#!/usr/bin/env python
#-*- coding:utf-8 -*-
##
## card.py
##
## Created on: Sep 26, 2017
## Author: Alexey S. Ignatiev
## E-mail: aignatiev@ciencias.ulisboa.pt
##
"""
===============
List of classes
===============
.. autosummary::
:nosignatures:
EncType
CardEnc
ITotalizer
==================
Module description
==================
This module provides access to various *cardinality constraint* [1]_
encodings to formulas in conjunctive normal form (CNF). These include
pairwise [2]_, bitwise [2]_, ladder/regular [3]_ [4]_, sequential counters
[5]_, sorting [6]_ and cardinality networks [7]_, totalizer [8]_, modulo
totalizer [9]_, and modulo totalizer for :math:`k`-cardinality [10]_, as
well as a *native* cardinality constraint representation supported by the
`MiniCard solver <https://github.com/liffiton/minicard>`__.
.. [1] Olivier Roussel, Vasco M. Manquinho. *Pseudo-Boolean and Cardinality
Constraints*. Handbook of Satisfiability. 2009. pp. 695-733
.. [2] Steven David Prestwich. *CNF Encodings*. Handbook of Satisfiability.
2009. pp. 75-97
.. [3] Carlos Ansótegui, Felip Manyà. *Mapping Problems with Finite-Domain
Variables to Problems with Boolean Variables*. SAT (Selected Papers)
2004. pp. 1-15
.. [4] Ian P. Gent, Peter Nightingale. *A New Encoding of Alldifferent Into
SAT*. In International workshop on modelling and reformulating
constraint satisfaction problems 2004. pp. 95-110
.. [5] Carsten Sinz. *Towards an Optimal CNF Encoding of Boolean
Cardinality Constraints*. CP 2005. pp. 827-831
.. [6] Kenneth E. Batcher. *Sorting Networks and Their Applications*.
AFIPS Spring Joint Computing Conference 1968. pp. 307-314
.. [7] Roberto Asin, Robert Nieuwenhuis, Albert Oliveras,
Enric Rodriguez-Carbonell. *Cardinality Networks and Their
Applications*. SAT 2009. pp. 167-180
.. [8] Olivier Bailleux, Yacine Boufkhad. *Efficient CNF Encoding of
Boolean Cardinality Constraints*. CP 2003. pp. 108-122
.. [9] Toru Ogawa, Yangyang Liu, Ryuzo Hasegawa, Miyuki Koshimura,
Hiroshi Fujita. *Modulo Based CNF Encoding of Cardinality Constraints
and Its Application to MaxSAT Solvers*. ICTAI 2013. pp. 9-17
.. [10] António Morgado, Alexey Ignatiev, Joao Marques-Silva. *MSCG: Robust
Core-Guided MaxSAT Solving*. System Description. JSAT 2015. vol. 9,
pp. 129-134
A cardinality constraint is a constraint of the form:
:math:`\sum_{i=1}^n{x_i}\leq k`. Cardinality constraints are ubiquitous in
practical problem formulations. Note that the implementation of the
pairwise, bitwise, and ladder encodings can only deal with AtMost1
constraints, e.g. :math:`\sum_{i=1}^n{x_i}\leq 1`.
Access to all cardinality encodings can be made through the main class of
this module, which is :class:`.CardEnc`.
Additionally, to the standard cardinality encodings that are basically
"static" CNF formulas, the module is designed to able to construct
*incremental* cardinality encodings, i.e. those that can be incrementally
extended at a later stage. At this point only the *iterative totalizer*
[11]_ encoding is supported. Iterative totalizer can be accessed with the
use of the :class:`.ITotalizer` class.
.. [11] Ruben Martins, Saurabh Joshi, Vasco M. Manquinho, Inês Lynce.
*Incremental Cardinality Constraints for MaxSAT*. CP 2014. pp. 531-548
==============
Module details
==============
"""
#
#==============================================================================
import math
from pysat.formula import CNF, CNFPlus, IDPool
from pysat._utils import MainThread
import pycard
import signal
#
#==============================================================================
class NoSuchEncodingError(Exception):
"""
This exception is raised when creating an unknown an AtMostk, AtLeastK,
or EqualK constraint encoding.
