Recursion vs' Iteration

1
Recursion vs. Iteration

• The original Lisp language was truly a functional
  language
• Everything was expressed as functions
• No local variables
• No iteration
• You had to use recursion to get around these
  problems
• Although they werent considered problems,
  functions are usually described recursively, and
  since the vision of Lisp was to model
  mathematical functions in a language, this was
  appropriate
• But recursion is hard, in CL why should we use
  it?
• Cant we just use iteration?

2
Should we avoid recursive?

• Recursion is hard
• It is hard to conceptualize how a problem can be
  solved recursively
• Once implemented, it is often very difficult to
  debug a recursive program
• When reading recursive code, it is sometimes hard
  to really see how it solves the problem
• Recursion is inefficient
• Every time we recurse, we are doing another
  function call, this results in manipulating the
  run-time stack in memory, passing parameters, and
  transferring control
• So recursion costs us both in time and memory
  usage
• Consider the example on the next slide which
  compares iterative and recursive factorial
  solutions

3
A Simple Comparison
(defun ifact (n) (let ((product 1))
(do ((j 0 ( 1 j))) (( j n))
(setf product ( product ( j 1))))
product)) (defun rfact (n) (if (lt n 1)
1 ( n (rfact (- n 1)))))
Here, the function is called once, there are
two local variables The loop does a
comparison and if the terminating condition is
not yet true, we branch back up to the top
Total instructions n 5 3
Here, we have less code, no local variables (only
a parameter) and fewer total instructions in the
code, each iteration has a comparison and then
either returns 1 or performs 3 function calls
(-, rfact, in that order) But we arent seeing
the stack manipulations which require pushing a
new n, space for the functions return value, and
updating the stack pointer register, and popping
off the return value and n when done
4
Why Recursion?

• If recursion is harder to understand and less
  efficient, why use it?
• It leads to elegant solutions less code, less
  need for local variables, etc
• If we can define a function mathematically, the
  solution is easy to codify
• Some problems require recursion
• Tree traversals
• Graph traversals
• Search problems
• Some sorting algorithms (quicksort, mergesort)
• Note this is not strictly speaking true, we can
  accomplish a solution without recursion by using
  iteration and a stack, but in effect we would be
  simulating recursion, so why not use it?
• In some cases, an algorithm with a recursive
  solution leads to a lesser computational
  complexity than an algorithm without recursion
• Compare Insertion Sort to Merge Sort for example

5
Lisp is Set Up For Recursion

• As stated earlier, the original intention of Lisp
  was to model mathematical functions so the
  language calls for using recursion
• Basic form
• The components here are to test for a base case
  and if true, return the base cases value,
  otherwise recurse passing the function the
  parameter(s) manipulated for the next level

(defun name (params) (if (terminating
condition) return-base-case-value
(name (manipulate params))))
6
What Happens During Recursion

• You should have studied this in 262, but as a
  refresher
• We use the run-time stack to coordinate recursion
• The stack gives us a LIFO access
• Imagine that function1 calls function2 which
  calls function3
• When function3 ends, where do we return to?
• the run-time stack stores the location in
  function2 to return to
• When function2 ends, where do we return to?
• the run-time stack stores the location in
  function1 to return to
• etc
• Using a stack makes it easy to backtrack to the
  proper location when a method ends
• Notice that we want this behavior whether we are
  doing normal function calls or recursion

Run-time stack main calls m1 m1 calls m2 m2
calls m3 m3 calls m4 We are currently in m4
Main m1 m2 m3 m4
stack pointer
7
More On the Run-time Stack

• For each active function, the run-time stack
  stores an activation record instance
• This is a description of the functions execution
  and stores
• Local variables, Parameters, Return value
• Return pointer (where to return to in the calling
  function upon function termination)
• Every time a function (or method in Java) is
  called
• the run-time stack is manipulated by pushing a
  new activation record instance onto it
• proper memory space is allocated on the stack for
  all local variables and parameters
• the return pointer is set up
• the stack pointer register is adjusted
• Every time a function terminates
• the run-time stack has the top activation record
  instance popped off of it, returning the value
  that the function returns
• the PC (program counter register) is adjusted to
  the proper location in the calling function
• the stack pointer register is adjusted

8
An Example
(defun fact (n) (if (lt n 1) 1 ( n
(fact (- n 1))))) The activation record instance
(AR) for fact stores three things, n, the return
value, and the pointer of where to return to
in the next AR when fact terminates We start
with (fact 3)
AR for factorial n 3 return value ___
return to interpreter
AR for factorial n 1 return value ___
return to (fact 2)
AR for factorial n 2 return value ___
return to (fact 3)
AR for factorial n 2 return value ___
return to (fact 3)
AR for factorial n 3 return value ___
return to interpreter
AR for factorial n 3 return value ___
return to interpreter
9
Example Continued
AR for factorial n 1 return value 1 return
to (fact 2)
AR for factorial n 2 return value 2 return
to (fact 3)
AR for factorial n 2 return value return
to (fact 3)
AR for factorial n 3 return value ___
return to interpreter
AR for factorial n 3 return value 6 return
to interpreter
AR for factorial n 3 return value ___
return to interpreter
6 is returned and printed in the interpreter
10
Lisp Makes Recursion Easy

