A library for performing combinatorial exploration.

## Project description

The comb_spec_searcher package contains code for combinatorial exploration.

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## Installing

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:

`./setup.py 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.

```
>>> from comb_spec_searcher import CombinatorialObject
>>> class Word(str, CombinatorialObject):
... def size(self):
... return str.__len__(self)
>>> w = Word("acbabcabbb")
>>> p = Word("abcab")
>>> p in w
True
```

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[Word]):
... def __init__(self, prefix, patterns, alphabet, just_prefix=False):
... self.alphabet = frozenset(alphabet)
... self.prefix = Word(prefix)
... self.patterns = frozenset(map(Word, 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):
... prefix = self.prefix if self.prefix else '""'
... if self.just_prefix:
... return "The word {}".format(prefix)
... return ("Words over {{{}}} avoiding {{{}}} with prefix {}"
... "".format(", ".join(l for l in self.alphabet),
... ", ".join(p for p in self.patterns),
... prefix))
...
... 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 size 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'])))
```

We also add some methods that we will need to get the enumerations of the objects later.

```
... def is_atom(self):
... """Return True if the class contains a single word."""
... return self.just_prefix
...
... def minimum_size_of_object(self):
... """Return the size of the smallest object in the class."""
... return len(self.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 AtomStrategy, StrategyPack
>>> pack = StrategyPack(initial_strats=[],
... inferral_strats=[],
... expansion_strats=[],
... ver_strats=[AtomStrategy()],
... 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 - ver_strats: returns a rule when the count of a class is known.

In our pack we have already added the AtomStrategy. This will verify any combinatorial class that is an atom, in particular this is determined by the is_atom method we implemented on CombinatorialClass. To get the enumeration at the end, the strategy also uses the method minimum_size_of_object on CombinatorialClass. As we’ve implemented these two methods already, we are free to use the AtomStrategy.

Now we will create our first strategy. 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 DisjointUnionStrategy. Although in this case thereis 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 DisjointUnionStrategy
>>> class ExpansionStrategy(DisjointUnionStrategy[AvoidingWithPrefix, Word]):
... def decomposition_function(self, avoiding_with_prefix):
... if not avoiding_with_prefix.just_prefix:
... alphabet, prefix, patterns = (
... avoiding_with_prefix.alphabet,
... avoiding_with_prefix.prefix,
... avoiding_with_prefix.patterns,
... )
... children = [AvoidingWithPrefix(prefix, patterns, alphabet, True)]
... for a in alphabet:
... ends_with_a = AvoidingWithPrefix(prefix + a, patterns, alphabet)
... children.append(ends_with_a)
... return tuple(children)
...
... def formal_step(self):
... return "Either just the prefix, or append a letter from the alphabet"
...
... def forward_map(self, avoiding_with_prefix, word, children=None):
... """
... The backward direction of the underlying bijection used for object
... generation and sampling.
... """
... assert isinstance(word, Word)
... if children is None:
... children = self.decomposition_function(avoiding_with_prefix)
... assert children is not None
... if len(word) == len(avoiding_with_prefix.prefix):
... return (word,) + tuple(None for i in range(len(children) - 1))
... for idx, child in enumerate(children[1:]):
... if word[: len(child.prefix)] == child.prefix:
... return (
... tuple(None for _ in range(idx + 1))
... + (word,)
... + tuple(None for _ in range(len(children) - idx - 1))
... )
...
... def __str__(self):
... return self.formal_step()
...
... def __repr__(self):
... return self.__class__.__name__ + "()"
...
... @classmethod
... def from_dict(cls, d):
... return cls()
```

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 CartesianProductStrategy
>>> class RemoveFrontOfPrefix(CartesianProductStrategy[AvoidingWithPrefix, Word]):
... def decomposition_function(self, avoiding_with_prefix):
... """If the k is the maximum size 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 not avoiding_with_prefix.just_prefix:
... safe = self.index_safe_to_remove_up_to(avoiding_with_prefix)
... if safe > 0:
... prefix, patterns, alphabet = (
... avoiding_with_prefix.prefix,
... avoiding_with_prefix.patterns,
... avoiding_with_prefix.alphabet,
... )
... start_prefix = prefix[:safe]
... end_prefix = prefix[safe:]
... start = AvoidingWithPrefix(start_prefix, patterns, alphabet, True)
... end = AvoidingWithPrefix(end_prefix, patterns, alphabet)
... return (start, end)
...
... def index_safe_to_remove_up_to(self, avoiding_with_prefix):
... prefix, patterns = (
... avoiding_with_prefix.prefix,
... avoiding_with_prefix.patterns,
... )
... # safe will be the index of the prefix in which we can remove upto without
... # affecting the avoidance conditions
... m = max(len(p) for p in patterns) if patterns else 1
... safe = max(0, len(prefix) - m + 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
... return safe
...
... def formal_step(self):
... return "removing redundant prefix"
...
... def backward_map(self, avoiding_with_prefix, words, children=None):
... """
... The forward direction of the underlying bijection used for object
... generation and sampling.
... """
... assert len(words) == 2
... assert isinstance(words[0], Word)
... assert isinstance(words[1], Word)
... if children is None:
... children = self.decomposition_function(avoiding_with_prefix)
... assert children is not None
... yield Word(words[0] + words[1])
...
... def forward_map(self, comb_class, word, children=None):
... """
... The backward direction of the underlying bijection used for object
... generation and sampling.
... """
... assert isinstance(word, Word)
... if children is None:
... children = self.decomposition_function(comb_class)
... assert children is not None
... return Word(children[0].prefix), Word(word[len(children[0].prefix) :])
...
... @classmethod
... def from_dict(cls, d):
... return cls()
...
... def __str__(self):
... return self.formal_step()
...
... def __repr__(self):
... return self.__class__.__name__ + "()"
```

