
1

Note

This function which is part of the PolyBoRi upstream API works with a current global ring. This notion is avoided in Sage.

get_order_code()

EXAMPLES:

sage: B.<a,b,c,d,e,f> = BooleanPolynomialRing()
sage: B.get_order_code()
0

sage: B.<a,b,c,d,e,f> = BooleanPolynomialRing(order='deglex')
sage: B.get_order_code()
1

Note

This function which is part of the PolyBoRi upstream API works with a current global ring. This notion is avoided in Sage.

has_degree_order()

Return checks whether the order code corresponds to a degree ordering.

EXAMPLES:

sage: P.<x,y> = BooleanPolynomialRing(2)
sage: P.has_degree_order()
False

id()

Return a unique identifier for this boolean polynomial ring.

EXAMPLES:

sage: P.<x,y> = BooleanPolynomialRing(2)
sage: print("id: {}".format(P.id()))
id: ...

sage: P = BooleanPolynomialRing(10, 'x')
sage: Q = BooleanPolynomialRing(20, 'x')

sage: P.id() != Q.id()
True

ideal(*gens, **kwds)

Create an ideal in this ring.

INPUT:

• gens - list or tuple of generators

• coerce - bool (default: True) automatically coerce the given polynomials to this ring to form the ideal

EXAMPLES:

sage: P.<x,y,z> = BooleanPolynomialRing(3)
sage: P.ideal(x+y)
Ideal (x + y) of Boolean PolynomialRing in x, y, z

sage: P.ideal(x*y, y*z)
Ideal (x*y, y*z) of Boolean PolynomialRing in x, y, z

sage: P.ideal([x+y, z])
Ideal (x + y, z) of Boolean PolynomialRing in x, y, z

interpolation_polynomial(zeros, ones)

Return the lexicographically minimal boolean polynomial for the given sets of points.

Given two sets of points zeros - evaluating to zero - and ones - evaluating to one -, compute the lexicographically minimal boolean polynomial satisfying these points.

INPUT:

• zeros - the set of interpolation points mapped to zero

• ones - the set of interpolation points mapped to one

EXAMPLES:

First we create a random-ish boolean polynomial.

sage: B.<a,b,c,d,e,f> = BooleanPolynomialRing(6)
sage: f = a*b*c*e + a*d*e + a*f + b + c + e + f + 1

Now we find interpolation points mapping to zero and to one.

sage: zeros = set([(1, 0, 1, 0, 0, 0), (1, 0, 0, 0, 1, 0),
....:              (0, 0, 1, 1, 1, 1), (1, 0, 1, 1, 1, 1),
....:              (0, 0, 0, 0, 1, 0), (0, 1, 1, 1, 1, 0),
....:              (1, 1, 0, 0, 0, 1), (1, 1, 0, 1, 0, 1)])
sage: ones = set([(0, 0, 0, 0, 0, 0), (1, 0, 1, 0, 1, 0),
....:             (0, 0, 0, 1, 1, 1), (1, 0, 0, 1, 0, 1),
....:             (0, 0, 0, 0, 1, 1), (0, 1, 1, 0, 1, 1),
....:             (0, 1, 1, 1, 1, 1), (1, 1, 1, 0, 1, 0)])
sage: [f(*p) for p in zeros]
[0, 0, 0, 0, 0, 0, 0, 0]
sage: [f(*p) for p in ones]
[1, 1, 1, 1, 1, 1, 1, 1]

Finally, we find the lexicographically smallest interpolation polynomial using PolyBoRi .

sage: g = B.interpolation_polynomial(zeros, ones); g
b*f + c + d*f + d + e*f + e + 1

sage: [g(*p) for p in zeros]
[0, 0, 0, 0, 0, 0, 0, 0]
sage: [g(*p) for p in ones]
[1, 1, 1, 1, 1, 1, 1, 1]

Alternatively, we can work with PolyBoRi’s native BooleSet’s. This example is from the PolyBoRi tutorial:

sage: B = BooleanPolynomialRing(4,"x0,x1,x2,x3")
sage: x = B.gen
sage: V=(x(0)+x(1)+x(2)+x(3)+1).set(); V
{{x0}, {x1}, {x2}, {x3}, {}}
sage: f=x(0)*x(1)+x(1)+x(2)+1
sage: z = f.zeros_in(V); z
{{x1}, {x2}}
sage: o = V.diff(z); o
{{x0}, {x3}, {}}
sage: B.interpolation_polynomial(z,o)
x1 + x2 + 1

ALGORITHM: Calls interpolate_smallest_lex as described in the PolyBoRi tutorial.

n_variables()

Return the number of variables in this boolean polynomial ring.

EXAMPLES:

sage: P.<x,y> = BooleanPolynomialRing(2)
sage: P.n_variables()
2

sage: P = BooleanPolynomialRing(1000, 'x')
sage: P.n_variables()
1000

Note

This is part of PolyBoRi’s native interface.

ngens()

Return the number of variables in this boolean polynomial ring.

