Language 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 main structure should go in the patch #12224. Up to now the code is a bit dissaminated everywhere in Sage:

• sage.categories.languages
• sage.categories.factorial_languages
• sage.categories.shifts
• sage.categories.examples.languages
• sage.monoids.free_monoid
• sage.combinat.languages.*
• sage.combinat.words.*
• sage.dynamics.symbolic.full_shift

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

What should we keep? What categories should 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

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)

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

other todos

• 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
• ...