# This lecture covers Chapter 8 of HMU: Properties of CFLs

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 This lecture covers Chpter 8 of HMU: Properties of CFLs Turing Mchine Extensions of Turing Mchines Restrictions of Turing Mchines Additionl Reding: Chpter 8 of HMU.

2 Turing Mchine: Informl Definition B B B B c b b b B B An tpe extending infinitely in both sides A reding hed tht cn edit tpe, move right or left. A finite control. A string is ccepted if finite control reches finl/ccepting stte 2 / 22

3 Turing Mchine: Forml Definition A Turing mchine M = (Q, Σ, Γ, δ, q 0, B, F ) such tht: Q: finite set of sttes Σ: finite set of input symbols Γ: finite set of tpe symbols such tht Σ Γ δ: trnsition function. δ is prtil function over Q Γ, where the first component is viewed s the present stte, nd the second is viewed s the tpe symbol red. If δ(q, X ) is defined, then Tpe symbol Next Stte Reding hed direction to move next Present stte (q;x) =(q 0 ;Y;D) The symbol replcing X B Γ \ Σ is the blnk symbol. All but finite number of tpe symbols re Bs. q 0: the initil stte of the TM. F : the set of finl/ccepting sttes fo the TM. Hed lwys moves to the left or right. Being sttionry is not n option. The Turing Mchine is deterministic. 3 / 22

4 Describing TMs Turing mchines cn be defined by describing δ using trnsition tble. They cn lso be defined using trnsition digrms (with lbels ppropritely ltered) A TM tht ccepts ny binry string tht does not contin 111 0=0! 0=0! 1=1! q 1 q 0 B=B! 1=1! B=B! 0=0! B=B! q f q 3 1=1! q 2 4 / 22

5 Instntneous Descriptions of TMs An instntneous description (or configurtion) of TM is complete description of the system tht enbles one to determine the trjectory of the TM s it opertes. The instntneous description or configurtion or ID of TM contins 3 prts: () The (finite, non-trivil) portion of tpe to the left of the reding hed; (b) the stte tht the TM is presently in; nd (c) the (finite, non-trivil) portion of the tpe to the right of the reding hed. Stte, Tpe contents, Reding hed loction 1 pple i pple q B B X 1 X 2 X 3 X i X B B ID segment to the strict left stte segment from the hed onwrds z } { z} { z } { X 1 X i 1 q X i X hed i Blnks z } { q B B X 1 X 2 X 3 X B B segment to the strict left z } { stte X 1 X B i 1 z} { q hed z i Blnks } { q B B X 1 X 2 X 3 X B B hed stte z} { q segment from the hed onwrds z } { B i X 1 X 5 / 22

6 Moves of TM Just s in the cse of PDA, we use to indicte single move of TM M, nd M M to indicte zero or finite number of moves of TM. Present ID Trnsition Next ID X 1 X i 1 qx i X (1 <i< ) X 1 X B i 1 q qb i X 1 :::X (q;x i )=(q 0 ;Y;R) (q;x i )=(q 0 ;Y;L) (q;b) =(q 0 ;Y;R) (q;b) =(q 0 ;Y;L) (q;b) =(q 0 ;Y;R) (q;b) =(q 0 ;Y;L) X 1 X i 1 Yq 0 X i+1 X X 1 X i 2 q 0 X i 1 YX i+1 X { X1 { { X 1 X B i 1 Yq 0 X 1 X 1 q 0 X Y i =1 X B i 2 q 0 BY i > 1 Yq 0 X 2 X i =0 Yq 0 B i 1 X 1 X i>0 q 0 BY X 2 X i =0 q 0 BY B i 1 X 1 X i>0 6 / 22

7 Lnguge ccepted by TM A string w is in the lnguge ccepted by TM M iff q 0w αpβ for some p F. M Another notion of cceptnce tht is common is to require TM to hlt (i.e., no further trnsitions re possible). It is lwys possible to design TM such tht the TM hlts when it reches finl stte without chnging the lnguge the TM ccepts. However, we cnnot require (ll) TMs to hlt for ll inputs. A lnguge L is sid to be recursively enumerble if it is ccepted by some TM. A lnguge L is sid to be recursive if both L nd L c re recursively enumerble. Regulr Context-free Recursive Recursively Enumerble (RE) 7 / 22