EXAMPLES:

sage: P.<x,y> = BooleanPolynomialRing(2)
sage: P.ngens()
2

sage: P = BooleanPolynomialRing(1000, 'x')
sage: P.ngens()
1000

one()

EXAMPLES:

sage: P.<x0,x1> = BooleanPolynomialRing(2)
sage: P.one()
1

random_element(degree=None, terms=None, choose_degree=False, vars_set=None)

Return a random boolean polynomial. Generated polynomial has the given number of terms, and at most given degree.

INPUT:

• degree - maximum degree (default: 2 for len(var_set) > 1, 1 otherwise)

• terms – number of terms requested (default: 5). If more terms are requested than exist, then this parameter is silently reduced to the maximum number of available terms.

• choose_degree - choose degree of monomials randomly first, rather than monomials uniformly random

• vars_set - list of integer indices of generators of self to use in the generated polynomial

EXAMPLES:

sage: P.<x,y,z> = BooleanPolynomialRing(3)
sage: P.random_element(degree=3, terms=4)
x*y*z + x*z + x + y*z

sage: P.random_element(degree=1, terms=2)
z + 1

In corner cases this function will return fewer terms by default:

sage: P = BooleanPolynomialRing(2,'y')
sage: P.random_element()
y0*y1 + y0

sage: P = BooleanPolynomialRing(1,'y')
sage: P.random_element()
y

We return uniformly random polynomials up to degree 2:

sage: B.<a,b,c,d> = BooleanPolynomialRing()
sage: B.random_element(terms=Infinity)
a*b + a*c + a*d + b*c + b*d + d

remove_var(order=None, *var)

Remove a variable or sequence of variables from this ring.

If order is not specified, then the subring inherits the term order of the original ring, if possible.

EXAMPLES:

sage: R.<x,y,z,w> = BooleanPolynomialRing()
sage: R.remove_var(z)
Boolean PolynomialRing in x, y, w
sage: R.remove_var(z,x)
Boolean PolynomialRing in y, w
sage: R.remove_var(y,z,x)
Boolean PolynomialRing in w

Removing all variables results in the base ring:

sage: R.remove_var(y,z,x,w)
Finite Field of size 2

If possible, the term order is kept:

sage: R.<x,y,z,w> = BooleanPolynomialRing(order=’deglex’) sage: R.remove_var(y).term_order() Degree lexicographic term order

sage: R.<x,y,z,w> = BooleanPolynomialRing(order=’lex’) sage: R.remove_var(y).term_order() Lexicographic term order

Be careful with block orders when removing variables:

sage: R.<x,y,z,u,v> = BooleanPolynomialRing(order='deglex(2),deglex(3)')
sage: R.remove_var(x,y,z)
Traceback (most recent call last):
...
ValueError: impossible to use the original term order (most likely because it was a block order). Please specify the term order for the subring
sage: R.remove_var(x,y,z, order='deglex')
Boolean PolynomialRing in u, v

variable(i=0)

Return the i-th generator of this boolean polynomial ring.

INPUT:

• i - an integer or a boolean monomial in one variable

EXAMPLES:

sage: P.<x,y,z> = BooleanPolynomialRing(3)
sage: P.variable()
x
sage: P.variable(2)
z
sage: m = x.monomials()[0]
sage: P.variable(m)
x

zero()

EXAMPLES:

sage: P.<x0,x1> = BooleanPolynomialRing(2)
sage: P.zero()
0

class sage.rings.polynomial.pbori.pbori.BooleanPolynomialVector

Bases: object

A vector of boolean polynomials.

EXAMPLES:

sage: B.<a,b,c,d,e,f> = BooleanPolynomialRing()
sage: from sage.rings.polynomial.pbori.pbori import BooleanPolynomialVector
sage: l = [B.random_element() for _ in range(3)]
sage: v = BooleanPolynomialVector(l)
sage: len(v)
3
sage: v[0]
a*b + a + b*e + c*d + e*f
sage: list(v)
[a*b + a + b*e + c*d + e*f, a*d + c*d + d*f + e + f, a*c + a*e + b*c + c*f + f]

append(el)

Append the element el to this vector.

EXAMPLES:

sage: B.<a,b,c,d,e,f> = BooleanPolynomialRing()
sage: from sage.rings.polynomial.pbori.pbori import BooleanPolynomialVector
sage: v = BooleanPolynomialVector()
sage: for i in range(5):
....:   v.append(B.random_element())

sage: list(v)
[a*b + a + b*e + c*d + e*f, a*d + c*d + d*f + e + f, a*c + a*e + b*c + c*f + f, a*c + a*d + a*e + a*f + b*e, b*c + b*d + c*d + c + 1]

class sage.rings.polynomial.pbori.pbori.BooleanPolynomialVectorIterator

Bases: object

class sage.rings.polynomial.pbori.pbori.CCuddNavigator

Bases: object

constant()

else_branch()

terminal_one()

then_branch()

value()

class sage.rings.polynomial.pbori.pbori.FGLMStrategy

Bases: object

Strategy object for the FGLM algorithm to translate from one Groebner basis with respect to a term ordering A to another Groebner basis with respect to a term ordering B.

main()

Execute the FGLM algorithm.

