Keisan

Keisan (計算, to calculate) is a Ruby library for parsing equations into an abstract syntax tree. This allows for safe evaluation of string representations of mathematical/logical expressions. It has support for variables, functions, conditionals, and loops, making it a Turing complete programming language.

Installation

Add this line to your application's Gemfile:

gem 'keisan'

And then execute:

$ bundle

Or install it yourself as:

$ gem install keisan

Usage

REPL

To try keisan out locally, clone this repository and run the executable bin/keisan to open up an interactive REPL. The commands you type in to this REPL are relayed to an internal Keisan::Calculator class and displayed back to you.

Calculator class

This library is interacted with primarily through the Keisan::Calculator class. The evaluate method evaluates an expression by parsing it into an abstract syntax tree (AST), and evaluating it. There is also a simplify method that allows undefined variables and functions to exist, and will just return the simplified AST.

calculator = Keisan::Calculator.new
calculator.evaluate("15 + 2 * (1 + 3)")
#=> 23
calculator.simplify("1*(0*2+x*g(t))").to_s
#=> "x*g(t)"

For users who want access to the parsed abstract syntax tree, you can use the ast method to parse any given expression.

calculator = Keisan::Calculator.new
ast = calculator.ast("x**2+1")
ast.class
#=> Keisan::AST::Plus
ast.to_s
#=> "(x**2)+1"
ast.children.map(&:to_s)
#=> ["x**2", "1"]

Caching AST results

Computing the AST from a string takes some non-zero amount of time. For applications of this gem that evaluate some set of fixed expressions (possibly with different variable values, but with fixed ASTs), it might be worthwhile to cache the ASTs for faster computation. To accomplish this, you can use the Keisan::AST::Cache class. Passing an instance of this class into the Calculator will mean everytime a new expression is encountered it will compute the AST and store it in this cache for retrieval next time the expression is encountered.

cache = Keisan::AST::Cache.new
# Note: if you don't want to create the Cache instance, you can just pass `cache: true` here as well
calculator = Keisan::Calculator.new(cache: cache)
calculator.evaluate("exp(-x/T)", x: 1.0, T: 10)
#=> 0.9048374180359595
# This call will use the cached AST for "exp(-x/T)"
calculator.evaluate("exp(-x/T)", x: 2.0, T: 10)
#=> 0.8187307530779818

If you just want to pre-populate the cache with some predetermined values, you can call #fetch_or_build on the Cache for each instance, freeze the cache, then use this frozen cache in your calculator. A cache that has been frozen will only fetch from the cache, never write new values to it.

cache = Keisan::AST::Cache.new
cache.fetch_or_build("f(x) + diff(g(x), x)")
cache.freeze
# This calculator will never write new values to the cache, but when
# evaluating `"f(x) + diff(g(x), x)"` will fetch this cached AST.
calculator = Keisan::Calculator.new(cache: cache)

Specifying variables

Passing in a hash of variable (name, value) pairs to the evaluate method is one way of defining variables

calculator = Keisan::Calculator.new
calculator.evaluate("3*x + y**2", x: -2.5, y: 3)
#=> 1.5

It is also possible to define variables in the string expression itself using the assignment = operator

calculator = Keisan::Calculator.new
calculator.evaluate("x = 10*n", n: 2)
calculator.evaluate("3*x + 1")
#=> 61

To perform multiple assignments, lists can be used

calculator = Keisan::Calculator.new
calculator.evaluate("x = [1, 2]")
calculator.evaluate("[x[1], y] = [11, 22]")
calculator.evaluate("x")
#=> [1, 11]
calculator.evaluate("y")
#=> 22

Specifying functions

Just like variables, functions can be defined by passing a Proc object as follows

calculator = Keisan::Calculator.new
calculator.evaluate("2*f(1+2) + 4", f: Proc.new {|x| x**2})
#=> 22

Note that functions work in both regular (f(x)) and postfix (x.f()) notation, where for example a.f(b,c) is translated internally to f(a,b,c). The postfix notation requires the function to take at least one argument, and if there is only one argument to the function then the braces can be left off: x.f.

calculator = Keisan::Calculator.new
calculator.evaluate("[1,3,5,7].size")
#=> 4
calculator.define_function!("f", Proc.new {|x| [[x-1,x+1], [x-2,x,x+2]]})
calculator.evaluate("4.f[0]")
#=> [3,5]
calculator.evaluate("4.f[1].size")
#=> 3

