Combinatorial specification searcher

Christian and Henning

The comb_spec_searcher package contains code for combinatorial exploration.


To install comb_spec_searcher on your system, run:

pip install comb_spec_searcher

It is also possible to install comb_spec_searcher in development mode to work on the source code, in which case you run the following after cloning the repository:

./ develop

Combinatorial exploration

A (combinatorial) class is a set of objects with a notion of size such that there are finitely many objects of each size. One of the primary goals of enumerative combinatorics is to count how many objects of each size there are in a class. One method for doing this is to find a (combinatorial) specification, which is a collection of (combinatorial) rules that describe how to build a class from other classes using well-defined constructors. Such a specification can then be used to count the number of objects of each size.

Combinatorial exploration is a systematic application of strategies to create rules about a class of interest, until a specification can be found. This package can be used to perform this process automatically. See the Combinatorial Exploration project and Christian Bean’s PhD thesis for more details.

The remainder of this README will be an example of how to use this package for performing combinatorial exploration on a specific class, namely words avoiding consecutive patterns.

Avoiding consecutive patterns in words

A word w over an alphabet Σ is a string consisting of letters from Σ. We say that w contains the word p if there is a consecutive sequence of letters in w equal to p. We say w avoids p if it does not contain p. In this context, we call p a pattern. In python, this containment check can be checked using in.

>>> w = "acbabcabbb"
>>> p = "abcab"
>>> p in w

For an alphabet Σ and a set of patterns P we define Σ*(P) to be the set of words over Σ that avoid every pattern in P. These are the classes that we will count. Of course, these all form regular languages, but it will serve as a good example of how to use the comb_spec_searcher package.

The first step is to create the classes that will be used for discovering the underlying structure of the class of interest. In this case, considering the prefix of the words is what we need. We then create a new python class representing this that inherits from CombinatorialClass which can be imported from comb_spec_searcher.

>>> from comb_spec_searcher import CombinatorialClass

>>> class AvoidingWithPrefix(CombinatorialClass):
...     def __init__(self, prefix, patterns, alphabet, just_prefix=False):
...         self.alphabet = frozenset(alphabet)
...         self.prefix = prefix
...         self.patterns = frozenset(patterns)
...         self.just_prefix = just_prefix # this will be needed later

Inheriting from CombinatorialClass requires you to implement a few functions for combinatorial exploration: is_empty, to_jsonable, __eq__, __hash__, __repr__, and __str__.

We will start by implementing the dunder methods (the ones with double underscores) required. The __eq__ method is particularly important as the CombinatorialSpecificationSearcher will use it to recognise if the same class appears multiple times.

...     # The dunder methods required to perform combinatorial exploration
...     def __eq__(self, other):
...         return (self.alphabet == other.alphabet and
...                 self.prefix == other.prefix and
...                 self.patterns == other.patterns and
...                 self.just_prefix == other.just_prefix)
...     def __hash__(self):
...         return hash(hash(self.prefix) + hash(self.patterns) +
...                     hash(self.alphabet) + hash(self.just_prefix))
...     def __str__(self):
...         if self.just_prefix:
...             return "The word {}".format(self.prefix)
...         return ("Words over } avoiding } with prefix {}"
...                 "".format(", ".join(l for l in self.alphabet),
...                           ", ".join(p for p in self.patterns),
...                           self.prefix if self.prefix else '""'))
...     def __repr__(self):
...         return "AvoidingWithPrefix({}, {}, {}".format(repr(self.prefix),
...                                                       repr(self.patterns),
...                                                       repr(self.alphabet))

Perhaps the most important function to be implemented is the is_empty function. This should return True if there are no objects of any length in the class, otherwise False. If it is not correctly implemented it may lead to tautological specifications. For example, in our case the class is empty if and only if the prefix contains a pattern to be avoided.

...     def is_empty(self):
...         return any(p in self.prefix for p in self.patterns)

The final function required is to_jsonable. This is primarily for the output, and only necessary for saving the output. It should be in a format that can be interpretated by json. What is important is that the from_dict function is written in such a way that for any class c we have CombinatorialClass.from_dict(c.to_jsonable()) == c.