EXAMPLES:

sage: from sage.rings.polynomial.pbori.pbori import *
sage: B.<x,y,z> = BooleanPolynomialRing()
sage: ideal = BooleanPolynomialVector([x+z, y+z])
sage: list(ideal)
[x + z, y + z]
sage: old_ring = B
sage: new_ring = B.clone(ordering=dp_asc)
sage: list(FGLMStrategy(old_ring, new_ring, ideal).main())
[y + x, z + x]

class sage.rings.polynomial.pbori.pbori.GroebnerStrategy

Bases: object

A Groebner strategy is the main object to control the strategy for computing Groebner bases.

Note

This class is mainly used internally.

add_as_you_wish(p)

Add a new generator but let the strategy object decide whether to perform immediate interreduction.

INPUT:

• p - a polynomial

EXAMPLES:

sage: from sage.rings.polynomial.pbori.pbori import *
sage: B.<a,b,c,d,e,f> = BooleanPolynomialRing()
sage: gbs = GroebnerStrategy(B)
sage: gbs.add_as_you_wish(a + b)
sage: list(gbs)
[a + b]
sage: gbs.add_as_you_wish(a + c)

Note that nothing happened immediately but that the generator was indeed added:

sage: list(gbs)
[a + b]

sage: gbs.symmGB_F2()
sage: list(gbs)
[a + c, b + c]

add_generator(p)

Add a new generator.

INPUT:

• p - a polynomial

EXAMPLES:

sage: from sage.rings.polynomial.pbori.pbori import *
sage: B.<a,b,c,d,e,f> = BooleanPolynomialRing()
sage: gbs = GroebnerStrategy(B)
sage: gbs.add_generator(a + b)
sage: list(gbs)
[a + b]
sage: gbs.add_generator(a + c)
Traceback (most recent call last):
...
ValueError: strategy already contains a polynomial with same lead

add_generator_delayed(p)

Add a new generator but do not perform interreduction immediately.

INPUT:

• p - a polynomial

EXAMPLES:

sage: from sage.rings.polynomial.pbori.pbori import *
sage: B.<a,b,c,d,e,f> = BooleanPolynomialRing()
sage: gbs = GroebnerStrategy(B)
sage: gbs.add_generator(a + b)
sage: list(gbs)
[a + b]
sage: gbs.add_generator_delayed(a + c)
sage: list(gbs)
[a + b]

sage: list(gbs.all_generators())
[a + b, a + c]

all_generators()

EXAMPLES:

sage: from sage.rings.polynomial.pbori.pbori import *
sage: B.<a,b,c,d,e,f> = BooleanPolynomialRing()
sage: gbs = GroebnerStrategy(B)
sage: gbs.add_as_you_wish(a + b)
sage: list(gbs)
[a + b]
sage: gbs.add_as_you_wish(a + c)

sage: list(gbs)
[a + b]

sage: list(gbs.all_generators())
[a + b, a + c]

all_spolys_in_next_degree()

clean_top_by_chain_criterion()

contains_one()

Return True if 1 is in the generating system.

EXAMPLES:

We construct an example which contains 1 in the ideal spanned by the generators but not in the set of generators:

sage: B.<a,b,c,d,e,f> = BooleanPolynomialRing()
sage: from sage.rings.polynomial.pbori.pbori import GroebnerStrategy
sage: gb = GroebnerStrategy(B)
sage: gb.add_generator(a*c + a*f + d*f + d + f)
sage: gb.add_generator(b*c + b*e + c + d + 1)
sage: gb.add_generator(a*f + a + c + d + 1)
sage: gb.add_generator(a*d + a*e + b*e + c + f)
sage: gb.add_generator(b*d + c + d*f + e + f)
sage: gb.add_generator(a*b + b + c*e + e + 1)
sage: gb.add_generator(a + b + c*d + c*e + 1)
sage: gb.contains_one()
False

Still, we have that:

sage: from sage.rings.polynomial.pbori import groebner_basis
sage: groebner_basis(gb)
[1]

faugere_step_dense(v)

Reduces a vector of polynomials using linear algebra.

INPUT:

• v - a boolean polynomial vector

EXAMPLES:

sage: B.<a,b,c,d,e,f> = BooleanPolynomialRing()
sage: from sage.rings.polynomial.pbori.pbori import GroebnerStrategy
sage: gb = GroebnerStrategy(B)
sage: gb.add_generator(a*c + a*f + d*f + d + f)
sage: gb.add_generator(b*c + b*e + c + d + 1)
sage: gb.add_generator(a*f + a + c + d + 1)
sage: gb.add_generator(a*d + a*e + b*e + c + f)
sage: gb.add_generator(b*d + c + d*f + e + f)
sage: gb.add_generator(a*b + b + c*e + e + 1)
sage: gb.add_generator(a + b + c*d + c*e + 1)

sage: from sage.rings.polynomial.pbori.pbori import BooleanPolynomialVector
sage: V= BooleanPolynomialVector([b*d, a*b])
sage: list(gb.faugere_step_dense(V))
[b + c*e + e + 1, c + d*f + e + f]

implications(i)

Compute “useful” implied polynomials of i-th generator, and add them to the strategy, if it finds any.