Like variables, it is also possible to define functions in the string expression itself using the assignment operator =

calculator = Keisan::Calculator.new
calculator.evaluate("f(x) = n*x", n: 10) # n is local to this definition only
calculator.evaluate("f(3)")
#=> 30
calculator.evaluate("f(0-a)", a: 2)
#=> -20
calculator.evaluate("n") # n only exists in the definition of f(x)
#=> Keisan::Exceptions::UndefinedVariableError: n
calculator.evaluate("includes(a, element) = a.reduce(false, found, x, found || (x == element))")
calculator.evaluate("[3, 9].map(x, [1, 3, 5].includes(x))").value
#=> [true, false]

This form even supports recursion, but you must explicitly allow it.

calculator = Keisan::Calculator.new
calculator = Keisan::Calculator.new(allow_recursive: false)
calculator.evaluate("my_fact(n) = if (n > 1, n*my_fact(n-1), 1)")
#=> Keisan::Exceptions::InvalidExpression: Unbound function definitions are not allowed by current context

calculator = Keisan::Calculator.new(allow_recursive: true)
calculator.evaluate("my_fact(n) = if (n > 1, n*my_fact(n-1), 1)")
calculator.evaluate("my_fact(4)")
#=> 24
calculator.evaluate("my_fact(5)")
#=> 120

Multiple lines and blocks

Keisan understands strings which contain multiple lines. It will evaluate each line separately, and the last line will be the the result of the total evaluation. Lines can be separated by newlines or semi-colons.

calculator = Keisan::Calculator.new
calculator.evaluate("x = 2; y = 5\n x+y")
#=> 7

The use of curly braces {} can be used to create block which has a new closure where variable definitions are local to the block itself. Inside a block, external variables are still visible and re-assignable, but new variable definitions remain local.

calculator = Keisan::Calculator.new
calculator.evaluate("x = 10; y = 20")
calculator.evaluate("{a = 100; x = 15; a+x+y}")
#=> 135
calculator.evaluate("x")
#=> 15
calculator.evaluate("a")
#=> Keisan::Exceptions::UndefinedVariableError: a

By default assigning to a variable or function will bubble up to the first definition available in the parent scopes. To assign to a local variable instead of modifying an existing variable out of the closure, you can use the let keyword. The difference is illustrated below.

calculator = Keisan::Calculator.new
calculator.evaluate("x = 1; {x = 2}; x")
#=> 2
calculator.evaluate("x = 11; {let x = 12}; x")
#=> 11

Comments

When working with multi-line blocks of code, sometimes comments are useful to include. Comments are parts of a string from the # character to the end of a line (indicated by a newline character "\n").

calculator = Keisan::Calculator.new
calculator.evaluate("""
  # This is a comment
  x = 'foo'
  x += '#bar' # Notice that `#` inside strings is not part of the comment
  x # Should print 'foo#bar'
""")
#=> "foo#bar"

Lists

Just like in Ruby, lists can be defined using square brackets, and indexed using square brackets

calculator = Keisan::Calculator.new
calculator.evaluate("[2, 3, 5, 8]")
#=> [2, 3, 5, 8]
calculator.evaluate("[[1,2,3],[4,5,6],[7,8,9]][1][2]")
#=> 6
calculator.evaluate("a = [1,2,3]")
calculator.evaluate("a[1] += 10*a[2]")
calculator.evaluate("a")
#=> [1, 32, 3]

They can also be concatenated using the + operator

calculator = Keisan::Calculator.new
calculator.evaluate("[3, 5] + [x, x+1]", x: 10)
#=> [3, 5, 10, 11]

Keisan also supports the following useful list methods,

calculator = Keisan::Calculator.new
calculator.evaluate("[1,3,5].size")
#=> 3
calculator.evaluate("[1,3,5].max")
#=> 5
calculator.evaluate("[1,3,5].min")
#=> 1
calculator.evaluate("[1,3,5].reverse")
#=> [5,3,1]
calculator.evaluate("[[1,2],[3,4]].flatten")
#=> [1,2,3,4]
calculator.evaluate("range(5)")
#=> [0,1,2,3,4]
calculator.evaluate("range(5,10)")
#=> [5,6,7,8,9]
calculator.evaluate("range(0,10,2)")
#=> [0,2,4,6,8]

Hashes

Keisan also supports associative arrays (hashes), which maps keys to values.

calculator = Keisan::Calculator.new
calculator.evaluate("my_hash = {777: 3*4, \"bar\": \"hello world\"}")
calculator.evaluate("my_hash[777]")
#=> 12
calculator.evaluate("s = 'ba'")
calculator.evaluate("my_hash[s + 'r']")
#=> "hello world"
calculator.evaluate("my_hash['baz']")
#=> nil
calculator.evaluate("my_hash['baz'] = 999")
calculator.evaluate("my_hash['baz']")
#=> 999

There is also a to_h method which converts a list of key value pairs into a hash.

