#+Title: Simple Algebraic Manipulation #+Subtitle: An approach to allow for the solving and simplification of expressions #+AUTHOR: Samuel W. Flint #+EMAIL: swflint@flintfam.org #+DATE: \today #+INFOJS_OPT: view:info toc:nil path:http://flintfam.org/org-info.js #+OPTIONS: toc:nil H:5 ':t *:t todo:nil stat:nil d:nil #+PROPERTY: noweb no-export #+PROPERTY: comments noweb #+LATEX_HEADER: \usepackage[margins=0.75in]{geometry} #+LATEX_HEADER: \parskip=5pt #+LATEX_HEADER: \parindent=0pt #+LATEX_HEADER: \lstset{texcl=true,breaklines=true,columns=fullflexible,basicstyle=\ttfamily,frame=lines,literate={<=}{$\leq$}1 {>=}{$\geq$}1} #+LATEX_CLASS_OPTIONS: [10pt,twoside] #+LATEX_HEADER: \pagestyle{headings} * COMMENT Export #+Caption: Export Document #+Name: export-document #+BEGIN_SRC emacs-lisp :exports none :results none (save-buffer) (let ((org-confirm-babel-evaluate (lambda (lang body) (declare (ignorable lang body)) nil))) (org-latex-export-to-pdf)) #+END_SRC * COMMENT Tangle #+Caption: Tangle Document #+Name: tangle-document #+BEGIN_SRC emacs-lisp :exports none :results none (save-buffer) (let ((python-indent-offset 4)) (org-babel-tangle)) #+END_SRC * DONE Introduction :nonum: CLOSED: [2016-05-01 Sun 14:33] :PROPERTIES: :CREATED: <2016-04-30 Sat 22:55> :END: As a part of my lisp-based Computer Algebra System, an algebraic manipulation toolkit is required. This will be used to simplify equations, or for that matter solve them. This creates this toolkit, but does not create a complete simplifier or solver. It does this by providing manipulators and automatic rewriters. These together will provide simplification and solving utilities. * TOC :ignore: :PROPERTIES: :CREATED: <2016-04-30 Sat 22:55> :END: #+TOC: headlines 3 #+TOC: listings * DONE Expression Typing [7/7] :PROPERTIES: :CREATED: <2016-04-30 Sat 23:15> :ID: c6921b1e-d269-4243-acff-5a77685c331e :END: To accomplish the goal of providing a complete system to manipulate algebraic expressions, a way to determine the type of expression is important. This will allow for a form of "generic programming" to be used in the development of the manipulator functions, as a way to ensure that the correct manipulator is chosen. This includes a form of storage, the classification definition macro, a way to check a classification, an expression classifier, and all possible classifications. #+Caption: Determine Expression Type #+Name: determine-expression-type #+BEGIN_SRC lisp <> <> <> <> <> <> <> #+END_SRC ** DONE Define Classification CLOSED: [2016-05-04 Wed 19:30] :PROPERTIES: :CREATED: <2016-05-02 Mon 13:56> :ID: d8826a51-50b8-467a-9e52-158502bd4138 :END: This is the classification definition macro, ~define-classification~. It takes one symbol argument, ~name~ (the name of the classification), and a body, which is encapsulated within a defun, and binds the following variables: - ~expression~ :: the expression which is to be classified - ~length~ :: the length of the expression if the expression is a list, or 0 if it is not. Aside from defining the classification, it also pushes the classification name and the classifier onto the stack, which can be used for direct classification checking or to completely classify an expression. #+Caption: Define Classification #+Name: define-classification #+BEGIN_SRC lisp (defmacro define-classification (name &body body) (check-type name symbol) (let ((classifier-name (symbolicate name '-classifier))) `(progn (defun ,classifier-name (expression &aux (length (if (listp expression) (length expression) 0))) (declare (ignorable length)) ,@body) (pushnew '(,name . ,classifier-name) *classifications*) ',name))) #+END_SRC ** DONE Check Classification CLOSED: [2016-05-04 Wed 19:37] :PROPERTIES: :CREATED: <2016-05-02 Mon 13:56> :ID: 6505b0b1-ffd8-4dd6-b81a-3e49483d8437 :END: To check a classification, the classifier is obtained, unless the specified classifier is ~*~, in which case, ~t~ is always returned. If the classification is not, the classifier function is called on the expression, the result of which is returned. #+Caption: Check Classification #+Name: check-classification #+BEGIN_SRC lisp (defun classified-as-p (expression