18 Recursive Definitions

8/18/2019 18 Recursive Definitions

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Recursive DefinitionsRecursive Definitions

and Structural Inductionand Structural InductionCS/APMA 202CS/APMA 202

Rosen section 3.4Rosen section 3.4 Aaron Bloomfield Aaron Bloomfield


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RecursionRecursion

Recursion means definin somet!in"Recursion means definin somet!in"

suc! as a function" in terms of itself suc! as a function" in terms of itself #or e$am%le" let#or e$am%le" let f f && x x ' (' ( x x ))

*e can define*e can define f f && x x ' as' as f f && x x ' (' ( x x + f&+ f& x x ,-',-'


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Recursion e$am%leRecursion e$am%le

Rosen" section 3.4" uestion -Rosen" section 3.4" uestion - #ind#ind f f &-'"&-'" f f &2'"&2'" f f &3'" and&3'" and f f &4'" !ere&4'" !ere f f &0' ( -&0' ( -

a'a' etet f f &&nn1-' (1-' ( f f &&nn' 1 2' 1 2

f f &-' (&-' ( f f &0' 1 2 ( - 1 2 ( 3&0' 1 2 ( - 1 2 ( 3f f &2' (&2' ( f f &-' 1 2 ( 3 1 2 ( &-' 1 2 ( 3 1 2 (

f f &3' (&3' ( f f &2' 1 2 ( 1 2 ( &2' 1 2 ( 1 2 (

f f &4' (&4' ( f f &3' 1 2 ( 1 2 ( &3' 1 2 ( 1 2 (

5'5' etet f f &&nn1-' ( 31-' ( 3f f &&nn''f f &-' ( 3 +&-' ( 3 + f f &0' ( 3+- ( 3&0' ( 3+- ( 3

f f &2' ( 3 +&2' ( 3 + f f &-' ( 3+3 ( &-' ( 3+3 (

f f &3' ( 3 +&3' ( 3 + f f &2' ( 3+ ( 2&2' ( 3+ ( 2

f f &4' ( 3 +&4' ( 3 + f f &3' ( 3+2 ( 6-&3' ( 3+2 ( 6-


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Recursion e$am%leRecursion e$am%le

Rosen" section 3.4" uestion -Rosen" section 3.4" uestion - #ind#ind f f &-'"&-'" f f &2'"&2'" f f &3'" and&3'" and f f &4'" !ere&4'" !ere f f &0' ( -&0' ( -

c'c' etet f f &&nn1-' ( 21-' ( 2f f &&nn''

f f &-' ( 2&-' ( 2f f &0'&0' ( 2( 2-- ( 2( 2f f &2' ( 2&2' ( 2f f &-'&-' ( 2( 222 ( 4( 4

f f &3' ( 2&3' ( 2f f &2'&2' ( 2( 244 ( -7( -7

f f &4' ( 2&4' ( 2f f &3'&3' ( 2( 2-7-7 ( 737( 737

d'd' etet f f &&nn1-' (1-' ( f f &&nn''22

11 f f &&nn' 1 -' 1 -f f &-' (&-' ( f f &0'&0'22 11 f f &0' 1 - ( -&0' 1 - ( -22 1 - 1 - ( 31 - 1 - ( 3

f f &2' (&2' ( f f &-'&-'22 11 f f &0' 1 - ( 3&0' 1 - ( 322 1 3 1 - ( -31 3 1 - ( -3

f f &3' (&3' ( f f &2'&2'22 11 f f &0' 1 - ( -3&0' 1 - ( -322 1 -3 1 - ( -631 -3 1 - ( -63

f f &4' (&4' ( f f &3'&3'22 11 f f &0' 1 - ( -63&0' 1 - ( -6322 1 -63 1 - ( 33731 -63 1 - ( 3373


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#ractals#ractals

A fractal is a %attern t!at uses recursion A fractal is a %attern t!at uses recursion 8!e %attern itself re%eats indefinitel98!e %attern itself re%eats indefinitel9


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FractalsFractals


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#i5onacci seuence#i5onacci seuence

Definition of t!e #i5onacci seuenceDefinition of t!e #i5onacci seuence

:on,recursive;:on,recursive;

Recursive;Recursive; F F &&nn' (' ( F F &&nn,-' 1,-' 1 F F &&nn,2',2'

or;or; F F &&nn1-' (1-' ( F F &&nn' 1' 1 F F &&nn,-',-'

Must ala9s s%ecif9 5ase case&s')Must ala9s s%ecif9 5ase case&s') F F &-' ( -"&-' ( -" F F &2' ( -&2' ( -

:ote t!at some ill use:ote t!at some ill use F F &0' ( -"&0' ( -" F F &-' ( -&-' ( -

( ) ( )n

nn

n F

25

5151)(

⋅

−−+=


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#i5onacci seuence in


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Recursion definitionRecursion definition

#rom =8!e >ac?er@s Dictionar9;#rom =8!e >ac?er@s Dictionar9;

recursionrecursion nn. See. See recursionrecursion. See also. See alsotail recursiontail recursion..