"""
pass
#
#==============================================================================
class EncType(object):
"""
This class represents a C-like ``enum`` type for choosing the
cardinality encoding to use. The values denoting the encodings are:
::
pairwise = 0
seqcounter = 1
sortnetwrk = 2
cardnetwrk = 3
bitwise = 4
ladder = 5
totalizer = 6
mtotalizer = 7
kmtotalizer = 8
native = 9
The desired encoding can be selected either directly by its integer
identifier, e.g. ``2``, or by its alphabetical name, e.g.
``EncType.sortnetwrk``.
Note that while most of the encodings are produced as a list of
clauses, the "native" encoding of `MiniCard
<https://github.com/liffiton/minicard>`__ is managed as one clause.
Given an AtMostK constraint :math:`\sum_{i=1}^n{x_i\leq k}`, the native
encoding represents it as a pair ``[lits, k]``, where ``lits`` is a
list of size ``n`` containing literals in the sum.
"""
pairwise = 0
seqcounter = 1
sortnetwrk = 2
cardnetwrk = 3
bitwise = 4
ladder = 5
totalizer = 6
mtotalizer = 7
kmtotalizer = 8
native = 9 # native representation used by Minicard
#
#==============================================================================
class CardEnc(object):
"""
This abstract class is responsible for the creation of cardinality
constraints encoded to a CNF formula. The class has three *class
methods* for creating AtMostK, AtLeastK, and EqualsK constraints. Given
a list of literals, an integer bound and an encoding type, each of
these methods returns an object of class :class:`pysat.formula.CNFPlus`
representing the resulting CNF formula.
Since the class is abstract, there is no need to create an object of
it. Instead, the methods should be called directly as class methods,
e.g. ``CardEnc.atmost(lits, bound)`` or ``CardEnc.equals(lits,
bound)``. An example usage is the following:
.. code-block:: python
>>> from pysat.card import *
>>> cnf = CardEnc.atmost(lits=[1, 2, 3], encoding=EncType.pairwise)
>>> print(cnf.clauses)
[[-1, -2], [-1, -3], [-2, -3]]
>>> cnf = CardEnc.equals(lits=[1, 2, 3], encoding=EncType.pairwise)
>>> print(cnf.clauses)
[[1, 2, 3], [-1, -2], [-1, -3], [-2, -3]]
"""
@classmethod
def _update_vids(cls, cnf, vpool):
"""
Update variable ids in the given formula and id pool.
:param cnf: a list of literals in the sum.
:param vpool: the value of bound :math:`k`.
:type cnf: :class:`.formula.CNFPlus`
:type vpool: :class:`.formula.IDPool`
"""
top, vmap = vpool.top, {} # current top and variable mapping
# creating a new variable mapping, taking into
# account variables marked as "occupied"
while top < cnf.nv:
top += 1
vpool.top += 1
while vpool._occupied and vpool.top >= vpool._occupied[0][0]:
if vpool.top <= vpool._occupied[0][1] + 1:
vpool.top = vpool._occupied[0][1] + 1
vpool._occupied.pop(0)
vmap[top] = vpool.top
# updating the clauses
for cl in cnf.clauses:
cl[:] = map(lambda l: int(math.copysign(vmap[abs(l)], l)) if abs(l) in vmap else l, cl)
# updating the number of variables
cnf.nv = vpool.top
@classmethod
def atmost(cls, lits, bound=1, top_id=None, vpool=None,
encoding=EncType.seqcounter):
"""
This method can be used for creating a CNF encoding of an AtMostK
constraint, i.e. of :math:`\sum_{i=1}^{n}{x_i}\leq k`. The method
shares the arguments and the return type with method
:meth:`CardEnc.atleast`. Please, see it for details.
"""
if encoding < 0 or encoding > 9:
raise(NoSuchEncodingError(encoding))
assert not top_id or not vpool, \
'Use either a top id or a pool of variables but not both.'
