λ λ CS 251 Fall 2019 CS 251 Spring 2020 Principles of Programming Languages Principles of Programming Languages Ben Wood Ben Wood Tail Recursion +tail.rkt https://cs.wellesley.edu/~cs251/s20/ 1 Tail Recursion
Topics Recursion is an elegant and natural match for many computations and data structures. • Natural recursion with immutable data can be space- inefficient compared to loop iteration with mutable data. • Ta Tail recursion eliminates the space inefficiency with a simple, general pattern. • Recursion over immutable data expresses iteration more clearly than loop iteration with mutable state. • More higher-order patterns: fold Tail Recursion 2
Naturally recursive factorial (define (fact n) (if (= n 0) 1 (* n (fact (- n 1))))) Sp Space: e: O( ) How efficient is this implementation? Ti Time: O( ) Tail Recursion 3
CS 240-style machine model Co Code St Stack Re Registers Call frame fixed size, general purpose Call frame Call frame Heap Hea arguments, variables, return address per function call Program Counter cons cells, data structures, … Stack Pointer Tail Recursion 4
Evaluation (define (fact n) example (if (= n 0) 1 (* n (fact (- n 1))))) Call stacks at each step (fact 3) (fact 3): 3*_ (fact 3): 3*_ (fact 3): 3*_ (fact 2) (fact 2): 2*_ (fact 2): 2*_ (fact 1) (fact 1): 1*_ Remember : n ↦ 2 ; and (fact 0) “rest of function” for this call. (fact 3): 3*_ (fact 3): 3*_ (fact 3): 3*_ (fact 3): 3*2 (fact 2): 2*_ (fact 2): 2*_ (fact 2): 2*1 Sp Space: e: O( ) (fact 1): 1*_ (fact 1): 1*1 Ti Time: O( ) (fact 0): 1 Tail Recursion 5
Naturally recursive factorial (define (fact n) Compute result so far (if (= n 0) after/from recursive call. af Base case returns 1 base result. (* n (fact (- n 1))))) Recursive case returns Compute remaining argument result so far. be before/for recursive call. Tail Recursion 6
Tail recursive factorial Accumulator parameter provides result so far. (define (fact n) (define (fact-tail n acc) Compute result so far (if (= n 0) before/for recursive call. be Base case returns acc full result. (fact-tail (- n 1) (* n acc)))) Recursive case returns Compute remaining argument full result. be before/for recursive call. (fact-tail n 1)) Initial accumulator provides base result. Tail Recursion 7
Common patterns of work Natural recursion: Tail recursion: Argument Full result Argument Base result Deeper recursive calls Re Reduce Re Reduce argument argument Accumulate Ac Ac Accumulate result result so far so far Base case Base result Base case Full result Tail Recursion 8
ar argume ment full fu ll result lt Natural (fact 4): 24 recursion -1 * accumulate (fact 3): 6 reduce -1 * Recursive case: Compute result (fact 2): 2 in terms of argument and accumulated recursive result. -1 * (fact 1): 1 -1 * (fact 0): 1 (define (fact n) ba base case ba base result (if (= n 0) 1 (* n (fact (- n 1))))) Tail Recursion 9
argume ar ment base result ba Tail (fact-tail 4 1): 24 recursion -1 * accumulate reduce (fact-tail 3 4): 24 -1 * Recursive case: Compute recursive argument (fact-tail 2 12): 24 in terms of argument and accumulator. -1 * (fact-tail 1 24): 24 -1 * (fact-tail 0 24): 24 (define (fact n) ba base case full fu ll result lt (define (fact-tail n acc) (if (= n 0) acc (fact-tail (- n 1) (* n acc)))) (fact-tail n 1)) Tail Recursion 10
Evaluation example Nothing useful remembered here. Call stacks at each step (fact 3) (fact 3): _ (fact 3): _ (fact 3): _ (ft 3 1) (ft 3 1):_ (ft 3 1):_ (ft 2 3) (ft 2 3):_ ft = fact-tail (ft 1 6) (fact 3): _ (fact 3): _ (fact 3): _ (fact 3): _ (ft 3 1):_ (ft 3 1):_ (ft 3 1):_ (ft 3 1):_ (ft 2 3):_ (ft 2 3):_ (ft 2 3):_ (ft 2 3):6 (ft 1 6):_ (ft 1 6):_ (ft 6 1):6 etc. (ft 0 6) (ft 0 6):6 Tail Recursion 11
Tail-call optimization Space: Sp e: O( ) (define (fact n) (define (fact-tail n acc) (if (= n 0) Ti Time: O( ) acc (fact-tail (- n 1) (* n acc)))) (fact-tail n 1)) (fact 3) (ft 3 1) (ft 2 3) (ft 1 6) (ft 0 6) Language implementation recognizes tail calls. • Caller frame never needed again. • Reuse same space for every recursive tail call. • Low-level: acts just like a loop. Racket, ML, most “functional” languages, but not Java, C, etc. Tail Recursion 12