8 Extensions of TMs Extensions of TMs 8 / 22

9 Extensions of TMs Multiple-Trck TMs Multiple-trck TM There re k trcks, ech hving symbols written on them. The mchine cn only red symbols from ech tpe corresponding to one loction, i.e., ll symbols in column t ny one time. A k-trck TM with tpe lphbet Γ hs the sme lnguge-cceptnce power s TM with tpe lphbet Γ k. X 1 X 2. X k 9 / 22

10 Extensions of TMs Multi-tpe TMs Multiple-tpe TM There re k tpes, ech hving symbols written on them. The mchine cn ech tpe independently, i.e., the symbols red from ech tpe need not correspond to the sme loction After red of ech tpes, ech reding hed cn move independently to the right, left, or sty sttionry. X 1 X k X / 22

11 Extensions of TMs Multi-tpe TMs Theorem Every lnguge tht is ccepted by multi-tpe TM is lso recursively enumerble (i.e., ccepted by some stndrd TM). Proof of Theorem Let L be ccepted by k-tpe TM M. We ll devise 2k-trck TM M tht ccepts L. Every even tpe of M hs the sme lphbet s tht of the k-tpe TM. The 2i th trck of M contins exctly the sme contents s the i th tpe of M. Every odd trck hs n lphbet {B,, }, nd contins single or ; the 2i 1 th trck of M contins or t the loction where the i th reding hed of M is locted. M M / 22

12 Extensions of TMs Multi-tpe TMs Proof of Theorem The stte of M hs 3 components: () the stte of M; (b) the number of s to its strict left; nd (c) vector of length k with ech component tking vlue in Γ {?}. Ech move of M corresponds to multiple moves of M. Ech move of M corresponds of sweep of the tpe from the loction of the leftmost dgger to tht of the rightmost dgger nd bck performing the chnges to trcks tht M would do to its corresponding tpes. At the beginning of the sweep, the hed of M is t loction where the leftmost is nd the stte of M is (q, 0, [?,,?]). The hed moves to the right uncovering s nd the corresponding trck symbols. The right sweep ends when the second component is k M Stte: (q ; 0; [0; 1; 1]) Stte: q 12 / 22 M

13 Extensions of TMs Multi-tpe TMs Proof of Theorem At this stge, M knows the input symbols M will hve red, nd knows wht ctions to tke. It then sweeps left mking pproprite chnges to the trcks (just like M does to its tpe) ech time is encountered. M lso moves the s ccordingly nd lters it to to indicte tht it hs processed this trck. The left sweep ends when the second component is zero. At this time, M would hve completed moving the s nd the trck contents; they ll now mtch those of M. The TM then sweeps right nd returns reverting ech bck to. M then moves the stte to (q, 0, [?,,?]) nd strt the next sweep if q is not finl stte. Note tht M mimics M nd hence the lnguges ccepted re identicl. 13 / 22

14 Extensions of TMs Multi-tpe TMs The running time of TM M with input w is the number of moves M mkes before it hlts. (If it does not, the running time is ). The time complexity T M : {0, 1,...} {0, 1,...} of TM M is defined s follows: T M (n) := mximum running time of M for n input w of length n symbols. Theorem The time tken for M in Theorem to process n moves of M is O(n 2 ). Outline of Proof of Theorem After n moves of M, ny two heds of M cn be t most 2n loctions prt. Ech sweep then requires 8n moves of M. Ech trck updte requires finite number of moves. Totlly, to updte the trcks, Θ(k) time steps re needed. Loction of tpe heds n 0 n!!!! 0 Moves of M n 2n prt 14 / 22

15 Extensions of TMs Non-deterministic TMs Non-deterministic TM: δ(q, X ) is set of triples representing possible moves. Theorem For every non-deterministic TM N, there is TM M such tht L(M) = L(N). Outline of Proof of Theorem ID 1 (N does Bredth-First explortion of IDs of M) ID 2;1 ID 2;2 ID 2;k ID 3;1 ID 3;2 ID 3;3 ID 3;4 ID 3; Tpe 1 ID 1 ID 1 ID 2;1 ID 2;2 ID 2;k (If M does not hlt t ID1) (If M does not hlt t ID1 nd ID2;1) ID 1 ID 2;1 ID 2;2 ID 2;k ID 3;1 ID 3;2 (If M does not hlt t ID1, ID2;1 nd ID2;2) ID 1 ID 2;1 ID 2;2 ID 2;k ID 3;1 ID 3;2 ID 3;3 ID 3;4 15 / 22