classification) (if (eq '* classification) t (funcall (cdr (assoc classification *classifications*)) expression))) #+END_SRC ** DONE Classify Expression CLOSED: [2016-05-04 Wed 19:44] :PROPERTIES: :CREATED: <2016-05-02 Mon 14:09> :ID: 82d75d54-1d33-400b-86a3-7d16af938ac8 :END: To completely classify an expression, the ~*classifications*~ alist is mapped over, checking to see if each classification is applicable to the expression, if so, the name being returned, otherwise ~nil~. All nils are removed, leaving the complete classification, which is returned for use. #+Caption: Classify Expression #+Name: classify-expression #+BEGIN_SRC lisp (defun classify (expression) (let ((classifications '())) (dolist (possible ,*classifications* (reverse classifications)) (let ((name (car possible)) (checker (cdr possible))) (when (funcall checker expression) (push name classifications)))))) #+END_SRC ** DONE Classification Case CLOSED: [2016-05-30 Mon 18:17] :PROPERTIES: :CREATED: <2016-05-20 Fri 14:15> :ID: 19a4e467-baa0-47eb-9267-93ff3801b1fd :END: Following the case pattern, and to allow for cleaner code, I've defined the classification case macro. It does this by taking a variable name and a list of cases. These are then mapped over, producing clauses suitable for a ~cond~ expression, to which this macro finally expands, binding the complete classification of the given expression to ~the-classification~. #+Caption: Classification Case #+Name: classification-case #+BEGIN_SRC lisp (defmacro classification-case (var &rest cases) (declare (slime-indent (as case))) (let ((conditions (map 'list #'(lambda (case) (destructuring-bind (type &body body) case (if (eq type 't) `((classified-as-p ,var '*) ,@body) `((classified-as-p ,var ',type) ,@body)))) cases))) `(let ((the-classification (classify ,var))) (declare (ignorable the-classification)) (cond ,@conditions)))) #+END_SRC ** DONE When Classified CLOSED: [2016-05-30 Mon 19:18] :PROPERTIES: :CREATED: <2016-05-30 Mon 18:31> :ID: 5c7c3e0b-9170-48e9-a414-6ac4528f9ac3 :END: The ~when-classified-as~ macro takes a classification, variable and a body. It expands to a ~when~ form, with the classification and variable put into a ~classified-as-p~ call becoming the predicate, determining whether or not the body is run. #+Caption: When Classified #+Name: when-classified #+BEGIN_SRC lisp (defmacro when-classified-as (classification variable &body body) `(when (classified-as-p ,variable ',classification) ,@body)) #+END_SRC ** DONE Classifications [13/13] :PROPERTIES: :CREATED: <2016-05-02 Mon 13:56> :ID: dcce4a6b-1b2d-4638-a82b-0c4917b0698a :END: I must define several different classifications, ranging from simple numeric expressions (numbers) to trigonometric expressions ($\sin$, $\cos$ and the lot). They are as follows: - Numbers - Variables - Non-Atomics - Additives - Subtractives - Powers - Exponentials - Multiplicatives - Logarithmics - Rationals - Polynomial Terms - Polynomials - Trigonometrics #+Caption: Possible Classifications #+Name: possible-classifications #+BEGIN_SRC lisp <> <> <> <> <> <> <> <> <> <> <> <> <> #+END_SRC *** DONE Numbers CLOSED: [2016-05-04 Wed 19:56] :PROPERTIES: :CREATED: <2016-05-02 Mon 14:26> :ID: 42081153-7cc5-42ff-a17f-53e171c6d1a7 :END: Check to see if a given expression is a number using ~numberp~. #+Caption: Classify Numbers #+Name: classify-numbers #+BEGIN_SRC lisp (define-classification numeric (numberp expression)) #+END_SRC *** DONE Variables CLOSED: [2016-05-04 Wed 19:57] :PROPERTIES: :CREATED: <2016-05-02 Mon 14:26> :ID: 4c676754-ef9a-485f-91a2-8f1bd83c7659 :END: Check to see if a given expression is a variable, that is to say a symbol, using ~symbolp~. #+Caption: Classify Variables #+Name: classify-variables #+BEGIN_SRC lisp (define-classification variable (symbolp expression)) #+END_SRC *** DONE Non Atomics CLOSED: [2016-05-04 Wed 19:59] :PROPERTIES: :CREATED: <2016-05-04 Wed 19:52> :ID: 414df063-0be1-4849-8b9f-d71aa828be2a :END: Check to see if a given expression is a non-atomic (any expression other than a number or a variable) using ~listp~. #+Caption: Classify Non-Atomics #+Name: classify-non-atomics #+BEGIN_SRC lisp (define-classification non-atomic (listp expression)) #+END_SRC *** DONE Additives CLOSED: [2016-05-04 Wed 20:01] :PROPERTIES: :CREATED: <2016-05-02 Mon 14:26> :ID: 736d79dc-f34c-4247-b592-690d7f2fddd9 :END: Check to see whether or not an expression is an additive by ensuring that it is non-atomic and the first element is the symbol ~+~. #+Caption: Classify Additives #+Name: classify-additives #+BEGIN_SRC lisp (define-classification additive (when-classified-as non-atomic expression (eq '+ (first expression)))) #+END_SRC *** DONE Subtractive CLOSED: [2016-05-04 Wed 20:02] :PROPERTIES: :CREATED: <2016-05-02 Mon 14:26> :ID: c59d086f-2f49-485a-8f96-57d85e774f60 :END: Check to see whether a given expression is a subtractive by ensuring it is non-atomic and the first element is the symbol ~-~. #+Caption: Classify Subtractives #+Name: classify-subtractives #+BEGIN_SRC lisp (define-classification subtractive (when-classified-as non-atomic expression (eq '- (first expression)))) #+END_SRC *** DONE Powers CLOSED: [2016-05-04 Wed 20:07] :PROPERTIES: :CREATED: <2016-05-02 Mon 14:27> :ID: cc15dd10-7cc0-4370-9e69-daf903b30ad5 :END: This is used to classify "powers", that is to say, equations of the form $x^n$, where $n$ is any numeric. It does so by first ensuring that the expression is non-atomic, following that, it checks to see if the first element in the expression is the symbol ~expt~, the second is a variable and the third a numeric. #+Caption: Classify Powers #+Name: classify-powers #+BEGIN_SRC lisp (define-classification power (when-classified-as non-atomic expression (and (eq 'expt (first expression)) (classified-as-p (second expression) 'variable) (classified-as-p (third expression) 'numeric)))) #+END_SRC *** DONE Exponentials CLOSED: [2016-05-30 Mon 18:24] :PROPERTIES: :CREATED: <2016-05-02 Mon 15:04> :ID: a11fdd94-d56c-4749-bb22-dca75159dbcb :END: This classifies both natural and non-natural exponentials. It does so by ensuring that natural exponentials ($e^x$) are of the form ~(exp x)~, and non-natural exponentials ($a^x$) are of the form ~(expt base power)~. #+Caption: Classify Exponentials #+Name: classify-exponentials #+BEGIN_SRC lisp (define-classification natural-exponential (when-classified-as non-atomic expression (and (= 2 length) (eq 'exp (first expression))))) (define-classification exponential (when-classified-as non-atomic expression (and (= 3 length) (eq 'expt (first expression))))) #+END_SRC *** DONE Multiplicatives CLOSED: [2016-05-30 Mon 18:55] :PROPERTIES: :CREATED: <2016-05-02 Mon 14:27> :ID: feb85a20-93e3-45a1-be01-9893ecc07c53 :END: To classify multiplicative expressions, it is first ensured that they are non-atomic, and then, the first element is tested to see if it is equal to the symbol ~*~. #+Caption: Classify Multiplicatives #+Name: classify-multiplicatives #+BEGIN_SRC lisp (define-classification multiplicative (when-classified-as non-atomic expression (eq '* (first expression)))) #+END_SRC *** DONE Logarithmics CLOSED: [2016-05-30 Mon 18:30] :PROPERTIES: :CREATED: <2016-05-02 Mon 14:27> :ID: 0b733d75-e1ab-413f-8f8a-6a8a47db409c :END: This defines the classifications for logarithmic expressions, for both natural and non-natural bases. For natural bases ($\ln x$), it ensures that expressions are of the form ~(log x)~, and for non-natural bases ($\log_{b}x$) are of the form ~(log expression base-expression)~. #+Caption: Classify Lograthmics #+Name: classify-logarithmics #+BEGIN_SRC lisp (define-classification natural-logarithmic (when-classified-as non-atomic expression (and (= 2 length) (eq 'log (first expression))))) (define-classification logarithmic (when-classified-as non-atomic expression (and (= 3 length) (eq 'log (first expression))))) #+END_SRC *** DONE Rationals CLOSED: [2016-05-30 Mon 18:58] :PROPERTIES: :CREATED: <2016-05-02 Mon 14:28> :ID: a4505a66-c249-4438-a6df-81e21718e23e :END: Rationals are classified similarly to multiplicatives, checking to see whether or not they are non-atomic and checking whether or not the first element is ~/~, but rationals are also defined as only having three elements, the operation and two following operands, and thus, the length