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@ntrang086
ntrang086 committed Apr 21, 2018
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46 changes: 22 additions & 24 deletions Sympy_Intro.ipynb
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"editable": true
},
"source": [
"# Warmup - Introduction to SymPy\n",
"This lab exercise uses the [SymPy](http://www.sympy.org/en/index.html) symbolic math library to model constraints in the problem. To do that, we will use symbols (`sympy.Symbol`), functions (`sympy.Function`), and expressions (`sympy.Expr`) from sympy, and then we'll combine the function and expression classes to make constraints -- evaluatable symbolic functions.\n",
"# Introduction to SymPy\n",
"This project uses the [SymPy](http://www.sympy.org/en/index.html) symbolic math library to model constraints in a problem. To do that, we will use symbols (`sympy.Symbol`), functions (`sympy.Function`), and expressions (`sympy.Expr`) from sympy, and then we'll combine the function and expression classes to make constraints -- evaluatable symbolic functions.\n",
"\n",
"In this warmup, you will be introduced to the syntax and functionality of SymPy:\n",
"We will look at examples to understand the syntax and functionality of SymPy:\n",
"- [Example 1](#Example-1:-Symbols): Creating [symbols](http://docs.sympy.org/dev/modules/core.html#module-sympy.core.symbol)\n",
"- [Example 2](#Example-2:-Expressions): Creating & manipulating [expressions](http://docs.sympy.org/dev/modules/core.html#id16) with [arithmetic & logical operators](http://docs.sympy.org/dev/modules/core.html#sympy-core)\n",
"- [Example 3](#Example-3:-Symbolic-substitution-and-expression-evaluation): Symbolic [substitution & evaluation](http://docs.sympy.org/dev/modules/core.html#subs)\n",
"- [Exercises](#SymPy-Exercises): Creating & manipulating constraints & [functions](http://docs.sympy.org/dev/modules/functions/index.html)\n",
"\n",
"(See a list of common \"gotchas\" for sympy in their documentation: http://docs.sympy.org/dev/gotchas.html)\n",
"\n",
"Start by reading and running the example cells, then complete the steps in the warmup cell."
"(See a list of common \"gotchas\" for sympy in their documentation: http://docs.sympy.org/dev/gotchas.html)"
]
},
{
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},
"source": [
"## Example 1: Symbols\n",
"**Sympy provides the `Symbol` class to create symbolic variables. Create individual symbols by calling the constructor with a symbol name.** (Tip: Use the `display()` function to pretty-print symbolic terms.)"
"**Sympy provides the `Symbol` class to create symbolic variables. Create individual symbols by calling the constructor with a symbol name.**"
]
},
{
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"editable": true
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"source": [
"**You can also create symbols from an iterable sequence using the `symbols()` function.**"
"**We can also create symbols from an iterable sequence using the `symbols()` function.**"
]
},
{
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"source": [
"## Example 2: Expressions\n",
"\n",
"**A symbol is the most basic expression.** (Tip: Jupyter notebooks show information about objects\n",
"using the `?` magic function)"
"**A symbol is the most basic expression.** Jupyter notebooks show information about objects\n",
"using the `?` magic function,"
]
},
{
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"source": [
"**You can also define expressions with relations between symbols.** (However, notice that expressions have no _names_...)"
"**We can also define expressions with relations between symbols.**"
]
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{
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"source": [
"x, y = symbols(\"x y\")\n",
"y = x # now y references the same symbolic object as x\n",
"display(y) # display(y) == x ??!"
"display(y)"
]
},
{
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"editable": true
},
"source": [
"**Use `sympy.Eq` for symbolic equality expressions:** (Tip: there are lots of expressions in the [sympy docs](http://docs.sympy.org/dev/modules/core.html#sympy-core))"
"**Use `sympy.Eq` for symbolic equality expressions:** There are lots of expressions in the [sympy docs](http://docs.sympy.org/dev/modules/core.html#sympy-core)."
