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For a course project I got to write a program in lisp.
The program should contain the most important lisp functions, their input and output parameters and maybe optional parameters.
For example: function - first, input - list, output - object (first member of list).
The program should work in 2 different ways:
You give the program the name of a function and it should return the function parameters.
You enter function parameters and if a function with these parameters exists, it should return the name of the function.
My questions:
What would be the right way to approach a task like this in lisp? I think maybe a tree would be a way to handle it? (make a tree with all functions and parameters and then write a program which handles it).
Does anyone have a better idea than that to approach this task? Or some suggestions where / how to start? Or Tutorials containing any info?
At the moment I'm a little lost how to start. Any help you can give would be highly appreciated.
English isn't my first language, so I hope everything is understandable.
Greetings.
First of all take a look to prepare your common lisp development environment. After that I think that you should, investigate:
create functions with defun,
declare types.
and things like that. Ffter that take a look to two common lisp functions:
documentation
describe
Here is a little example:
CL-USER> (defun my-sum (a b) "Add my-sum parameters A and B." (+ a b))
MY-SUM
CL-USER> (my-sum 2 3)
5 (3 bits, #x5, #o5, #b101)
CL-USER> (describe #'my-sum)
#<FUNCTION MY-SUM>
[compiled function]
Lambda-list: (A B)
Derived type: (FUNCTION (T T) (VALUES NUMBER &OPTIONAL))
Documentation:
Add my-sum parameters A and B.
Source form:
(SB-INT:NAMED-LAMBDA MY-SUM
(A B)
"Add my-sum parameters A and B."
(BLOCK MY-SUM (+ A B)))
; No values
CL-USER> (documentation 'my-sum 'function)
"Add my-sum parameters A and B."
CL-USER> (defun my-sum (a b) "Add my-sum parameters A and B." (declare (type fixnum a b)) (+ a b))
WARNING: redefining COMMON-LISP-USER::MY-SUM in DEFUN
MY-SUM
CL-USER> (describe #'my-sum)
#<FUNCTION MY-SUM>
[compiled function]
Lambda-list: (A B)
Derived type: (FUNCTION (FIXNUM FIXNUM)
(VALUES
(INTEGER -9223372036854775808 9223372036854775806)
&OPTIONAL))
Documentation:
Add my-sum parameters A and B.
Source form:
(SB-INT:NAMED-LAMBDA MY-SUM
(A B)
"Add my-sum parameters A and B."
(DECLARE (TYPE FIXNUM A B))
(BLOCK MY-SUM (+ A B)))
; No values
Finally, one last tip to work with strings from the output of describe:
CL-USER> (with-output-to-string (*standard-output*)
(describe #'my-sum))
"#<FUNCTION MY-SUM>
[compiled function]
Lambda-list: (A B)
Derived type: (FUNCTION (FIXNUM FIXNUM)
(VALUES
(INTEGER -9223372036854775808 9223372036854775806)
&OPTIONAL))
Documentation:
Add my-sum parameters A and B.
Source form:
(SB-INT:NAMED-LAMBDA MY-SUM
(A B)
\"Add my-sum parameters A and B.\"
(DECLARE (TYPE FIXNUM A B))
(BLOCK MY-SUM (+ A B)))
"
At face value, the task seems to be the construction of a simple symbolic database in memory, which is searchable in two ways. Entries in the database are understood to be functions. The "output parameters" can probably be understood as one or more return values. These things are not named in ANSI Lisp. A useful interpretation of the task is to give return values symbolic labels anyway. Moreover, we can perhaps use type symbols for the return values as well as parameters. So for instance, a database entry for the cons function might look like:
(cons (t t) cons) ;; function named cons takes two objects, returns a cons
The type t is the supertype of all types in ANSI Lisp; it means "any value".
A list of such records can be put into some global variable. Then we write a function that is perhaps named get-params-by-name such that:
(get-params-by-name 'cons) -> (t t)
and another one: get-names-by-params:
(get-names-by-params '(t t)) -> (cons)
This function returns all the matching functions, as a list. More than one function could have this signature.
