A Replace Function in Lisp That Duplicates Mathematica Functionality - haskell

What is the easiest way to accomplish the following in a Mathematica clone or in any version of Lisp(any language is probably okay actually even Haskell)? It doesn't appear any lisps have a similar replace function.
Replace[{
f[{x, "[", y, "]"}],
f#f[{x, "[", y, y2, "]"}]
}
, f[{x_, "[", y__, "]"}] :> x[y],
Infinity]
and a return value of {x[y], f[x[y, y2]]}
It replaces all instances of f[{x_, "[", y__, "]"}] in args where x_ represents a single variable and y__ represents one or more variables.
In lisp the function and replacement would probably be the equivalent(forgive me I am not the best with Lisp). I'm looking for a function of the form (replace list search replace).
(replace
'(
(f (x "[" y "]"))
(f (f '(x "[" y y2 "]")))
)
'(f (x_ "[" y__ "]"))
'(x y)
)
and get a return value of ((x y) (f (x y y2))).

Let's give it another try.
First, install quicklisp and use it to fetch, install and load optima and alexandria.
(ql:quickload :optima)
(ql:quickload :alexandria)
(use-package :alexandria)
The functions from alexandria referenced below are ensure-list and last-elt. If you don't have them installed, you can use the following definitions:
(defun ensure-list (list) (if (listp list) list (list list)))
(defun last-elt (list) (car (last list)))
We define rules as functions from one form to another.
Below, the function tries to destructure the input as (f (<X> "[" <ARGS> "]"), where <ARGS> is zero or more form. If destructuring fails, we return NIL (we expect non-matching filters to return NIL hereafter).
(defun match-ugly-funcall (form)
(optima:match form
((list 'f (cons x args))
(unless (and (string= "[" (first args))
(string= "]" (last-elt args)))
(optima:fail))
`(,x ,#(cdr (butlast args))))))
(match-ugly-funcall '(f (g "[" 1 3 5 4 8 "]")))
; => (G 1 3 5 4 8)
Then, we mimic Mathematica's Replace with this function, which takes a form and a list of rules to be tried. It is possible to pass a single rule (thanks to ensure-list). If a list of list of rules is given, a list of matches should be returned (to be done).
(defun match-replace (form rules &optional (levelspec '(0)))
(setf rules (ensure-list rules))
(multiple-value-bind (match-levelspec-p recurse-levelspec-p)
(optima:ematch levelspec
((list n1 n2) (if (some #'minusp (list n1 n2))
(optima:fail)
(values (lambda (d) (<= n1 d n2))
(lambda (d) (< d n2)))))
((list n) (if (minusp n)
(optima:fail)
(values (lambda (d) (= d n))
(lambda (d) (< d n)))))
(:infinity (values (constantly t) (constantly t))))
(labels
((do-replace (form depth)
(let ((result
(and (funcall match-levelspec-p depth)
(some (lambda (r) (funcall r form)) rules))))
(cond
(result (values result t))
((and (listp form)
(funcall recurse-levelspec-p depth))
(incf depth)
(do (newlist
(e (pop form) (pop form)))
((endp form) (values form nil))
(multiple-value-bind (result matchedp) (do-replace e depth)
(if matchedp
(return (values (nconc (nreverse newlist)
(list* result form)) t))
(push e newlist)))))
(t (values form nil))))))
(do-replace form 0))))
And a test:
(match-replace '(a b (f (x "[" 1 2 3 "]")) c d)
#'match-ugly-funcall
:infinity)
; => (A B (X 1 2 3) C D)
; T
In order to replace all expressions instead of the first matching one, use this instead:
(defun match-replace-all (form rules &optional (levelspec '(0)))
(setf rules (ensure-list rules))
(multiple-value-bind (match-levelspec-p recurse-levelspec-p)
(optima:ematch levelspec
((list n1 n2) (if (some #'minusp (list n1 n2))
(optima:fail)
(values (lambda (d) (<= n1 d n2))
(lambda (d) (< d n2)))))
((list n) (if (minusp n)
(optima:fail)
(values (lambda (d) (= d n))
(lambda (d) (< d n)))))
(:infinity (values (constantly t) (constantly t))))
(labels
((do-replace (form depth)
(let ((result
(and (funcall match-levelspec-p depth)
(some (lambda (r) (funcall r form)) rules))))
(cond
(result result)
((and (listp form)
(funcall recurse-levelspec-p depth))
(incf depth)
(mapcar (lambda (e) (do-replace e depth)) form))
(t form)))))
(do-replace form 0))))