INPUT:

• i - an index

ll_reduce_all()

Use the built-in ll-encoded BooleSet of polynomials with linear lexicographical leading term, which coincides with leading term in current ordering, to reduce the tails of all polynomials in the strategy.

minimalize()

Return a vector of all polynomials with minimal leading terms.

Note

Use this function if strat contains a GB.

minimalize_and_tail_reduce()

Return a vector of all polynomials with minimal leading terms and do tail reductions.

Note

Use that if strat contains a GB and you want a reduced GB.

next_spoly()

nf(p)

Compute the normal form of p with respect to the generating set.

INPUT:

• p - a boolean polynomial

EXAMPLES:

sage: P = PolynomialRing(GF(2),10, 'x')
sage: B = BooleanPolynomialRing(10,'x')
sage: I = sage.rings.ideal.Cyclic(P)
sage: I = B.ideal([B(f) for f in I.gens()])
sage: gb = I.groebner_basis()

sage: from sage.rings.polynomial.pbori.pbori import GroebnerStrategy

sage: G = GroebnerStrategy(B)
sage: _ = [G.add_generator(f) for f in gb]
sage: G.nf(gb[0])
0
sage: G.nf(gb[0] + 1)
1
sage: G.nf(gb[0]*gb[1])
0
sage: G.nf(gb[0]*B.gen(1))
0

Note

The result is only canonical if the generating set is a Groebner basis.

npairs()

reduction_strategy

select(m)

Return the index of the generator which can reduce the monomial m.

INPUT:

• m - a BooleanMonomial

EXAMPLES:

sage: B.<a,b,c,d,e> = BooleanPolynomialRing()
sage: f = B.random_element()
sage: g = B.random_element()
sage: from sage.rings.polynomial.pbori.pbori import GroebnerStrategy
sage: strat = GroebnerStrategy(B)
sage: strat.add_generator(f)
sage: strat.add_generator(g)
sage: strat.select(f.lm())
0
sage: strat.select(g.lm())
1
sage: strat.select(e.lm())
-1

small_spolys_in_next_degree(f, n)

some_spolys_in_next_degree(n)

suggest_plugin_variable()

symmGB_F2()

Compute a Groebner basis for the generating system.

Note

This implementation is out of date, but it will revived at some point in time. Use the groebner_basis() function instead.

top_sugar()

variable_has_value(v)

Computes, whether there exists some polynomial of the form \(v+c\) in the Strategy – where c is a constant – in the list of generators.

INPUT:

• v - the index of a variable

EXAMPLES:

sage: B.<a,b,c,d,e,f> = BooleanPolynomialRing()
sage: from sage.rings.polynomial.pbori.pbori import GroebnerStrategy
sage: gb = GroebnerStrategy(B)
sage: gb.add_generator(a*c + a*f + d*f + d + f)
sage: gb.add_generator(b*c + b*e + c + d + 1)
sage: gb.add_generator(a*f + a + c + d + 1)
sage: gb.add_generator(a*d + a*e + b*e + c + f)
sage: gb.add_generator(b*d + c + d*f + e + f)
sage: gb.add_generator(a*b + b + c*e + e + 1)
sage: gb.variable_has_value(0)
False

sage: from sage.rings.polynomial.pbori import groebner_basis
sage: g = groebner_basis(gb)
sage: list(g)
[a, b + 1, c + 1, d, e + 1, f]

sage: gb = GroebnerStrategy(B)
sage: _ = [gb.add_generator(f) for f in g]
sage: gb.variable_has_value(0)
True

class sage.rings.polynomial.pbori.pbori.MonomialConstruct

Bases: object

Implements PolyBoRi’s Monomial() constructor.

class sage.rings.polynomial.pbori.pbori.MonomialFactory

Bases: object

Implements PolyBoRi’s Monomial() constructor. If a ring is given is can be used as a Monomial factory for the given ring.

EXAMPLES:

sage: from sage.rings.polynomial.pbori.pbori import *
sage: B.<a,b,c> = BooleanPolynomialRing()
sage: fac = MonomialFactory()
sage: fac = MonomialFactory(B)

class sage.rings.polynomial.pbori.pbori.PolynomialConstruct

Bases: object

Implements PolyBoRi’s Polynomial() constructor.

lead(x)

Return the leading monomial of boolean polynomial x, with respect to the order of parent ring.

EXAMPLES:

sage: from sage.rings.polynomial.pbori.pbori import *
sage: B.<a,b,c> = BooleanPolynomialRing()
sage: PolynomialConstruct().lead(a)
a

class sage.rings.polynomial.pbori.pbori.PolynomialFactory

Bases: object

Implements PolyBoRi’s Polynomial() constructor and a polynomial factory for given rings.

lead(x)

Return the leading monomial of boolean polynomial x, with respect to the order of parent ring.

EXAMPLES:

sage: from sage.rings.polynomial.pbori.pbori import *
sage: B.<a,b,c> = BooleanPolynomialRing()
sage: PolynomialFactory().lead(a)
a

class sage.rings.polynomial.pbori.pbori.ReductionStrategy

Bases: object

Functions and options for boolean polynomial reduction.

add_generator(p)

Add the new generator p to this strategy.

INPUT:

• p - a boolean polynomial.

