I'm very new with functional programming, lisp and lambda calculus. Im trying to implement the AND operator with Common Lisp Lambda Calc style.
From Wikipedia:
AND := λp.λq.p q p
So far this is my code:
(defvar TRUE #'(lambda(x)#'(lambda(y)x)))
(defvar FALSE #'(lambda(x)#'(lambda(y)y)))
(defun OPAND (p q)
#'(lambda(f)
#'(lambda(p) #'(lambda(q) (funcall p (funcall q(funcall p))))))
)
I found this 2 conversion functions:
(defun church2int(numchurch)
(funcall (funcall numchurch #'(lambda (x) (+ x 1))) 0)
)
(defun int2church(n)
(cond
((= n 0) #'(lambda(f) #'(lambda(x)x)))
(t #'(lambda(f) #'(lambda(x) (funcall f
(funcall(funcall(int2church (- n 1))f)x))))))
)
If I do:
(church2int FALSE)
I've got 0. If I do this:
(church2int TRUE)
I have
#<FUNCTION :LAMBDA (X) (+ X 1)>
Which I think it's ok. But if I do this:
(church2int (OPAND FALSE FALSE))
I've got:
#<FUNCTION :LAMBDA (Q) (FUNCALL P (FUNCALL Q (FUNCALL P)))>
Where I should have 0. Is there something wrong with my code? Or am I missing something?
Thanks
If you want to define opand as a function with 2 parameters, like you are trying to, you need to do this:
(defun OPAND (p q)
(funcall (funcall p q) p) )
and then:
(opand false false)
#<FUNCTION :LAMBDA (X) #'(LAMBDA (Y) Y)> ;; which is FALSE
(opand true true)
#<FUNCTION :LAMBDA (X) #'(LAMBDA (Y) X)> ;; which is TRUE
This is my implementation, based on the original paper http://www.utdallas.edu/~gupta/courses/apl/lambda.pdf, of the and operator λxy.xyF
(defvar OPAND
#'(lambda(x)
#'(lambda(y)
(funcall (funcall x y) FALSE) ) ) )
And if you do
(funcall (funcall opand false) false)
#<FUNCTION :LAMBDA (X) #'(LAMBDA (Y) Y)> ;; which is FALSE
(funcall (funcall opand true) true)
#<FUNCTION :LAMBDA (X) #'(LAMBDA (Y) X)> ;; which is TRUE
Related
I am trying to do the exercises on this tutorial about CLOS using SBCL and Slime (Emacs).
I have this class, instance, and function to set values for the slots:
(defclass point ()
(x y z))
(defvar my-point
(make-instance 'point))
(defun with-slots-set-point-values (point a b c)
(with-slots (x y z) point (setf x a y b z c)))
Using the REPL, it works fine:
CL-USER> (with-slots-set-point-values my-point 111 222 333)
333
CL-USER> (describe my-point)
#<POINT {1003747793}>
[standard-object]
Slots with :INSTANCE allocation:
X = 111
Y = 222
Z = 333
; No value
Now, the exercises indicates that using the symbol-macrolet I need to implement my version of with-slots.
I have a partial implementation of my with-slots (I still need to insert add the operation):
(defun partial-my-with-slots (slot-list object)
(mapcar #'(lambda (alpha beta) (list alpha beta))
slot-list
(mapcar #'(lambda (var) (slot-value object var)) slot-list)))
It works when calling it:
CL-USER> (partial-my-with-slots '(x y z) my-point)
((X 111) (Y 222) (Z 333))
Since this use of symbol-macrolet works:
CL-USER> (symbol-macrolet ((x 111) (y 222) (z 333))
(+ x y z))
666
I tried doing:
CL-USER> (symbol-macrolet (partial-my-with-slots '(x y z) my-point)
(+ x y z))
But, for some reason that I do not know, Slime throws the error:
malformed symbol/expansion pair: PARTIAL-MY-WITH-SLOTS
[Condition of type SB-INT:SIMPLE-PROGRAM-ERROR]
Why does this happen? How can I fix this?
You can't write with-slots as a function which is called at run time. Instead it needs to be a function which takes source code as an argument and returns other source code. In particular if given this argument
(my-with-slots (x ...) <something> <form> ...)