calculator = Keisan::Calculator.new
calculator.evaluate("range(1, 6).map(x, [x, x**2]).to_h")
#=> {1 => 1, 2 => 4, 3 => 9, 4 => 16, 5 => 25}

Date and time objects

Keisan supports date and time objects like in Ruby. You create a date object using either the method date (either a string to be parsed, or year, month, day numerical arguments) or today. They support methods year, month, day, weekday, strftime, and to_time to convert to a time object. epoch_days computes the number of days since Unix epoch (Jan 1, 1970).

calculator = Keisan::Calculator.new
calculator.evaluate("x = 11")
calculator.evaluate("(5 + date(2018, x, 2*x)).day")
#=> 27
calculator.evaluate("today() > date(2018, 11, 1)")
#=> true
calculator.evaluate("date('1999-12-31').to_time + 10")
#=> Time.new(1999, 12, 31, 0, 0, 10)
calculator.evaluate("date(1970, 1, 15).epoch_days")
#=> 14

Time objects are created using time (either a string to be parsed, or year, month, day, hour, minute, second arguments) or now. They support methods year, month, day, hour, minute, second, weekday, strftime, and to_date to convert to a date object. epoch_seconds computes the number of seconds since Unix epoch (00:00:00 on Jan 1, 1970).

calculator = Keisan::Calculator.new
calculator.evaluate("time(2018, 11, 22, 12, 0, 0).to_date <= date(2018, 11, 22)")
#=> true
calculator.evaluate("time('2000-4-15 12:34:56').minute")
#=> 34
calculator.evaluate("time('5000-10-10 20:30:40').strftime('%b %d, %Y')")
#=> "Oct 10, 5000"
calculator.evaluate("time(1970, 1, 1, 2, 3, 4).epoch_seconds")
#=> 7384

Functional programming methods

Keisan also supports the basic functional programming operators map (or collect), filter (or select), and reduce (or inject).

calculator = Keisan::Calculator.new
calculator.evaluate("map([1,3,5], x, 2*x)")
#=> [2,6,10]
calculator.simplify("{'a': 1, 'b': 3, 'c': 5}.collect(k, v, y*v**2)").to_s
#=> "[y,9*y,25*y]"

calculator.evaluate("[1,2,3,4].select(x, x % 2 == 0)")
#=> [2,4]
calculator.evaluate("filter({'a': 1, 'bb': 4, 'ccc': 9}, k, v, k.size == 2)")
#=> {"bb" => 4}

calculator.evaluate("[1,2,3,4,5].inject(1, total, x, total*x)")
#=> 120
calculator.evaluate("{'foo': 'hello', 'bar': ' world'}.reduce('', res, k, v, res + v)")
#=> "hello world"

Logical operations

keisan understands basic boolean logic operators, like <, <=, >, >=, &&, ||, !, so calculations like the following are possible

calculator = Keisan::Calculator.new
calculator.evaluate("1 > 0")
#=> true
calculator.evaluate("!!!true")
#=> false
calculator.evaluate("x >= 0 && x < 10", x: 5)
#=> true

There is also a useful ternary if function defined

calculator = Keisan::Calculator.new
calculator.evaluate("2 + if(1 > 0, 10, 29)")
#=> 12

For looping, you can use the basic while loop, which has an expression that evaluates to a boolean as the first argument, and any expression in the second argument. One can use the keywords break and continue to control loop flow as well.

calculator = Keisan::Calculator.new
calculator.evaluate("my_sum(a) = {let i = 0; let total = 0; while(i < a.size, {total += a[i]; i += 1}); total}")
calculator.evaluate("my_sum([1,3,5,7,9])")
#=> 25
calculator.evaluate("has_element(a, x) = {let i=0; let found=false; while(i<a.size, if(a[i] == x, found = true; break); i+=1); found}")
calculator.evaluate("[2, 3, 7, 11].has_element(11)")
#=> true

Bitwise operations

The basic bitwise operations, NOT ~, OR |, XOR ^, AND &, and left/right bitwise shifts (<< and >>) are also available for use

calculator = Keisan::Calculator.new
calculator.evaluate("0b00001111 & 0b10101010")
#=> 10

String

keisan also can parse in strings, and access the characters by index

calculator = Keisan::Calculator.new
calculator.evaluate("'hello'[1]")
#=> "e"

Binary, octal, and hexadecimal numbers

Using the prefixes 0b, 0o, and 0x (standard in Ruby) indicates binary, octal, and hexadecimal numbers respectively.

calculator = Keisan::Calculator.new
calculator.evaluate("0b1100")
#=> 12
calculator.evaluate("0o775")
#=> 509
calculator.evaluate("0x1f0")
#=> 496