is also checked. #+Caption: Classify Rationals #+Name: classify-rationals #+BEGIN_SRC lisp (define-classification rational (when-classified-as non-atomic expression (and (= 3 length) (eq '/ (first expression))))) #+END_SRC *** DONE Polynomial Terms CLOSED: [2016-05-30 Mon 19:13] :PROPERTIES: :CREATED: <2016-05-02 Mon 14:28> :ID: 37da52b7-98a0-4a16-8a17-a62fcff2ba59 :END: To classify a polynomial term, The expression is checked to see if it satisfies one of the following: - Numeric - Variable - Power - Multiplicative that composed of a numeric and a power or variable. #+Caption: Classify Polynomial Term #+Name: classify-polynomial-term #+BEGIN_SRC lisp (define-classification polynomial-term (or (classified-as-p expression 'numeric) (classified-as-p expression 'variable) (classified-as-p expression 'power) (and (classified-as-p expression 'multiplicative) (= (length (rest expression)) 2) (or (and (classified-as-p (second expression) 'numeric) (or (classified-as-p (third expression) 'power) (classified-as-p (third expression) 'variable))) (and (classified-as-p (third expression) 'numeric) (or (classified-as-p (second expression) 'power) (classified-as-p (second expression) 'variable))))))) #+END_SRC *** DONE Polynomials CLOSED: [2016-05-08 Sun 16:46] :PROPERTIES: :CREATED: <2016-05-02 Mon 14:28> :ID: 8cd9045b-81dd-4571-930a-a852f81969c9 :END: This determines whether or not a given expression is a polynomial, that is to say it is either ~additive~ or ~subtractive~, and each and every term is classified as ~polynomial-term~, that is to say, a ~numeric~, ~power~, or a ~multiplicative~ consisting of a ~numeric~ followed by a ~power~. #+Caption: Classify Polynomials #+Name: classify-polynomials #+BEGIN_SRC lisp (define-classification polynomial (when-classified-as non-atomic expression (and (or (eq '- (first expression)) (eq '+ (first expression))) (reduce #'(lambda (a b) (and a b)) (map 'list #'(lambda (the-expression) (classified-as-p the-expression 'polynomial-term)) (rest expression)))))) #+END_SRC *** DONE Trigonometrics CLOSED: [2016-05-30 Mon 19:15] :PROPERTIES: :CREATED: <2016-05-04 Wed 13:38> :ID: 6f433cad-4b81-4a6f-ab65-981f4a924812 :END: Trigonometrics are classified as many others are, they are first checked to see if they are non-atomic, and then the first element is checked, with the following being valid symbols: - ~sin~ - ~cos~ - ~tan~ - ~csc~ - ~sec~ - ~cot~ #+Caption: Classify Trigonometrics #+Name: classify-trigonometrics #+BEGIN_SRC lisp (define-classification trigonometric (when-classified-as non-atomic expression (member (first expression) '(sin cos tan csc sec cot)))) (define-classification sin (when-classified-as non-atomic expression (eq 'sin (first expression)))) (define-classification cos (when-classified-as non-atomic expression (eq 'cos (first expression)))) (define-classification tan (when-classified-as non-atomic expression (eq 'tan (first expression)))) (define-classification csc (when-classified-as non-atomic expression (eq 'csc (first expression)))) (define-classification sec (when-classified-as non-atomic expression (eq 'sec (first expression)))) (define-classification cot (when (classified-as-p expression 'non-atomic) (eq 'cot (first expression)))) #+END_SRC ** DONE Classification Storage CLOSED: [2016-05-04 Wed 19:49] :PROPERTIES: :CREATED: <2016-05-02 Mon 13:55> :ID: ff35cd33-3c10-4a45-a2c5-32bc3fdc1acc :END: The storage of classifications is simple, they are stored as an alist in the form of ~(name . classifier)~, in the list ~*classifications*~. #+Caption: Classification Storage #+Name: classification-storage #+BEGIN_SRC lisp (defvar *classifications* '()) #+END_SRC * DONE Collect Variables CLOSED: [2016-05-31 Tue 18:54] :PROPERTIES: :CREATED: <2016-05-20 Fri 15:15> :ID: 6333322c-e12f-4ef6-8394-2fe219a72836 :END: Variable collection is somewhat important, and to accomplish this, I use a recursive algorithm. An expression is passed to the function, and if the expression is a variable, then the variable is collected and spit out; otherwise, if the expression is non-atomic, it is passed to the function recursively, and the returned variables are then merged into