It should return this result:
(let ((<invisible-variable> <something))
(symbol-macrolet ((x (slot-value <invisible-variable>)) ...)
<form> ...))
You need <invisible-variable> so you evaluate <object-form> only once.
Well, here is a function which does most of that:
(defun mws-expander (form)
(destructuring-bind (mws (&rest slot-names) object-form &rest forms) form
(declare (ignore mws))
`(let ((<invisible-variable> ,object-form))
(symbol-macrolet ,(mapcar (lambda (slot-name)
`(,slot-name (slot-value <invisible-variable>
',slot-name)))
slot-names)
,#forms))))
And you can check this:
> (mws-expander '(my-with-slots (x y) a (list x y)))
(let ((<invisible-variable> a))
(symbol-macrolet ((x (slot-value <invisible-variable> 'x))
(y (slot-value <invisible-variable> 'y)))
(list x y)))
So that's almost right, except the invisible variable really needs to be invisible:
(defun mws-expander (form)
(destructuring-bind (mws (&rest slot-names) object-form &rest forms) form
(declare (ignore mws))
(let ((<invisible-variable> (gensym)))
`(let ((,<invisible-variable> ,object-form))
(symbol-macrolet ,(mapcar (lambda (slot-name)
`(,slot-name (slot-value ,<invisible-variable>
',slot-name)))
slot-names)
,#forms)))))
And now:
> (mws-expander '(my-with-slots (x y) a (list x y)))
(let ((#:g1509 a))
(symbol-macrolet ((x (slot-value #:g1509 'x))
(y (slot-value #:g1509 'y)))
(list x y)))
Well, a function which takes source code as an argument and returns other source code is a macro. So, finally, we need to install this function as a macroexpander, arranging to ignore the second argument that macro functions get:
(setf (macro-function 'mws)
(lambda (form environment)
(declare (ignore environment))
(mws-expander form)))
And now:
> (macroexpand '(mws (x y) a (list x y)))
(let ((#:g1434 a))
(symbol-macrolet ((x (slot-value #:g1434 'x)) (y (slot-value #:g1434 'y)))
(list x y)))
This would be more conventionally written using defmacro, of course:
(defmacro mws ((&rest slot-names) object-form &rest forms)
(let ((<invisible-variable> (gensym)))
`(let ((,<invisible-variable> ,object-form))
(symbol-macrolet ,(mapcar (lambda (slot-name)
`(,slot-name (slot-value ,<invisible-variable> ',slot-name)))
slot-names)
,#forms))))
However the two definitions are equivalent (modulo needing some eval-whenery to make the first work properly with the compiler).
You need to return expressions that will call slot-value when substituted into the macro expansion, rather than calling the function immediately. Backquote is useful for this.
(defun partial-my-with-slots (slot-list object)
(mapcar #'(lambda (alpha beta) (list alpha beta))
slot-list
(mapcar #'(lambda (var) `(slot-value ,object ',var)) slot-list)))
> (partial-my-with-slots '(x y z) 'my-point)
((x (slot-value my-point 'x)) (y (slot-value my-point 'y)) (z (slot-value my-point 'z)))
You use this in your with-slots macro like this:
(defmacro my-with-slots ((&rest slot-names) instance-form &body body)
`(symbol-macrolet ,(partial-my-with-slots slot-names instance-form)
,#body))
> (macroexpand '(my-with-slots (x y z) point (setf x a y b z c)))
(SYMBOL-MACROLET ((X (SLOT-VALUE POINT 'X))
(Y (SLOT-VALUE POINT 'Y))
(Z (SLOT-VALUE POINT 'Z)))
(SETF X A
Y B
Z C))
I have the following function in lisp:
(defun F(F)
#'(lambda (n)
(if (zerop n)
(funcall F n)
(1+ (funcall (F F) (1- n))))))
How does this code behaves if I call:
(funcall (F #'(lambda (x) (+ 2 x))) 2)
I dont understand why the output is 4.