Random numbers

keisan has a couple methods for doing random operations, rand and sample. For example,

calculator = Keisan::Calculator.new
(0...10).include? calculator.evaluate("rand(10)")
#=> true
[2,4,6,8].include? calculator.evaluate("sample([2, 4, 6, 8])")
#=> true

If you want reproducibility, you can pass in your own Random object to the calculator's context.

calculator1 = Keisan::Calculator.new(context: Keisan::Context.new(random: Random.new(1234)))
calculator2 = Keisan::Calculator.new(context: Keisan::Context.new(random: Random.new(1234)))
5.times.map {calculator1.evaluate("rand(1000)")}
#=> [815, 723, 294, 53, 204]
5.times.map {calculator2.evaluate("rand(1000)")}
#=> [815, 723, 294, 53, 204]

Builtin variables and functions

keisan includes all standard methods given by the Ruby Math class.

calculator = Keisan::Calculator.new
calculator.evaluate("log10(1000)")
#=> 3.0

Furthermore, the constants PI, E, I, and INF are included.

calculator = Keisan::Calculator.new
calculator.evaluate("E**(I*PI)+1")
#=> (0.0+0.0i)

There is a replace method that can replace instances of a variable in an expression with another expression. The form is replace(original_expression, variable_to_replace, replacement_expression). Before the replacement is carried out, the original_expression and replacement_expression are evaluated, then instances in the original expression of the given variable are replaced by the replacement expression.

calculator = Keisan::Calculator.new
calculator.evaluate("replace(x**2, x, 3)")
#=> 9

When using Calculator class, all variables must be replaced before an expression can be calculated, but the ability to replace any expression is useful when working directly with the AST.

ast = Keisan::AST.parse("replace(replace(x**2 + y**2, x, sin(theta)), y, cos(theta))")
ast.evaluate.to_s
#=> "(sin(theta)**2)+(cos(theta)**2)"

The derivative operation is also builtin to Keisan as the diff function.

calculator = Keisan::Calculator.new
calculator.evaluate("diff(4*x, x)")
#=> 4
calculator.evaluate("replace(diff(4*x**2, x), x, 3)")
#=> 24

This also works intelligently with user defined functions.

calculator = Keisan::Calculator.new
calculator.evaluate("f(x, y) = x**2 + y")
calculator.simplify("diff(f(2*t, t+1), t)").to_s
#=> "1+(8*t)"
calculator.evaluate("replace(diff(f(2*t, t+1), t), t, 3)")
#=> 1+8*3

There is also a puts function that can be used to output the result of an expression to STDOUT.

calculator = Keisan::Calculator.new
calculator.evaluate("x = 5")
calculator.evaluate("puts x**2") # prints "25\n" to STDOUT

Supported elements/operators

keisan supports the following operators and elements.

Numbers, variables, brackets, functions, lists, hashes

• 150, -5.67, 6e-5: regular numbers
• x, _myvar1: variables
• ( and ): round brackets for grouping parts to evaluate first
• f(x,y,z), my_function(max([2.5, 5.5])), [2,4,6,8].size: functions using ( ) brackets (optional if using postfix notation and only takes a single argument)
• [0, 3, 6, 9]: square brackets with comma separated values to denote lists
• {'foo': 11, 'bar': 22}: curly brackets containing key/value pairs creates a hash

Arithmetic operators

• +, -, *, /: regular arithmetic operators
• **: Ruby style exponent notation (to avoid conflict with bitwise xor ^)
• %: Ruby modulo operator, sign of a % b is same as sign of b
• +, -: Unary plus and minus

Logical operators

• <, >, <=, >=: comparison operators
• == and !=: logical equality check operators
• && and ||: logical operators, and and or
• !: unary logical not

Bitwise operators

• &, |, ^: bitwise and, or, xor operators
• <<, >> bitwise shift operators
• ~: unary bitwise not

Indexing of arrays/hashes

• list[i]: for accessing elements in an array
• hash[k]: for accessing elements in a hash

Assignment

• =: can be used to define variables and functions
• +=: can be used in combination with operators above

Development

After checking out the repository, run bin/setup to install dependencies. Then, run rake spec to run the tests. You can also run bin/console for an interactive prompt that will allow you to experiment with the library pre-loaded.

To install this gem onto your local machine, run bundle exec rake install. To release a new version, update the version number in version.rb, and then run bundle exec rake release, which will create a git tag for the version, push git commits and tags, and push the .gem file to rubygems.org.

Contributing

Bug reports and pull requests are welcome on GitHub at https://github.com/project-eutopia/keisan. If there is any functionality you would like (e.g. new functions), feel free to open a new issue.