EXAMPLES:

sage: from sage.rings.polynomial.pbori.pbori import *
sage: B.<x,y,z> = BooleanPolynomialRing()
sage: red = ReductionStrategy(B)
sage: red.add_generator(x)
sage: list([f.p for f in red])
[x]

can_rewrite(p)

Return True if p can be reduced by the generators of this strategy.

EXAMPLES:

sage: from sage.rings.polynomial.pbori.pbori import *
sage: B.<a,b,c,d> = BooleanPolynomialRing()
sage: red = ReductionStrategy(B)
sage: red.add_generator(a*b + c + 1)
sage: red.add_generator(b*c + d + 1)
sage: red.can_rewrite(a*b + a)
True
sage: red.can_rewrite(b + c)
False
sage: red.can_rewrite(a*d + b*c + d + 1)
True

cheap_reductions(p)

Perform ‘cheap’ reductions on p.

INPUT:

• p - a boolean polynomial

EXAMPLES:

sage: from sage.rings.polynomial.pbori.pbori import *
sage: B.<a,b,c,d> = BooleanPolynomialRing()
sage: red = ReductionStrategy(B)
sage: red.add_generator(a*b + c + 1)
sage: red.add_generator(b*c + d + 1)
sage: red.add_generator(a)
sage: red.cheap_reductions(a*b + a)
0
sage: red.cheap_reductions(b + c)
b + c
sage: red.cheap_reductions(a*d + b*c + d + 1)
b*c + d + 1

head_normal_form(p)

Compute the normal form of p with respect to the generators of this strategy but do not perform tail any reductions.

INPUT:

• p – a polynomial

EXAMPLES:

sage: from sage.rings.polynomial.pbori.pbori import *
sage: B.<x,y,z> = BooleanPolynomialRing()
sage: red = ReductionStrategy(B)
sage: red.opt_red_tail = True
sage: red.add_generator(x + y + 1)
sage: red.add_generator(y*z + z)

sage: red.head_normal_form(x + y*z)
y + z + 1

sage: red.nf(x + y*z)
y + z + 1

nf(p)

Compute the normal form of p w.r.t. to the generators of this reduction strategy object.

EXAMPLES:

sage: from sage.rings.polynomial.pbori.pbori import *
sage: B.<x,y,z> = BooleanPolynomialRing()
sage: red = ReductionStrategy(B)
sage: red.add_generator(x + y + 1)
sage: red.add_generator(y*z + z)
sage: red.nf(x)
y + 1

sage: red.nf(y*z + x)
y + z + 1

reduced_normal_form(p)

Compute the normal form of p with respect to the generators of this strategy and perform tail reductions.

INPUT:

• p - a polynomial

EXAMPLES:

sage: from sage.rings.polynomial.pbori.pbori import *
sage: B.<x,y,z> = BooleanPolynomialRing()
sage: red = ReductionStrategy(B)
sage: red.add_generator(x + y + 1)
sage: red.add_generator(y*z + z)
sage: red.reduced_normal_form(x)
y + 1

sage: red.reduced_normal_form(y*z + x)
y + z + 1

sage.rings.polynomial.pbori.pbori.TermOrder_from_pb_order(n, order, blocks)

class sage.rings.polynomial.pbori.pbori.VariableBlock

Bases: object

class sage.rings.polynomial.pbori.pbori.VariableConstruct

Bases: object

Implements PolyBoRi’s Variable() constructor.

class sage.rings.polynomial.pbori.pbori.VariableFactory

Bases: object

Implements PolyBoRi’s Variable() constructor and a variable factory for given ring

sage.rings.polynomial.pbori.pbori.add_up_polynomials(v, init)

Add up all entries in the vector v.

INPUT:

• v - a vector of boolean polynomials

EXAMPLES:

sage: from sage.rings.polynomial.pbori.pbori import *
sage: B.<a,b,c,d> = BooleanPolynomialRing()
sage: v = BooleanPolynomialVector()
sage: l = [B.random_element() for _ in range(5)]
sage: _ = [v.append(e) for e in l]
sage: add_up_polynomials(v, B.zero())
a*d + b*c + b*d + c + 1
sage: sum(l)
a*d + b*c + b*d + c + 1

sage.rings.polynomial.pbori.pbori.contained_vars(m)

sage.rings.polynomial.pbori.pbori.easy_linear_factors(p)

sage.rings.polynomial.pbori.pbori.gauss_on_polys(inp)

Perform Gaussian elimination on the input list of polynomials.