The trick is then finding a good representation of optional and rest parameters. It could just be the same notation that the language uses:
(list (&rest t) list) ;; list takes rest arguments of any type, returns list
Since we are only interested in exact matches, we don't have to actually parse the &rest notation. When the user queries by parameter, their query object will be literally (&rest t), in that same syntax.
The equal function can be used to tell whether two lists of symbols are identical:
(equal '(&rest t) '(&rest t)) -> t
(equal '(t t) '(t t)) -> nil
So the exercise is not difficult: just mapping through lists, looking for matches.
(defun get-name-by-params (database params)
(let ((matching-entries (remove-if-not (lambda (entry)
(equal (second entry) params))
database)))
(mapcar #'first matching-entries))) ;; just the names, please
Here, the function takes the database list as a parameter, instead of referring to a global variable. The overall program into which we integrate this can provide alternative interfaces, but this is our low-level lookup function.
Test:
[1]> (get-name-by-params '((cons (t t) cons) (list (&rest t) list)) '(integer string))
NIL
[3]> (get-name-by-params '((cons (t t) cons) (list (&rest t) list)) '(t t))
(CONS)
[4]> (get-name-by-params '((cons (t t) cons) (list (&rest t) list)) '(&rest t))
(LIST)
I'd get clarification from the instructor whether this is the right interpretation of the vague requirements, before the assignment is due.
Given that this is a course project I'm going to provide an incomplete answer, and leave you to fill in the blanks.
What the program should do
My interpretation of what you're being asked to do is to provide a utility which will
given the name of a function return its argument list (called a 'lambda list' below);
given a lambda list return all the functions with that lambda list.
So, first of all you need to decide whether two lambda lists are the same or not. As an example is (x) the same as (y), as a lambda list? Yes, it is: the names of formal parameters only matter in the implementation of a function and you generally won't know them: both of these lambda lists mean 'function of one argument'.
The interestring thing is optional arguments of various kinds: (a &optional b) is clearly not the same as (a), but is the same as (b &optional c) but is it the same as (a &optional (b 1 bp))? In this code I say that yes, it is the same: default values and present parameters for optional arguments don't alter whether lambda lists are the same. That's because very often these are implementation details of functions.
A package
We'll put it into a package so it's clear what the interface is:
(defpackage :com.stackoverflow.lisp.fdesc-search
(:use :cl)
(:export
#:defun/recorded
#:record-function-description
#:clear-recorded-functions
#:name->lambda-list
#:lambda-list->names))
(in-package :com.stackoverflow.lisp.fdesc-search)
Recording information
So, to start with we need a mechanism of recording information about functions. We'll do this with a macro which is like defun but records information, which I'll call defun/recorded. We want to be able to record information about things even before the program exists & we do this by having defun/recorded stash 'pending' records on a list which, once the program exists, it will pull off and record properly. That lets us use defun/recorded throughout this code.
;;; These define whether there is a recorder, and if not where pending
;;; records should be stashed
;;;
(defvar *function-description-recorder* nil)
(defvar *pending-function-records* '())
(defmacro defun/recorded (name lambda-list &body forms)
"Like DEFUN but record function information."
;; This deals with bootstrapping by, if there is not yet a recording
;; function, stashing pending records in *PENDING-FUNCTION-RECORDS*,
;; which gets replayed into the recorder at the point it becomes
;; available.
`(progn
;; do the DEFUN first, which ensures that the LAMBDA-LIST is OK
(defun ,name ,lambda-list ,#forms)
(if *function-description-recorder*
(progn
(dolist (p (reverse *pending-function-records*))
(funcall *function-description-recorder*
(car p) (cdr p)))
(setf *pending-function-records* '())
(funcall *function-description-recorder*
',name ',lambda-list))
(push (cons ',name ',lambda-list)
*pending-function-records*))
',name))
Matching lambda lists, first steps
Now we want to be able to match lambda lists. Since we're obviously going to store things indexed by lambda list in some kind of tree we only really need to be able to deal with matching elements of them. And (see above) we don't care about things like default values. I've chosen to do this by first of all simplifying lambda lists to remove them and then matching the simplifies elements: there are other approaches.
simplify-lambda-list does the simplification and argument-matches-p tells you if two arguments match: the interesting bit is that it needs to know about lambda list keywords, which must match exactly, while everything else matches anything. The lambda-list-keywords constant is conveniently provided by the CL standard.