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Bad recursive definitionsBad recursive definitions

Consider;Consider; f f &0' ( -&0' ( -

f f &&nn' ( - 1' ( - 1 f f &&nn,2',2'

*!at is*!at is f f &-'&-'

Consider;Consider; f f &0' ( -&0' ( -

f f &n' ( -1&n' ( -1f f &,n'&,n'

*!at is f&-'*!at is f&-'


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--

Definin sets via recursionDefinin sets via recursion

Same to %arts;Same to %arts; Base case &or 5asis ste%'Base case &or 5asis ste%'

Recursive ste%Recursive ste%

$am%le; t!e set of %ositive inteers$am%le; t!e set of %ositive inteers

Basis ste%Basis ste%; -; - ∈∈ SS Recursive ste%Recursive ste%; if; if x x ∈∈ SS" t!en" t!en x x 1-1- ∈∈ SS


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Definin sets via recursionDefinin sets via recursion

Rosen" section 3.4" uestion 24; ive recursiveRosen" section 3.4" uestion 24; ive recursive

definitions for;definitions for;

a'a' 8!e set of odd %ositive inteers8!e set of odd %ositive inteers

-- ∈∈ SS IfIf x x ∈∈ SS" t!en" t!en x x 1212 ∈∈ SS

5'5' 8!e set of %ositive inteer %oers of 38!e set of %ositive inteer %oers of 3 33 ∈∈ SS

IfIf x x ∈∈ SS" t!en 3+" t!en 3+ x x ∈∈ SSc'c' 8!e set of %ol9nomials it! inteer coefficients8!e set of %ol9nomials it! inteer coefficients

00 ∈∈ SS

IfIf p p&& x x '' ∈∈ SS" t!en" t!en p p&& x x ' 1 c$' 1 c$nn ∈∈ SS c c ∈∈ ZZ"" nn ∈∈ ZZ andand nn ≥ 0≥ 0


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Definin strins via recursionDefinin strins via recursion

8erminolo98erminolo9 λλ is t!e em%t9 strin; =is t!e em%t9 strin; =

ΣΣ is t!e set of all letters; a" 5" c" E" F Gis t!e set of all letters; a" 5" c" E" F G

8!e set of letters can c!ane de%endin on t!e %ro5lem8!e set of letters can c!ane de%endin on t!e %ro5lem

*e can define a set of strins*e can define a set of strins ΣΣ+ as follos+ as follos Base ste%;Base ste%; λλ ∈∈ ΣΣ++

IfIf w w ∈∈ ΣΣ+ and+ and x x ∈∈ ΣΣ" t!en $" t!en $ ∈∈ ΣΣ++

8!us"8!us" ΣΣ+ s t!e set of all t!e %ossi5le strins t!at can+ s t!e set of all t!e %ossi5le strins t!at can

5e enerated it! t!e al%!a5et5e enerated it! t!e al%!a5et

Is t!is counta5l9 infinite or uncounta5l9 infiniteIs t!is counta5l9 infinite or uncounta5l9 infinite


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Definin strins via recursionDefinin strins via recursion

etet ΣΣ ( 0" - G( 0" - G

8!us"8!us" ΣΣ+ is t!e set of all 5inar9 num5ers+ is t!e set of all 5inar9 num5ers

Hr all 5inar9 strinsHr all 5inar9 strins Hr all %ossi5le com%uter filesHr all %ossi5le com%uter files


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Strin lent! via recursionStrin lent! via recursion

>o to define strin lent! recursivel9>o to define strin lent! recursivel9 Basis ste%;Basis ste%; l l &&λλ' ( 0' ( 0 Recursive ste%;Recursive ste%; l l &&wx wx ' (' ( l l &&w w ' 1 - if' 1 - if w w ∈∈ ΣΣ+ and+ and x x

∈∈ ΣΣ

$am%le;$am%le; l l &=aaa'&=aaa' l l &=aaa' (&=aaa' ( l l &=aa' 1 -&=aa' 1 -

l l &=aa' (&=aa' ( l l &=a' 1 -&=a' 1 - l l &=a' (&=a' ( l l &=' 1 -&=' 1 - l l &=' ( 0&=' ( 0 Result; 3Result; 3


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Strins via recursion e$am%leStrins via recursion e$am%le

Rosen" section 3.4" uestion 36; ive aRosen" section 3.4" uestion 36; ive a

recursive definition for t!e set of strin t!atrecursive definition for t!e set of strin t!at

are %alindromesare %alindromes *e ill define set*e ill define set P P " !ic! is t!e set of all" !ic! is t!e set of all

%alindromes%alindromes

Basis ste%Basis ste%;; λλ ∈∈ P P Second 5asis ste%;Second 5asis ste%; x x ∈∈ P P !en!en x x ∈∈ ΣΣ