Ta Tail call intuition: “nothing left for caller to do after call”, Tail position “callee result is immediate caller result” Recursive definition of ta tail il p positio ition: – In (lambda (x1 … xn) e ) , the body e is in tail position. – If If (if e1 e2 e3 ) is is in in tail il posit itio ion , then e2 and e3 are in tail position (but e1 is not). – If If (let ([x1 e1] … [xn en]) e ) is is in in tail il posit itio ion , then e is in tail position (but the binding expressions are not). No Note: • If a non-lambda expression is not in tail position, then no subexpressions are. • Critically, in a function call expression (e1 e2) , not in tail position. subexpressions e1 and e2 are no A ta tail call is a function call in tail position. A function is ta tail-re recursive if it uses a recursive tail call. Tail Recursion 13
Tail recursion transformation Common pattern for transforming naturally recursive functions to tail-recursive form. Works for functions that do commutative operations (order of steps doesn't matter). (define (fact n) na natur ural rec ecur ursion (if (= n 0) 1 (* n (fact (- n 1)) ) )) tail ta il r recursio ion (define (fact n) (define (fact-tail n acc ) (if (= n 0) Ac Accumulator acc becomes be (fact-tail (- n 1) (* n acc ) ))) base result. ba (fact-tail n 1 )) Recursive Re ve step applied to ac accumulat ator in instead d of re recursive re result. Base resul Bas ult becomes init in itia ial l accumula lator. Tail Recursion 14
Practice: use the transformation ;; Naturally recursive sum (define (sum-natural xs) (if (null? xs) 0 (+ (car xs) (sum-natural (cdr xs))))) ;; Tail-recursive sum (define (sum-tail xs) (define (sum-onto xs acc) (if (null? xs) acc (sum-onto (cdr xs) (+ (car xs) acc))) (sum-onto xs 0)) Tail Recursion 15
(order matters) Transforming non-commutative steps (define (reverse-natural-slow xs) (if (null? xs) null (append (reverse-natural-slow (cdr xs)) (list (car xs))))) ✘ (define (reverse-tail-just-kidding xs) (define (rev xs acc) (if (null? xs) acc (rev (cdr xs) (append acc (list (car xs)))))) (rev xs null)) ✓ (define (reverse-tail-slow xs) (define (rev xs acc) (if (null? xs) acc (rev (cdr xs) (append (list (car xs)) acc)))) (rev xs null)) Tail Recursion 16
The transformation is not always ideal. O( ) (define (reverse-tail-slow xs) (define (rev xs acc) (if (null? xs) acc (rev (cdr xs) (append (list (car xs)) acc)))) (rev xs null)) (define (reverse-tail-good xs) O(n) (define (rev xs acc) (if (null? xs) acc (rev (cdr xs) (cons (car xs) acc)))) (rev xs null)) • Append-based recursive reverse is O(n 2 ): each recursive call must traverse to end of list and build a fully new list. – 1+2+…+(n-1) is almost n*n/2 – Moral: beware append, especially within recursion Tail-recursive reverse can avoid append in O(n). What about map, filter? • – Cons is O(1), done n times. Tail Recursion 17
Tail recursion ≠ accumulator pattern ; mutually tail recursive (define (even n) (or (zero? n) (odd (- n 1)))) (define (odd n) (or (not (zero? n)) (even (- n 1)))) ; tail recursive (define (even2 n) (cond [(= 0 n) #t] [(= 1 n) #f] [#t (even2 (- n 2))])) • Tail recursion and the accumulator pattern are co commonl nly y ther . They are no onyms . used toge us geth not sy synon – Natural recursion may use an accumulator. – Tail recursion does not necessarily involve an accumulator. Tail Recursion 18
Why tail recursion instead of loops with mutation? 1. Simpler language, but just as efficient. 2. Explicit dependences for easier reasoning. – Especially with HOFs like fold! Tail Recursion 19
Identify dependences between ________. Ra Racket: immutable le natural l recursion (define (fib n) re recursiv ive (if (< n 2) calls ca n (+ (fib (- n 1)) (fib (- n 2))))) Ra Racket: immutable le tail l recursion (define (fib n) (define (fib-tail n fibi fibi+1) (if (= 0 n) fibi (fib-tail (- n 1) fibi+1 (+ fibi fibi+1)))) (fib n 0 1)) def fib(n): Py Python: loop iteration with mutation fib_i = 0 fib_i_plus_1 = 1 lo loop for i in range(n): fib_i_prev = fib_i it iterat ratio ions fib_i = fib_i_plus_1 fib_i_plus_1 = fib_i_prev + fib_i_plus_1 return fib_i Tail Recursion 20
What must we inspect to Wh to Identify dependences between ________. Ra Racket: immutable le natural l recursion (define (fib n) recursiv re ive (if (< n 2) calls ca n (+ (fib (- n 1)) (fib (- n 2))))) Ra Racket: immutable le tail l recursion (define (fib n) (define (fib-tail n fibi fibi+1) (if (= 0 n) fibi (fib-tail (- n 1) fibi+1 (+ fibi fibi+1)))) (fib n 0 1)) def fib(n): Python: loop iteration with mutation Py fib_i = 0 fib_i_plus_1 = 1 lo loop for i in range(n): fib_i_prev = fib_i it iterat ratio ions fib_i = fib_i_plus_1 fib_i_plus_1 = fib_i_prev + fib_i_plus_1 return fib_i Tail Recursion 21
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