(defun/recorded simplify-lambda-list (ll)
;; Simplify a lambda list by replacing optional arguments with inits
;; by their names. This does not validate the list
(loop for a in ll
collect (etypecase a
(symbol a)
(list (first a)))))
(defun/recorded argument-matches-p (argument prototype)
;; Does an argument match a prototype.
(unless (symbolp argument)
(error "argument ~S isn't a symbol" argument))
(unless (symbolp prototype)
(error "prototype ~S isn't a symbol" prototype))
(if (find-if (lambda (k)
(or (eq argument k) (eq prototype k)))
lambda-list-keywords)
(eq argument prototype)
t))
Function descriptions (partial)
Information about functions is stored in objects called fdescs: the definition of these objects is not given here, but one question we need to answer is 'do two fdescs refer to versions of the same function?' Well, they do if the names of the functions are the same. Remember that function names do not have to be symbols ((defun (setf x) (...) ...) is allowed), so we must compare with equal not eql:
(defun/recorded fdescs-equivalent-p (fd1 fd2)
;; do FD1 & FD2 refer to the same function?
(equal (fdesc-name fd1)
(fdesc-name fd2)))
Storing fdescs indexed by lambda list (partial)
To index things efficiently by lambda list we build a tree. The nodes in this tree are called lambda-list-tree-nodes and their definition is not given here.
There are functions which intern a fdesc in a tree, and which return a list of fdescs indexed by a given lambda list. Neither have an implementation here, but this is what they look like:
(defun/recorded intern-lambda-list (lambda-list tree-node fdesc)
;; return the node where it was interned
...)
(defun/recorded lambda-list-fdescs (lambda-list tree-node)
;; Return a list of fdescs for a lambda list & T if there were any
;; or NIL & NIL if there were not (I don't think () & T is possible,
;; but it might be in some future version)
...)
The implementation of these functions will probably need to use use argument-matches-p and fdescs-equivalent-p.
The top-level databases (slightly partial)
Now we can define the top-level database objects: the root of the tree for indexing by lambda list, and a hashtable for indexing by name
(defvar *lambda-list-tree* (make-lambda-list-tree-node))
(defvar *tree-nodes-by-name* (make-hash-table :test #'equal))
Note that *tree-nodes-by-name* maps from names to the node where the information about that function is stored: that's done to make redefinition easier, as seen in the following function:
(defun/recorded record-function-description (name lambda-list)
"Record information about a function called NAME with lambda list LAMBDA-LIST.
Replace any existing information abot NAME. Return NAME."
(let ((fdesc (make-fdesc :name name :lambda-list lambda-list)))
;; First of all remove any existing information
(multiple-value-bind (node foundp) (gethash name *tree-nodes-by-name*)
(when foundp
(setf (lambda-list-tree-node-values node)
(delete fdesc (lambda-list-tree-node-values node)
:test #'fdescs-equivalent-p))))
(setf (gethash name *tree-nodes-by-name*)
(intern-lambda-list lambda-list *lambda-list-tree* fdesc)))
name)
Note that this function first of all looks up any existing information for name, and if it exists it removes it from the node where it was found. This makes sure that function redefinition does not leave obsolete information in the tree.
This function is the actual recorder which defun/recorded wants to know about, so tell it that:
(setf *function-description-recorder*
#'record-function-description)
Now the next time we invoke defun/recorded it will bootstrap the system by inserting all the the pending definitions.
record-function-description is part of the API to the package: it can be used to record information about functions we don't define.
User-interface functions
Apart from defun/recorded & record-function-description we want some functions which let us make enquiries into the database, as well as one which resets things:
(defun/recorded clear-recorded-functions ()
"Clear function description records. Return no values"
(setf *lambda-list-tree* (make-lambda-list-tree-node)
*tree-nodes-by-name* (make-hash-table :test #'equal))
(values))
(defun/recorded name->lambda-list (name)
"Look up a function by name.