Thanks in advance
Since we know the argument, we can simplify the if statement in the function:
(funcall (F #'(lambda (x) (+ 2 x))) 2)
(1+ (funcall (F #'(lambda (x) (+ 2 x))) 1))
(1+ (1+ (funcall #'(lambda (x) (+ 2 x)) 0)))
(1+ (1+ 2))
4
The first 2 transformations replace (if false A B) with B, while the 3rd replaces (if true A B) with A.
First, untangle the two F:
(defun foo (fun)
#'(lambda (n)
(if (zerop n)
(funcall fun n)
(1+ (funcall (foo fun) (1- n))))))
Now, you call:
(funcall (foo #'(lambda (x) (+ 2 x))) 2)
We can give the inner lambda a name, I'll call it add-2.
(funcall (foo #'add-2) 2)
(Foo #'add-2) then returns the function
(lambda (n)
(if (zerop n)
(funcall #'add-2 n) ; n is always 0 here
(1+ (funcall (foo #'add-2) (1- n)))))
This gets called with 2, which is not zerop, so it is:
(1+ (funcall (foo #'add-2) 1))
We already know what (foo #'add-2) returns, and it gets called with 1 now, which still is not zerop:
(1+ (1+ (funcall (foo #'add-2) 0)))
Now the argument is 0, so we get to the base case:
(1+ (1+ (funcall #'add-2 0)))
We now can see that foo creates a function that adds n to the result of calling (fun 0).
This program produces an error:
define: unbound identifier;
also, no #%app syntax transformer is bound in: define
When pasted into the REPL (to be exact, the last line: (displayln (eval-clause clause state))), it works. When run in definition window, it fails. I don't know why.
#lang racket
(define *state* '((a false) (b true) (c true) (d false)))
(define *clause* '(a (not b) c))
(define (eval-clause clause state)
(for ([x state])
(eval `(define ,(first x) ,(second x))))
(eval (cons 'or (map eval clause))))
(displayln (eval-clause *clause* *state*))
This too:
(define (eval-clause clause state)
(eval `(let ,state ,(cons 'or clause))))
produces
let: unbound identifier;
also, no #%app syntax transformer is bound in: let
This was my attempt to translate the following Common Lisp program: Common Lisp wins here?
; (C) 2013 KIM Taegyoon
; 3-SAT problem
; https://groups.google.com/forum/#!topic/lisp-korea/sVajS0LEfoA
(defvar *state* '((a nil) (b t) (c t) (d nil)))
(defvar *clause* '(a (not b) c))
(defun eval-clause (clause state)
(dolist (x state)
(set (car x) (nth 1 x)))
(some #'identity (mapcar #'eval clause)))
(print (eval-clause *clause* *state*))
And in Paren:
(set *state* (quote ((a false) (b false) (c true) (d false))))
(set *clause* (quote (a (! b) c)))
(defn eval-clause (clause state)
(for i 0 (dec (length state)) 1
(set x (nth i state))
(eval (list set (nth 0 x) (nth 1 x))))
(eval (cons || clause)))
(eval-clause *clause* *state*)
eval is tricky in Racket. As per Racket Guide, 15.1.2, you need to hook into the current namespace as follows
(define-namespace-anchor anc)
(define ns (namespace-anchor->namespace anc))
and then add ns to every call to eval:
(define (eval-clause clause state)
(for ([x state])
(eval `(define ,(first x) ,(second x)) ns))
(eval (cons 'or (map (curryr eval ns) clause)) ns))
Note that this is not necessary in the REPL, as explained in the document referenced above.
However, it's probably a better idea to create a specific namespace for your definitions so that they don't get mixed up with your own module's definitions:
(define my-eval
(let ((ns (make-base-namespace)))
(lambda (expr) (eval expr ns))))
(define *state* '((a #f) (b #t) (c #t) (d #f)))
(define *clause* '(a (not b) c))
(define (eval-clause clause state)
(for ([x state])
(my-eval `(define ,(first x) ,(second x))))
(my-eval (cons 'or (map my-eval clause))))
(displayln (eval-clause *clause* *state*))
or, if you want to continue using true and false from racket/bool, define my-eval as follows;
(define my-eval
(let ((ns (make-base-namespace)))
(parameterize ((current-namespace ns))
(namespace-require 'racket/bool))
(lambda (expr) (eval expr ns))))
I would write the Common Lisp version slightly simpler:
(defun eval-clause (clause state)
(loop for (var value) in state
do (set var value))
(some #'eval clause))
The LOOP form is more descriptive (since we can get rid of CAR and NTH) and EVAL can be directly used in the SOME function.