16 Extensions of TMs Outline of Proof of Theorem We cn devise 2-tpe TM M tht simultes N. M first replces the content of the first tpe by followed by the ID tht N is initilly in, which is then followed by specil symbol, which serves s ID seprtor. (M uses the second tpe s scrtch tpe to enble this opertion). If the ID corresponds to finl stte, N hlts (s would M). If not, M then identifies ll possible choices for the next IDs for N nd enters ech one of them followed by t the right end of it s first tpe. (Agin, M uses the second tpe s scrtch tpe to enble this opertion) M then serches for to the right of, chnges the to (to signify tht it is processing the succeeding ID), nd processes tht ID in the similr wy. M hlts t n ID it iff M would t tht ID. 16 / 22

17 Restrictions of TMs Restrictions of TMs 17 / 22

18 Restrictions of TMs TM Semi-infinite Tpe Theorem Every recursively enumerble lnguge is lso ccepted by TM with semi-infinite tpe. Outline of Proof of Theorem Given TM M tht ccepts lnguge L, construct two-trck TM M s follows. The first nd second trcks of M re the R nd L semi-infinite prts of the tpe of M. First, write specil symbol, sy t the leftmost prt of the second trck; this indictes to M tht left move is not to be ttempted t this loction. At ny time, M keeps trck of whether M is to the right or left of its strt loction. If M is to the strict right of its strt loction, M mimics M on the first trck. If M is to the strict left of its strt loction, M mimics M on second trck, but with the hed directions reversed. M detects the strt by the symbol. It cn be formlly shown tht M ccepts string iff M ccepts it. M L \$ R B B 2 L \$ R b b b M b B B 1 2 L \$ R R \$ L 18 / 22

19 Restrictions of TMs Multi-stck Mchines A multistck mchne is PDA with severl independent stcks (i.e., one cn be popping symbol, while the other is writing symbol). Theorem Every recursively enumerble lnguge is ccepted by two-stck PDA Outline of Proof of Theorem TM PDA PDA PDA B b B b b b S S 1 2 S b 3 b R semi-infinite portion of TM s tpe Strict L semi-infinite portion of TM s tpe indictes the end of the stck content (to prevent PDA from hlting) If TM moves right chnging tpe symbol X to Y nd stte from q to q, PDA moves from stte q to q popping X from left stck nd pushing Y to the right stck. 19 / 22

20 Restrictions of TMs Counter Mchines A counter mchine is multi-stck mchine whose stck lphbet contins two symbols: Z 0 (stck end mrker) nd X Z 0 is initilly in the stck. Z 0 my be replced by X i Z 0 for ny i 0 X my be replced by X i for ny i 0. A counter mchine effectively stores non-negtive number. X X X Z 0 X X X X X Z 0 20 / 22

21 Restrictions of TMs Counter Mchines Theorem Every recursively enumerble lnguge is ccepted by three-counter mchine Outline of Proof of Theorem We know two-stck PDA cn simulte ny TM. We ll show tht 3-counter mchine cn simulte ny PDA. WLOG, let the stck lphbet of Γ = {0, 1,..., r}. Suppose the first stck contins Y 1(top),..., Y k. Then the first counter stores Y 1 + ry r k 1 Y k. Similrly for the second stck. The third counter is used to chnge the two stck contents. Popping the top symbol stck (sy A) = finding quotient when Y 1 + ry r k 1 Y k is divided by r. pop r X s from stck A, nd push single X on the third stck. Repet until ll X s re exhusted on the stck where popping is performed. Now empty stck A nd copy the third stck contents onto stck A. Chnge Y 1 to some Y 1 requires dding or subtrcting, which is done by popping or pushing the corresponding number of X s. 21 / 22