Oh boy, how Mathematica manages to obfuscate everything by applying its renown NIH approach.
Basically, you're looking for a function to perform string replacement according to some pattern. In most languages, this is accomplished with regular expressions.
For instance, in Common Lisp using the cl-ppcre library it will look something like this:
(cl-ppcre:regex-replace-all
;; regular expression you match against with groups
"f\\[{(x[^ ]*), \"\\[\", ((y[^ ]* ?)+), \"\\]\"}\\]"
;; your string
"{f[{x, \"[\", y, \"]\"}], f#f[{x, \"[\", y, y2, \"]\"}]}"
;; substitution expression using groups 1 & 2
"\\1[\\2]")
Surely, you can write a specialized 20-line function for this problem of matching and substituting subtrees using subst and recursion, but if all that you want is cases similar to the presented one you can get away with a simple regex-based approach.

Related

How do I find the number of characters in a string using scheme programming language?

I used string-length to get the number of characters but I am having difficulties in defining a recursive function. Should I convert the string to a list and then count the elements?
There's no useful way of doing this recursively (or even tail recursively): strings in Scheme are objects which know how long they are. There would be such an approach in a language like C where strings don't know how long they are but are delimited by some special marker. So for instance if (special-marker? s i) told you whether the i'th element of s was the special marker object, then you could write a function to know how long the string was:
(define (silly-string-length s)
(let silly-string-length-loop ([i 1])
(if (special-marker? s i)
(- i 1)
(silly-string-length-loop (+ i 1)))))
But now think about how you would implement special-marker? in Scheme: in particular here's the obvious implementation:
(define (special-marker? s i)
(= i (+ (string-length s) 1)))
And you can see that silly-string-length is now just a terrible version of string-length.
Well, if you wanted to make it look even more terrible, you could, as you suggest, convert a string to a list and then compute the length of the lists. Lists are delimited by a special marker object, () so this approach is reasonable:
(define (length-of-list l)
(let length-of-list-loop ([i 0]
[lt l])
(if (null? lt)
i
(length-of-list-loop (+ i 1) (rest lt)))))
So you could write
(define (superficially-less-silly-string-length s)
(length-of-list
(turn-string-into-list s)))
But, wait, how do you write turn-string-into-list? Well, something like this perhaps:
(define (turn-string-into-list s)
(let ([l (string-length s)])
(let loop ([i 0]
[r '()])
(if (= i l)
(reverse r)
(loop (+ i 1)
(cons (string-ref s i) r))))))
And this ... uses string-length.
What is the problem with?
(string-length string)
If the question is a puzzle "count characters in a string without using string-length",
then maybe:
(define (my-string-length s)
(define (my-string-length t n)
(if (string=? s t) n
(my-string-length
(string-append t (string (string-ref s n))) (+ n 1))))
(my-string-length "" 0))
or:
(define (my-string-length s)
(define (my-string-length n)
(define (try thunk)
(call/cc (lambda (k)
(with-exception-handler (lambda (x)
(k n))
thunk))))
(try (lambda ()
(string-ref s n)
(my-string-length (+ n 1)))))
(my-string-length 0))
(but of course string-ref will be using the base string-length or equivalent)

How to modify string by index in Clojure?