INPUT:

• inp – an iterable

EXAMPLES:

sage: B.<a,b,c,d,e,f> = BooleanPolynomialRing()
sage: from sage.rings.polynomial.pbori.pbori import *
sage: l = [B.random_element() for _ in range(B.ngens())]
sage: A, v = Sequence(l, B).coefficient_matrix()
sage: A
[1 0 0 0 0 1 0 0 1 1 0 0 0 0 1 0 0 0]
[0 0 1 0 0 0 0 0 0 1 0 0 0 1 0 1 1 0]
[0 1 0 1 0 0 1 0 0 0 1 0 0 0 0 0 1 0]
[0 1 1 1 1 0 0 0 1 0 0 0 0 0 0 0 0 0]
[0 0 0 0 0 0 1 1 0 1 0 1 0 0 0 0 0 1]
[0 1 0 0 1 0 1 0 0 0 0 0 1 0 0 0 0 1]

sage: e = gauss_on_polys(l)
sage: E, v = Sequence(e, B).coefficient_matrix()
sage: E
[1 0 0 0 0 1 0 0 1 1 0 0 0 0 1 0 0 0]
[0 1 0 0 0 0 0 0 1 1 1 0 1 1 0 1 0 1]
[0 0 1 0 0 0 0 0 0 1 0 0 0 1 0 1 1 0]
[0 0 0 1 0 0 0 1 1 0 0 1 1 1 0 1 1 0]
[0 0 0 0 1 0 0 1 1 0 1 1 0 1 0 1 0 1]
[0 0 0 0 0 0 1 1 0 1 0 1 0 0 0 0 0 1]

sage: A.echelon_form()
[1 0 0 0 0 1 0 0 1 1 0 0 0 0 1 0 0 0]
[0 1 0 0 0 0 0 0 1 1 1 0 1 1 0 1 0 1]
[0 0 1 0 0 0 0 0 0 1 0 0 0 1 0 1 1 0]
[0 0 0 1 0 0 0 1 1 0 0 1 1 1 0 1 1 0]
[0 0 0 0 1 0 0 1 1 0 1 1 0 1 0 1 0 1]
[0 0 0 0 0 0 1 1 0 1 0 1 0 0 0 0 0 1]

sage.rings.polynomial.pbori.pbori.get_var_mapping(ring, other)

Return a variable mapping between variables of other and ring. When other is a parent object, the mapping defines images for all variables of other. If it is an element, only variables occurring in other are mapped.

Raises NameError if no such mapping is possible.

EXAMPLES:

sage: P.<x,y,z> = BooleanPolynomialRing(3)
sage: R.<z,y> = QQ[]
sage: sage.rings.polynomial.pbori.pbori.get_var_mapping(P,R)
[z, y]
sage: sage.rings.polynomial.pbori.pbori.get_var_mapping(P, z^2)
[z, None]

sage: R.<z,x> = BooleanPolynomialRing(2)
sage: sage.rings.polynomial.pbori.pbori.get_var_mapping(P,R)
[z, x]
sage: sage.rings.polynomial.pbori.pbori.get_var_mapping(P, x^2)
[None, x]

sage.rings.polynomial.pbori.pbori.if_then_else(root, a, b)

The opposite of navigating down a ZDD using navigators is to construct new ZDDs in the same way, namely giving their else- and then-branch as well as the index value of the new node.

INPUT:

• root - a variable

• a - the if branch, a BooleSet or a BoolePolynomial

• b - the else branch, a BooleSet or a BoolePolynomial

EXAMPLES:

sage: from sage.rings.polynomial.pbori.pbori import if_then_else
sage: B = BooleanPolynomialRing(6,'x')
sage: x0,x1,x2,x3,x4,x5 = B.gens()
sage: f0 = x2*x3+x3
sage: f1 = x4
sage: if_then_else(x1, f0, f1)
{{x1,x2,x3}, {x1,x3}, {x4}}

sage: if_then_else(x1.lm().index(),f0,f1)
{{x1,x2,x3}, {x1,x3}, {x4}}

sage: if_then_else(x5, f0, f1)
Traceback (most recent call last):
...
IndexError: index of root must be less than the values of roots of the branches.

sage.rings.polynomial.pbori.pbori.interpolate(zero, one)

Interpolate a polynomial evaluating to zero on zero and to one on ones.

INPUT:

• zero - the set of zero

• one - the set of ones

EXAMPLES:

sage: B = BooleanPolynomialRing(4,"x0,x1,x2,x3")
sage: x = B.gen
sage: from sage.rings.polynomial.pbori.interpolate import *
sage: V=(x(0)+x(1)+x(2)+x(3)+1).set()

sage: V
{{x0}, {x1}, {x2}, {x3}, {}}

sage: f=x(0)*x(1)+x(1)+x(2)+1
sage: nf_lex_points(f, V)
x1 + x2 + 1

sage: z=f.zeros_in(V)
sage: z
{{x1}, {x2}}

sage: o=V.diff(z)
sage: o
{{x0}, {x3}, {}}

sage: interpolate(z,o)
x0*x1*x2 + x0*x1 + x0*x2 + x1*x2 + x1 + x2 + 1

sage.rings.polynomial.pbori.pbori.interpolate_smallest_lex(zero, one)

Interpolate the lexicographical smallest polynomial evaluating to zero on zero and to one on ones.

INPUT:

• zero - the set of zeros

• one - the set of ones

EXAMPLES:

Let V be a set of points in \(\GF{2}^n\) and f a Boolean polynomial. V can be encoded as a BooleSet. Then we are interested in the normal form of f against the vanishing ideal of V : I(V).