Recursive ste%Recursive ste%;; xpx xpx ∈∈ P P ifif x x ∈∈ ΣΣ andand p p ∈∈ P P


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Strins and induction e$am%leStrins and induction e$am%le

8!is reuires structural induction" !ic!8!is reuires structural induction" !ic!

ill 5e covered later in t!is slide setill 5e covered later in t!is slide set


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Recursion %rosRecursion %ros

as9 to %roramas9 to %roram

as9 to understandas9 to understand


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Recursion consRecursion cons

Consider t!e recursive #i5onacci enerator Consider t!e recursive #i5onacci enerator

>o man9 recursive calls does it ma?e>o man9 recursive calls does it ma?e F F &-'; -&-'; - F F &2'; -&2'; -

F F &3'; 3&3'; 3 F F &4'; &4'; F F &'; &'; F F &-0'; -0&-0'; -0 F F &20'; -3"2&20'; -3"2 F F &30'; -"774"0&30'; -"774"0 F F &40'; 204"776"30&40'; 204"776"30 F F &0'; 2"-2"36"04&0'; 2"-2"36"04 F F &-00'; 06"44"77"36"23"630"-4&-00'; 06"44"77"36"23"630"-4 ≈≈ + -0 + -02020

At - 5illion recursive calls %er second &enerous'" t!is ould ta?e over At - 5illion recursive calls %er second &enerous'" t!is ould ta?e over22"000 9ears22"000 9ears

But t!at ould also ta?e ell over -0But t!at ould also ta?e ell over -0-2-2 5 of memor9)5 of memor9)


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8rees8rees

Rooted trees;Rooted trees; A ra%! containin nodes and edes A ra%! containin nodes and edes

Cannot contain a c9cle)Cannot contain a c9cle)

C9cle not alloed in a tree


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Rooted treesRooted trees

Recursive definition;Recursive definition; Basis ste%Basis ste%; A sinle verte$; A sinle verte$ r r is a rooted treeis a rooted tree

Recursive ste%Recursive ste%;;

etet T T --"" T T 22" E"" E" T T nn 5e rooted trees5e rooted trees#orm a ne tree it! a ne root#orm a ne tree it! a ne root r r t!at contains an edet!at contains an ede

to t!e root of eac! of t!e treesto t!e root of eac! of t!e trees T T --"" T T 22" E"" E" T T nn


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&$tended' Binar9 trees&$tended' Binar9 trees

Recursive definitionRecursive definition Basis ste%Basis ste%;; 8!e em%t9 set is an e$tended 5inar9 tree8!e em%t9 set is an e$tended 5inar9 tree


etet T T --" and" and T T 22 5e e$tended 5inar9 trees5e e$tended 5inar9 trees#orm a ne tree it! a ne root#orm a ne tree it! a ne root r r

#orm a ne tree suc! t!at#orm a ne tree suc! t!at T T -- is t!e left su5tree"is t!e left su5tree"

andand T T 22 is t!e ri!t su5treeis t!e ri!t su5tree


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#ull 5inar9 trees#ull 5inar9 trees

Recursive definitionRecursive definition Basis ste%;Basis ste%; A full 5inar9 tree consistin onl9 of t!e verte$ A full 5inar9 tree consistin onl9 of t!e verte$ r r


etet T T --" and" and T T 22 5e e$tended 5inar9 trees5e e$tended 5inar9 trees

#orm a ne tree it! a ne root#orm a ne tree it! a ne root r r

#orm a ne tree#orm a ne tree T T suc! t!atsuc! t!at T T -- is t!e left su5tree"is t!e left su5tree"

andand T T 22 is t!e ri!t su5treeis t!e ri!t su5tree

8!is is denoted 598!is is denoted 59 T T (( T T --JJT T 22

:ote t!e onl9 difference 5eteen a reular:ote t!e onl9 difference 5eteen a reular

5inar9 tree and a full one is t!e 5asis ste%5inar9 tree and a full one is t!e 5asis ste%


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Binar9 tree !ei!tBinar9 tree !ei!t

hh&&T T ' denotes t!e !ei!t of tree' denotes t!e !ei!t of tree T T

Recursive definition;Recursive definition; Basis ste%Basis ste%; 8!e !ei!t of a tree it!; 8!e !ei!t of a tree it!

onl9 one nodeonl9 one node r r is 0is 0 Recursive ste%Recursive ste%;;

etet T T -- andand T T 22 5e 5inar9 trees5e 5inar9 trees

8!e 5inar9 tree8!e 5inar9 tree T T (( T T --JJT T

22 !as !ei!t!as !ei!t

hh&&T T ' ( - 1 ma$ &' ( - 1 ma$ & hh&&T T --'"'" hh&&T T 22' '' '

8!is definition can 5e eneraliFed to8!is definition can 5e eneraliFed tonon,5inar9 treesnon,5inar9 trees