Return either its lambda list & T if it is found, or NIL & NIL if not."
(multiple-value-bind (node foundp) (gethash name *tree-nodes-by-name*)
(if foundp
(values
(fdesc-lambda-list
(find-if (lambda (fd)
(equal (fdesc-name fd) name))
(lambda-list-tree-node-values node)))
t)
(values nil nil))))
(defun/recorded lambda-list->names (lambda-list)
"find function names matching a lambda-list.
Return a list of name & T if there are any, or NIL & NIL if none.
Note that lambda lists are matched so that argument names do not match, and arguments with default values or presentp parameters match just on the argument."
(multiple-value-bind (fdescs foundp) (lambda-list-fdescs lambda-list
*lambda-list-tree*)
(if foundp
(values (mapcar #'fdesc-name fdescs) t)
(values nil nil))))
And that's it.
Examples
After compiling, loading & using the package (with the missing bits added) we can first inject some useful extra functions into it (this is just a random scattering)
> (dolist (x '(car cdr null))
(record-function-description x '(thing)))
nil
> (dolist (x '(car cdr))
(record-function-description `(setf ,x) '(new thing)))
nil
> (record-function-description 'cons '(car cdr))
cons
> (record-function-description 'list '(&rest args))
Now we can make some enquiries:
> (lambda-list->names '(x))
(null cdr
car
lambda-list->names
name->lambda-list
com.stackoverflow.lisp.fdesc-search::simplify-lambda-list)
t
> (lambda-list->names '(&rest anything))
(list)
t
> (name->lambda-list 'cons)
(car cdr)
t
An example of storing things in trees
Below is some code which demonstrates one approach to storing information in trees (often known as tries). This is not usable above for a lot of reasons, but reading it might help implement the missing parts.
;;;; Storing things in trees of nodes
;;;
;;; Node protocol
;;;
;;; Nodes have values which may or may not be bound, and which may be
;;; assigned. Things may be interned in (trees of) nodes with a
;;; value, and the value associated with a thing may be retrieved
;;; along with an indicator as to whether it is present in the tree
;;; under the root.
;;;
(defgeneric node-value (node)
;; the immediate value of a node
)
(defgeneric (setf node-value) (new node)
;; Set the immediate value of a node
)
(defgeneric node-value-boundp (node)
;; Is a node's value bound?
)
(defgeneric intern-thing (root thing value)
;; intern a thing in a root, returning the value
(:method :around (root thing value)
;; Lazy: this arround method just makes sure that primary methods
;; don't need to beother returning the value
(call-next-method)
value))
(defgeneric thing-value (root thing)
;; return two values: the value of THING in ROOT and T if is it present, or
;; NIL & NIL if not
)
;;; Implementatation for STRING-TRIE-NODEs, which store strings
;;;
;;; The performance of these will be bad if large numbers of strings
;;; with characters from a large alphabet are stored: how might you
;;; fix this without making the nodes enormous?
;;;
(defclass string-trie-node ()
;; a node in a string trie. This is conceptually some kind of
;; special case of an abstract 'node' class, but that doesn't
;; actually exist.