# we are going to return this formula
ret = CNFPlus()
# if the list of literals is empty, return empty formula
if not lits:
return ret
# obtaining the top id from the variable pool
if vpool:
top_id = vpool.top
# choosing the maximum id among the current top and the list of literals
top_id = max(map(lambda x: abs(x), lits + [top_id if top_id != None else 0]))
# MiniCard's native representation is handled separately
if encoding == 9:
ret.atmosts, ret.nv = [(lits, bound)], top_id
return ret
res = pycard.encode_atmost(lits, bound, top_id, encoding,
int(MainThread.check()))
if res:
ret.clauses, ret.nv = res
# updating vpool if necessary
if vpool:
if vpool._occupied and vpool.top <= vpool._occupied[0][0] <= ret.nv:
cls._update_vids(ret, vpool)
else:
vpool.top = ret.nv - 1
vpool._next()
return ret
@classmethod
def atleast(cls, lits, bound=1, top_id=None, vpool=None,
encoding=EncType.seqcounter):
"""
This method can be used for creating a CNF encoding of an AtLeastK
constraint, i.e. of :math:`\sum_{i=1}^{n}{x_i}\geq k`. The method
takes 1 mandatory argument ``lits`` and 3 default arguments can be
specified: ``bound``, ``top_id``, ``vpool``, and ``encoding``.
:param lits: a list of literals in the sum.
:param bound: the value of bound :math:`k`.
:param top_id: top variable identifier used so far.
:param vpool: variable pool for counting the number of variables.
:param encoding: identifier of the encoding to use.
:type lits: iterable(int)
:type bound: int
:type top_id: integer or None
:type vpool: :class:`.IDPool`
:type encoding: integer
Parameter ``top_id`` serves to increase integer identifiers of
auxiliary variables introduced during the encoding process. This
is helpful when augmenting an existing CNF formula with the new
cardinality encoding to make sure there is no collision between
identifiers of the variables. If specified, the identifiers of the
first auxiliary variable will be ``top_id+1``.
Instead of ``top_id``, one may want to use a pool of variable
identifiers ``vpool``, which is automatically updated during the
method call. In many circumstances, this is more convenient than
using ``top_id``. Also note that parameters ``top_id`` and
``vpool`` **cannot** be specified *simultaneusly*.
The default value of ``encoding`` is :attr:`Enctype.seqcounter`.
The method *translates* the AtLeast constraint into an AtMost
constraint by *negating* the literals of ``lits``, creating a new
bound :math:`n-k` and invoking :meth:`CardEnc.atmost` with the
modified list of literals and the new bound.
:raises CardEnc.NoSuchEncodingError: if encoding does not exist.
:rtype: a :class:`.CNFPlus` object where the new \
clauses (or the new native atmost constraint) are stored.
"""
if encoding < 0 or encoding > 9:
raise(NoSuchEncodingError(encoding))
assert not top_id or not vpool, \
'Use either a top id or a pool of variables but not both.'
# we are going to return this formula
ret = CNFPlus()
# if the list of literals is empty, return empty formula
if not lits:
return ret
# obtaining the top id from the variable pool
if vpool:
top_id = vpool.top
# choosing the maximum id among the current top and the list of literals
top_id = max(map(lambda x: abs(x), lits + [top_id if top_id != None else 0]))
# Minicard's native representation is handled separately
if encoding == 9:
ret.atmosts, ret.nv = [([-l for l in lits], len(lits) - bound)], top_id
return ret
res = pycard.encode_atleast(lits, bound, top_id, encoding,
int(MainThread.check()))
if res:
ret.clauses, ret.nv = res
# updating vpool if necessary
if vpool:
if vpool._occupied and vpool.top <= vpool._occupied[0][0] <= ret.nv:
cls._update_vids(ret, vpool)
else:
vpool.top = ret.nv - 1
vpool._next()
return ret
@classmethod
def equals(cls, lits, bound=1, top_id=None, vpool=None,
encoding=EncType.seqcounter):
"""
This method can be used for creating a CNF encoding of an EqualsK
constraint, i.e. of :math:`\sum_{i=1}^{n}{x_i}= k`. The method
makes consecutive calls of both :meth:`CardEnc.atleast` and
:meth:`CardEnc.atmost`. It shares the arguments and the return type
with method :meth:`CardEnc.atleast`. Please, see it for details.