I want to modify a string by applying a function to some of its chars (by starting index and length).
For example, I want to increment the ascii representation of the string "aaaaa" from the 2nd index to the 4th.
[start=1 length=3]
"aaaaa" => "abbba"
The only way I could think of is applying map, but it goes over all the sequence.
You could use subs to get the portions you do and don't want to modify. After modification use str to concatenate the result together:
(defn replace-in-str [f in from len]
(let [before (subs in 0 from)
after (subs in (+ from len))
being-replaced (subs in from (+ from len))
replaced (f being-replaced)]
(str before replaced after)))
You can call it:
(replace-in-str
(fn [sub-str] (apply str (map #(char (+ 1 (int %))) sub-str)))
"aaaaa"
1
3)
Indeed map applies the function to every element in the sequence. One way to get around that is to start with map-indexed. Unlike map, map-indexed passes the element's index as the first argument to the mapping function. When we have element's index, we can use it to choose if we need to perform the operation or just return the element as is.
A solution might look like this:
(defn inc-char [c]
(char (inc (long c))))
(defn if-in-range [from to f]
(fn [i x & args]
(if (<= from i (dec to))
(apply f x args)
x)))
(defn map-subs [from to f s]
(apply str (map-indexed (if-in-range from to f) s)))
(map-subs 1 4 inc-char "aaaaa")
;; "abbba"
I thought of using map-index to execute the operation only on the specified index:
((fn [op start length] (map-indexed (fn [i m] (if (<= start i length)
(op m)
m)) "aaaaa"))
#(char (+ 1 (int %)))
1
3)
=> (\a \b \b \b \a)
Here you go:
(defn replace-str
[s start-i end-i]
(apply str (map-indexed (fn [index val]
(if (and (>= index start-i)
(<= index end-i))
(char (+ (int val) 1))
val))
s)))
(replace-str "aaaa" 1 2)
;=> "abba"

Coloring N segments using M colors with dynamic programming

Problem: I have N contiguous segments numbered from 1 to N and M colors also numbered from 1 to M.
Now, there are two numbers U and V defined as:
U = color(i) + color(j)
V = color(j) + color(k)
U, V are coprime.
where 1 <= i,j,k <= N and
j = i+1, k=j+1
Problem is to find the number of ways that all N segments can be colored such that the above property holds for all i,j,k.
Is there a dynamic programming solution to this problem? What is it?
I have a recursive but non-[dynamic programming] implementation of this that should help get you pointed in the right direction. It's implemented in Common Lisp since there's no language specified.
The way to extend it to be a dynamic programming solution would be to add a cache.
count-all-coprime-triple-colorings constructs all the colorings in memory and then checks each of them for satisfying the coprime triple condition.
count-all-coprime-triple-colorings-lazy tries to aggressively prune the colorings we even consider by ruling out colorings with a prefix that doesn't satisfy the coprime condition.
This approach could be improved by noting that only the last two elements of the prefix are relevant, so you could use that to populate the cache.
(defun coprime-p (a b)
"check whether a and b are coprime"
(= (gcd a b) 1))
(defun coprime-triple-p (a b c)
"check whether (a+b) and (b+c) are coprime"
(coprime-p (+ a b) (+ b c)))
(defun coprime-triple-sequence-p (seq)
"check whether seq is a sequence of corpime triples"
(cond
;; if the length is less than 2 then
;; every triple is trivially coprime
((<= (length seq) 2) t)
(t (let
((a (nth 0 seq))
(b (nth 1 seq))
(c (nth 2 seq))
(tail (cdr seq)))
(if (coprime-triple-p a b c)
(coprime-triple-sequence-p tail)
nil)))))
(defun curry-cons (x)
"curried cons operator"
(lambda (list) (cons x list)))
(defun all-colorings (sections colors)
"generate all possible #colors-colorings of sections"
(assert (>= sections 0))
(assert (>= colors 1))
(cond
;; if there are no sections
;; then there are no colorings
((= sections 0) ())
;; when we have one section there is one coloring
;; for each color
((= sections 1) (loop for i from 1 upto colors collecting (list i)))
(t
;; wildly inefficient
(loop for i from 1 upto colors appending
(mapcar (curry-cons i) (all-colorings (1- sections) colors))))))
(defun count-all-coprime-triple-colorings (sections colors)
"count all the colorings that have coprime triples"
(loop for i in (all-colorings sections colors) counting (coprime-triple-sequence-p i)))
(defun coprime-triple-check-boundary (reversed-prefix suffix)
"prefix = [...a, b] ; suffix = [c,...] ; check
gcd(a+b, b+c) != 1"
;; if there aren't enough elements in reversed-prefix and suffix
;; then we admit the list
(if (and (nth 1 reversed-prefix) (nth 0 suffix))
(let
((b (nth 0 reversed-prefix)) (a (nth 1 reversed-prefix)) (c (nth 0 suffix)))
(coprime-triple-p a b c))
t))
(defun count-all-coprime-triple-colorings-lazy (sections colors reversed-prefix)
"count the number of sequences with coprime triples with a particular number
of sections and colors with a particular reversed-prefix."
(let
((sections-- (1- sections)))
(cond
((= sections 0) 1)
(t (loop for i from 1 upto colors summing
(if (coprime-triple-check-boundary reversed-prefix (list i))
(count-all-coprime-triple-colorings-lazy sections-- colors (cons i reversed-prefix))
0))))))
(defun summarize-coloring (i j)
"summarize the given coloring number"
(print (list "triples" i "colors" j
(count-all-coprime-triple-colorings-lazy i j nil))))
(loop for i from 1 upto 9 doing
(loop for j from 1 upto 9 doing (summarize-coloring i j)))

