Computation. Models of Computation. x 2. Example: memory CPU. temporary memory. temporary memory. input memory CPU.

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1 Computtion Models of Computtion CPU memory Exmple: f ( x) x temporry memory temporry memory input memory input memory CPU CPU Progrm memory output memory Progrm memory compute x x output memory compute x x 4 temporry memory f ( x) x input memory x temporry memory z * 4 f ( x) z * 8 f ( x) x input memory x CPU CPU Progrm memory compute x x output memory Progrm memory compute x x output memory compute x x compute x x 5 6

2 temporry memory z * 4 f ( x) z * 8 f ( x) x input memory x Automton temporry memory Automton input memory Progrm memory compute CPU x x f ( x) 8 output memory CPU Progrm memory output memory compute x x 7 8 Different Kinds of Automt Automt re distinguished y the temporry memory Finite Automton temporry memory Finite Automt: no temporry memory Pushdown Automt: Turing Mchines: stck rndom ccess memory Finite Automton input memory output memory 9 Exmple: Vending Mchines (smll computing power) Pushdown Automton Turing Mchine Stck Push, Pop Rndom Access Memory Pushdown input memory Turing input memory Automton output memory Mchine output memory Exmple: Compilers for Progrmming Lnguges Exmples: Any Algorithm (medium computing power) (highest computing power)

3 Power of Automt Finite Automt Pushdown Automt Turing Mchine Lnguges Less power Solve more More power computtionl prolems 4 A lnguge is set of strings Alphets nd Strings We will use smll lphets:, String: A seuence of letters Exmples: ct, dog, house, Defined over n lphet:,, c,, z Strings u v w 5 6 String Opertions w v m n w n Conctention Reverse wv n m w R n 7 8

4 Length: Exmples: String Length w w n n 4 Exmple: Length of Conctention uv u v u, u v, v 5 uv 8 uv u v Empty String A string with no letters: Sustring Sustring of string: suseuence of consecutive chrcters Oservtions: w w w String Sustring Prefix nd Suffix Prefixes Suffixes w uv prefix suffix Exmple: Definition: Another Opertion n w ww w n w 4 4

5 * * The * Opertion : the set of ll possile strings from lphet,,,,,,,,,, Lnguges A lnguge is ny suset of * Exmple: *,,,,,,,,, 5 Lnguges:,, {,,,,, } 6 Sets Note tht: { } { } Another Exmple An infinite lnguge n n L { : n } Set size Set size String length {} { } L L 7 8 Opertions on Lnguges The usul set opertions,,, {,,,,, { },,,, Complement: L * L,,,,,,,, } Definition: Exmples: L R Reverse R { w : w L} R,,,, L { L R n n n { : n } n : n } 9 5

6 Definition: Exmple: L Conctention L xy x L,,,, : y L,,,,, Another Opertion Definition: n L LL L n,,,,,,,,, Specil cse: L,,,, More Exmples Str-Closure (Kleene *) n n L { : n } Definition: L* L L L L n n m m { : n, m } L Exmple:,,,, *,,,,,,,, 4 Finite Automton Finite Automt Input String Finite Automton Output Accept or Reject 5 6 6

7 initil stte Trnsition Grph 4 stte trnsition, 5, ccepting stte Initil Configurtion Input String, 5, Reding the Input,, 5, 4 5, 4 9 4,, 5, 4 5,

8 Input finished Rejection,, 5, 4 5, 4 ccept 4 44,, 5, 4 5, Input finished, 5, 4, 5 4 reject,

9 Another Rejection,, 5, 4 5, 4 reject 49 5 Another Exmple,,,, 5 5,,,,

10 Input finished Rejection Exmple ccept,,,, 55 56,,,, Input finished,,,, reject 59 6

11 FA Lnguges Accepted y FAs M M Exmple L M Definition: The lnguge LM contins ll input strings ccepted y L M = { strings tht ring M to n ccepting stte} M, 5, 4 ccept 6 6 Exmple M, L, 5 M,, 4 ccept ccept ccept L Exmple n M { : n },, ccept trp stte 6 64 Exmple L M = { ll strings with prefix }, ccept L M Exmple = { ll strings without sustring },, 65 66

12 Exmple L( M ) w : w, * 4 Regulr Lnguges Definition: A lnguge L is regulr if there is FA such tht M L LM Oservtion: All lnguges ccepted y FAs form the fmily of regulr lnguges, Exmples of regulr lnguges:,, w : w, * { n : n } { ll strings with prefix } { ll strings without sustring } Non-Deterministic Finite Automt There exist utomt tht ccept these Lnguges Nondeterministic Finite Automton (NFA) Alphet = {} Alphet = {} Two choices 7 7

13 First Choice Alphet = {} Two choices No trnsition No trnsition 7 74 First Choice First Choice First Choice Second Choice All input is consumed ccept 77 78

14 Second Choice Second Choice No trnsition: the utomton hngs 79 8 Second Choice Input cnnot e consumed reject An NFA ccepts string: when there is computtion of the NFA tht ccepts the string There is computtion: ll the input is consumed nd the utomton is in n ccepting stte 8 8 Exmple Rejection exmple is ccepted y the NFA: ccept ecuse this computtion ccepts reject

15 First Choice First Choice reject Second Choice Second Choice Second Choice An NFA rejects string: when there is no computtion of the NFA tht ccepts the string. reject For ech computtion: All the input is consumed nd the utomton is in non finl stte OR The input cnnot e consumed

16 Exmple Rejection exmple is rejected y the NFA: reject reject All possile computtions led to rejection 9 9 First Choice First Choice No trnsition: the utomton hngs 9 94 First Choice Second Choice Input cnnot e consumed reject

17 Second Choice Second Choice No trnsition: the utomton hngs Second Choice Input cnnot e consumed is rejected y the NFA: reject reject reject 99 All possile computtions led to rejection Lnguge ccepted: L {} One pth from to n ccepting stte suffices w L M w w w i k j 7

18 Lmd Trnsitions 4 (red hed does not move) 5 6 ll input is consumed ccept String is ccepted 7 8 8

19 Rejection Exmple 9 (red hed doesn t move) No trnsition: the utomton hngs Input cnnot e consumed Lnguge ccepted: L {} reject String is rejected 4 9

20 Another NFA Exmple Another String ccept 9

21 4 5 6

22 Lnguge ccepted ccept L,,, Another NFA Exmple, Lnguge ccepted { } L(M ) = λ,,,,... = { }*, (redundnt stte) 9 Remrks: The symol never ppers on the input tpe Lnguges ccepted y NFAs Theorem: NFAs nd FAs hve the sme computtion power Regulr Lnguges Lnguges ccepted y FAs

23 We cn show: Proof-Step Lnguges ccepted y NFAs Regulr Lnguges Lnguges ccepted y NFAs Regulr Lnguges Lnguges ccepted y NFAs Regulr Lnguges Proof: Every FA is trivilly n NFA L Any lnguge ccepted y FA is lso ccepted y n NFA 4 Proof-Step Lnguges ccepted y NFAs Regulr Lnguges Proof: Any NFA cn e converted to n euivlent FA Properties of Regulr Lnguges L Any lnguge ccepted y n NFA is lso ccepted y FA 5 6 For regulr lnguges L nd L : We sy: Regulr lnguges re closed under Union: L L Union: L L Conctention: L L Conctention: L L Str: Reversl: L * R L Are regulr Lnguges Str: Reversl: L * R L Complement: L Complement: L Intersection: L L 7 Intersection: L L 8