22 Restrictions of TMs Counter Mchines Outline of Proof of Theorem pushing symbol Z onto stck (sy A) = compute rc + Z where C is the number presently stored in the stck A. pop one X from stck A, nd push r X s on the third stck. Finlly push Z X s onto the third stck. Now empty stck A nd copy the third stck contents onto stck A. Since the bove three re the only opertions needed to simulte TM on two-stck PDA, we cn stimulte 2-stck PDA nd hence TM using 3-counter mchine. Theorem Every recursively enumerble lnguge is ccepted by two-counter mchine Outline of Proof of Theorem The key ide: simulte three counters using one, nd use the other for mnipultions. The first counter stores 2 i 3 j 5 k where i, j, k re the contents of the 3-counter mchine. Updtes to the stck involve: () divide by 2,3, or 5; (b) multiply by 2,3, or 5; or (c) identify if i or j or k is zero (check divisibility). Ech opertion cn be esily seen to be done with spre counter. 22 / 22

### 5.1 Definitions and Examples 5.2 Deterministic Pushdown Automata

CSC4510 AUTOMATA 5.1 Definitions nd Exmples 5.2 Deterministic Pushdown Automt Definitions nd Exmples A lnguge cn be generted by CFG if nd only if it cn be ccepted by pushdown utomton. A pushdown utomton

### CS S-12 Turing Machine Modifications 1. When we added a stack to NFA to get a PDA, we increased computational power

CS411-2015S-12 Turing Mchine Modifictions 1 12-0: Extending Turing Mchines When we dded stck to NFA to get PDA, we incresed computtionl power Cn we do the sme thing for Turing Mchines? Tht is, cn we dd

### Recursively Enumerable and Recursive. Languages

Recursively Enumerble nd Recursive nguges 1 Recll Definition (clss 19.pdf) Definition 10.4, inz, 6 th, pge 279 et S be set of strings. An enumertion procedure for Turing Mchine tht genertes ll strings

### Section: Other Models of Turing Machines. Definition: Two automata are equivalent if they accept the same language.

Section: Other Models of Turing Mchines Definition: Two utomt re equivlent if they ccept the sme lnguge. Turing Mchines with Sty Option Modify δ, Theorem Clss of stndrd TM s is equivlent to clss of TM

### Chapter 2 Finite Automata

Chpter 2 Finite Automt 28 2.1 Introduction Finite utomt: first model of the notion of effective procedure. (They lso hve mny other pplictions). The concept of finite utomton cn e derived y exmining wht

### State Minimization for DFAs

Stte Minimiztion for DFAs Red K & S 2.7 Do Homework 10. Consider: Stte Minimiztion 4 5 Is this miniml mchine? Step (1): Get rid of unrechle sttes. Stte Minimiztion 6, Stte is unrechle. Step (2): Get rid

### CS 373, Spring Solutions to Mock midterm 1 (Based on first midterm in CS 273, Fall 2008.)

CS 373, Spring 29. Solutions to Mock midterm (sed on first midterm in CS 273, Fll 28.) Prolem : Short nswer (8 points) The nswers to these prolems should e short nd not complicted. () If n NF M ccepts

### Lexical Analysis Finite Automate

Lexicl Anlysis Finite Automte CMPSC 470 Lecture 04 Topics: Deterministic Finite Automt (DFA) Nondeterministic Finite Automt (NFA) Regulr Expression NFA DFA A. Finite Automt (FA) FA re grph, like trnsition

### 1 Nondeterministic Finite Automata

1 Nondeterministic Finite Automt Suppose in life, whenever you hd choice, you could try oth possiilities nd live your life. At the end, you would go ck nd choose the one tht worked out the est. Then you

### CSCI FOUNDATIONS OF COMPUTER SCIENCE

1 CSCI- 2200 FOUNDATIONS OF COMPUTER SCIENCE Spring 2015 My 7, 2015 2 Announcements Homework 9 is due now. Some finl exm review problems will be posted on the web site tody. These re prcqce problems not

### a,b a 1 a 2 a 3 a,b 1 a,b a,b 2 3 a,b a,b a 2 a,b CS Determinisitic Finite Automata 1

CS4 45- Determinisitic Finite Automt -: Genertors vs. Checkers Regulr expressions re one wy to specify forml lnguge String Genertor Genertes strings in the lnguge Deterministic Finite Automt (DFA) re nother

### 12.1 Nondeterminism Nondeterministic Finite Automata. a a b ε. CS125 Lecture 12 Fall 2016

CS125 Lecture 12 Fll 2016 12.1 Nondeterminism The ide of nondeterministic computtions is to llow our lgorithms to mke guesses, nd only require tht they ccept when the guesses re correct. For exmple, simple