]
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{
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"editable": true
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"source": [
"**Symbolic variables can be replaced by other variables, or by concrete values.** (Tip: use positional arguments in the `subs()` method to replace one symbol)"
"**Symbolic variables can be replaced by other variables, or by concrete values.** "
]
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{
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"editable": true
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"source": [
"**But keep in mind that substitution returns a _copy_ of the expression -- it doesn't operate in-place.** (Tip: as a result, you can use substitution on one expression bound to generic variables to generate new instances bound to specific variables.)\n",
"**But substitution returns a _copy_ of the expression -- it doesn't operate in-place.** As a result, we can use substitution on one expression bound to generic variables to generate new instances bound to specific variables.\n",
"\n",
"Look at what happens when we bind new variables to our equality relation:"
]
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"editable": true
},
"source": [
"**Symbol substitution returns an expression.** (Recall that Symbols _are_ expressions)."
"**Symbol substitution returns an expression.**"
]
},
{
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"editable": true
},
"source": [
"**But substituting values for all symbols returns a value type.** (Tip: You can substitute multiple symbols in the `subs()` command by providing a mapping (dict) from current symbols to new symbols or values.)"
"**But substituting values for all symbols returns a value type.** We can substitute multiple symbols in the `subs()` command by providing a mapping (dict) from current symbols to new symbols or values."
]
},
{
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"source": [
"## Example 4: Constraints\n",
"\n",
"**Constraints are a construct of this lab exercise (not part of sympy) that combine symbolic Functions with Expressions for evaluation. The `constraint()` function (defined in the `util` module) takes a name and an expression and returns a \"named expression\" -- a constraint.**"
"**Constraints are a construct of this project (not part of sympy) that combine symbolic Functions with Expressions for evaluation. The `constraint()` function (defined in the `util` module) takes a name and an expression and returns a \"named expression\" -- a constraint.**"
]
},
{
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},
"source": [
"## SymPy Exercises\n",
"Complete the following exercises to check your understanding of sympy symbols, expressions, and constraints:"
"The following exercises check our understanding of sympy symbols, expressions, and constraints:"
]
},
{
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"source": [
"A = symbols(\"A:3\")\n",
"\n",
"# test for completion\n",
"# Test for completion\n",
"assert(len(A) == 3)\n",
"assert(all([type(v) == Symbol for v in A]))\n",
"print(\"All tests passed!\")"
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"source": [
"E = Xor(\"a\", \"b\")\n",
"\n",
"# test for completion\n",
"# Test for completion\n",
"_vars = E.free_symbols\n",
"assert(len(_vars) == 2)\n",
"xor_table = {(0, 0): False, (0, 1): True, (1, 0): True, (1, 1): False}\n",
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"maxAbsDiff = constraint(\"MaxAbsDiff\", LessThan(abs(a - b), c))\n",
"maxAbsDiff_copy = maxAbsDiff.subs({a: A[0], b: A[1], c: A[2]})\n",
"\n",
"# test for completion\n",
"# Test for completion\n",
"assert(maxAbsDiff.free_symbols != maxAbsDiff_copy.free_symbols)\n",
"assert(len(maxAbsDiff_copy.free_symbols) == len(maxAbsDiff_copy.args))\n",
"inputs = {(0, 6, 7): True, (6, 0, 7): True, (7, 6, 0): False}\n",
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"editable": true
},
"source": [
"**(Optional) Question 4:** Create a constraint AllDiff accepting the symbols in A as arguments, returning True if they are all different, and False if any pair is the same"
"**Question 4:** Create a constraint AllDiff accepting the symbols in A as arguments, returning True if they are all different, and False if any pair is the same"
]
},
{
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"name": "python",
"nbconvert_exporter": "python",
"pygments_lexer": "ipython3",
"version": "3.6.0"
"version": "3.6.3"
}
},
"nbformat": 4,
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