• Well, strictly speaking, recursion in Lisp is
  similar to recursion in any language
• What Lisp can do for us is give us easy access to
  the debugger
• You can insert a (break) instruction which forces
  the evaluation step of the REPL cycle to stop
  executing, leaving us in the debugger
• Or, if you have a run-time error, you are
  automatically placed into the debugger
• From the debugger you can
• inspect the run-time stack to see what values are
  there
• return to a previous level of recursive call
• provide a value to be returned
• Thus, you can either determine
• why you got an error by inspecting the stack
• see what is going on in the program by inspecting
  the stack
• return from an error by inserting a partial or
  complete solution
• CL can also make a recursive program more
  efficient (to be explained later)

11
Examples of Recursive Code

• Every List function in CL can be implemented
  recursively
• whether they are or not Im not sure, but
  probably they are
• Here we start with 3 versions of last

(defun last (lis) (cond ((null (cdr lis))
(car lis)) (t (last (cdr
lis))))) (defun last2 (lis) (cond
((and (listp lis) ( (length lis) 1)) lis)
(t (last2 (cdr lis))))) (defun last3 (lis)
(cond ((atom lis) (list lis)) ((and
(listp lis) ( (length lis) 1)) lis)
(t (last3 (cdr lis)))))
The top definition returns an atom, the bottom
definition can handle atoms and Lists, CLs last
is probably last2
12
Butlast With and Without Recursion
(defun butlast1 (lis) (cdr (reverse (cdr
lis)))) (defun mybutlast (lis) (cond ((null
(cdr lis)) nil) (t (cons (car lis)
(mybutlast (cdr lis)))))) (defun reverse1 (lis)
(let (temp (size (length lis))) (dotimes
(a size) (setf temp
(append temp
(list (nth (- (- size a) 1) lis)))))
temp) (defun reverse2 (lis) (if (null
lis) nil (append (reverse2 (cdr
lis)) (list (car lis)))))
The iterative version of reverse builds a list
iteratively using a local variable The
recursive version, while being harder to
understand, contains far less code
13
Member

• In actuality, member does not work as indicated
  here because member only tests top level items
  using eql instead of equal
• So
• (member2 '(1 2) '(1 (1 2) 2)) returns ((1 2) 2)
• While
• (member '(1 2) '(1 (1 2) 2)) returns nil

(defun member1 (a lis) (dotimes (i (length
lis)) (if (equal a (car lis))
(return lis) (setf lis (cdr
lis))))) (defun member2 (a lis) (cond
((null lis) nil) ((equal a (car
lis)) lis) (t (member2 a (cdr
lis)))))
14
Nth and Nthcdr
(defun nth (n lis) (dotimes (a n)
(setf lis (cdr lis))) (car lis)) (defun
nth2 (a lis) (cond (( a 0) (car lis))
(t (nth2 (- a 1) (cdr lis))))) (defun
nth3 (a lis) (cond ((lt a 0) nil)
(( a 0) (car lis)) (t (nth3 (- a
1) (cdr lis)))))
(defun nthcdr1 (n lis) (dotimes (a n)
(setf lis (cdr lis))) lis) (defun
nthcdr2 (n lis) (cond ((lt n 0) nil)
(( n 0) lis) (t
(nthcdr2 (- n 1)
(cdr lis)))))
15
Remove from a List
(defun remove1 (a lis) (let ((temp nil))
(dolist (i lis)
(if (not (equal a i))
(setf temp
(append temp (list i))))) temp)) (defun
remove2 (a lis) (cond ((null lis) nil)
((equal a (car lis)) (remove2 a (cdr
lis))) (t (cons (car lis)
(remove2 a (cdr lis)))))) (defun remove-first (a
lis) (cond ((null lis) nil)
((equal a (car lis)) (cdr lis))
(t (cons (car lis) (remove-first a (cdr lis))))))
What would remove-last look like?
16
Substitute Item In List
(defun subs1 (a b lis) (let ((temp nil))
(dolist (i lis) (if
(equal a i) (setf temp
(append temp (list b)))
(setf temp (append temp (list i)))))
temp)) (defun subs2 (a b lis) (cond ((null
lis) nil) ((equal a (car lis))
(cons b (subs2 a b (cdr lis))))
(t (cons (car lis) (subs2 a b (cdr
lis)))))) (defun sub-first (a b lis) (cond
((null lis) nil) ((equal a (car
lis)) (cons b (cdr lis))) (t
(cons (car lis) (sub-first a b (cdr lis))))))
17
Flattening a List

• Now consider the problem of delistifying a list
• That is, taking all of the items in sublists and
  moving them into the top-level list
• The recursive version is fairly straightforward
• If the parameter is nil, return the empty list
• If the parameter is an atom, return the atom as a
  list
• Otherwise, append what we get back by recursively
  calling this function with the car of the
  parameter (null, an atom, or a list) and the cdr
  of the parameter (null or a list)
• Since sublists may contain subsublists, etc, an
  iterative version would be extremely complicated!

