expresso

A clojure library for symbolic manipulation of Algebraic Expressions.

(solve 'blue
  (ex (= pencils (+ green white blue red)))
  (ex (= (/ pencils 10) green))
  (ex (= (/ pencils 2) white))
  (ex (= (/ pencils 4) blue))
  (ex (= red 45))) ;=> #{{blue 75N}}

Objectives

expresso aims to be a general library for manipulating mathematical expressions. This are the key objectives:

• Enable mathematical expressions to be encoded
• Provide a powerful facility for general expression manipulation (aka term rewriting)
• Provide a range of very useful manipulations, which include
  • simplifying mathematical expressions
  • solving a (set of) equations in regard to unknowns
  • differentiating expressions
  • optimizing of an expression for performance
  • compiling an expression to a clojure function
• Be extensible through adding domain knowledge
• Full compatibility with core.matrix

Getting started

Add the following line to your leiningen dependencies:

[expresso "0.2.3"]

For an in-depth tutorial and showcase of expresso, see the expresso tutorial

Defining expressions

expresso's expressions are just normal clojure s-expressions. expresso has various convenience functions/macros for creating expressions:

;;the ex macro constructs an expression with automatic quoting of variables
(let [x 3]
  (ex (+ x ~x))) ;=> (+ x 3)

;;the ex' macro constructs an expression with explicit quoting of variables
(let [x 3]
  (ex' (+ x 'x))) ;=> (+ 3 x)

;;all functions in the core namespace also support pure s-exp as parameters
(solve '[x] '(= (+ 1 x) 3)) ;=> #{2}

manipulations of algebraic expressions

(use 'numeric.expresso.core)

(simplify (ex (+ (* 4 a) (* 3 a) (* -1 (* 7 a))))) 
=> 0

(def F1 (ex (= Y (+ X Z)))
(def F2 (ex (= X [1 2 3]))
(def F3 (ex (= Z (* 2.0 X)))

(solve [Y] F1 F2 F3)
=> #{[3.0 6.0 9.0]}        

(def opt (optimize (ex (+ b (* (+ 5 b) (** y (+ a b)) (** z (+ b a)))))))
opt
=> (let [local478813 (+ a b)] (+ (* b 6) (** y local478813) (** z local478813)))

(def f (compile-expr [a b y z] opt))

(f 1 2 3 4)
=> 103.0

The public API is in numeric.expresso.core - go test it out!

General Manipulation of Expressions

Expresso supports a powerful way to manipulate expressions with rewrite rules, which are built ontop of core.logic. Here are a few example rules

(use 'numeric.expresso.rules)

;;?&* matches zero or more items
(def r [(rule (ex (* 0 ?&*)) :=> 0)
        (rule (ex (* 1 ?&*)) :=> (ex (* ?&*)))
        ;;supports optional guard with :if
        (rule (ex (/ ?x ?x)) :=> 1 :if (guard (not= ?x 0)))])

;;rules match semantically. because * is commutative the rules match regardless of the order of arguments
(transform-expression r (ex (+ (* 2 (/ 4 4) 3) a (* 4 0))))
=> (+ (* 2 3) a 0)

;;The right hand side and guard of a rule can be 'arbitrary core.logic relations'.
;;The trans and guard macro create suitable core.logic relations out of normal clojure code.
;;==> is a shorthand for trans

(apply-rule (rule (ex (map ?f ?coll)) :==> (map ?f ?coll) :if (guard (coll? ?coll)))
  	    (ex (map ~inc [1 2 3]))) ;=> (2 3 4)

Status

This library is ready to use, but still in an early state so you might find some bugs. Any bug reports/feature recommendations are very welcome.

Future Development

Currently, expresso is well suited for manipulations of symbolic expressions in clojure, it also has some support for solving equations, etc what you excpect from a computer algebra system. However it is still far from a full featured CAS System like Maxima. Any contributions to help it become a real clojure Computer Algebra System are very welcome!

In the short term I am hoping to get symbolic matrices working soon. That means having an expression as a core.matrix implementation. This will be possible with the generic core.matrix api which is currently in development.

Also I am planning to replace the current rule based engine with a more faster one and to have proper compilation from rules to fast clojure functions. Kovas Boguta has made some very interesting experiments in this direction with his combinator project.