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Binar9 tree siFeBinar9 tree siFe

nn&&T T ' denotes t!e num5er of vertices in tree' denotes t!e num5er of vertices in tree T T

Recursive definition;Recursive definition; Basis ste%Basis ste%; 8!e num5er of vertices of an; 8!e num5er of vertices of an

em%t9 tree is 0em%t9 tree is 0 Basis ste%Basis ste%; 8!e num5er of vertices of a tree; 8!e num5er of vertices of a tree

it! onl9 one nodeit! onl9 one node r r is -is - Recursive ste%Recursive ste%;;

etet T T -- andand T T 22 5e 5inar9 trees5e 5inar9 trees

8!e num5er of vertices in 5inar9 tree8!e num5er of vertices in 5inar9 tree T T (( T T --JJT T 22 is;is;

n&n&T T ' ( - 1' ( - 1 nn&&T T --' 1' 1 nn&&T T 22''

8!is definition can 5e eneraliFed to non,8!is definition can 5e eneraliFed to non,5inar9 trees5inar9 trees


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A bit of humor: ComputerA bit of humor: Computer

terminologyterminology


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Recursion vs. inductionRecursion vs. induction

Consider t!e recursive definition forConsider t!e recursive definition for

factorial;factorial;

f f &0' ( -&0' ( -

f f &n' ( n +&n' ( n + f f &n,-'&n,-'

Sort of li?e inductionSort of li?e induction

Base case

8!e =ste%


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Rosen" section 3.4" e$am%le &%ae 272'Rosen" section 3.4" e$am%le &%ae 272'

Consider t!e set of all inteers t!at are multi%lesConsider t!e set of all inteers t!at are multi%les

of 3of 3 3" 7" " -2" -" E G 3" 7" " -2" -" E G

x x KK x x ( 3( 3k k andand k k ∈∈ ZZ11 GG

Recursive definition;Recursive definition; Basis ste%Basis ste%; 3; 3 ∈∈ SS

Recursive ste%Recursive ste%; If; If x x ∈∈ SS andand y y ∈∈ SS" t!en" t!en x x 11y y ∈∈ SS


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Proof via induction; %rove t!atProof via induction; %rove t!at SS contains all t!e inteerscontains all t!e inteerst!at are divisi5le 59 3t!at are divisi5le 59 3 etet A A 5e t!e set of all ints divisi5le 59 35e t!e set of all ints divisi5le 59 3 S!o t!atS!o t!at SS (( A A

8o %arts;8o %arts; S!o t!atS!o t!at SS ⊆⊆ A A

etet P P &&nn' ( 3' ( 3nn ∈∈ SS

Base caseBase case;; P P &-' ( 3+-&-' ( 3+- ∈∈ SS B9 t!e 5asis ste% of t!e recursive definitionB9 t!e 5asis ste% of t!e recursive definition

Inductive !9%ot!esisInductive !9%ot!esis; assume; assume P P &&k k ' ( 3+' ( 3+k k ∈∈ SS is trueis true

Inductive ste%Inductive ste%; s!o t!at; s!o t!at P P &&k k 1-' ( 3+&1-' ( 3+&k k 1-' is true1-' is true 3+&3+&k k 1-' ( 31-' ( 3k k 1313 33k k ∈∈ SS 59 t!e inductive !9%ot!esis59 t!e inductive !9%ot!esis 33 ∈∈ SS 59 t!e 5ase case59 t!e 5ase case 8!us" 38!us" 3k k 1313 ∈∈ SS 59 t!e recursive definition59 t!e recursive definition

S!o t!atS!o t!at A A ⊆⊆ SSDone in t!e te$t" %ae 27 &not re%roduced !ere'Done in t!e te$t" %ae 27 &not re%roduced !ere'


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*!at did e Lust do*!at did e Lust do

:otice !at e did;:otice !at e did; S!oed t!e 5ase caseS!oed t!e 5ase case

Assumed t!e inductive !9%ot!esis Assumed t!e inductive !9%ot!esis

#or t!e inductive ste%" e;#or t!e inductive ste%" e;

S!oed t!at eac! of t!e =%arts ere inS!oed t!at eac! of t!e =%arts ere in SS 8!e %arts 5ein 38!e %arts 5ein 3k k and 3and 3

S!oed t!at since 5ot! %arts ere inS!oed t!at since 5ot! %arts ere in SS" 59 t!e" 59 t!erecursive definition" t!e com5ination of t!ose %artsrecursive definition" t!e com5ination of t!ose %arts

is inis in SS i.e." 3i.e." 3k k 1313 ∈∈ SS

8!is is called8!is is called structural inductionstructural induction


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8ree structural induction e$am%le8ree structural induction e$am%le