### CS 311 Homework 3 due 16:30, Thursday, 14 th October 2010

CS 311 Homework 3 due 16:30, Thursdy, 14 th Octoer 2010 Homework must e sumitted on pper, in clss. Question 1. [15 pts.; 5 pts. ech] Drw stte digrms for NFAs recognizing the following lnguges:. L = {w

### How to simulate Turing machines by invertible one-dimensional cellular automata

How to simulte Turing mchines by invertible one-dimensionl cellulr utomt Jen-Christophe Dubcq Déprtement de Mthémtiques et d Informtique, École Normle Supérieure de Lyon, 46, llée d Itlie, 69364 Lyon Cedex

### Harvard University Computer Science 121 Midterm October 23, 2012

Hrvrd University Computer Science 121 Midterm Octoer 23, 2012 This is closed-ook exmintion. You my use ny result from lecture, Sipser, prolem sets, or section, s long s you quote it clerly. The lphet is

### Nondeterminism. Nondeterministic Finite Automata. Example: Moves on a Chessboard. Nondeterminism (2) Example: Chessboard (2) Formal NFA

Nondeterminism Nondeterministic Finite Automt Nondeterminism Subset Construction A nondeterministic finite utomton hs the bility to be in severl sttes t once. Trnsitions from stte on n input symbol cn

### Automata and Languages

Automt nd Lnguges Prof. Mohmed Hmd Softwre Engineering Lb. The University of Aizu Jpn Grmmr Regulr Grmmr Context-free Grmmr Context-sensitive Grmmr Regulr Lnguges Context Free Lnguges Context Sensitive

### Non-deterministic Finite Automata

Non-deterministic Finite Automt From Regulr Expressions to NFA- Eliminting non-determinism Rdoud University Nijmegen Non-deterministic Finite Automt H. Geuvers nd J. Rot Institute for Computing nd Informtion

### 1 From NFA to regular expression

Note 1: How to convert DFA/NFA to regulr expression Version: 1.0 S/EE 374, Fll 2017 Septemer 11, 2017 In this note, we show tht ny DFA cn e converted into regulr expression. Our construction would work

### Name Ima Sample ASU ID

Nme Im Smple ASU ID 2468024680 CSE 355 Test 1, Fll 2016 30 Septemer 2016, 8:35-9:25.m., LSA 191 Regrding of Midterms If you elieve tht your grde hs not een dded up correctly, return the entire pper to

### Regular Expressions (RE) Regular Expressions (RE) Regular Expressions (RE) Regular Expressions (RE) Kleene-*

Regulr Expressions (RE) Regulr Expressions (RE) Empty set F A RE denotes the empty set Opertion Nottion Lnguge UNIX Empty string A RE denotes the set {} Alterntion R +r L(r ) L(r ) r r Symol Alterntion

### CHAPTER 1 Regular Languages. Contents

Finite Automt (FA or DFA) CHAPTE 1 egulr Lnguges Contents definitions, exmples, designing, regulr opertions Non-deterministic Finite Automt (NFA) definitions, euivlence of NFAs nd DFAs, closure under regulr

### Types of Finite Automata. CMSC 330: Organization of Programming Languages. Comparing DFAs and NFAs. NFA for (a b)*abb.

CMSC 330: Orgniztion of Progrmming Lnguges Finite Automt 2 Types of Finite Automt Deterministic Finite Automt () Exctly one sequence of steps for ech string All exmples so fr Nondeterministic Finite Automt

### Chapter 1, Part 1. Regular Languages. CSC527, Chapter 1, Part 1 c 2012 Mitsunori Ogihara 1

Chpter 1, Prt 1 Regulr Lnguges CSC527, Chpter 1, Prt 1 c 2012 Mitsunori Ogihr 1 Finite Automt A finite utomton is system for processing ny finite sequence of symols, where the symols re chosen from finite

### The University of Nottingham

The University of Nottinghm SCHOOL OF COMPUTR SCINC AND INFORMATION TCHNOLOGY A LVL 1 MODUL, SPRING SMSTR 2004-2005 MACHINS AND THIR LANGUAGS Time llowed TWO hours Cndidtes must NOT strt writing their

