Languages and tilings

This page gathers ideas for refactorization of sage.combinat.words and implementation of tilings.

You can subscribe to the associated mailing-list to discuss about this.

Structure

The refactorization of the current code should go in the patch #12224 which is almost done. Up to now the code is a bit dissaminated everywhere in Sage:

• sage.categories: Most of the generic code is contained there.
  • .languages: A language is a subset of A^* where A is a set called alphabet. It is naturally graded by N and the grading is called the length.
  • .factorial_languages: category of factorial languages (= language stable under taking factors)
  • .shifts: the category of shift A^G where G is almost anything and A is a set called alphabet
• sage.combinat.words
  • data structure for finite and infinite words
• sage.monoids
  • .free_monoid: the free monoid (replaces part of sage.combinat.words.words.Words)

  • .free_monoid_morphism: (replaces sage.combinat.words.morphism.WordMorphism)

• sage.dynamics.symbolic
  • .full_shift: an implementation of the full shift (replaces part of sage.combinat.words.words.Words)
• sage.combinat.languages
  • implementation of different languages (balanced, language of a finite word, ...)
  • specific data structure (suffix tree/trie, rauzy graph, return tree, ...)

What is bad/nice with categories:

• inheritance of generic code
• a bit confusing for the user who want to find the implementation of a method
• confusing for the person who writes the code and ask "where should I put this ?"
• ...

What do we keep? What categories do we create?

Behavior of algorithms with infinite input data

What to do for equality of infinite words ?

What should do

sage: w1 == w2

Two possibilities:

1. test the first XXX letters for finding a difference. If find one then returns False otherwise raise an error, "seems to be equal use .is_equal(force=True) to launch the infinite test".
2. test all letters and never return True

Other suggestions ?

Subprojects

Finite languages and factor set

Most of it was implemented by Franco (suffix tree and suffix trie). We would like to enhance it and make a specific data structure (called Rauzy castle) for FiniteFactorialLanguages. See #12225.

Substitutive and adic languages

There are many algorithms for languages described by a sequence of substitutions (called a directive word). The particular case of morphic and purely morphic languages correspond respectively to periodic and purely_periodic directive words.

• Enumeration of factors, desubstitution (#12227)

• Factor complexity for purely morphic languages (#12231)

• Equality for purely morphic languages (following J. Honkala, CANT, chapter 10)

Eventually periodic languages / words

They will be useful to define eventually periodic directive words for adic languages. See #12228.

TODO list

which should go in the main trac ticket

• words path (currently in sage.combinat.words.paths) which have to be modified to fit with the new implementation
• backward compatibility with the previous implementation (in particular with respect to pickling)
• make difference between finite/infinite/enumerated/ordered alphabet (in particular with respect to the category initialization)

other todos for other tickets

• specific data structure rauzy graphs and return tree (Thierry)
• 1-dim subshift of finite type / sofic
• n-dim finite words and n-dimensional shifts
• n-dim subshifts of finite type
• n-dim substitutive subshift
• cellular automata

• ... add your whishes