the variables list. Upon termination (no further sub-expressions), all variables are returned. (See Figure [[fig:variable-collection]].) #+Caption: Variable Collection #+Name: variable-collection #+BEGIN_SRC dot :file "variable-collection.png" :export results :cache yes digraph { start [label = "Start"]; stop [label = "Stop"]; collect [label = "Collect"]; if_var [label = "If Variable", shape = rectangle]; recurse_collect [label = "Iterate, Recurse and Collect Results"]; start -> if_var; if_var -> collect [label = "True"]; collect -> stop; if_var -> recurse_collect [label = "Non-atomic"]; recurse_collect -> start; } #+END_SRC #+Caption: Variable Collection Algorithm #+Name: fig:variable-collection #+ATTR_LATEX: :width 8cm #+RESULTS[e1586dc50921f7ba260f125e7221a978d489bd34]: variable-collection [[file:variable-collection.png]] #+Caption: Collect Variables #+Name: collect-variables #+BEGIN_SRC lisp (defun collect-variables (expression) (let ((variables '())) (flet ((merge-variables (variable) (pushnew variable variables))) (classification-case expression (variable (merge-variables expression)) (non-atomic (map 'list #'(lambda (expr) (dolist (variable (collect-variables expr)) (merge-variables variable))) (rest expression))))) (reverse variables))) #+END_SRC * WORKING Term Collection :noexport: :PROPERTIES: :CREATED: <2016-04-30 Sat 22:59> :ID: c1856735-914b-4f73-8825-3e5a062113d2 :END: Foo #+Caption: Collect Terms #+Name: collect-terms #+BEGIN_SRC lisp (defun collect-terms (expression &aux (terms (rest expression))) (let ((numerics '()) (variables '()) (additives '()) (subtractives '()) (multiplicatives '()) (polynomial-terms '()) (rationals '()) (powers '()) (natural-exponentials '()) (exponentials '()) (natural-logarithmics '()) (trigonometrics '())) (dolist (term terms) (classification-case term (numeric (pushnew term numerics)) (variable (pushnew term variables)) (power (pushnew term powers)) (additive (pushnew term additives)) (subtractive (pushnew term subtractives)) (polynomial-term (pushnew term polynomial-terms)) (multiplicative (pushnew term multiplicatives)) (rational (pushnew term rationals)) (power (pushnew term powers)) (natural-exponential (pushnew term natural-exponentials)) (exponential (pushnew term exponentials)) (natural-logarithmic (pushnew term natural-logarithmics)) (trigonometric (pushnew term trigonometrics)))) (remove-if #'(lambda (expr) (null (cdr expr))) (list (cons :numerics numerics) (cons :variables variables) (cons :powers powers) (cons :additives additives) (cons :subtractives subtractives) (cons :multiplicatives multiplicatives) (cons :polynomial-terms polynomial-terms) (cons :rationals rationals) (cons :powers powers) (cons :natural-exponentials natural-exponentials) (cons :exponentials exponentials) (cons :natural-logarithmics natural-logarithmics) (cons :trigonometrics trigonometrics))))) #+END_SRC * WORKING Polynomial Related Functions [0/6] :PROPERTIES: :CREATED: <2016-05-01 Sun 12:29> :ID: 984d0f52-4c52-4bfa-a150-f3289d25bdf1 :END: #+Caption: Polynomial Related Functions #+Name: polynomial-related-functions #+BEGIN_SRC lisp <> <> <> <> <> <> #+END_SRC ** TODO Get Coefficient :PROPERTIES: :CREATED: <2016-05-31 Tue 19:08> :ID: cbc927fc-ae5e-46bf-a028-2872b5c31831 :END: #+Caption: Get Coefficient #+Name: get-coefficient #+BEGIN_SRC lisp (defun coefficient (term) (when (classified-as-p term 'polynomial-term) (classification-case term (variable 1) (power 1) (multiplicative (second term)) (numeric term)))) #+END_SRC ** TODO Get Term Variables :PROPERTIES: :CREATED: <2016-05-31 Tue 19:08> :ID: 55729698-bd51-48af-ab42-197871c54dbb :END: #+Caption: Get Term Variable #+Name: get-term-variable #+BEGIN_SRC lisp (defun term-variable (term) (when (classified-as-p term 'polynomial-term) (classification-case term (power (second term)) (multiplicative (if (listp (third term)) (second (third term)) (third term))) (numeric nil)))) #+END_SRC ** TODO Get Power :PROPERTIES: :CREATED: <2016-05-31 Tue 19:08> :ID: 7d5a10da-bb30-496f-b285-470057a46db0 :END: #+Caption: Get Power #+Name: get-power #+BEGIN_SRC lisp (defun get-power (term) (classification-case term (numeric 0) (variable 