"""
if vpool:
res1 = cls.atleast(lits, bound=bound, vpool=vpool, encoding=encoding)
res2 = cls.atmost(lits, bound=bound, vpool=vpool, encoding=encoding)
else:
res1 = cls.atleast(lits, bound=bound, top_id=top_id, encoding=encoding)
res2 = cls.atmost(lits, bound=bound, top_id=res1.nv, encoding=encoding)
# merging together AtLeast and AtMost constraints
res1.nv = max(res1.nv, res2.nv)
res1.clauses.extend(res2.clauses)
res1.atmosts.extend(res2.atmosts)
return res1
#
#==============================================================================
class ITotalizer(object):
"""
This class implements the iterative totalizer encoding [11]_. Note that
:class:`ITotalizer` can be used only for creating AtMostK constraints.
In contrast to class :class:`EncType`, this class is not abstract and
its objects once created can be reused several times. The idea is that
a *totalizer tree* can be extended, or the bound can be increased, as
well as two totalizer trees can be merged into one.
The constructor of the class object takes 3 default arguments.
:param lits: a list of literals to sum.
:param ubound: the largest potential bound to use.
:param top_id: top variable identifier used so far.
:type lits: iterable(int)
:type ubound: int
:type top_id: integer or None
The encoding of the current tree can be accessed with the use of
:class:`.CNF` variable stored as ``self.cnf``. Potential bounds **are
not** imposed by default but can be added as unit clauses in the final
CNF formula. The bounds are stored in the list of Boolean variables as
``self.rhs``. A concrete bound :math:`k` can be enforced by considering
a unit clause ``-self.rhs[k]``. **Note** that ``-self.rhs[0]`` enforces
all literals of the sum to be *false*.
An :class:`ITotalizer` object should be deleted if it is not needed
anymore.
Possible usage of the class is shown below:
.. code-block:: python
>>> from pysat.card import ITotalizer
>>> t = ITotalizer(lits=[1, 2, 3], ubound=1)
>>> print(t.cnf.clauses)
[[-2, 4], [-1, 4], [-1, -2, 5], [-4, 6], [-5, 7], [-3, 6], [-3, -4, 7]]
>>> print(t.rhs)
[6, 7]
>>> t.delete()
Alternatively, an object can be created using the ``with`` keyword. In
this case, the object is deleted automatically:
.. code-block:: python
>>> from pysat.card import ITotalizer
>>> with ITotalizer(lits=[1, 2, 3], ubound=1) as t:
... print(t.cnf.clauses)
[[-2, 4], [-1, 4], [-1, -2, 5], [-4, 6], [-5, 7], [-3, 6], [-3, -4, 7]]
... print(t.rhs)
[6, 7]
"""
def __init__(self, lits=[], ubound=1, top_id=None):
"""
Constructor.
"""
# internal totalizer object
self.tobj = None
# its characteristics
self.lits = []
self.ubound = 0
self.top_id = 0
# encoding result
self.cnf = CNF() # CNF formula encoding the totalizer object
self.rhs = [] # upper bounds on the number of literals (rhs)
# number of new clauses
self.nof_new = 0
# this newly created totalizer object is not yet merged in any other
self._merged = False
if lits:
self.new(lits=lits, ubound=ubound, top_id=top_id)
def new(self, lits=[], ubound=1, top_id=None):
"""
The actual constructor of :class:`ITotalizer`. Invoked from
``self.__init__()``. Creates an object of :class:`ITotalizer` given
a list of literals in the sum, the largest potential bound to
consider, as well as the top variable identifier used so far. See
the description of :class:`ITotalizer` for details.
"""
self.lits = list(lits)
self.ubound = ubound
self.top_id = max(map(lambda x: abs(x), self.lits + [top_id if top_id != None else 0]))
# creating the object
self.tobj, clauses, self.rhs, self.top_id = pycard.itot_new(self.lits,
self.ubound, self.top_id, int(MainThread.check()))
# saving the result
self.cnf.clauses = clauses
self.cnf.nv = self.top_id
# for convenience, keeping the number of clauses
self.nof_new = len(clauses)
def delete(self):
"""
Destroys a previously constructed :class:`ITotalizer` object.
Internal variables ``self.cnf`` and ``self.rhs`` get cleaned.
