Class 11: Deeper List Procedures

Class 11: Deeper List Procedures

cs1120 Fall 2011David Evans16 September 2011

Plan

Accumulator vs. CruncherThe Story so Far: 3 main themes and what’s

coming nextPS3

Reading handout today: excerpt from Gödel, Escher, Bach: An Eternal Golden Braid by Douglas Hofstadter (also on line). You are not required to read it (i.e., there will not be any questions about it on quizzes or exams), but it will embiggen your mind and help you on the path to true enlightenment!

Recap: list-accumulator

(define (list-accumulate f base p) (if (null? p) base (f (car p) (list-accumulate f base (cdr p))))

(define (list-map f p) (list-accumulate (lambda (el rest) (cons (f el) rest)) null p))

(define (list-sum f p) (list-accumulate + 0 p))

(define (list-length f p) (list-accumulate (lambda (el rest) (+ 1 rest)) 0 p))

list-cruncher(define (list-cruncher base carproc combiner p) (if (null? p) base (combiner (carproc (car p)) (list-cruncher baseres carproc combiner (cdr p)))))

Is list-cruncher as powerful as list-accumulator?

Using list-cruncher(define (list-cruncher base carproc combiner p) (if (null? p) base (combiner (carproc (car p)) (list-cruncher baseres carproc combiner (cdr p)))))

(define (list-sum p) (list-cruncher

Can list-cruncher crunch length?(define (list-cruncher baseres carproc combiner p) (if (null? p) baseres (combiner (carproc (car p)) (list-cruncher baseres carproc combiner (cdr p)))))

(define (list-length p) (if (null? p) 0 (+ 1 (list-length (cdr p)))))

Can list-cruncher crunch length?(define (list-cruncher baseres carproc combiner p) (if (null? p) baseres (combiner (carproc (car p)) (list-cruncher baseres carproc combiner (cdr p)))))

(define (list-length p) (if (null? p) 0 (+ 1 (list-length (cdr p)))))

(define (list-length p) (list-cruncher 0 (lambda (x) 1) + p))

Can list-cruncher crunch is-list??(define (list-cruncher baseres carproc combiner p) (if (null? p) baseres (combiner (carproc (car p)) (list-cruncher baseres carproc combiner (cdr p)))))

(define (is-list? p) (if (null? p) true (if (pair? p) (is-list? (cdr p)) false)))

No! If p is non-null, (car p) is always evaluated by list-cruncher so it produces an error if p is not a list.

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Crunchers vs. Accumulators

(define (list-cruncher baseres carproc combiner p) (if (null? p) baseres (combiner (carproc (car p)) (list-cruncher baseres carproc combiner (cdr p)))))

(define (list-accumulate f base p) (if (null? p) base (f (car p) (list-accumulate f base (cdr p)))))

(define (list-sum p) (list-accumulate + 0 p))

(define (list-sum p) (list-cruncher 0 (lambda (x) x) + p))

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Is there any procedure that can be defined using list accumulate‑ that can’t be defined using list-cruncher?

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No! Proof-by-construction:(define (list-accumulate f base p) (list-cruncher base (lambda (x) x) f p))

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Gold star bonus: Is there any procedure that can be defined using list cruncher‑ that can’t be defined using list accumulate‑ ? (convincing proof required)

Story So Far

Theme Classes 1-11

Recursive Definitions

Universality

Abstraction

Theme Classes 1-11 Next Week Later

Recursive Definitions

Languages: BNFProceduresData: List

ProceduresData: Trees, etc. Interpreters

Universality Procedures as Inputs and Outputs

Computers as Universal Machines

Proving Universality

AbstractionDefinitions

Procedural AbstractionData Abstraction

Abstract MachinesDigital Abstraction

Objects

Language Abstraction

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PS3: Lindenmayer System Fractals

Aristid Lindenmayer (1925-1989)

Purple Arrow by Rachel Lathbury and Andrea Yoon

Broccoli Fallout by Paul DiOrio, Rachel Phillips

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L-Systems

CommandSequence ::= ( CommandList )CommandList ::= Command CommandListCommandList ::= εCommand ::= FCommand ::= RAngleCommand ::= OCommandSequence

Cherry Blossom by Ji Hyun Lee, Wei Wang

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L-System Rewriting

Start: (F)Rewrite Rule:

F (F O(R30 F) F O(R-60 F) F)

Work like BNF replacement rules, except replace all instances at once!

Why is this a better model for biological systems?

CommandSequence ::= ( CommandList )CommandList ::= Command CommandListCommandList ::=Command ::= FCommand ::= RAngleCommand ::= OCommandSequence

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Maps are Useful!

PS3 Question 5:

(define (rewrite-lcommands lcommands replacement) (flatten-commands (map ; Procedure to apply to each command lcommands)))

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Drawing Functions

x

y

y = (la

mbda (x) x)

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Drawing Arbitrary Curves

Rose Bush by Jacintha Henry and Rachel Kay Tie Dye by Bill Ingram

(x, y) = (lambda (t) (make-point …))

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Curves

A curve c is the set of points that results from applying c to every real number t value between 0.0 and 1.0.

How many points?Infinitely many!

We can draw an approximate representation of the curve by sampling some of the points.

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Defining Curves

x

y

y = (la

mbda (x) x)

(lambda (t)

(make-point t

t))

1.00.0

1.0

(define diag-line (lambda (t) (make-point t t))

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Drawing Curves

(define (draw-curve-points curve n) (define (draw-curve-worker curve t step) (if (<= t 1.0) (begin (window-draw-point (curve t)) (draw-curve-worker curve (+ t step) step)))) (draw-curve-worker curve 0.0 (/ 1 n)))

Does this recursive definition have a base case?

Deep List Mapping

Return PS2

PS2 Comments will be posted soon

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Charge

Quiz 2 MondaySee notes for what it coversPlease do the reading!

where the red fern grows by Jessica Geist, Ellen Clarke

Technology

Class 11: Deeper List Procedures