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

```
>>> pack = StrategyPack(initial_strats=[RemoveFrontOfPrefix()],
... inferral_strats=[],
... expansion_strats=[[ExpansionStrategy()]],
... ver_strats=[AtomStrategy()],
... 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 CombinatorialSpecification assuming one is found (which in this case always will).

```
>>> from comb_spec_searcher import CombinatorialSpecificationSearcher
>>> searcher = CombinatorialSpecificationSearcher(start_class, pack)
>>> spec = searcher.auto_search()
>>> # spec.show() will display the specification in your browser
```

Now that we have a CombinatorialSpecification, 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 methods on CombinatorialSpecification.

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

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

With these in place if we then call the get_genf function

```
>>> spec.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 the generating function is F = -(x**7 + x**5 + x**4 + x**3 + x**2 + 1)/(x**6 + x**3 - x**2 + 2*x - 1).

Moreover, we can get directly the number of objects by size with the method count_objects_of_size.

```
>>> [spec.count_objects_of_size(i) for i in range(11)]
[1, 2, 4, 8, 15, 27, 48, 87, 157, 283, 511]
```

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

Now we will demonstrate how a bijection can be found between classes. We will first need a couple of imports.

```
>>> from comb_spec_searcher.bijection import ParallelSpecFinder
>>> from comb_spec_searcher.isomorphism import Bijection
```

We start by defining our two classes that we wish to find a bijection between.

```
>>> prefix1 = ''
>>> patterns1 = ["00"]
>>> alphabet1 = ['0', '1']
>>> class1 = AvoidingWithPrefix(prefix1, patterns1, alphabet1)
>>> prefix2 = ''
>>> patterns2 = ["bb"]
>>> alphabet2 = ['a', 'b']
>>> class2 = AvoidingWithPrefix(prefix2, patterns2, alphabet2)
```

To find a bijection we expand the universe given a pack for both classes and try to construct specifications that are parallel.

```
>>> searcher1 = CombinatorialSpecificationSearcher(class1, pack)
>>> searcher2 = CombinatorialSpecificationSearcher(class2, pack)
```

We get two parallel specs if successful, None otherwise.

`>>> specs = ParallelSpecFinder(searcher1, searcher2).find()`

We then construct the bijection from the parallel specifications.

`>>> bijection = Bijection.construct(*specs)`

We can use the Bijection object to map (either way) sampled objects from the sepcifications.

```
>>> for i in range(10):
... for w in bijection.domain.generate_objects_of_size(i):
... assert w == bijection.inverse_map(bijection.map(w))
... for w in bijection.codomain.generate_objects_of_size(i):
... assert w == bijection.map(bijection.inverse_map(w))
...
```

Whether we find a bijection or not (when one exists) is highly dependent on the packs chosen.

### Citing

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