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term-simplifier.lisp
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term-simplifier.lisp
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;;;; -*- Lisp -*-
;;; The term simplifier
;;; Ripped off from Norvigs PAIP
(in-package :hokuspokus)
;; (proclaim '(optimize speed))
;; turning this on gives some nice compiler warnings where to optimize
;;;; Code from Paradigms of AI Programming
;;;; Copyright (c) 1991 Peter Norvig
(defun starts-with (list x)
"Is x a list whose first element is x?"
(and (consp list) (eql (first list) x)))
(defun length=1 (x)
"Is x a list of length 1?"
(and (consp x) (null (cdr x))))
;;;; PATTERN MATCHING FACILITY
(defconstant fail nil)
(defvar no-bindings '((t . t)))
(defun match-variable (var input bindings)
"Does VAR match input? Uses (or updates) and returns bindings."
(let ((binding (get-binding var bindings)))
(cond ((not binding) (extend-bindings var input bindings))
((equal input (binding-val binding)) bindings)
(t fail))))
(defun make-binding (var val) (cons var val))
(defun binding-var (binding)
"Get the variable part of a single binding."
(car binding))
(defun binding-val (binding)
"Get the value part of a single binding."
(cdr binding))
(defun get-binding (var bindings)
"Find a (variable . value) pair in a binding list."
(assoc var bindings))
(defun lookup (var bindings)
"Get the value part (for var) from a binding list."
(binding-val (get-binding var bindings)))
(defun extend-bindings (var val bindings)
"Add a (var . value) pair to a binding list."
(cons (cons var val)
;; Once we add a "real" binding,
;; we can get rid of the dummy no-bindings
(if (eq bindings no-bindings)
nil
bindings)))
(defun variable-p (x)
"Is x a variable (a symbol beginning with `?')?"
(and (symbolp x) (equal (elt (symbol-name x) 0) #\?)))
;;;; File pat-match.lisp: Pattern matcher from section 6.2
;;; A bug fix By Richard Fateman, rjf@cs.berkeley.edu October 92.
(defun pat-match (pattern input &optional (bindings no-bindings))
"Match pattern against input in the context of the bindings"
(cond ((eq bindings fail) fail)
((variable-p pattern)
(match-variable pattern input bindings))
((eql pattern input) bindings)
((segment-pattern-p pattern)
(segment-matcher pattern input bindings))
((single-pattern-p pattern) ; ***
(single-matcher pattern input bindings)) ; ***
((and (consp pattern) (consp input))
(pat-match (rest pattern) (rest input)
(pat-match (first pattern) (first input)
bindings)))
(t fail)))
(setf (get '?is 'single-match) 'match-is)
(setf (get '?or 'single-match) 'match-or)
(setf (get '?and 'single-match) 'match-and)
(setf (get '?not 'single-match) 'match-not)
(setf (get '?* 'segment-match) 'segment-match)
(setf (get '?+ 'segment-match) 'segment-match+)
(setf (get '?? 'segment-match) 'segment-match?)
(setf (get '?if 'segment-match) 'match-if)
(defun segment-pattern-p (pattern)
"Is this a segment-matching pattern like ((?* var) . pat)?"
(and (consp pattern) (consp (first pattern))
(symbolp (first (first pattern)))
(segment-match-fn (first (first pattern)))))
(defun single-pattern-p (pattern)
"Is this a single-matching pattern?
E.g. (?is x predicate) (?and . patterns) (?or . patterns)."
(and (consp pattern)
(single-match-fn (first pattern))))
(defun segment-matcher (pattern input bindings)
"Call the right function for this kind of segment pattern."
(funcall (segment-match-fn (first (first pattern)))
pattern input bindings))
(defun single-matcher (pattern input bindings)
"Call the right function for this kind of single pattern."
(funcall (single-match-fn (first pattern))
(rest pattern) input bindings))
(defun segment-match-fn (x)
"Get the segment-match function for x,
if it is a symbol that has one."
(when (symbolp x) (get x 'segment-match)))
(defun single-match-fn (x)
"Get the single-match function for x,
if it is a symbol that has one."
(when (symbolp x) (get x 'single-match)))
(defun match-is (var-and-pred input bindings)
"Succeed and bind var if the input satisfies pred,
where var-and-pred is the list (var pred)."
(let* ((var (first var-and-pred))
(pred (second var-and-pred))
(new-bindings (pat-match var input bindings)))
(if (or (eq new-bindings fail)
(not (funcall pred input)))
fail
new-bindings)))
(defun match-and (patterns input bindings)
"Succeed if all the patterns match the input."