...     def to_jsonable(self):
...         return {"prefix": self.prefix,
...                 "patterns": tuple(sorted(self.patterns)),
...                 "alphabet": tuple(sorted(self.alphabet)),
...                 "just_prefix": int(self.just_prefix)}
...     @classmethod
...     def from_dict(cls, data):
...         return cls(data['prefix'],
...                    data['patterns'],
...                    data['alphabet'],
...                    bool(int(data['just_prefix'])))

Our CombinatorialClass is now ready. What is left to do is create the strategies that the CombinatorialSpecificationSearcher will use for performing combinatorial exploration. This is given in the form of a StrategyPack which can be imported from comb_spec_searcher that we will populate in the remainder of this example.

>>> from comb_spec_searcher import StrategyPack
>>> pack = StrategyPack(initial_strats=[],
...                     inferral_strats=[],
...                     expansion_strats=[],
...                     ver_strats=[],
...                     name=("Finding specification for words avoiding "
...                           "consecutive patterns."))

Strategies are functions that take as input a class C and produce rules about C. The types of strategies are as follows: -initial_strats: yields rules for classes - inferral_strats: returns a single equivalence rule - expansion_strats: yields rules for classes

For example, every word over the alphabet Σ starting with prefix p is either just p or has prefix pa for some a in Σ. This rule is splitting the original into disjoint subsets. We call a rule using disjoint union a BatchRule. Although in this case there is a unique rule created by the strategy, strategies are assumed to create multiple rules, and as such should be implemented as generators.

>>> from comb_spec_searcher import BatchRule

>>> def expansion(avoiding_with_prefix, **kwargs):
...     if avoiding_with_prefix.just_prefix:
...         return
...     alphabet, prefix, patterns = (avoiding_with_prefix.alphabet,
...                                   avoiding_with_prefix.prefix,
...                                   avoiding_with_prefix.patterns)
...     # either just p
...     comb_classes = [AvoidingWithPrefix(prefix, patterns, alphabet, True)]
...     for a in alphabet:
...         # or has prefix pa for some a in Σ.
...         ends_with_a = AvoidingWithPrefix(prefix + a, patterns, alphabet)
...         comb_classes.append(ends_with_a)
...     yield BatchRule(("The next letter in the prefix is one of }"
...                      "".format(", ".join(l for l in alphabet))),
...                     comb_classes)

The classes that we will verify are those that consist of just the prefix. To verify these we create a new strategy that returns a VerificationRule when this is the case.

>>> from comb_spec_searcher import VerificationRule

>>> def only_prefix(avoiding_with_prefix, **kwargs):
...     if avoiding_with_prefix.just_prefix:
...         return VerificationRule(("The set contains only the word {}"
...                                  "".format(avoiding_with_prefix.prefix)))

The final strategy we will need is one that peels off much as possible from the front of the prefix p such that the avoidance conditions are unaffected. This should then give a rule that is a cartesian product of the part that is peeled off together with the words whose prefix is that of the remainder of the original prefix. We call rules whose constructor is cartesian product a DecompositionRule.

>>> from comb_spec_searcher import DecompositionRule

>>> def remove_front_of_prefix(avoiding_with_prefix, **kwargs):
...     """If the k is the maximum length of a pattern to be avoided, then any
...     occurrence using indices further to the right of the prefix can use at
...     most the last k - 1 letters in the prefix."""
...     if avoiding_with_prefix.just_prefix:
...         return
...     prefix, patterns, alphabet = (avoiding_with_prefix.prefix,
...                                   avoiding_with_prefix.patterns,
...                                   avoiding_with_prefix.alphabet)
...     # safe will be the index of the prefix in which we can remove upto without
...     # affecting the avoidance conditions
...     safe = max(0, len(prefix) - max(len(p) for p in patterns) + 1)
...     for i in range(safe, len(prefix)):
...         end = prefix[i:]
...         if any(end == patt[:len(end)] for patt in patterns):
...             break
...         safe = i + 1
...     if safe > 0:
...         start_prefix = prefix[:safe]
...         end_prefix = prefix[safe:]
...         start = AvoidingWithPrefix(start_prefix, patterns, alphabet, True)
...         end = AvoidingWithPrefix(end_prefix, patterns, alphabet)
...         yield DecompositionRule("Remove up to index {} of prefix".format(safe),
...                                 [start, end])

With these three strategies we are now ready to perform combinatorial exploration using the following pack.

