The Vcsn platform relies on a central concept: "contexts". They denote typing information about automata, rational expressions, etc. This information is alike a function type: an input type (the label), and an output type (the weight).

Contexts are created by the vcsn.context function which takes a string as input. This string follows the following syntax:

<context> ::= <labelset> , <weightset>

i.e., a context name is composed of a labelset name, then a comma, then a weightset name.


Different LabelSets model multiple variations on labels, members of a monoid:

  • letterset< genset >
    Fully defined by an alphabet $A$, its labels being just letters. It is simply denoted by $A$. It corresponds to the usual definition of an NFA.

  • nullableset< labelset >
    Denoted by $A^?$, also defined by an alphabet $A$, its labels being either letters or the empty word. This corresponds to what is often called $\varepsilon$-NFAs.

  • wordset< genset >
    Denoted by $A^*$, also defined by an alphabet $A$, its labels being (possibly empty) words on this alphabet.

  • oneset
    Denoted by $\{1\}$, containing a single label: 1, the empty word.

  • tupleset< labelset1 , labelset2 , ..., labelsetn >
    Cartesian product of LabelSets, $L_1 \times \cdots \times L_n$. This type implements the concept of transducers with an arbitrary number of "tapes". The concept is developed more in-depth here: Transducers.


The gensets define the types of the letters, and sets of the valid letters. There is currently a single genset type.

  • char_letters
    Specify that the letters are implemented as char. Any char will be accepted. The genset is said to be "open".

  • char_letters(abc...)
    Specify that the letters are implemented as char, and the genset is closed to {a, b, c}. Any other char will be rejected.

Abbreviations for Labelsets

There are a few abbreviations that are accepted.

  • lal_char: letterset<char_letters>
  • lal_char(abc): letterset<char_letters(abc)>
  • lan_char: nullableset<letterset<char_letters>>
  • law_char: wordset<letterset<char_letters>>


The WeightSets define the semiring of the weights. Builtin weights include:

  • b
    The classical Booleans: $\langle \mathbb{B}, \vee, \wedge, \bot, \top \rangle$

  • z
    The integers coded as ints: $\langle \mathbb{Z}, +, \times, 0, 1 \rangle$

  • q
    The rationals, coded as pairs of ints: $\langle \mathbb{Q}, +, \times, 0, 1 \rangle$

  • qmp
    The rationals, with support for multiprecision: $\langle \mathbb{Q}_\text{mp}, +, \times, 0, 1 \rangle$

  • r
    The reals, coded as doubles: $\langle \mathbb{R}, +, \times, 0, 1 \rangle$

  • zmin
    The tropical semiring, coded as ints: $\langle \mathbb{Z} \cup \{\infty\}, \min, +, \infty, 0 \rangle$

  • rmin
    The tropical semiring, coded as floatss: $\langle \mathbb{R} \cup \{\infty\}, \min, +, \infty, 0 \rangle$

  • log
    The log semiring, coded as doubles: $\langle \mathbb{R} \cup \{-\infty, +\infty\}, \oplus_\mathrm{log}, +, +\infty, 0 \rangle$ (where $\oplus_\mathrm{log}$ denotes $x, y \rightarrow - \mathrm{log}(\exp(-x) + \exp(-y))$.

  • f2
    The field: $\langle \mathbb{F}_2, \oplus, \wedge, 0, 1 \rangle$ (where $\oplus$ denotes the "exclusive or").

  • tupleset
    Cartesian product of WeightSets, $W_1 \times \cdots \times W_n$.


The usual framework for automaton is to use letters as labels, and Booleans as weights:

In [1]:
import vcsn
vcsn.context('lal_char(abc), b')
$\{a, b, c\}\rightarrow\mathbb{B}$

If instead of a simple accepter that returns "yes" or "no", you want to compute an integer, work in $\mathbb{Z}$:

In [2]:
vcsn.context('lal_char(abc), z')
$\{a, b, c\}\rightarrow\mathbb{Z}$

To use words on the usual alphabet as labels:

In [3]:
vcsn.context('law_char(a-z), z')
$\{a, b, c, d, e, f, g, h, i, j, k, l, m, n, o, p, q, r, s, t, u, v, w, x, y, z\}^*\rightarrow\mathbb{Z}$

To create a "classical" two-tape automaton:

In [4]:
vcsn.context('lat<lal_char(a-f), lal_char(A-F)>, b')
$\{a, b, c, d, e, f\} \times \{A, B, C, D, E, F\}\rightarrow\mathbb{B}$

To compute a Boolean and an integer:

In [5]:
vcsn.context('lal_char(ab), lat<b, z>')
$\{a, b\}\rightarrow\mathbb{B} \times \mathbb{Z}$

The interpretation of the following monster is left to the reader as an exercise:

In [6]:
             +' lat<expressionset<lan<lat<lan_char(fe),lan_char(hg)>>, lat<r, q>>, lat<b, z>>')
$(\{a, b\} \times (\{u, v\})^? \times \{x, y, z\}^*)^?\rightarrow\mathsf{}[(\{e, f\})^? \times (\{g, h\})^?\rightarrow\mathbb{R} \times \mathbb{Q}] \times \mathbb{B} \times \mathbb{Z}$