Rosen" section 3.4" uestion 43Rosen" section 3.4" uestion 43

S!o t!atS!o t!at nn&&T T '' ≥≥ 22hh&&T T ' 1 -' 1 -

Basis ste%Basis ste%; et; et T T 5e t!e full 5inar95e t!e full 5inar9

tree of Lust one nodetree of Lust one node r r hh&&T T ' ( 0' ( 0

nn&&T T ' ( -' ( - nn&&T T '' ≥≥ 22hh&&T T ' 1 -' 1 -

-- ≥≥ 2+0 1 -2+0 1 -

-- ≥≥ --


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8ree structural induction e$am%le8ree structural induction e$am%le

S!o t!atS!o t!at nn&&T T '' ≥≥ 22hh&&T T ' 1 -' 1 -

Inductive !9%ot!esisInductive !9%ot!esis;; etet T T -- andand T T 22 5e full 5inar9 trees5e full 5inar9 trees

Assume t!at Assume t!at nn&&T T 11'' ≥≥ 22hh&&T T 11' 1 - for some tree' 1 - for some tree T T 11 Assume t!at Assume t!at nn&&T T 2 2 '' ≥≥ 22hh&&T T 2 2 ' 1 - for some tree' 1 - for some tree T T 2 2

Recursive ste%Recursive ste%;; etet T T (( T T -- JJ T T 22>ere t!e J o%erator means creatin a ne tree it! a root note>ere t!e J o%erator means creatin a ne tree it! a root note r r and su5treesand su5treesT T -- andand T T 22

:e element is:e element is T T B9 t!e definition of !ei!t and siFe" e ?no;B9 t!e definition of !ei!t and siFe" e ?no;

nn&&T T ' ( - 1' ( - 1 nn&&T T --' 1' 1 nn&&T T 22''

hh&&T T ' ( - 1 ma$ &' ( - 1 ma$ & hh&&T T --'"'" hh&&T T 22' '' ' 8!erefore;8!erefore;

nn&&T T ' ( - 1' ( - 1 nn&&T T --' 1' 1 nn&&T T 22''

≥≥ - 1 2- 1 2hh&&T T --' 1 - 1 2' 1 - 1 2hh&&T T 22' 1 -' 1 -

≥≥ - 1 2+ma$ &- 1 2+ma$ & hh&&T T --'"'" hh&&T T 22' '' ' t!e sum of to non,ne N@s is at leastt!e sum of to non,ne N@s is at least

as lare as t!e larer of t!e toas lare as t!e larer of t!e to

( - 1 2+( - 1 2+hh&&T T '' 8!us"8!us" nn&&T T '' ≥≥ 22hh&&T T ' 1 -' 1 -


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Strin structural induction e$am%leStrin structural induction e$am%le

Rosen" section 3.4" uestion 32Rosen" section 3.4" uestion 32

Part &a'; ive t!e definition forPart &a'; ive t!e definition for ones(s)ones(s)" !ic! counts t!e" !ic! counts t!enum5er of ones in a 5it strinnum5er of ones in a 5it strin ss

etet ΣΣ ( 0" - G( 0" - G

Basis ste%Basis ste%;; onesones&&λλ'' == 00

Recursive ste%Recursive ste%;; ones(wx) = ones(w) + x ones(wx) = ones(w) + x *!ere*!ere x x ∈∈ ΣΣ andand w w ∈∈ ΣΣ++ :ote t!at:ote t!at x x is a 5it; eit!er 0 or -is a 5it; eit!er 0 or -


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Strin structural induction e$am%leStrin structural induction e$am%le

Part &5'; se structural induction to %rove t!atPart &5'; se structural induction to %rove t!at ones(st) = ones(s) +ones(st) = ones(s) +ones(t)ones(t)

Basis ste%Basis ste%;; t t (( λλ ones (sones (sJJλ λ ) = ones(s) = ones(s)+0 = ones(s) + ones( ) = ones(s) = ones(s)+0 = ones(s) + ones( λ λ ) )

Inductive !9%ot!esisInductive !9%ot!esis; Assume; Assume ones(s∙t) = ones(s) + ones(t)ones(s∙t) = ones(s) + ones(t)

Recursive ste%Recursive ste%; *ant to s!o t!at; *ant to s!o t!at ones(s∙t∙x) = ones(s) + ones(t∙x)ones(s∙t∙x) = ones(s) + ones(t∙x) *!ere*!ere ss"" tt ∈∈ ΣΣ+ and+ and x x ∈∈ ΣΣ :e element is:e element is ones(s∙t∙x)ones(s∙t∙x) ones (s∙t∙x) = ones ((s∙t)∙x))ones (s∙t∙x) = ones ((s∙t)∙x)) 59 associativit9 of concatenation59 associativit9 of concatenation = x+ones(s∙t)= x+ones(s∙t) 59 recursive definition59 recursive definition = x + ones(s) + ones(t)= x + ones(s) + ones(t) 59 inductive !9%ot!esis59 inductive !9%ot!esis = ones(s) + (x + ones(t))= ones(s) + (x + ones(t)) 59 commutativit9 and assoc. of 159 commutativit9 and assoc. of 1 = ones(s) + ones(t∙x)= ones(s) + ones(t∙x) 59 recursive definition59 recursive definition Proven)Proven)


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Induction met!ods com%aredInduction met!ods com%ared

WeakWeakmathematicalmathematical

StrongStrongMathematicalMathematical StructuralStructural

Used for Used for suall9suall9

formulaeformulaesuall9 formulae notsuall9 formulae not

%rova5le via%rova5le viamat!ematical inductionmat!ematical induction

Hnl9 t!ins definedHnl9 t!ins definedvia recursionvia recursion

AssumptionAssumption Assume P&?' Assume P&?' Assume P&-'" P&2'" E" Assume P&-'" P&2'" E"P&?'P&?'