((children-map :accessor string-trie-node-children-map
:initform '())
(value :accessor node-value)))
(defmethod node-value-boundp ((node string-trie-node))
(slot-boundp node 'value))
(defmethod intern-thing ((root string-trie-node) (thing string) value)
;; intern a string into a STRING-TRIE-NODE, storing VALUE
(let ((pmax (length thing)))
(labels ((intern-loop (node p)
(if (= p pmax)
(setf (node-value node) value)
(let ((next-maybe (assoc (char thing p)
(string-trie-node-children-map node)
:test #'char=)))
(if next-maybe
(intern-loop (cdr next-maybe) (1+ p))
(let ((next (cons (char thing p)
(make-instance (class-of node)))))
(push next (string-trie-node-children-map node))
(intern-loop (cdr next) (1+ p))))))))
(intern-loop root 0))))
(defmethod thing-value ((root string-trie-node) (thing string))
;; Return the value associated with a string in a node & T or NIL &
;; NIL if there is no value for this string
(let ((pmax (length thing)))
(labels ((value-loop (node p)
(if (= p pmax)
(if (node-value-boundp node)
(values (node-value node) t)
(values nil nil))
(let ((next (assoc (char thing p)
(string-trie-node-children-map node)
:test #'char=)))
(if next
(value-loop (cdr next) (1+ p))
(values nil nil))))))
(value-loop root 0))))
;;; Draw node trees in LW
;;;
#+LispWorks
(defgeneric graph-node-tree (node))
(:method ((node string-trie-node))
(capi:contain
(make-instance 'capi:graph-pane
:roots `((nil . ,node))
:children-function (lambda (e)
(string-trie-node-children-map (cdr e)))
:edge-pane-function (lambda (pane parent child)
(declare (ignore pane parent))
(make-instance
'capi:labelled-line-pinboard-object
:text (format nil "~A" (car child))))
:print-function (lambda (n)
(let ((node (cdr n)))
(format nil "~A"
(if (node-value-boundp node)
(node-value node)
""))))))))
I'm new to macros and struggling with a requirement in a macro of JSON-RPC.
It is asking for a type and I don't know how to enter it correctly.
(defmacro defun-json-rpc (name type lambda-list &body body)
"Defines a function and registers it as a json-rpc target."
(unless (json-rpc-encoding-p type)
(error "New version of defun-json-rpc requires a TYPE argument"))
`(progn
(defun ,name ,lambda-list ,#body)
(export-as-json-rpc ',name (lisp-to-camel-case (symbol-name ',name)) ,type)))
A piece of sample code I found follows, but it does not contain the type argument:
(json-rpc:defun-json-rpc add (x y)
(+ x y))
How would I enter the type?
Found it. It is a JSON-RPC specific keyword, not a lisp type. I just needed to look at json-rpc-encoding-p to see that only 3 keywords are valid.
(defgeneric json-rpc-encoding-p (keyword)
(:documentation "Is KEYWORD a valid JSON-RPC value encoding?")
(:method (keyword)
"Default is no."
(declare (ignore keyword))
nil)
;;; built-in methods
(:method ((keyword (eql :guessing)))
t)
(:method ((keyword (eql :streaming)))
t)
(:method ((keyword (eql :explicit)))
t))
Is it possible to remove the duplicates (as in nub) from a list of functions in Haskell?
Basically, is it possible to add an instance for (Eq (Integer -> Integer))
In ghci:
let fs = [(+2), (*2), (^2)]
let cs = concat $ map subsequences $ permutations fs
nub cs
<interactive>:31:1:
No instance for (Eq (Integer -> Integer))
arising from a use of `nub'
Possible fix:
add an instance declaration for (Eq (Integer -> Integer))
In the expression: nub cs
In an equation for `it': it = nub cs
Thanks in advance.
...
Further, based on larsmans' answer, I am now able to do this
> let fs = [AddTwo, Double, Square]
> let css = nub $ concat $ map subsequences $ permutations fs
in order to get this
> css
[[],[AddTwo],[Double],[AddTwo,Double],[Square],[AddTwo,Square],[Double,Square],[AddTwo,Double,Square],[Double,AddTwo],[Double,AddTwo,Square],[Square,Double],[Square,AddTwo],[Square,Double,AddTwo],[Double,Square,AddTwo],[Square,AddTwo,Double],[AddTwo,Square,Double]]
and then this
> map (\cs-> call <$> cs <*> [3,4]) css
[[],[5,6],[6,8],[5,6,6,8],[9,16],[5,6,9,16],[6,8,9,16],[5,6,6,8,9,16],[6,8,5,6],[6,8,5,6,9,16],[9,16,6,8],[9,16,5,6],[9,16,6,8,5,6],[6,8,9,16,5,6],[9,16,5,6,6,8],[5,6,9,16,6,8]]
, which was my original intent.
No, this is not possible. Functions cannot be compared for equality.