(cond ((eq bindings fail) fail)
((null patterns) bindings)
(t (match-and (rest patterns) input
(pat-match (first patterns) input
bindings)))))
(defun match-or (patterns input bindings)
"Succeed if any one of the patterns match the input."
(if (null patterns)
fail
(let ((new-bindings (pat-match (first patterns)
input bindings)))
(if (eq new-bindings fail)
(match-or (rest patterns) input bindings)
new-bindings))))
(defun match-not (patterns input bindings)
"Succeed if none of the patterns match the input.
This will never bind any variables."
(if (match-or patterns input bindings)
fail
bindings))
(defun segment-match (pattern input bindings &optional (start 0))
"Match the segment pattern ((?* var) . pat) against input."
(let ((var (second (first pattern)))
(pat (rest pattern)))
(if (null pat)
(match-variable var input bindings)
(let ((pos (first-match-pos (first pat) input start)))
(if (null pos)
fail
(let ((b2 (pat-match
pat (subseq input pos)
(match-variable var (subseq input 0 pos)
bindings))))
;; If this match failed, try another longer one
(if (eq b2 fail)
(segment-match pattern input bindings (+ pos 1))
b2)))))))
(defun first-match-pos (pat1 input start)
"Find the first position that pat1 could possibly match input,
starting at position start. If pat1 is non-constant, then just
return start."
(cond ((and (atom pat1) (not (variable-p pat1)))
(position pat1 input :start start :test #'equal))
((<= start (length input)) start) ;*** fix, rjf 10/1/92 (was <)
(t nil)))
(defun segment-match+ (pattern input bindings)
"Match one or more elements of input."
(segment-match pattern input bindings 1))
(defun segment-match? (pattern input bindings)
"Match zero or one element of input."
(let ((var (second (first pattern)))
(pat (rest pattern)))
(or (pat-match (cons var pat) input bindings)
(pat-match pat input bindings))))
(defun match-if (pattern input bindings)
"Test an arbitrary expression involving variables.
The pattern looks like ((?if code) . rest)."
;; *** fix, rjf 10/1/92 (used to eval binding values)
(and (progv (mapcar #'car bindings)
(mapcar #'cdr bindings)
(eval (second (first pattern))))
(pat-match (rest pattern) input bindings)))
(defun pat-match-abbrev (symbol expansion)
"Define symbol as a macro standing for a pat-match pattern."
(setf (get symbol 'expand-pat-match-abbrev)
(expand-pat-match-abbrev expansion)))
(defun expand-pat-match-abbrev (pat)
"Expand out all pattern matching abbreviations in pat."
(cond ((and (symbolp pat) (get pat 'expand-pat-match-abbrev)))
((atom pat) pat)
(t (cons (expand-pat-match-abbrev (first pat))
(expand-pat-match-abbrev (rest pat))))))
(defun rule-based-translator
(input rules &key (matcher 'pat-match)
(rule-if #'first) (rule-then #'rest) (action #'sublis))
"Find the first rule in rules that matches input,
and apply the action to that rule."
(some
#'(lambda (rule)
(let ((result (funcall matcher (funcall rule-if rule)
input)))
(if (not (eq result fail))
(funcall action result (funcall rule-then rule)))))
rules))
;;;; File macsyma.lisp: The implementation of MACSYMA in Chapter 8
;; (defun variable-p (exp)
;; "Variables are the symbols M through Z."
;; ;; put x,y,z first to find them a little faster
;; (member exp '(x y z m n o p q r s t u v w)))
;;; From student.lisp:
(defstruct (rule (:type list)) pattern response)
(defstruct (expr (:type list)
(:conc-name exp-)
(:constructor mkexp (lhs op rhs)))
op lhs rhs)
(defun exp-p (x) (consp x))
(defun exp-args (x) (rest x))
(defun binary-exp-p (x)
(and (exp-p x) (= (length (exp-args x)) 2)))
(defun prefix->infix (exp)
"Translate prefix to infix expressions."
(if (atom exp) exp
(mapcar #'prefix->infix
(if (binary-exp-p exp)
(list (exp-lhs exp) (exp-op exp) (exp-rhs exp))
exp))))
;; Define x+ and y+ as a sequence:
(pat-match-abbrev 'x+ '(?+ x))
(pat-match-abbrev 'y+ '(?+ y))
;; Define n and m as numbers; s as a non-number:
(pat-match-abbrev 'n '(?is n numberp))
(pat-match-abbrev 'm '(?is m numberp))
(pat-match-abbrev 's '(?is s not-numberp))
(defparameter *infix->prefix-rules*
(mapcar #'expand-pat-match-abbrev
'(((x+ = y+) (= x y))
((- x+) (- x))
((+ x+) (+ x))
((x+ + y+) (+ x y))
((x+ - y+) (- x y))
((d y+ / d x) (d y x)) ;*** New rule
((Int y+ d x) (int y x)) ;*** New rule
((x+ * y+) (* x y))
((x+ / y+) (/ x y))
((x+ ^ y+) (^ x y)))))
(defun infix->prefix (exp)
"Translate an infix expression into prefix notation."