Assume statement is Assume statement istrue for some OoldOtrue for some OoldO

elementselements

What toWhat to

proveprove8rue for8rue for

P&?1-'P&?1-' 8rue for P&?1-'8rue for P&?1-'

Statement is true forStatement is true forsome OneO elementssome OneO elements

created it! OoldOcreated it! OoldOelementselements

Step 1Step 1calledcalled

Base caseBase case Base caseBase case Basis ste%Basis ste%

Step 3Step 3

calledcalled

Inductive ste%Inductive ste% Inductive ste%Inductive ste% Recursive ste%Recursive ste%


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Induction t9%es com%aredInduction t9%es com%ared

S!o t!atS!o t!at F F &&nn' 2' 2nn *!ere*!ere F F &&nn' is t!e' is t!e nnt!t! #i5onacci num5er #i5onacci num5er Actuall9 Actuall9 F F &&nn' 2' 20.3+0.3+nn" 5ut e on@t %rove t!at !ere" 5ut e on@t %rove t!at !ere

#i5onacci definition;#i5onacci definition; Basis ste%Basis ste%;; F F &-' ( - and&-' ( - and F F &2' ( -&2' ( - Recursive ste%Recursive ste%;; F F &&nn' (' ( F F &&nn,-' 1,-' 1 F F &&nn,2',2'

Base case &or 5asis ste%'Base case &or 5asis ste%'; S!o true for; S!o true for F F &-'&-'andand F F &2'&2' #&-' ( - 2#&-' ( - 2-- ( 2( 2 #&2' ( - 2#&2' ( - 222 ( 4( 4


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Qia ea? mat!ematical inductionQia ea? mat!ematical induction

Inductive !9%ot!esisInductive !9%ot!esis; Assume; Assume F F &&k k ' 2' 2k k

Inductive ste%Inductive ste%; Prove; Prove F F &&k k 1-' 21-' 2k k 1-1- F F &&k k 1-' (1-' ( F F &&k k ' 1' 1 F F &&k k ,-',-'

*e ?no*e ?no F F &&k k ' 2' 2k k

59 t!e inductive !9%ot!esis59 t!e inductive !9%ot!esis ac! term is less t!an t!e ne$t" t!ereforeac! term is less t!an t!e ne$t" t!erefore

F F &&k k ' ' F F &&k k ,-',-'8!us"8!us" F F &&k k ,-' ,-' F F &&k k ' 2' 2k k

8!erefore"8!erefore" F F &&k k 1-' (1-' ( F F &&k k ' 1' 1 F F &&k k ,-' 2,-' 2k k 1 21 2k k ( 2( 2k k 1-1-

Proven)Proven)


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Qia stron mat!ematical inductionQia stron mat!ematical induction

Inductive !9%ot!esisInductive !9%ot!esis; Assume; Assume F F &-' 2&-' 2--""

F F &2' 2&2' 222" E"" E" F F &&k-k--' 2-' 2k k ,-,-"" F F &&k k ' 2' 2k k

Inductive ste%Inductive ste%; Prove; Prove F F &&k k 1-' 21-' 2k k 1-1-

F F &&k k 1-' (1-' ( F F &&k k ' 1' 1 F F &&k k ,-',-'

*e ?no*e ?no F F &&k k ' 2' 2k k 59 t!e inductive !9%ot!esis59 t!e inductive !9%ot!esis

*e ?no*e ?no F F &&k k ,-' 2,-' 2k k ,-,- 59 t!e inductive59 t!e inductive

!9%ot!esis!9%ot!esis 8!erefore"8!erefore" F F &&k k ' 1' 1 F F &&k k ,-' 2,-' 2k k 1 21 2k k ,-,- 2 2k k 1-1-

Proven)Proven)


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44

Qia structural inductionQia structural induction

Inductive !9%ot!esisInductive !9%ot!esis; Assume; Assume F F &&nn' 2' 2nn

Recursive ste%Recursive ste%;; S!o true for =ne element;S!o true for =ne element; F F &&nn1-'1-' *e ?no*e ?no F F &&nn' 2' 2nn 59 t!e inductive !9%ot!esis59 t!e inductive !9%ot!esis ac! term is less t!an t!e ne$t" t!ereforeac! term is less t!an t!e ne$t" t!erefore