>>> pack = StrategyPack(initial_strats=[remove_front_of_prefix],
...                     inferral_strats=[],
...                     expansion_strats=[[expansion]],
...                     ver_strats=[only_prefix],
...                     name=("Finding specification for words avoiding "
...                           "consecutive patterns."))

First we need to create the combinatorial class we want to count. For example, consider the words over the alphabet {a, b} that avoid ababa and babb. This class can be created using our initialise function.

>>> prefix = ''
>>> patterns = ['ababa', 'babb']
>>> alphabet = ['a', 'b']
>>> start_class = AvoidingWithPrefix(prefix, patterns, alphabet)

We can then initialise our CombinatorialSpecificationSearcher, and use the auto_search function which will return a ProofTree object that represents a specification assuming one is found (which in this case always will).

>>> from comb_spec_searcher import CombinatorialSpecificationSearcher

>>> searcher = CombinatorialSpecificationSearcher(start_class, pack)
>>> tree = searcher.auto_search()

Now that we have a ProofTree i.e., a specification, the obvious thing we want to do is find the generating function for the class that counts the number of objects of each size. This can be done by using the get_genf or get_min_poly methods on ProofTree. To use these methods we will need to go back and implement a few functions in our CombinatorialClass.

When you verify a class, this tells the ProofTree class that it can get the generating function by calling the get_genf (and/or the get_min_poly) function on CombinatorialClass. In our case, we verified exactly when the class was only the prefix, say p. The generating function of this is clearly x**len(p). We add these methods to our class.

>>> from sympy import abc, var

>>> def get_genf(self, **kwargs):
...     """Return the generating function when only a prefix."""
...     if self.just_prefix:
...         if self.is_empty():
...             return 0
...         else:
...             return abc.x**len(self.prefix)
>>> AvoidingWithPrefix.get_genf = get_genf
>>> def get_min_poly(self, *args, **kwargs):
...     """Return the minimum polynomial satisfied by the generating function
...     of the combinatorial class (in terms of F)."""
...     if self.just_prefix:
...         if self.is_empty():
...             return 0
...         else:
...             return var('F') - abc.x**len(self.prefix)
>>> AvoidingWithPrefix.get_min_poly = get_min_poly

Finally, in order to get initial terms, you will also need to implement the objects_of_length function which should yield all of the objects of a given length in the class.

>>> from itertools import product

>>> def objects_of_length(self, length):
...     """Yield the words of given length that start with prefix and avoid the
...     patterns. If just_prefix, then only yield that word."""
...     def possible_words():
...         """Yield all words of given length over the alphabet with prefix"""
...         for letters in product(self.alphabet,
...                                 repeat=length - len(self.prefix)):
...             yield self.prefix + "".join(a for a in letters)
...     if self.just_prefix:
...         if length == len(self.prefix) and not self.is_empty():
...             yield self.prefix
...         return
...     for word in possible_words():
...         if all(patt not in word for patt in self.patterns):
...             yield word
>>> AvoidingWithPrefix.objects_of_length = objects_of_length

With these in place if we then call the get_min_poly function with the flag solve=True

>>> tree.get_min_poly()
F*x**6 + F*x**3 - F*x**2 + 2*F*x - F + x**7 + x**5 + x**4 + x**3 + x**2 + 1
>>> tree.get_genf()
-(x + 1)*(x**2 - x + 1)**2*(x**2 + x + 1)/(x**6 + x**3 - x**2 + 2*x - 1)

we see that the minimum polynomial satisfied by the generating function F is F*(x**6 + x**3 - x**2 + 2*x - 1) + x**7 + x**5 + x**4 + x**3 + x**2 + 1 and moreover F = -(x**7 + x**5 + x**4 + x**3 + x**2 + 1)/(x**6 + x**3 - x**2 + 2*x - 1).

You can now try this yourself using the file, which can count any set of words avoiding consecutive patterns.