;; Note we cannot do implicit multiplication in this system
(cond ((atom exp) exp)
((= (length exp) 1) (infix->prefix (first exp)))
((rule-based-translator exp *infix->prefix-rules*
:rule-if #'rule-pattern :rule-then #'rule-response
:action
#'(lambda (bindings response)
(sublis (mapcar
#'(lambda (pair)
(cons (first pair)
(infix->prefix (rest pair))))
bindings)
response))))
((symbolp (first exp))
(list (first exp) (infix->prefix (rest exp))))
(t (error "Illegal exp"))))
(defvar *simplification-rules* nil) ;Rules are in file macsymar.lisp
(defun ^ (x y) "Exponentiation" (expt x y))
(defun simplifier ()
"Read a mathematical expression, simplify it, and print the result."
(loop
(print 'simplifier>)
(print (simp (read)))))
(defun simp (inf) (prefix->infix (simplify (infix->prefix inf))))
(defun simplify (exp)
"Simplify an expression by first simplifying its components."
(if (atom exp) exp
(simplify-exp (mapcar #'simplify exp))))
;;; simplify-exp is redefined below
;(defun simplify-exp (exp)
; "Simplify using a rule, or by doing arithmetic."
; (cond ((rule-based-translator exp *simplification-rules*
; :rule-if #'exp-lhs :rule-then #'exp-rhs
; :action #'(lambda (bindings response)
; (simplify (sublis bindings response)))))
; ((evaluable exp) (eval exp))
; (t exp)))
(defun evaluable (exp)
"Is this an arithmetic expression that can be evaluated?"
(and (every #'numberp (exp-args exp))
(or (member (exp-op exp) '(+ - * /))
(and (eq (exp-op exp) '^)
(integerp (second (exp-args exp)))))))
(defun not-numberp (x) (not (numberp x)))
(defun simp-rule (rule)
"Transform a rule into proper format."
(let ((exp (infix->prefix rule)))
(mkexp (expand-pat-match-abbrev (exp-lhs exp))
(exp-op exp) (exp-rhs exp))))
(defun simp-fn (op) (get op 'simp-fn))
(defun set-simp-fn (op fn) (setf (get op 'simp-fn) fn))
(defun simplify-exp (exp)
"Simplify using a rule, or by doing arithmetic,
or by using the simp function supplied for this operator."
(cond ((simplify-by-fn exp)) ;***
((rule-based-translator exp *simplification-rules*
:rule-if #'exp-lhs :rule-then #'exp-rhs
:action #'(lambda (bindings response)
(simplify (sublis bindings response)))))
((evaluable exp) (eval exp))
(t exp)))
(defun simplify-by-fn (exp)
"If there is a simplification fn for this exp,
and if applying it gives a non-null result,
then simplify the result and return that."
(let* ((fn (simp-fn (exp-op exp)))
(result (if fn (funcall fn exp))))
(if (null result)
nil
(simplify result))))
(defun factorize (exp)
"Return a list of the factors of exp^n,
where each factor is of the form (^ y n)."
(let ((factors nil)
(constant 1))
(labels
((fac (x n)
(cond
((numberp x)
(setf constant (* constant (expt x n))))
((starts-with x '*)
(fac (exp-lhs x) n)
(fac (exp-rhs x) n))
((starts-with x '/)
(fac (exp-lhs x) n)
(fac (exp-rhs x) (- n)))
((and (starts-with x '-) (length=1 (exp-args x)))
(setf constant (- constant))
(fac (exp-lhs x) n))
((and (starts-with x '^) (numberp (exp-rhs x)))
(fac (exp-lhs x) (* n (exp-rhs x))))
(t (let ((factor (find x factors :key #'exp-lhs
:test #'equal)))
(if factor
(incf (exp-rhs factor) n)
(push `(^ ,x ,n) factors)))))))
;; Body of factorize:
(fac exp 1)
(case constant
(0 '((^ 0 1)))
(1 factors)
(t `((^ ,constant 1) .,factors))))))
(defun unfactorize (factors)
"Convert a list of factors back into prefix form."