F F &&nn' ' F F &&nn,-',-'

8!us"8!us"

F F

&&nn

,-' ,-'

F F

&&nn

' 2' 2nn

8!erefore"8!erefore" F F &&nn' 1' 1 F F &&nn,-' 2,-' 2nn 1 21 2nn ( 2( 2nn1-1-

Proven)Proven)


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4

Anot!er a9 via structural induction Anot!er a9 via structural induction

Inductive !9%ot!esisInductive !9%ot!esis; Assume; Assume F F &&nn' 2' 2nn andandF F &&nn,-' 2,-' 2nn,-,- 8!e difference !ere is e are usin to =old8!e difference !ere is e are usin to =old

elements versus one" as in t!e last slideelements versus one" as in t!e last slide

Recursive ste%Recursive ste%;; S!o true for =ne element;S!o true for =ne element; F F &&nn1-'1-' F F &&nn1-' (1-' ( F F &&nn' 1' 1 F F &&nn,-',-'

*e ?no*e ?no

F F

&&nn

' 2' 2nn

59 t!e inductive !9%ot!esis59 t!e inductive !9%ot!esis

*e ?no*e ?no F F &&nn,-' 2,-' 2nn,-,- 59 t!e inductive !9%ot!esis59 t!e inductive !9%ot!esis 8!erefore"8!erefore" F F &&nn' 1' 1 F F &&nn,-' 2,-' 2k k 1 21 2k k ,-,- 2 2k k 1-1-

Proven)Proven)


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But ait)But ait)

In t!is e$am%le" t!e structural induction %roofIn t!is e$am%le" t!e structural induction %roof

as essentiall9 t!e same as t!e ea? or stronas essentiall9 t!e same as t!e ea? or stron

mat!ematical induction %roof mat!ematical induction %roof

It@s !ard to find an e$am%le t!at or?s ell for all ofIt@s !ard to find an e$am%le t!at or?s ell for all oft!e induction t9%est!e induction t9%es

Structural induction ill or? on some recursiveStructural induction ill or? on some recursive

%ro5lems !ic! ea? or stron mat!ematical%ro5lems !ic! ea? or stron mat!ematicalinduction ill notinduction ill not 8rees" strins" etc.8rees" strins" etc.


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Section 3.4" uestion 6Section 3.4" uestion 6

ive t!e recursive definition of t!e folloin seuencesive t!e recursive definition of t!e folloin seuences :ote t!at man9 ansers are %ossi5le):ote t!at man9 ansers are %ossi5le)

a)a) aann ( 4( 4nn 2 2 8erms; 2" 7" -0" -4" -7" etc.8erms; 2" 7" -0" -4" -7" etc. aa11 ( 2( 2 aann (( aan-1n-1 1 41 4

)) aann ( - 1 &,-'( - 1 &,-'nn 8erms; 0" 2" 0" 2" 0" 2" etc.8erms; 0" 2" 0" 2" 0" 2" etc. aa11 ( 0"( 0" aa2 2 ( 2( 2 aann (( aan-2 n-2

c)c) aann (( nn&&nn1-'1-' 8erms; 2" 7" -2" 20" 30" 42" etc.8erms; 2" 7" -2" 20" 30" 42" etc. aa11 ( 2( 2 aann (( aan-1n-1 1 2+n1 2+n

!)!) aann (( nn22 8erms; -" 4" " -7" 2" 37" 4" etc.8erms; -" 4" " -7" 2" 37" 4" etc. aa11 ( -( -

aann (( aan-1n-1 1 21 2nn , -, -


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4

Section 3.4" uestion -2Section 3.4" uestion -2

S!o t!atS!o t!at f f --22 11 f f 2222 11 f f 3322 1 E 11 E 1 f f nn22 (( f f nnf f nn1-1-

Base caseBase case;; nn ( -( - f f --22 (( f f 11f f 2 2 --22 ( -+-( -+-

Inductive !9%ot!esisInductive !9%ot!esis; Assume; Assume f f --22 11 f f 2222 11 f f 3322 1 E 11 E 1 f f k k 22 (( f f k k f f k k 1-1-

Inductive ste%Inductive ste%; Prove; Prove f f --22 11 f f 2222 11 f f 3322 1 E 11 E 1 f f k k 22 11 f f k+1k+122 (( f f k+1k+1f f k k 1212


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0


Inductive !9%ot!esisInductive !9%ot!esis; Assume; Assume f f --22 11 f f 2222 11 f f 3322 1 E 11 E 1 f f k k 22 (( f f k k f f k k 1-1-