"""
if self.tobj:
if not self._merged:
pycard.itot_del(self.tobj)
# otherwise, this totalizer object is merged into a larger one
# therefore, this memory should be freed in its destructor
self.tobj = None
self.lits = []
self.ubound = 0
self.top_id = 0
self.cnf = CNF()
self.rhs = []
self.nof_new = 0
def __enter__(self):
"""
'with' constructor.
"""
return self
def __exit__(self, exc_type, exc_value, traceback):
"""
'with' destructor.
"""
self.delete()
def __del__(self):
"""
Destructor.
"""
self.delete()
def increase(self, ubound=1, top_id=None):
"""
Increases a potential upper bound that can be imposed on the
literals in the sum of an existing :class:`ITotalizer` object to a
new value.
:param ubound: a new upper bound.
:param top_id: a new top variable identifier.
:type ubound: int
:type top_id: integer or None
The top identifier ``top_id`` applied only if it is greater than
the one used in ``self``.
This method creates additional clauses encoding the existing
totalizer tree up to the new upper bound given and appends them to
the list of clauses of :class:`.CNF` ``self.cnf``. The number of
newly created clauses is stored in variable ``self.nof_new``.
Also, a list of bounds ``self.rhs`` gets increased and its length
becomes ``ubound+1``.
The method can be used in the following way:
.. code-block:: python
>>> from pysat.card import ITotalizer
>>> t = ITotalizer(lits=[1, 2, 3], ubound=1)
>>> print(t.cnf.clauses)
[[-2, 4], [-1, 4], [-1, -2, 5], [-4, 6], [-5, 7], [-3, 6], [-3, -4, 7]]
>>> print(t.rhs)
[6, 7]
>>>
>>> t.increase(ubound=2)
>>> print(t.cnf.clauses)
[[-2, 4], [-1, 4], [-1, -2, 5], [-4, 6], [-5, 7], [-3, 6], [-3, -4, 7], [-3, -5, 8]]
>>> print(t.cnf.clauses[-t.nof_new:])
[[-3, -5, 8]]
>>> print(t.rhs)
[6, 7, 8]
>>> t.delete()
"""
self.top_id = max(self.top_id, top_id if top_id != None else 0)
# do nothing if the bound is set incorrectly
if ubound <= self.ubound or self.ubound >= len(self.lits):
self.nof_new = 0
return
else:
self.ubound = ubound
# updating the object and adding more variables and clauses
clauses, self.rhs, self.top_id = pycard.itot_inc(self.tobj,
self.ubound, self.top_id, int(MainThread.check()))
# saving the result
self.cnf.clauses.extend(clauses)
self.cnf.nv = self.top_id
# keeping the number of newly added clauses
self.nof_new = len(clauses)
def extend(self, lits=[], ubound=None, top_id=None):
"""
Extends the list of literals in the sum and (if needed) increases a
potential upper bound that can be imposed on the complete list of
literals in the sum of an existing :class:`ITotalizer` object to a
new value.
:param lits: additional literals to be included in the sum.
:param ubound: a new upper bound.
:param top_id: a new top variable identifier.
:type lits: iterable(int)
:type ubound: int
:type top_id: integer or None
The top identifier ``top_id`` applied only if it is greater than
the one used in ``self``.
This method creates additional clauses encoding the existing
totalizer tree augmented with new literals in the sum and updating
the upper bound. As a result, it appends the new clauses to the
list of clauses of :class:`.CNF` ``self.cnf``. The number of newly
created clauses is stored in variable ``self.nof_new``.
Also, if the upper bound is updated, a list of bounds ``self.rhs``
gets increased and its length becomes ``ubound+1``. Otherwise, it
is updated with new values.