(cond ((null factors) 1)
((length=1 factors) (first factors))
(t `(* ,(first factors) ,(unfactorize (rest factors))))))
(defun divide-factors (numer denom)
"Divide a list of factors by another, producing a third."
(let ((result (mapcar #'copy-list numer)))
(dolist (d denom)
(let ((factor (find (exp-lhs d) result :key #'exp-lhs
:test #'equal)))
(if factor
(decf (exp-rhs factor) (exp-rhs d))
(push `(^ ,(exp-lhs d) ,(- (exp-rhs d))) result))))
(delete 0 result :key #'exp-rhs)))
(defun free-of (exp var)
"True if expression has no occurrence of var."
(not (find-anywhere var exp)))
(defun find-anywhere (item tree)
"Does item occur anywhere in tree? If so, return it."
(cond ((eql item tree) tree)
((atom tree) nil)
((find-anywhere item (first tree)))
((find-anywhere item (rest tree)))))
(defun integrate (exp x)
;; First try some trivial cases
(cond
((free-of exp x) `(* ,exp x)) ; Int c dx = c*x
((starts-with exp '+) ; Int f + g =
`(+ ,(integrate (exp-lhs exp) x) ; Int f + Int g
,(integrate (exp-rhs exp) x)))
((starts-with exp '-)
(ecase (length (exp-args exp))
(1 (integrate (exp-lhs exp) x)) ; Int - f = - Int f
(2 `(- ,(integrate (exp-lhs exp) x) ; Int f - g =
,(integrate (exp-rhs exp) x))))) ; Int f - Int g
;; Now move the constant factors to the left of the integral
((multiple-value-bind (const-factors x-factors)
(partition-if #'(lambda (factor) (free-of factor x))
(factorize exp))
(identity ;simplify
`(* ,(unfactorize const-factors)
;; And try to integrate:
,(cond ((null x-factors) x)
((some #'(lambda (factor)
(deriv-divides factor x-factors x))
x-factors))
;; <other methods here>
(t `(int? ,(unfactorize x-factors) ,x)))))))))
(defun partition-if (pred list)
"Return 2 values: elements of list that satisfy pred,
and elements that don't."
(let ((yes-list nil)
(no-list nil))
(dolist (item list)
(if (funcall pred item)
(push item yes-list)
(push item no-list)))
(values (nreverse yes-list) (nreverse no-list))))
(defun deriv-divides (factor factors x)
(assert (starts-with factor '^))
(let* ((u (exp-lhs factor)) ; factor = u^n
(n (exp-rhs factor))
(k (divide-factors
factors (factorize `(* ,factor ,(deriv u x))))))
(cond ((free-of k x)
;; Int k*u^n*du/dx dx = k*Int u^n du
;; = k*u^(n+1)/(n+1) for n/=1
;; = k*log(u) for n=1
(if (= n -1)
`(* ,(unfactorize k) (log ,u))
`(/ (* ,(unfactorize k) (^ ,u ,(+ n 1)))
,(+ n 1))))
((and (= n 1) (in-integral-table? u))
;; Int y'*f(y) dx = Int f(y) dy
(let ((k2 (divide-factors
factors
(factorize `(* ,u ,(deriv (exp-lhs u) x))))))
(if (free-of k2 x)
`(* ,(integrate-from-table (exp-op u) (exp-lhs u))
,(unfactorize k2))))))))
(defun deriv (y x) (simplify `(d ,y ,x)))
(defun integration-table (rules)
(dolist (i-rule rules)
;; changed infix->prefix to simp-rule - norvig Jun 11 1996
(let ((rule (simp-rule i-rule)))
(setf (get (exp-op (exp-lhs (exp-lhs rule))) 'int)
rule))))
(defun in-integral-table? (exp)
(and (exp-p exp) (get (exp-op exp) 'int)))
(defun integrate-from-table (op arg)
(let ((rule (get op 'int)))
(subst arg (exp-lhs (exp-lhs (exp-lhs rule))) (exp-rhs rule))))
(set-simp-fn 'Int #'(lambda (exp)
(unfactorize
(factorize
(integrate (exp-lhs exp) (exp-rhs exp))))))
;;;; File macsymar.lisp: The rewrite rules for MACSYMA in Chapter 8