The problem with flet is that the functions bound therein must be defined inline. In other words, there's no way to do this:
(new-flet ((a (lambda (f x)
(funcall f (* x 2))))
(b (function-generator)))
(a #'b 10))
I considered defining such a macro myself, but the problem is that flet seems to be the only way to set local function values. symbol-function always gets the global definition only, and function can't be used with setf. Anyone have an idea how this can be done fairly cleanly, if at all?
You can easily build a trampoline
(defun function-generator (x)
(lambda (y) (* x y)))
(let ((fg (function-generator 42)))
(flet ((a (f x) (funcall f (* x 2)))
(b (x) (funcall fg x)))
(a #'b 10)))
A macro implementation of new-flet with this approach is
(defmacro new-flet (bindings &body body)
(let ((let-bindings (list))
(flet-bindings (list))
(args (gensym)))
(dolist (binding bindings)
(let ((name (gensym)))
(push `(,name ,(second binding))
let-bindings)
(push `(,(first binding) (&rest ,args)
(apply ,name ,args))
flet-bindings)))
`(let ,(nreverse let-bindings)
(flet ,(nreverse flet-bindings)
,#body))))
that expands in your example case as
(macroexpand-1 '(new-flet ((a (lambda (f x) (funcall f (* x 2))))
(b (function-generator)))
(a #'b 10)))
==> (LET ((#:G605 (LAMBDA (F X)
(FUNCALL F (* X 2))))
(#:G606 (FUNCTION-GENERATOR)))
(FLET ((A (&REST #:G604)
(APPLY #:G605 #:G604))
(B (&REST #:G604)
(APPLY #:G606 #:G604)))
(A #'B 10)))
Is
(let* ((a (lambda (f x) (funcall f (* x 2))))
(b (function-generator)))
(funcall a b 10))
a fairly clean solution to your problem?
How about binding the variables with let, so that they're setfable, and then using an flet as the body of the let so that they're funcallable and (function …)-able, too. E.g., where I've given a silly little function instead of (generate-function):
(let ((a (lambda (f x)
(funcall f (* x 2))))
(b (lambda (&rest args)
(print (list* 'print-from-b args)))))
(flet ((a (&rest args)
(apply a args))
(b (&rest args)
(apply b args)))
(a #'b 10)))
We can wrap this up in a macro relatively easily:
(defmacro let/flet (bindings &body body)
(let ((args (gensym (string '#:args-))))
`(let ,bindings
(flet ,(loop :for (name nil) :in bindings
:collect `(,name (&rest ,args) (apply ,name ,args)))
,#body))))
Now
(let/flet ((a (lambda (f x)
(funcall f (* x 2))))
(b (lambda (&rest args)
(print (list* 'print-from-b args)))))
(a #'b 10))
expands into the first block of code. Note that you can also use (a b 10) in the body as well, since the binding of b is the same as the value of #'b. You can use setf on the variable as well:
(let/flet ((a (lambda (x)
(print (list 'from-a x)))))
(a 23)
(setf a (lambda (x)
(print (list 'from-new-a x x))))
(a 23))
prints
(FROM-A 23)
(FROM-NEW-A 23 23)
If anyone's interested in a labels equivalent, here it is:
(defmacro my-labels ((&rest definitions) &rest body)
(let ((gensyms (loop for d in definitions collect (gensym)))
(names (loop for d in definitions collect (car d)))
(fdefs (loop for f in definitions collect (cadr f)))
(args (gensym)))
`(let (,#(loop for g in gensyms collect (list g)))
(labels (,#(loop for g in gensyms for n in names
collect `(,n (&rest ,args) (apply ,g ,args))))
,#(loop for g in gensyms for f in fdefs
collect `(setf ,g ,f))
,#body))))
This is sort of like Scheme's letrec.