The method can be used in the following way:
.. code-block:: python
>>> from pysat.card import ITotalizer
>>> t = ITotalizer(lits=[1, 2], ubound=1)
>>> print(t.cnf.clauses)
[[-2, 3], [-1, 3], [-1, -2, 4]]
>>> print(t.rhs)
[3, 4]
>>>
>>> t.extend(lits=[5], ubound=2)
>>> print(t.cnf.clauses)
[[-2, 3], [-1, 3], [-1, -2, 4], [-5, 6], [-3, 6], [-4, 7], [-3, -5, 7], [-4, -5, 8]]
>>> print(t.cnf.clauses[-t.nof_new:])
[[-5, 6], [-3, 6], [-4, 7], [-3, -5, 7], [-4, -5, 8]]
>>> print(t.rhs)
[6, 7, 8]
>>> t.delete()
"""
# preparing a new list of distinct input literals
lits = sorted(set(lits).difference(set(self.lits)))
if not lits:
# nothing to merge with -> just increase the bound
if ubound:
self.increase(ubound=ubound, top_id=top_id)
return
self.top_id = max(map(lambda x: abs(x), self.lits + [self.top_id, top_id if top_id != None else 0]))
self.ubound = max(self.ubound, ubound if ubound != None else 0)
# updating the object and adding more variables and clauses
self.tobj, clauses, self.rhs, self.top_id = pycard.itot_ext(self.tobj,
lits, self.ubound, self.top_id, int(MainThread.check()))
# saving the result
self.cnf.clauses.extend(clauses)
self.cnf.nv = self.top_id
self.lits.extend(lits)
# for convenience, keeping the number of new clauses
self.nof_new = len(clauses)
def merge_with(self, another, ubound=None, top_id=None):
"""
This method merges a tree of the current :class:`ITotalizer`
object, with a tree of another object and (if needed) increases a
potential upper bound that can be imposed on the complete list of
literals in the sum of an existing :class:`ITotalizer` object to a
new value.
:param another: another totalizer to merge with.
:param ubound: a new upper bound.
:param top_id: a new top variable identifier.
:type another: :class:`ITotalizer`
:type ubound: int
:type top_id: integer or None
The top identifier ``top_id`` applied only if it is greater than
the one used in ``self``.
This method creates additional clauses encoding the existing
totalizer tree merged with another totalizer tree into *one* sum
and updating the upper bound. As a result, it appends the new
clauses to the list of clauses of :class:`.CNF` ``self.cnf``. The
number of newly created clauses is stored in variable
``self.nof_new``.
Also, if the upper bound is updated, a list of bounds ``self.rhs``
gets increased and its length becomes ``ubound+1``. Otherwise, it
is updated with new values.
The method can be used in the following way:
.. code-block:: python
>>> from pysat.card import ITotalizer
>>> with ITotalizer(lits=[1, 2], ubound=1) as t1:
... print(t1.cnf.clauses)
[[-2, 3], [-1, 3], [-1, -2, 4]]
... print(t1.rhs)
[3, 4]
...
... t2 = ITotalizer(lits=[5, 6], ubound=1)
... print(t1.cnf.clauses)
[[-6, 7], [-5, 7], [-5, -6, 8]]
... print(t1.rhs)
[7, 8]
...
... t1.merge_with(t2)
... print(t1.cnf.clauses)
[[-2, 3], [-1, 3], [-1, -2, 4], [-6, 7], [-5, 7], [-5, -6, 8], [-7, 9], [-8, 10], [-3, 9], [-4, 10], [-3, -7, 10]]
... print(t1.cnf.clauses[-t1.nof_new:])
[[-6, 7], [-5, 7], [-5, -6, 8], [-7, 9], [-8, 10], [-3, 9], [-4, 10], [-3, -7, 10]]
... print(t1.rhs)
[9, 10]
...
... t2.delete()
"""
self.top_id = max(self.top_id, top_id if top_id != None else 0, another.top_id)
self.ubound = max(self.ubound, ubound if ubound != None else 0, another.ubound)
# extending the list of input literals
self.lits.extend(another.lits)
# updating the object and adding more variables and clauses
self.tobj, clauses, self.rhs, self.top_id = pycard.itot_mrg(self.tobj,
another.tobj, self.ubound, self.top_id, int(MainThread.check()))
# saving the result
self.cnf.clauses.extend(another.cnf.clauses)
self.cnf.clauses.extend(clauses)
self.cnf.nv = self.top_id
# for convenience, keeping the number of new clauses
self.nof_new = len(another.cnf.clauses) + len(clauses)
# memory deallocation should not be done for the merged tree
another._merged = True