It turns out, that the computation of the normal form can be done by the computation of a minimal interpolation polynomial, which takes the same values as f on V:

sage: B = BooleanPolynomialRing(4,"x0,x1,x2,x3")
sage: x = B.gen
sage: from sage.rings.polynomial.pbori.interpolate import *
sage: V=(x(0)+x(1)+x(2)+x(3)+1).set()

We take V = {e0,e1,e2,e3,0}, where ei describes the i-th unit vector. For our considerations it does not play any role, if we suppose V to be embedded in \(\GF{2}^4\) or a vector space of higher dimension:

sage: V
{{x0}, {x1}, {x2}, {x3}, {}}

sage: f=x(0)*x(1)+x(1)+x(2)+1
sage: nf_lex_points(f, V)
x1 + x2 + 1

In this case, the normal form of f w.r.t. the vanishing ideal of V consists of all terms of f with degree smaller or equal to 1.

It can be easily seen, that this polynomial forms the same function on V as f. In fact, our computation is equivalent to the direct call of the interpolation function interpolate_smallest_lex, which has two arguments: the set of interpolation points mapped to zero and the set of interpolation points mapped to one:

sage: z=f.zeros_in(V)
sage: z
{{x1}, {x2}}

sage: o=V.diff(z)
sage: o
{{x0}, {x3}, {}}

sage: interpolate_smallest_lex(z,o)
x1 + x2 + 1

sage.rings.polynomial.pbori.pbori.ll_red_nf_noredsb(p, reductors)

Redude the polynomial p by the set of reductors with linear leading terms.

INPUT:

• p - a boolean polynomial

• reductors - a boolean set encoding a Groebner basis with linear leading terms.

EXAMPLES:

sage: from sage.rings.polynomial.pbori.pbori import ll_red_nf_noredsb
sage: B.<a,b,c,d> = BooleanPolynomialRing()
sage: p = a*b + c + d + 1
sage: f,g  = a + c + 1, b + d + 1
sage: reductors = f.set().union( g.set() )
sage: ll_red_nf_noredsb(p, reductors)
b*c + b*d + c + d + 1

sage.rings.polynomial.pbori.pbori.ll_red_nf_noredsb_single_recursive_call(p, reductors)

Redude the polynomial p by the set of reductors with linear leading terms.

ll_red_nf_noredsb_single_recursive() call has the same specification as ll_red_nf_noredsb(), but a different implementation: It is very sensitive to the ordering of variables, however it has the property, that it needs just one recursive call.

INPUT:

• p - a boolean polynomial

• reductors - a boolean set encoding a Groebner basis with linear leading terms.

EXAMPLES:

sage: from sage.rings.polynomial.pbori.pbori import ll_red_nf_noredsb_single_recursive_call
sage: B.<a,b,c,d> = BooleanPolynomialRing()
sage: p = a*b + c + d + 1
sage: f,g  = a + c + 1, b + d + 1
sage: reductors = f.set().union( g.set() )
sage: ll_red_nf_noredsb_single_recursive_call(p, reductors)
b*c + b*d + c + d + 1

sage.rings.polynomial.pbori.pbori.ll_red_nf_redsb(p, reductors)

Redude the polynomial p by the set of reductors with linear leading terms. It is assumed that the set reductors is a reduced Groebner basis.

INPUT:

• p - a boolean polynomial

• reductors - a boolean set encoding a reduced Groebner basis with linear leading terms.

EXAMPLES:

sage: from sage.rings.polynomial.pbori.pbori import ll_red_nf_redsb
sage: B.<a,b,c,d> = BooleanPolynomialRing()
sage: p = a*b + c + d + 1
sage: f,g  = a + c + 1, b + d + 1
sage: reductors = f.set().union( g.set() )
sage: ll_red_nf_redsb(p, reductors)
b*c + b*d + c + d + 1

sage.rings.polynomial.pbori.pbori.map_every_x_to_x_plus_one(p)

Map every variable x_i in this polynomial to x_i + 1.

EXAMPLES:

sage: B.<a,b,z> = BooleanPolynomialRing(3)
sage: f = a*b + z + 1; f
a*b + z + 1
sage: from sage.rings.polynomial.pbori.pbori import map_every_x_to_x_plus_one
sage: map_every_x_to_x_plus_one(f)
a*b + a + b + z + 1
sage: f(a+1,b+1,z+1)
a*b + a + b + z + 1

sage.rings.polynomial.pbori.pbori.mod_mon_set(a_s, v_s)

sage.rings.polynomial.pbori.pbori.mod_var_set(a, v)

sage.rings.polynomial.pbori.pbori.mult_fact_sim_C(v, ring)

sage.rings.polynomial.pbori.pbori.nf3(s, p, m)

sage.rings.polynomial.pbori.pbori.parallel_reduce(inp, strat, average_steps, delay_f)

sage.rings.polynomial.pbori.pbori.random_set(variables, length)

Return a random set of monomials with length elements with each element in the variables variables.

EXAMPLES:

sage: from sage.rings.polynomial.pbori.pbori import random_set, set_random_seed
sage: B.<a,b,c,d,e> = BooleanPolynomialRing()
sage: (a*b*c*d).lm()
a*b*c*d
sage: set_random_seed(1337)
sage: random_set((a*b*c*d).lm(),10)
{{a,b,c,d}, {a,b}, {a,c,d}, {a,c}, {b,c,d}, {b,d}, {b}, {c,d}, {c}, {d}}

sage.rings.polynomial.pbori.pbori.recursively_insert(n, ind, m)

sage.rings.polynomial.pbori.pbori.red_tail(s, p)

Perform tail reduction on p using the generators of s.