1) (power (third term)) (multiplicative (if (listp (third term)) (third (third term)) 1)) (* 0))) #+END_SRC ** TODO Same Order :PROPERTIES: :CREATED: <2016-05-31 Tue 19:08> :ID: c56a1496-f4c2-4693-9448-5043570a752f :END: #+Caption: Same Order #+Name: same-order #+BEGIN_SRC lisp (defun same-order-p (term-a term-b) (= (get-power term-a) (get-power term-b))) #+END_SRC ** TODO Same Variable :PROPERTIES: :CREATED: <2016-05-31 Tue 19:08> :ID: 3806c97a-12fa-4488-b38c-d9ff3570c139 :END: #+Caption: Same Variable #+Name: same-variable #+BEGIN_SRC lisp (defun same-variable-p (term-a term-b) (eq (term-variable term-a) (term-variable term-b))) #+END_SRC ** TODO Is Combinable :PROPERTIES: :CREATED: <2016-05-31 Tue 19:08> :ID: db0410aa-bb12-4933-9be7-1a50d70ae90f :END: #+Caption: Is Combinable #+Name: is-combinable #+BEGIN_SRC lisp (defun single-term-combinable-p (term-a term-b) (and (same-order-p term-a term-b) (same-variable-p term-a term-b))) #+END_SRC * WORKING Expression Manipulators [2/8] :PROPERTIES: :CREATED: <2016-04-30 Sat 22:58> :ID: 4fe60cc1-be66-4d5e-8922-590554d99004 :END: Foo #+Caption: Expression Manipulation #+Name: expression-manipulation #+BEGIN_SRC lisp <> <> <> <> <> <> <> <> #+END_SRC ** DONE Manipulator Miscellaneous Functions CLOSED: [2016-05-08 Sun 10:34] :PROPERTIES: :CREATED: <2016-05-03 Tue 15:38> :ID: 20450528-d763-4c14-a085-5ac54d4d4b85 :END: This defines the ~*manipulator-map*~, where the manipulators for various functions are stored, and defines a function to generate an arguments list given a count of arguments. #+Caption: Misc Manipulator Functions #+Name: misc-manipulator-functions #+BEGIN_SRC lisp (defvar *manipulator-map* '()) (defun gen-args-list (count) (let ((letters '(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))) (let ((variables-list '())) (dotimes (i count) (pushnew (symbolicate 'expression- (nth i letters)) variables-list)) (reverse variables-list)))) #+END_SRC ** WORKING Define Expression Manipulator :PROPERTIES: :CREATED: <2016-04-30 Sat 22:57> :ID: 63909972-428d-47f3-9dc3-3e1fb213aa70 :END: #+Caption: Define Expression Manipulator #+Name: define-expression-manipulator #+BEGIN_SRC lisp (defmacro define-operation (name arity short) (declare (slime-indent (as defun))) (check-type name symbol) (check-type arity (integer 1 26)) (check-type short symbol) (let* ((args (gen-args-list arity)) (expression-types (map 'list #'(lambda (x) (symbolicate x '-type)) args)) (rules-name (symbolicate '*manipulators- name '*)) (base-manipulator-name (symbolicate name '-manipulator-)) (manipulator-define-name (symbolicate 'define- name '-manipulator)) (is-applicable-name (symbolicate name '-is-applicable-p)) (get-operations-name (symbolicate 'get- name '-manipulators)) (type-check-list (let ((i 0)) (loop for arg in args collect (prog1 `(classified-as-p ,arg (nth ,i types)) (incf i)))))) `(progn (push '(,short . ,name) *manipulator-map*) (defvar ,rules-name '()) (defun ,is-applicable-name (types ,@args) (and ,@type-check-list)) (defun ,get-operations-name (,@args) (remove-if #'null (map 'list #'(lambda (option) (let ((types (car option)) (name (cdr option))) (if (,is-applicable-name types ,@args) name))) ,rules-name))) (defun ,name (,@args) (funcall (first (,get-operations-name ,@args)) ,@args)) (defmacro ,manipulator-define-name ((,@expression-types) &body body) (declare (slime-indent (as defun))) (let ((manipulator-name (symbolicate ',base-manipulator-name ,@expression-types))) `(progn (setf ,',rules-name (append ,',rules-name '(((,,@expression-types) . ,manipulator-name)))) (defun ,manipulator-name ,',args ,@body))))))) #+END_SRC #+Caption: Expression Manipulation Example #+Name: ex-manip-example #+BEGIN_SRC lisp :results output raw :exports results :cache yes (load "manipulation") (in-package #:manipulator) (format t "#+Caption: Expression Manipulator Expansion~%#+Name: ex-manip-expansion~%#+BEGIN_SRC lisp :exports code~%~a~%#+END_SRC" (macroexpand-1 '(define-operation frobnicate 2 frob))) #+END_SRC #+RESULTS[8b2d6e575e0d168f96d4bba85d6dd90a56c5c5a6]: ex-manip-example #+Caption: Expression Manipulator Expansion #+Name: ex-manip-expansion #+BEGIN_SRC lisp :exports