{
"cells": [
{
"cell_type": "markdown",
"metadata": {},
"source": [
"# _`expression`_`.multiply`\n",
"\n",
"This function is overloaded, it supports these signatures:\n",
"\n",
"- _`expression`_`.multiply(`_`exp`_`)`\n",
"\n",
" The product (i.e., the concatenation) of two expressions: `a.multiply(b)` => `ab`.\n",
"\n",
"- _`expression`_`.multiply(`_`num`_`)`\n",
"\n",
" The repeated multiplication (concatenation) of an expression with itself: `a.multiply(3)` => `aaa`. Exponent `-1` denotes the infinity: the Kleene star.\n",
"\n",
"- _`expression`_`.multiply((`_`min`_`, `_`max`_`))`\n",
"\n",
" The sum of repeated multiplications of an expression: `a.multiply((2,4))` => `aa+aaa+aaaa`. When `min = -1`, it denotes `0`, when `max = -1`, it denotes the infinity.\n",
"\n",
"Preconditions:\n",
"- _`min`_ ` <= ` _`max`_\n",
"\n",
"See also:\n",
"- [automaton.multiply](automaton.multiply.ipynb)\n",
"- [polynomial.multiply](polynomial.multiply.ipynb)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Examples"
]
},
{
"cell_type": "code",
"execution_count": 1,
"metadata": {
"collapsed": false
},
"outputs": [],
"source": [
"import vcsn\n",
"exp = vcsn.context('law_char, q').expression"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### Simple Multiplication\n",
"Instead of `a.multiply(b)`, you may write `a * b`."
]
},
{
"cell_type": "code",
"execution_count": 2,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"text/latex": [
"${\\mathit{a}}^{*} \\, \\mathit{b} \\, \\mathit{a} \\, {\\mathit{b}}^{*}$"
],
"text/plain": [
"a*bab*"
]
},
"execution_count": 2,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"exp('a*b') * exp('ab*')"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Of course, trivial identities are applied."
]
},
{
"cell_type": "code",
"execution_count": 3,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"text/latex": [
"$ \\left\\langle 6 \\right\\rangle \\,\\mathit{a}$"
],
"text/plain": [
"<6>a"
]
},
"execution_count": 3,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"exp('<2>a') * exp('<3>\\e')"
]
},
{
"cell_type": "code",
"execution_count": 4,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"text/latex": [
"$\\emptyset$"
],
"text/plain": [
"\\z"
]
},
"execution_count": 4,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"exp('<2>a') * exp('\\z')"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"In the case of word labels, adjacent words are not fused: concatenation of two expressions behaves as if the expressions were parenthetized. Pay attention to the space between $a$ and $b$ below, admittedly too discreet."
]
},
{
"cell_type": "code",
"execution_count": 5,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"image/svg+xml": [
""
],
"text/plain": [
""
]
},
"execution_count": 5,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"(exp('a') * exp('b')).SVG() # Two one-letter words"
]
},
{
"cell_type": "code",
"execution_count": 6,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"image/svg+xml": [
""
],
"text/plain": [
""
]
},
"execution_count": 6,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"exp('ab').SVG() # One two-letter word"
]
},
{
"cell_type": "code",
"execution_count": 7,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"image/svg+xml": [
""
],
"text/plain": [
""
]
},
"execution_count": 7,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"exp('(a)(b)').SVG() # Two one-letter words"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### Repeated Multiplication\n",
"Instead of `a.multiply(3)`, you may write `a ** 3`."
]
},
{
"cell_type": "code",
"execution_count": 8,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"text/latex": [
"$\\left(\\mathit{ab}\\right)^{3}$"
],
"text/plain": [
"(ab){3}"
]
},
"execution_count": 8,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"exp('ab') ** 3"
]
},
{
"cell_type": "code",
"execution_count": 9,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"text/latex": [
"$\\varepsilon$"
],
"text/plain": [
"\\e"
]
},
"execution_count": 9,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"exp('ab') ** 0"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Beware that `a * 3` actually denotes `a.rweight(3)`."
]
},
{
"cell_type": "code",
"execution_count": 10,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"text/latex": [
"$ \\left\\langle 3 \\right\\rangle \\,{\\mathit{a}}^{*}$"
],
"text/plain": [
"<3>a*"
]
},
"execution_count": 10,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"exp('a*') * 3"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Use the exponent -1 to mean `infinity`. Alternatively, you may invoke `a.star` instead of `a ** -1`."
]
},
{
"cell_type": "code",
"execution_count": 11,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"text/latex": [
"$\\left(\\mathit{ab}\\right)^{*}$"
],
"text/plain": [
"(ab)*"
]
},
"execution_count": 11,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"exp('ab') ** -1"
]
},
{
"cell_type": "code",
"execution_count": 12,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"text/latex": [
"$\\left(\\mathit{ab}\\right)^{*}$"
],
"text/plain": [
"(ab)*"
]
},
"execution_count": 12,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"exp('ab').star()"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### Sums of Repeated Multiplications\n",
"Instead of `a.multiply((2, 4))`, you may write `a ** (2, 4)`. Again, use exponent -1 to mean infinity."
]
},
{
"cell_type": "code",
"execution_count": 13,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"text/latex": [
"$\\left(\\mathit{ab}\\right)^{2}$"
],
"text/plain": [
"(ab){2}"
]
},
"execution_count": 13,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"exp('ab') ** (2, 2)"
]
},
{
"cell_type": "code",
"execution_count": 14,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"text/latex": [
"$\\left(\\mathit{ab}\\right)^{2} \\, \\left(\\varepsilon + \\mathit{ab} + \\left(\\mathit{ab}\\right)^{2}\\right)$"
],
"text/plain": [
"(ab){2}(\\e+ab+(ab){2})"
]
},
"execution_count": 14,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"exp('ab') ** (2, 4)"
]
},
{
"cell_type": "code",
"execution_count": 15,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"text/latex": [
"$\\varepsilon + \\mathit{ab} + \\left(\\mathit{ab}\\right)^{2}$"
],
"text/plain": [
"\\e+ab+(ab){2}"
]
},
"execution_count": 15,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"exp('ab') ** (-1, 2)"
]
},
{
"cell_type": "code",
"execution_count": 16,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"text/latex": [
"$\\left(\\mathit{ab}\\right)^{2} \\, \\left(\\mathit{ab}\\right)^{*}$"
],
"text/plain": [
"(ab){2}(ab)*"
]
},
"execution_count": 16,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"exp('ab') ** (2, -1)"
]
}
],
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