Inductive ste%Inductive ste%; Prove; Prove f f --22 11 f f 2222 11 f f 3322 1 E 11 E 1 f f k k 22 11 f f k+1k+122 (( f f k+1k+1f f k k 1212

f f k k f f k k 1-1- 11 f f k+1k+122 (( f f k+1k+1f f k k 1212

f f k k f f k k 1-1- 11 f f k+1k+122 (( f f k+1k+1 &&f f k k 11 f f k k 1-1-''

f f k k f f k k 1-1- 11 f f k+1k+122 (( f f k k f f k k 1-1- 11 f f k+1k+122


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-


S!o t!atS!o t!at f f -- 11 f f 22 11 f f 33 1 E 11 E 1 f f 22nn,-,- (( f f 22nn

Base caseBase case;; nn ( -( - f f -- (( f f 2+-2+- - ( -- ( -

Inductive !9%ot!esisInductive !9%ot!esis; Assume; Assume

f f -- 11 f f 22 11 f f 33 1 E 11 E 1 f f 22k k ,-,- (( f f 22k k

Inductive ste%Inductive ste%; Prove; Prove f f -- 11 f f 22 11 f f 33 1 E 11 E 1 f f 22k k ,-,- 11 f f 2&2&k k 1-',-1-',- (( f f 2&2&k k 1-'1-' f f -- 11 f f 22 11 f f 33 1 E 11 E 1 f f 22k k ,-,- 11 f f 22k k 1-1- (( f f 22k k 1212


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2


Inductive !9%ot!esisInductive !9%ot!esis; Assume; Assume f f -- 11 f f 22 11 f f 33 1 E 11 E 1 f f 22k k ,-,- (( f f 22k k

Inductive ste%Inductive ste%; Prove; Prove f f -- 11 f f 22 11 f f 33 1 E 11 E 1 f f 22k k ,-,- 11 f f 22k k 1-1- (( f f 22k k 1212

f f 22k k 11 f f 22k k 1-1- (( f f 22k k 1212

8rue 59 definition of8rue 59 definition of f f 22k k 1212


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3


S!o t!at t!e setS!o t!at t!e set SS defined 59defined 59 Basis ste%Basis ste%; -; - ∈∈ SS Recursive ste%Recursive ste%;; ss 11 t t ∈∈ SS !en!en ss ∈∈ SS andand t t ∈∈ SS

is t!e set of %ositive inteers;is t!e set of %ositive inteers; ZZ11 ( -" 2" 3" E G( -" 2" 3" E G

:ote t!e &some!at recursive' definition of t!e %ositive inteers;:ote t!e &some!at recursive' definition of t!e %ositive inteers; - is a %ositive inteer - is a %ositive inteer #or an9 ar5itrar9#or an9 ar5itrar9 nn t!at is a %ositive inteer"t!at is a %ositive inteer" nn1- is also a %ositive inteer 1- is also a %ositive inteer

Proof 59 structural inductionProof 59 structural inductionBasis ste%Basis ste%; -; - ∈∈ SS and -and - ∈∈ ZZ11

Inductive !9%ot!esisInductive !9%ot!esis; Assume; Assume k k ∈∈ SS

Recursive ste%Recursive ste%; S!o; S!o k k 1-1- ∈∈ SS k k ∈∈ SS 59 t!e inductive !9%ot!esis59 t!e inductive !9%ot!esis -- ∈∈ SS 59 t!e 5ase case59 t!e 5ase case k k 1-1- ∈∈ SS 59 t!e recursive ste% of t!e recursive definition a5ove59 t!e recursive ste% of t!e recursive definition a5ove


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ive a recursive definition of t!e reversal of aive a recursive definition of t!e reversal of astrinstrin

Basis ste%Basis ste%;; λλRR

(( λλ :ote t!at t!e su%erscri%ted R means reversal of a:ote t!at t!e su%erscri%ted R means reversal of a

strinstrin

Recursive ste%Recursive ste%; Consider a strin; Consider a strin w w ∈∈ ΣΣ++

ReriteRerite w w asas "y "y !ere!ere " " ∈∈ ΣΣ+ and+ and yy ∈∈ ΣΣ" " is t!e firstis t!e first nn,- c!aracters in,- c!aracters in w w

y y is t!e last c!aracter inis t!e last c!aracter in w w

w w RR (( y y &&" " RR''

Parent!eses are for our 5enefitParent!eses are for our 5enefit


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55

Quick survey Quick survey

I felt I understood the material in thisI felt I understood the material in this

slide set…slide set…

a)a) Very well Very well

b)b) With some review, I’ll be goodWith some review, I’ll be good

c)c) Not reallyNot really

d)d) Not at allNot at all


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The pace of the lecture for thisThe pace of the lecture for this

slide set was…slide set was…

a)a) astast

b)b) !bout right !bout right

c)c) ! little slow ! little slow

d)d) Too slowToo slow


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"ow interesting was the material in"ow interesting was the material in

this slide set# $e honest%this slide set# $e honest%

a)a) Wow% That was &'''''' cool%Wow% That was &'''''' cool%

b)b) &omewhat interesting&omewhat interesting

c)c) (ather borting(ather borting

d)d) ********************


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Today’s demotivators Today’s demotivators

Documents

18 Recursive Definitions