INPUT:

• s - a reduction strategy

• p - a polynomial

EXAMPLES:

sage: from sage.rings.polynomial.pbori.pbori import *
sage: B.<x,y,z> = BooleanPolynomialRing()
sage: red = ReductionStrategy(B)
sage: red.add_generator(x + y + 1)
sage: red.add_generator(y*z + z)
sage: red_tail(red,x)
x
sage: red_tail(red,x*y + x)
x*y + y + 1

sage.rings.polynomial.pbori.pbori.set_random_seed(seed)

Set the PolyBoRi random seed to seed.

EXAMPLES:

sage: from sage.rings.polynomial.pbori.pbori import random_set, set_random_seed
sage: B.<a,b,c,d,e> = BooleanPolynomialRing()
sage: (a*b*c*d).lm()
a*b*c*d
sage: set_random_seed(1337)
sage: random_set((a*b*c*d).lm(),2)
{{b}, {c}}
sage: random_set((a*b*c*d).lm(),2)
{{a,c,d}, {c}}

sage: set_random_seed(1337)
sage: random_set((a*b*c*d).lm(),2)
{{b}, {c}}
sage: random_set((a*b*c*d).lm(),2)
{{a,c,d}, {c}}

sage.rings.polynomial.pbori.pbori.substitute_variables(parent, vec, poly)

var(i) is replaced by vec[i] in poly.

EXAMPLES:

sage: B.<a,b,c> = BooleanPolynomialRing()
sage: f = a*b + c + 1
sage: from sage.rings.polynomial.pbori.pbori import substitute_variables
sage: substitute_variables(B, [a,b,c],f)
a*b + c + 1
sage: substitute_variables(B, [a+1,b,c],f)
a*b + b + c + 1
sage: substitute_variables(B, [a+1,b+1,c],f)
a*b + a + b + c
sage: substitute_variables(B, [a+1,b+1,B(0)],f)
a*b + a + b

Substitution is also allowed with different rings:

sage: B.<a,b,c> = BooleanPolynomialRing()
sage: f = a*b + c + 1
sage: B.<w,x,y,z> = BooleanPolynomialRing(order='deglex')

sage: from sage.rings.polynomial.pbori.pbori import substitute_variables
sage: substitute_variables(B, [x,y,z], f) * w
w*x*y + w*z + w

sage.rings.polynomial.pbori.pbori.top_index(s)

Return the highest index in the parameter s.

INPUT:

• s - BooleSet, BooleMonomial, BoolePolynomial

EXAMPLES:

sage: B.<x,y,z> = BooleanPolynomialRing(3)
sage: from sage.rings.polynomial.pbori.pbori import top_index
sage: top_index(x.lm())
0
sage: top_index(y*z)
1
sage: top_index(x + 1)
0

sage.rings.polynomial.pbori.pbori.unpickle_BooleanPolynomial(ring, string)

Unpickle boolean polynomials

EXAMPLES:

sage: T = TermOrder('deglex',2)+TermOrder('deglex',2)
sage: P.<a,b,c,d> = BooleanPolynomialRing(4,order=T)
sage: loads(dumps(a+b)) == a+b # indirect doctest
True

sage.rings.polynomial.pbori.pbori.unpickle_BooleanPolynomial0(ring, l)

Unpickle boolean polynomials.

EXAMPLES:

sage: T = TermOrder('deglex',2)+TermOrder('deglex',2)
sage: P.<a,b,c,d> = BooleanPolynomialRing(4,order=T)
sage: loads(dumps(a+b)) == a+b # indirect doctest
True

sage.rings.polynomial.pbori.pbori.unpickle_BooleanPolynomialRing(n, names, order)

Unpickle boolean polynomial rings.

EXAMPLES:

sage: T = TermOrder('deglex',2)+TermOrder('deglex',2)
sage: P.<a,b,c,d> = BooleanPolynomialRing(4,order=T)
sage: loads(dumps(P)) == P  # indirect doctest
True

sage.rings.polynomial.pbori.pbori.zeros(pol, s)

Return a BooleSet encoding on which points from s the polynomial pol evaluates to zero.

INPUT:

• pol - a boolean polynomial

• s - a set of points encoded as a BooleSet

EXAMPLES:

sage: B.<a,b,c,d> = BooleanPolynomialRing(4)
sage: f = a*b + a*c + d + b

Now we create a set of points:

sage: s = a*b + a*b*c + c*d + b*c
sage: s = s.set(); s
{{a,b,c}, {a,b}, {b,c}, {c,d}}

This encodes the points (1,1,1,0), (1,1,0,0), (0,0,1,1) and (0,1,1,0). But of these only (1,1,0,0) evaluates to zero.:

sage: from sage.rings.polynomial.pbori.pbori import zeros
sage: zeros(f, s)
{{a,b}}

For comparison we work with tuples:

sage: f.zeros_in([(1,1,1,0), (1,1,0,0), (0,0,1,1), (0,1,1,0)])
((1, 1, 0, 0),)