I'm trying to implement a Division function with clisp Lambda Calc. style
I read from this site that lambda expression of a division is:
Y (λgqab. LT a b (PAIR q a) (g (SUCC q) (SUB a b) b)) 0
These are TRUE and FALSE
(defvar TRUE #'(lambda(x)#'(lambda(y)x)))
(defvar FALSE #'(lambda(x)#'(lambda(y)y)))
These are conversion functions between Int and Church numbers
(defun church2int(numchurch)
(funcall (funcall numchurch #'(lambda (x) (+ x 1))) 0)
)
(defun int2church(n)
(cond
((= n 0) #'(lambda(f) #'(lambda(x)x)))
(t #'(lambda(f) #'(lambda(x) (funcall f
(funcall(funcall(int2church (- n 1))f)x))))))
)
This is my IF-THEN-ELSE Implementation
(defvar IF-THEN-ELSE
#'(lambda(c)
#'(lambda(x)
#'(lambda(y)
#'(lambda(acc1)
#'(lambda (acc2)
(funcall (funcall (funcall (funcall c x) y) acc1) acc2))))))
)
And this is my div implementation
(defvar division
#'(lambda (g)
#'(lambda (q)
#'(lambda (a)
#'(lambda (b)
(funcall (funcall (funcall (funcall (funcall IF-THEN-ELSE LT) a) b)
(funcall (funcall PAIR q)a))
(funcall (funcall g (funcall succ q)) (funcall (funcall sub a)b))
)))))
)
PAIR, SUCC and SUB functions work fine. I set my church numbers up like this
(set six (int2church 6))
(set two (int2church 2))
Then I do:
(setq D (funcall (funcall division six) two))
And I've got:
#<FUNCTION :LAMBDA (A)
#'(LAMBDA (B)
(FUNCALL (FUNCALL (FUNCALL (FUNCALL (FUNCALL IF-THEN-ELSE LT) A) B) (FUNCALL (FUNCALL PAR Q) A))
(FUNCALL (FUNCALL G (FUNCALL SUCC Q)) (FUNCALL (FUNCALL SUB A) B))))>
For what I understand, this function return a Church Pair. If I try to get the first element
with a function FRST (FRST works ok) like this:
(funcall frst D)
I've got
#<FUNCTION :LAMBDA (B)
(FUNCALL (FUNCALL (FUNCALL (FUNCALL (FUNCALL IF-THEN-ELSE LT) A) B) (FUNCALL (FUNCALL PAR Q) A))
(FUNCALL (FUNCALL G (FUNCALL SUCC Q)) (FUNCALL (FUNCALL SUB A) B)))>
If I try to get the int value with Church2int (Church2int works OK) like this:
(church2int (funcall frst D))
I've got
*** - +:
#<FUNCTION :LAMBDA (N)
#'(LAMBDA (F)
#'(LAMBDA (X)
(FUNCALL (FUNCALL (FUNCALL N #'(LAMBDA (G) #'(LAMBDA (H) (FUNCALL H (FUNCALL G F))))) #'(LAMBDA (U) X)) (LAMBDA (U) U))))>
is not a number
Where I expect to get 3
I think the problem is in DIVISION function, after the IF-THEN-ELSE, I tried to change it a little bit (I thought it was a nested parenthesis problem) but I got lots of errors.
Any help would be appreciated
Thanks
There are several problems with your definition.
DIVISION does not use the Y combinator, but the original definition does.
This is important, because the DIVISION function expects a copy of itself in the g
parameter.
However, even if you added the Y invocation, your code would still not work
but go into an infinite loop instead. That's because Common Lisp, like most of today's languages, is a call-by-value language. All arguments are evaluated before a function is called. This means that you cannot define conditional functions as elegantly as the traditional lambda calculus semantics would allow.
Here's one way of doing church number division in Common Lisp. I've taken the liberty of introducing some syntax to make this a bit more readable.
