Contexts.ipynb

Kernel: Python 3

Contexts

Contexts are central concept of Vcsn: 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.

Labelsets

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.

Gensets

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>>

Weightsets

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: $\langle \mathbb{R}, +, \times, 0, 1 \rangle$

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

• 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, $W_1 \times \cdots \times W_n$.

Examples

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

import vcsn
vcsn.context('lal<char(abc)>, b')

$\{a, b, c\}\to\mathbb{B}$

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

vcsn.context('lal<char(abc)>, z')

$\{a, b, c\}\to\mathbb{Z}$

To use words on the usual alphabet as labels:

vcsn.context('law<char(a-z)>, z')

$\{a, b, c, d, e, f, g, h, i, j, k, l, m, n, o, p, q, r, s, t, u, v, w, x, y, z\}^*\to\mathbb{Z}$

$k$-tape Automata

To create a "classical" two-tape automaton:

vcsn.context('lat<lal<char(a-f)>, lal<char(A-F)>>, b')

$\{a, b, c, d, e, f\} \times \{A, B, C, D, E, F\}\to\mathbb{B}$

Multiple Weights

To compute a Boolean and an integer:

vcsn.context('lal<char(ab)>, lat<b, z>')

$\{a, b\}\to\mathbb{B} \times \mathbb{Z}$

The following automaton is almost able to recognize $a^nb^n$: it accepts only words of $a^nb^m$ (aka $a^*b^*$) and return $(n, m)$. One still has to check that $n = m$.

zmin2 = vcsn.context('lal<char(ab)>, lat<zmin, zmin>')
zmin2

$\{a, b\}\to\mathbb{Z}_{\text{min}} \times \mathbb{Z}_{\text{min}}$

ab = zmin2.expression('(<1,0>a)*(<0,0>b)* & (<0,0>a)*(<0,1>b)*')
ab

$\left( \left\langle 1,0 \right\rangle \,a\right)^{*} \, {b}^{*} \& {a}^{*} \, \left( \left\langle 0,1 \right\rangle \,b\right)^{*}$

a = ab.automaton()
a

print(a.shortest(len = 4).format('list'))

<0,0>\e
<1,0>a
<0,1>b
<2,0>aa
<1,1>ab
<0,2>bb
<3,0>aaa
<2,1>aab
<1,2>abb
<0,3>bbb
<4,0>aaaa
<3,1>aaab
<2,2>aabb
<1,3>abbb
<0,4>bbbb

Boss

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

vcsn.context('''nullableset< lat< lal<char(ba)>,
                                  lat< lan<char(vu)>, law<char(x-z)> >
                             >
                >
                ,
                lat<expressionset<nullableset<lat<lan<char(fe)>, lan<char(hg)>>>,
                                  lat<r, q>>,
                    lat<b, z>
                >
''')

$(\{a, b\} \times (\{u, v\})^? \times \{x, y, z\}^*)^?\to\mathsf{RatE}[(\{e, f\})^? \times (\{g, h\})^?\to\mathbb{R} \times \mathbb{Q}] \times \mathbb{B} \times \mathbb{Z}$