The reason for this is:
Pointer comparison makes very little sense for Haskell functions, since then the equality of id and \x -> id x would change based on whether the latter form is optimized into id.
Extensional comparison of functions is impossible, since it would require a positive solution to the halting problem (both functions having the same halting behavior is a necessary requirement for equality).
The workaround is to represent functions as data:
data Function = AddTwo | Double | Square deriving Eq
call AddTwo = (+2)
call Double = (*2)
call Square = (^2)
No, it's not possible to do this for Integer -> Integer functions.
However, it is possible if you're also ok with a more general type signature Num a => a -> a, as your example indicates! One naïve way (not safe), would go like
{-# LANGUAGE FlexibleInstances #-}
{-# LANGUAGE NoMonomorphismRestriction #-}
data NumResLog a = NRL { runNumRes :: a, runNumResLog :: String }
deriving (Eq, Show)
instance (Num a) => Num (NumResLog a) where
fromInteger n = NRL (fromInteger n) (show n)
NRL a alog + NRL b blog
= NRL (a+b) ( "("++alog++ ")+(" ++blog++")" )
NRL a alog * NRL b blog
= NRL (a*b) ( "("++alog++ ")*(" ++blog++")" )
...
instance (Num a) => Eq (NumResLog a -> NumResLog a) where
f == g = runNumResLog (f arg) == runNumResLog (g arg)
where arg = NRL 0 "THE ARGUMENT"
unlogNumFn :: (NumResLog a -> NumResLog c) -> (a->c)
unlogNumFn f = runNumRes . f . (`NRL`"")
which works basically by comparing a "normalised" version of the functions' source code. Of course this fails when you compare e.g. (+1) == (1+), which are equivalent numerically but yield "(THE ARGUMENT)+(1)" vs. "(1)+(THE ARGUMENT)" and thus are indicated as non-equal. However, since functions Num a => a->a are essentially constricted to be polynomials (yeah, abs and signum make it a bit more difficult, but it's still doable), you can find a data type that properly handles those equivalencies.
The stuff can be used like this:
> let fs = [(+2), (*2), (^2)]
> let cs = concat $ map subsequences $ permutations fs
> let ncs = map (map unlogNumFn) $ nub cs
> map (map ($ 1)) ncs
[[],[3],[2],[3,2],[1],[3,1],[2,1],[3,2,1],[2,3],[2,3,1],[1,2],[1,3],[1,2,3],[2,1,3],[1,3,2],[3,1,2]]
I read about SAT and SMT. I always wonder how it can be applied in real programming scenario.
Here is an example:
Given that var a = 20; var b = a; are true, we want to know if b = 20 is true or false.
How should I turn it into boolean algebra expressions and apply SAT?
Here is the simplest example using SMT-LIB 2.0 standard supported by many SMT solvers:
(declare-fun a () Int)
(declare-fun b () Int)
(assert (= a b))
(assert (= a 20))
(assert (= b 20))
(check-sat)
You can use http://rise4fun.com/z3 to experiment with it. It will respond "sat" meaning the assertions can be satisfied, meaning then b can be 20.
Then you can replace (assert (= b 20)) with (assert (distinct b 20)). Z3 will respond with "unsat" meaning it's not possible for b to be anything else.
I have function 'my_a' in OCaml, which could have a very complicated return type:
exception Backtrack
exception Continue of (* How do I put the type of function 'my_a' here? *)
let my_a arg = try do_stuff (List.hd arg)
with
| Backtrack -> my_a (List.tl arg)
| Continue (found_answer) -> (try my_a (List.tl arg)
with
| Backtrack -> raise Continue(found_answer)
| Continue (other_answer) ->
raise Continue (compare_answer(found_answer,other_answer));;
(* the caller of my_a will handle the Continue exception to catch the found value
if something was found*)
This is my problem: I'm using backtrack to find a solution. When a backtrack exception is raised by do_stuff, there was no solution going that path. However, when it raises an exception of type Continue, it means it found a solution, but, it may not be the best solution there is, that's when I try again with a different path. If there is another exception, I want to return the answer it already had found.