(defun flatten (lis) (cond ((null lis)
nil) ((atom lis) (list lis)) (t (append
(flatten (car lis))
(flatten (cdr lis))))))
(flatten (a (b c (d e) f (g)) ((h) i))) (A B C D
E F G H I)
18
Counting List Items

• We can count the top level items using length
• (length (1 2 3 4)) ? 4 but (length (1 (2 3) 4))
  ? 3
• We can implement a counting function easily
  enough as
• Counting the total number of atoms in a list that
  might contain sublists requires flattening, so we
  instead would do this

(defun countitems (lis) (if (null lis) 0
( 1 (countitems (cdr lis)))))
(defun countallitems (lis) (cond ((null lis)
0) ((atom (car lis)) ( 1 (countallitems
(cdr lis)))) (t ( (countallitems (car lis))
(countallitems (cdr lis))))))
19
Remove All

• We might similarly want to remove all of an atom
  from the lists and sublists of a given list so
  again we turn to flattening

(defun removeall (a lis) (cond ((null lis)
nil) ((equal a (car lis)) (removeall a (cdr
lis))) ((listp (car lis))
(cons (removeall a (car lis))
(removeall a (cdr lis)))) (t (cons (car lis)
(removeall a (cdr lis))))))
(removeall 'a '(a b (a c) (d ((a) b) a) c a))
returns (B (C) (D (NIL B)) C) Notice the nil
inserted into the list because we replace (a)
with nil, can we fix this? If so, how?
20
Towers of Hanoi
Towers of Hanoi with 4 disks Start
Intermediate Final
Partial solution
(defun hanoi (n a b c) (cond (( n 1)
(print (list move n from a to c))
done) used so that the last
message is not repeated as the return value
of the function (t (hanoi (- n 1) a c b)
(print (list move n from a to c))
(hanoi (- n 1) b a c))))
21
Tail Recursion

• When writing recursive code, we typically write
  the recursive function call in a cond statement
  as
• (t (name (manipulate params)))
• If the last thing that this function does is call
  itself, then this is known as tail recursion
• Tail recursion is important because it can be
  implemented more efficiently
• Consider the following implementation of
  factorial, why isnt it tail recursive?

(defun fact (n) (cond ((lt n 1) 1)
(t ( n (fact (- n 1))))))
The last thing fact does is , not fact so this
is not tail recursive!
22
Writing Factorial with Tail Recursion

• If you look carefully, you can see that is done
  after we return from calling fact
• This seems like a necessity because we define
  factorial as f(n) n f(n 1)
• So we must subtract 1, then call f, and then do
  multiplication
• Can we somehow rearrange the code so that is
  not performed last? Yes, by also passing a
  partial product as follows

We call this function as (fact-with-tr n 1)
(defun fact-with-tr (n prod) (cond ((lt n
1) prod) (t (fact-with-tr (- n 1) ( n
prod)))))
The last thing this function does is call
fact-with-tr, not or
23
Why Bother With TR?

• Optimizing compilers available in Common Lisp can
  benefit from tail recursion if they detect it
• Rather than placing multiple Activation Record
  Instances on the stack, a single activation
  record instance is pushed onto the stack when the
  function is called the first time
• For each recursive call, the same activation
  record instance is manipulated
• in this case, n would be decremented and prod
  would be updated
• Since we are guaranteed in any single recursive
  call that we will never need to use the parameter
  again in this call, we can change it
• why are we guaranteed that a parameters value
  wont change in this call?
• And the return location is always to the same
  location in this function
• Once the function terminates, it is popped off
  the stack and we return to the calling functions
  location, or the interpreter
• Note we have a significant problem if an error
  arises and we are dropped into the debugger
  what is that problem?
• Aside from saving on memory usage and a bit of
  run-time memory allocation, this optimization
  doesnt do anything else for us, so we dont
  really have to worry about tail recursion

24
Search Problems

• Lisp was the primary language for AI research
• Many AI problems revolve around searching for an
  answer
• Consider chess you have to make a move, what
  move do you make?
• A computer program must search from all the
  possibilities to decide what move to make
• but you dont want to search by just looking 1
  move ahead
• if you look 2 moves ahead, you dont have twice
  as many possible moves, but the number of
  possible moves2
• if you look 3 moves ahead, number of possible
  moves3
• this can quickly get out of hand
• So we limit our search by evaluating a top-level
  move using a heuristic function
• If the function says dont make this move, we
  dont consider it and dont search any further
  along that path
• If the function says possibly a good move, then
  we recursively search
• by using recursion, we can backup and try
  another route if needed, this is known as
  backtracking