How would you assign a operator to a string in scheme? (ex: "A" == + , "S" == -, etc.) Im having trouble computing ("A" 1 2)

Im having trouble computing ("A" 1 2)
The canonical way to do this is with symbols, not strings:
> (define a +)
> (a 1 2)
3
> +
#<procedure:+>
> a
#<procedure:+>
Here symbol a gets bound to the same procedure that symbol + is (and remains) bound to, which means that both + and a can be used interchangeably.
If you need to use strings, you could use a procedure that, using a lookup table, converts a string into a procedure:
(define (s->p str)
(cdr (assoc str (list (cons "A" +) (cons "S" -)))))
used like so:
> ((s->p "A") 1 2)
3
If you are making your own interpreter then you'll have a sort of assoc list to map names to the actual implementation. Eg.
(define env `(("a" . ,+) ("m" . ,-) ("id" ,(lambda (x) x))))
(define (string->value sym env)
(let ((v (assoc sym env)))
(if v ; v is #f when not found
(cdr v)
(error "Variable not found"))))
((string->value "a" env) 4 5) ; ==> 9
It's actually more common to use symbols, but I can think of implementations where all values are strings.
#lang racket
(define (combine sign)
(cond ((equal? "A" sign) +)
((equal? "S" sign) -)))
((combine "A") 1 2)
alsoļ¼š
#lang racket
(define env
(list (cons "a" +) (cons "m" -)))
(define (combine target env)
(define (search env)
(if (null? env)
(begin (displayln "not found and the default operate is +") +)
(if (equal? target (car (car env)))
(cdr (car env))
(search (cdr env)))))
(search env))
((combine "2" env) 1 2 3 4)

"set! not an identifier" in Scheme

I am trying to modify the cdr of the car of an element of a list using set!, but I am getting an error: "set! not an identifier". Can anyone explain to me why this is happening? I am starting to work with objects by the way.
#lang racket
(define (multiset)
(let ((main-list '()))
(define (empty)
(eq? main-list '()))
(define (insert x)
(cond ((empty)
(set! main-list (cons (cons x 1) main-list)))
((= (car (car main-list)) x)
(begin (set! (cdr (car main-list))
(+ 1 (cdr (car main-list))))))
(else (cdr main-list))))
A multiset is represented as a list of pairs. For example, if I had a list '(1 1 1 2 2 2), the multiset representation would be '((1.3)(2.3))
The syntax of set! is
(set! <identifier> <expression>)
which is to say that the first form must be a symbol. In your code you are using:
(cdr (car main-list))
as the 'identifier' - hence the error.
Perhaps your background is CommonLisp and you are expecting set! to behave like setf? In Scheme there are separate functions for setting the 'car' and 'cdr' of a pair. Use set-car! and set-cdr!
> (define pair (cons 'car 'cdr))
> pair
(car . cdr)
> (set-car! pair 'foo)
> pair
(foo . cdr)
> (set-cdr! pair 'bar)
> pair
(foo . bar)
In R6RS (and probably R7RS) set-car! and set-cdr! can be found in the (rnrs mutable-pairs) library
GoZoner has given you the right explanation, but it is perfectly possible (and desirable) do avoid set! procedures. Here's an example of a procedure having the same result:
(define (rle lst)
(define (newpair c l res)
(if (> l 0) (cons (cons c l) res) res))
(let loop ((lst lst) (c #f) (l 0) (res '()))
(if (null? lst)
(reverse (newpair c l res))
(let ((n (car lst)))
(if (equal? c n)
(loop (cdr lst) c (add1 l) res)
(loop (cdr lst) n 1 (newpair c l res)))))))
such as
(rle '(1 1 1 2 2 2))
=> '((1 . 3) (2 . 3))

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