(defun init-simplification-rules ()
(setf *simplification-rules* (mapcar #'simp-rule '(
(x + 0 = x)
(0 + x = x)
;(x + x = 2 * x)
((x + x) - x = x) ;; added by mp
(x - 0 = x)
(0 - x = - x)
(x - x = 0)
(- - x = x)
(x * 1 = x)
(1 * x = x)
(x * 0 = 0)
(0 * x = 0)
;(x * x = x ^ 2)
(x / 0 = undefined)
(0 / x = 0)
(x / 1 = x)
(x / x = 1)
(0 ^ 0 = undefined)
(x ^ 0 = 1)
(0 ^ x = 0)
(1 ^ x = 1)
(x ^ 1 = x)
(x ^ -1 = 1 / x)
(x * (y / x) = y)
((y / x) * x = y)
((y * x) / x = y)
((x * y) / x = y)
((x * y) / (x * z) = y / z) ;; added by mp
((y * x) / (x * z) = y / z) ;; added by mp
((x * y) / (z * x) = y / z) ;; added by mp
((y * x) / (z * x) = y / z) ;; added by mp
(x / (x * y) = 1 / y) ;; added by mp
(x / (y * x) = 1 / y) ;; added by mp
(x / (x / y) = y) ;; added by mp
(x / (y / x) = 1 / y) ;; added by mp
((x / y) / x = 1 / y) ;; added by mp
((x / y) / x = y) ;; added by mp
((y / x) / x = y) ;; added by mp
((x / y) / (z / y) = x / z) ;; added by mp
((x / y) / (z * x) = 1 / (y * z)) ;; added by mp
((x / y) / (x / z) = z / y) ;; added by mp
((x / y) / (z / y) = x / z) ;; added by mp
(x * (x / y) = (x * x) / y) ;; added by mp
((x / y) * x = (x * x) / y) ;; added by mp
;; removed the line below again, because sometimes hokuspokus
;; only find features one way and not the other and this eliminates
;; one way
;; ((x / y) + (z / y) = (x + z) / y) ;; added by mp
((x / y) * z = (x * z) / y) ;; added by mp ; only changes order
(x + - x = 0)
((- x) + x = 0)
(x + y - x = y)
((x + y) - x = y) ;; added by mp
)))
(setf *simplification-rules*
(append *simplification-rules* (mapcar #'simp-rule
'((s * n = n * s)
(n * (m * x) = (n * m) * x)
(x * (n * y) = n * (x * y))
((n * x) * y = n * (x * y))
(n * (x / y) = (n * x) / y) ;; added by mp
(n + s = s + n)
((x + m) + n = x + n + m)
(x + (y + n) = (x + y) + n)
((x + n) + y = (x + y) + n)))))
(setf *simplification-rules*
(append *simplification-rules* (mapcar #'simp-rule '(
(log 1 = 0)
(log 0 = undefined)
(log e = 1)
(sin 0 = 0)
(sin pi = 0)
(cos 0 = 1)
(cos pi = -1)
(sin(pi / 2) = 1)
(cos(pi / 2) = 0)
(log (e ^ x) = x)
(e ^ (log x) = x)
((x ^ y) * (x ^ z) = x ^ (y + z))
((x ^ y) / (x ^ z) = x ^ (y - z))
(log x + log y = log(x * y))
(log x - log y = log(x / y))
((sin x) ^ 2 + (cos x) ^ 2 = 1)
))))
(setf *simplification-rules*
(append *simplification-rules* (mapcar #'simp-rule '(
(d x / d x = 1)
(d (u + v) / d x = (d u / d x) + (d v / d x))
(d (u - v) / d x = (d u / d x) - (d v / d x))
(d (- u) / d x = - (d u / d x))
(d (u * v) / d x = u * (d v / d x) + v * (d u / d x))
(d (u / v) / d x = (v * (d u / d x) - u * (d v / d x))
/ v ^ 2) ; [This corrects an error in the first printing]
(d (u ^ n) / d x = n * u ^ (n - 1) * (d u / d x))
(d (u ^ v) / d x = v * u ^ (v - 1) * (d u / d x)
+ u ^ v * (log u) * (d v / d x))
(d (log u) / d x = (d u / d x) / u)
(d (sin u) / d x = (cos u) * (d u / d x))
(d (cos u) / d x = - (sin u) * (d u / d x))
(d (e ^ u) / d x = (e ^ u) * (d u / d x))
(d u / d x = 0)))))
)
(integration-table
'((Int log(x) d x = x * log(x) - x)
(Int exp(x) d x = exp(x))
(Int sin(x) d x = - cos(x))
(Int cos(x) d x = sin(x))
(Int tan(x) d x = - log(cos(x)))
(Int sinh(x) d x = cosh(x))
(Int cosh(x) d x = sinh(x))
(Int tanh(x) d x = log(cosh(x)))
))