;;;; -*- coding: utf-8 -*-
;;;; --- preamble, define lambda calculus language
(cl:in-package #:cl-user)
(defpackage #:lambda-calc
;; note: not using common-lisp package
(:use)
(:export #:λ #:call #:define))
;; (lambda-calc:λ (x y) body)
;; ==> (cl:lambda (x) (cl:lambda (y) body))
(defmacro lambda-calc:λ ((arg &rest more-args) body-expr)
(labels ((rec (args)
(if (null args)
body-expr
`(lambda (,(car args))
(declare (ignorable ,(car args)))
,(rec (cdr args))))))
(rec (cons arg more-args))))
;; (lambda-calc:call f a b)
;; ==> (cl:funcall (cl:funcall f a) b)
(defmacro lambda-calc:call (func &rest args)
(labels ((rec (args)
(if (null args)
func
`(funcall ,(rec (cdr args)) ,(car args)))))
(rec (reverse args))))
;; Defines top-level lexical variables
(defmacro lambda-calc:define (name value)
(let ((vname (gensym (princ-to-string name))))
`(progn
(defparameter ,vname nil)
(define-symbol-macro ,name ,vname)
(setf ,name
(flet ((,vname () ,value))
(,vname))))))
;; Syntax: {f a b}
;; ==> (lambda-calc:call f a b)
;; ==> (cl:funcall (cl:funcall f a) b)
(eval-when (:compile-toplevel :load-toplevel :execute)
(set-macro-character #\{
(lambda (stream char)
(declare (ignore char))
`(lambda-calc:call
,#(read-delimited-list #\} stream t))))
(set-macro-character #\} (get-macro-character #\))))
;;;; --- end of preamble, fun starts here
(in-package #:lambda-calc)
;; booleans
(define TRUE
(λ (x y) x))
(define FALSE
(λ (x y) y))
(define NOT
(λ (bool) {bool FALSE TRUE}))
;; numbers
(define ZERO
(λ (f x) x))
(define SUCC
(λ (n f x) {f {n f x}}))
(define PLUS
(λ (m n) {m SUCC n}))
(define PRED
(λ (n f x)
{n (λ (g h) {h {g f}})
(λ (u) x)
(λ (u) u)}))
(define SUB
(λ (m n) {n PRED m}))
(define ISZERO
(λ (n) {n (λ (x) FALSE) TRUE}))
(define <=
(λ (m n) {ISZERO {SUB m n}}))
(define <
(λ (m n) {NOT {<= n m}}))
(define ONE {SUCC ZERO})
(define TWO {SUCC ONE})
(define THREE {SUCC TWO})
(define FOUR {SUCC THREE})
(define FIVE {SUCC FOUR})
(define SIX {SUCC FIVE})
(define SEVEN {SUCC SIX})
(define EIGHT {SUCC SEVEN})
(define NINE {SUCC EIGHT})
(define TEN {SUCC NINE})
;; combinators
(define Y
(λ (f)
{(λ (rec arg) {f {rec rec} arg})
(λ (rec arg) {f {rec rec} arg})}))
(define IF
(λ (condition if-true if-false)
{{condition if-true if-false} condition}))
;; pairs
(define PAIR
(λ (x y select) {select x y}))
(define FIRST
(λ (pair) {pair TRUE}))
(define SECOND
(λ (pair) {pair FALSE}))
;; conversion from/to lisp integers
(cl:defun int-to-church (number)
(cl:if (cl:zerop number)
zero
{succ (int-to-church (cl:1- number))}))
(cl:defun church-to-int (church-number)
{church-number #'cl:1+ 0})
;; what we're all here for
(define DIVISION
{Y (λ (recurse q a b)
{IF {< a b}
(λ (c) {PAIR q a})
(λ (c) {recurse {SUCC q} {SUB a b} b})})
ZERO})
If you put this into a file, you can do:
[1]> (load "lambdacalc.lisp")
;; Loading file lambdacalc.lisp ...
;; Loaded file lambdacalc.lisp
T
[2]> (in-package :lambda-calc)
#<PACKAGE LAMBDA-CALC>
LAMBDA-CALC[3]> (church-to-int {FIRST {DIVISION TEN FIVE}})
2
LAMBDA-CALC[4]> (church-to-int {SECOND {DIVISION TEN FIVE}})
0
LAMBDA-CALC[5]> (church-to-int {FIRST {DIVISION TEN FOUR}})
2
LAMBDA-CALC[6]> (church-to-int {SECOND {DIVISION TEN FOUR}})
2