code (PROGN (PUSH '(FROB . FROBNICATE) *MANIPULATOR-MAP*) (DEFVAR *MANIPULATORS-FROBNICATE* 'NIL) (DEFUN FROBNICATE-IS-APPLICABLE-P (TYPES EXPRESSION-A EXPRESSION-B) (AND (CLASSIFIED-AS-P EXPRESSION-A (NTH 0 TYPES)) (CLASSIFIED-AS-P EXPRESSION-B (NTH 1 TYPES)))) (DEFUN GET-FROBNICATE-MANIPULATORS (EXPRESSION-A EXPRESSION-B) (REMOVE-IF #'NULL (MAP 'LIST #'(LAMBDA (OPTION) (LET ((TYPES (CAR OPTION)) (NAME (CDR OPTION))) (IF (FROBNICATE-IS-APPLICABLE-P TYPES EXPRESSION-A EXPRESSION-B) NAME))) *MANIPULATORS-FROBNICATE*))) (DEFUN FROBNICATE (EXPRESSION-A EXPRESSION-B) (FUNCALL (FIRST (GET-FROBNICATE-MANIPULATORS EXPRESSION-A EXPRESSION-B)) EXPRESSION-A EXPRESSION-B)) (DEFMACRO DEFINE-FROBNICATE-MANIPULATOR ((EXPRESSION-A-TYPE EXPRESSION-B-TYPE) &BODY BODY) (DECLARE (SLIME-INDENT (AS DEFUN))) (LET ((MANIPULATOR-NAME (SYMBOLICATE 'FROBNICATE-MANIPULATOR- EXPRESSION-A-TYPE EXPRESSION-B-TYPE))) `(PROGN (SETF ,'*MANIPULATORS-FROBNICATE* (APPEND ,'*MANIPULATORS-FROBNICATE* '(((,EXPRESSION-A-TYPE ,EXPRESSION-B-TYPE) ,@MANIPULATOR-NAME)))) (DEFUN ,MANIPULATOR-NAME ,'(EXPRESSION-A EXPRESSION-B) ,@BODY))))) #+END_SRC ** DONE External Manipulator CLOSED: [2016-05-31 Tue 19:48] :PROPERTIES: :CREATED: <2016-05-01 Sun 14:33> :ID: 6419490c-3cb0-47e4-840a-c20af4bfb3d7 :END: The Expression Manipulators should not be touched outside of this package, as they are not designed to be used outside of it. Instead, they should be used through this simple function. It takes an action and a list of expressions. The function used to perform the action correctly is determined, and used to reduce the expressions. #+Caption: External Manipulator #+Name: external-manipulator #+BEGIN_SRC lisp (defun manipulate (action &rest expressions) (let ((the-manipulator (cdr (assoc action *manipulator-map*)))) (reduce the-manipulator expressions))) #+END_SRC ** WORKING Addition :PROPERTIES: :CREATED: <2016-04-30 Sat 23:08> :ID: b794486c-e493-408f-b80c-a440edae1bc8 :END: Foo #+Caption: Addition Manipulator #+Name: addition-manipulator #+BEGIN_SRC lisp (define-operation add 2 +) (define-add-manipulator (numeric numeric) (+ expression-a expression-b)) (define-add-manipulator (numeric additive) (let ((total expression-a) (remainder (rest expression-b)) (non-numeric '())) (dolist (element remainder) (if (classified-as-p element 'numeric) (incf total element) (push element non-numeric))) (cond ((null non-numeric) total) ((= 0 total) `(+ ,@non-numeric)) (t `(+ ,total ,@non-numeric))))) (define-add-manipulator (additive additive) (let ((total 0) (elements (append (rest expression-a) (rest expression-b))) (non-numeric '())) (dolist (element elements) (if (classified-as-p element 'numeric) (incf total element) (push element non-numeric))) (cond ((null non-numeric) total) ((= 0 total) `(+ ,@non-numeric)) (t `(+ ,total ,@non-numeric))))) (define-add-manipulator (numeric subtractive) (let ((total expression-a) (the-other (rest expression-b)) (non-numeric '())) (dolist (element the-other) (if (classified-as-p element 'numeric) (decf total element) (push element non-numeric))) (cond ((null non-numeric) total) ((= 0 total) `(+ ,@non-numeric)) (t `(+ ,total (-,@non-numeric)))))) (define-add-manipulator (numeric polynomial-term) `(+ ,expression-a ,expression-b)) (define-add-manipulator (polynomial-term polynomial-term) (if (single-term-combinable-p expression-a expression-b) (let ((new-coefficient (+ (coefficient expression-a) (coefficient expression-b))) (variable (term-variable expression-a)) (power (get-power expression-a))) `(* ,new-coefficient (expt ,variable ,power))) `(+ ,expression-a ,expression-b))) (define-add-manipulator (* numeric) (add expression-b expression-a)) #+END_SRC ** WORKING Subtraction :PROPERTIES: :CREATED: <2016-04-30 Sat 23:08> :ID: f675fd81-e995-41ee-9570-cc78261d9dc1 :END: Foo #+Caption: Subtraction Manipulator #+Name: subtraction-manipulator #+BEGIN_SRC lisp (define-operation subtract 2 -) (define-subtract-manipulator (numeric numeric) (- expression-a expression-b)) (define-subtract-manipulator (numeric subtractive) (let ((total expression-a) (elements (rest expression-b)) (non-numeric '())) (dolist (element elements) (if (classified-as-p element 'numeric) (decf total element) (push element non-numeric))) (cond ((null non-numeric) total) ((= 0 total) `(- ,@(reverse non-numeric))) (t `(- ,total ,@(reverse non-numeric)))))) (define-subtract-manipulator (* numeric) (subtract expression-b expression-a)) #+END_SRC ** WORKING Multiplication :PROPERTIES: :CREATED: <2016-04-30 Sat 23:08> :ID: cddffdaa-10dd-425f-9697-3f0617162953 :END: Foo #+Caption: Multiplication Manipulators #+Name: multiplication-manipulators #+BEGIN_SRC lisp (define-operation multiply 2 *) (define-multiply-manipulator (numeric numeric) (* expression-a expression-b)) (define-multiply-manipulator (numeric polynomial-term) (let ((new-coefficient (* expression-a (coefficient expression-b))) (variable (term-variable expression-b)) (power (get-power expression-b))) (if (= 1 power) `(* ,new-coefficient ,variable) `(* ,new-coefficient (expt ,variable ,power))))) (define-multiply-manipulator (polynomial-term polynomial-term) (let ((new-coefficient (* (coefficient expression-a) (coefficient expression-b))) (variable (term-variable expression-b)) (power (+ (get-power expression-a) (get-power expression-b)))) `(* ,new-coefficient (expt ,variable ,power)))) #+END_SRC ** WORKING Division :PROPERTIES: :CREATED: <2016-04-30 Sat 23:09> :ID: 4c4f7034-555a-46b0-85b9-56a08cf48f9b :END: Foo #+Caption: Division Manipulators #+Name: division-manipulators #+BEGIN_SRC lisp (define-operation division 2 /) (define-division-manipulator (numeric numeric) (/ expression-a expression-b)) #+END_SRC ** WORKING Trigonometric [0/6] :PROPERTIES: :CREATED: <2016-04-30 Sat 23:09> :ID: ba4acf37-9074-429b-a2c8-a23094e1c86b :END: Foo #+Caption: Trigonometric Manipulators #+Name: trigonometric-manipulators #+BEGIN_SRC lisp <> <> <> <> <> <> #+END_SRC *** WORKING Sine :PROPERTIES: :CREATED: <2016-05-08 Sun 16:22> :ID: c733c6b3-a44a-488f-8b6e-38346830b257 :END: #+Caption: Sine Manipulators #+Name: sine-manipulators #+BEGIN_SRC lisp (define-operation sine 1 sin) (define-sine-manipulator (numeric) (sin expression-a)) #+END_SRC *** WORKING Cosine :PROPERTIES: :CREATED: <2016-05-08 Sun 16:22> :ID: c2fbd362-6932-4483-8270-e3ad72a308fd :END: #+Caption: Cosine Manipulators #+Name: cosine-manipulators #+BEGIN_SRC lisp (define-operation cosine 1 cos) (define-cosine-manipulator (numeric) (cosine expression-a)) #+END_SRC *** WORKING Tangent :PROPERTIES: :CREATED: <2016-05-08 Sun 16:22> :ID: 07222206-1c22-411e-a8ab-13e1a627e9ef :END: #+Caption: Tangent Manipulators #+Name: tangent-manipulators #+BEGIN_SRC lisp (define-operation tangent 1 tan) (define-tangent-manipulator (numeric) (tan expression-a)) #+END_SRC *** WORKING Cosecant :PROPERTIES: :CREATED: <2016-05-08 Sun 16:22> :ID: e77c0317-7281-45ff-b86b-8d66fb8c38ef :END: #+Caption: Cosecant Manipulators #+Name: cosecant-manipulators #+BEGIN_SRC lisp (define-operation cosecant 1 csc) #+END_SRC *** WORKING Secant :PROPERTIES: :CREATED: <2016-05-08 Sun 16:23> :ID: 6c377c7d-ec84-4fcf-be94-db89b832c2d8 :END: #+Caption: Secant Manipulators #+Name: secant-manipulators #+BEGIN_SRC lisp (define-operation secant 1 sec) #+END_SRC *** WORKING Cotangent :PROPERTIES: :CREATED: <2016-05-08 Sun 16:23> :ID: 70a9fb76-7ca7-4c7d-b25b-0fa94d390b6c :END: #+Caption: Cotangent Manipulators #+Name: cotangent-manipulators #+BEGIN_SRC lisp (define-operation cotangent 1 cot) #+END_SRC * DONE Packaging CLOSED: [2016-05-05 Thu 21:21] :PROPERTIES: :CREATED: <2016-04-30 Sat 23:07> :ID: d487ed31-295b-4274-aef2-b45e4fa7bec2 :END: This assembles and packages the algebraic manipulation system into a single file and library. To do so, it must first define a package, import specific symbols from other packages, and export symbols from itself. It then includes the remainder of the functionality, placing it in the file ~manipulation.lisp~. #+Caption: Packaging #+Name: packaging #+BEGIN_SRC lisp :tangle "manipulation.lisp" (defpackage #:manipulator (:use #:cl) (:import-from #:alexandria #:symbolicate) (:export #:manipulate #:classify #:classified-as-p #:classification-case #:collect-variables #:collect-terms)) (in-package #:manipulator) (declaim (declaration slime-indent)) <> <> <> <> <> #+END_SRC