The thing is, to be able to use that feature of OCaml I need to to tell it what data type Continue will be carrying. What the OCaml top level returns when i define my_a:
'a * ('a -> ('a, 'b) symbol list list) ->
'b list -> ('a * ('a, 'b) symbol list) list * 'b list = <fun>
Does anyone have any idea of how to do that, or a different solution to that?
It's hard to tell exactly what you're asking. I think you might be asking how to get the type inside the Two exception to be set to the return type of A without having to specifically declare this type. I can't think of any way to do it.
Things might go better if you used option types instead of exceptions. Or you can just declare the return type of A explicitly. It might be good documentation.
A couple of side comments: (a) function names have to start with a lower case letter (b) this code looks quite convoluted and hard to follow. There might be a simpler way to structure your computation.
You are gaining nothing by using exceptions. Here is a possible solution.
(** There are many ways to implement backtracking in Ocaml. We show here one
possibility. We search for an optimal solution in a search space. The
search space is given by an [initial] state and a function [search] which
takes a state and returns either
- a solution [x] together with a number [a] describing how good [x] is
(larger [a] means better solution), or
- a list of states that need still to be searched.
An example of such a problem: given a number [n], express it as a sum
[n1 + n2 + ... + nk = n] such that the product [n1 * n2 * ... * nk] is
as large as possible. Additionally require that [n1 <= n2 <= ... <= nk].
The state of the search can be expressed as pair [(lst, s, m)] where
[lst] is the list of numbers in the sum, [s] is the sum of numbers in [lst],
and [m] is the next number we will try to add to the list. If [s = n] then
[lst] is a solution. Otherwise, if [s + m <= n] then we branch into two states:
- either we add [m] to the list, so the next state is [(m :: lst, m+s, m)], or
- we do not add [m] to the list, and the next state is [(lst, s, m+1)].
The return type of [search] is described by the following datatype:
*)
type ('a, 'b, 'c) backtrack =
| Solution of ('a * 'b)
| Branches of 'c list
(** The main function accepts an initial state and the search function. *)
let backtrack initial search =
(* Auxiliary function to compare two optional solutions, and return the better one. *)
let cmp x y =
match x, y with
| None, None -> None (* no solution *)
| None, Some _ -> y (* any solution is better than none *)
| Some _, None -> x (* any solution is better than none *)
| Some (_, a), Some (_, b) ->
if a < b then y else x
in
(* Auxiliary function which actually performs the search, note that it is tail-recursive.
The argument [best] is the best (optional) solution found so far, [branches] is the
list of branch points that still needs to be processed. *)
let rec backtrack best branches =
match branches with
| [] -> best (* no more branches, return the best solution found *)
| b :: bs ->
(match search b with
| Solution x ->
let best = cmp best (Some x) in
backtrack best bs
| Branches lst ->
backtrack best (lst # bs))
in
(* initiate the search with no solution in the initial state *)
match backtrack None [initial] with
| None -> None (* nothing was found *)
| Some (x, _) -> Some x (* the best solution found *)
(** Here is the above example encoded. *)
let sum n =
let search (lst, s, m) =
if s = n then
(* solution found, compute the product of [lst] *)
let p = List.fold_left ( * ) 1 lst in
Solution (lst, p)
else
if s + m <= n then
(* split into two states, one that adds [m] to the list and another
that increases [m] *)
Branches [(m::lst, m+s, m); (lst, s, m+1)]
else
(* [m] is too big, no way to proceed, return empty list of branches *)
Branches []
in
backtrack ([], 0, 1) search
;;
(** How to write 10 as a sum of numbers so that their product is as large as possible? *)
sum 10 ;; (* returns Some [3; 3; 2; 2] *)
OCaml happily informs us that the type of backtrack is
'a -> ('a -> ('b, 'c, 'a) backtrack) -> 'b option
This makes sense:
the first argument is the initial state, which has some type 'a
the second argument is the search function, which takes a state of type 'a and
returns either a Solution (x,a) where x has type 'b and a has type 'c,
or Branches lst where lst has type 'a list.