### Context-Free Grammars and Languages

Context-Free Grmmrs nd Lnguges (Bsed on Hopcroft, Motwni nd Ullmn (2007) & Cohen (1997)) Introduction Consider n exmple sentence: A smll ct ets the fish English grmmr hs rules for constructing sentences;

### Normal Forms for Context-free Grammars

Norml Forms for Context-free Grmmrs 1 Linz 6th, Section 6.2 wo Importnt Norml Forms, pges 171--178 2 Chomsky Norml Form All productions hve form: A BC nd A vrile vrile terminl 3 Exmples: S AS S AS S S

### input tape head moves current state

CPS 140 - Mthemticl Foundtions of CS Dr. Susn Rodger Section: Finite Automt (Ch. 2) (lecture notes) Things to do in clss tody (Jn. 13, 2004): ffl questions on homework 1 ffl finish chpter 1 ffl Red Chpter

### Homework 4. 0 ε 0. (00) ε 0 ε 0 (00) (11) CS 341: Foundations of Computer Science II Prof. Marvin Nakayama

CS 341: Foundtions of Computer Science II Prof. Mrvin Nkym Homework 4 1. UsetheproceduredescriedinLemm1.55toconverttheregulrexpression(((00) (11)) 01) into n NFA. Answer: 0 0 1 1 00 0 0 11 1 1 01 0 1 (00)

### CS 310 (sec 20) - Winter Final Exam (solutions) SOLUTIONS

CS 310 (sec 20) - Winter 2003 - Finl Exm (solutions) SOLUTIONS 1. (Logic) Use truth tles to prove the following logicl equivlences: () p q (p p) (q q) () p q (p q) (p q) () p q p q p p q q (q q) (p p)

### Homework 3 Solutions

CS 341: Foundtions of Computer Science II Prof. Mrvin Nkym Homework 3 Solutions 1. Give NFAs with the specified numer of sttes recognizing ech of the following lnguges. In ll cses, the lphet is Σ = {,1}.

### Grammar. Languages. Content 5/10/16. Automata and Languages. Regular Languages. Regular Languages

5//6 Grmmr Automt nd Lnguges Regulr Grmmr Context-free Grmmr Context-sensitive Grmmr Prof. Mohmed Hmd Softwre Engineering L. The University of Aizu Jpn Regulr Lnguges Context Free Lnguges Context Sensitive

### CS375: Logic and Theory of Computing

CS375: Logic nd Theory of Computing Fuhu (Frnk) Cheng Deprtment of Computer Science University of Kentucky 1 Tle of Contents: Week 1: Preliminries (set lger, reltions, functions) (red Chpters 1-4) Weeks

### 20 MATHEMATICS POLYNOMIALS

0 MATHEMATICS POLYNOMIALS.1 Introduction In Clss IX, you hve studied polynomils in one vrible nd their degrees. Recll tht if p(x) is polynomil in x, the highest power of x in p(x) is clled the degree of

### 3 Regular expressions

3 Regulr expressions Given n lphet Σ lnguge is set of words L Σ. So fr we were le to descrie lnguges either y using set theory (i.e. enumertion or comprehension) or y n utomton. In this section we shll

### SOLUTIONS FOR ADMISSIONS TEST IN MATHEMATICS, COMPUTER SCIENCE AND JOINT SCHOOLS WEDNESDAY 5 NOVEMBER 2014

SOLUTIONS FOR ADMISSIONS TEST IN MATHEMATICS, COMPUTER SCIENCE AND JOINT SCHOOLS WEDNESDAY 5 NOVEMBER 014 Mrk Scheme: Ech prt of Question 1 is worth four mrks which re wrded solely for the correct nswer.

### Chapter 4 Contravariance, Covariance, and Spacetime Diagrams

Chpter 4 Contrvrince, Covrince, nd Spcetime Digrms 4. The Components of Vector in Skewed Coordintes We hve seen in Chpter 3; figure 3.9, tht in order to show inertil motion tht is consistent with the Lorentz

### The Wave Equation I. MA 436 Kurt Bryan

1 Introduction The Wve Eqution I MA 436 Kurt Bryn Consider string stretching long the x xis, of indeterminte (or even infinite!) length. We wnt to derive n eqution which models the motion of the string

### Lecture 9: LTL and Büchi Automata

Lecture 9: LTL nd Büchi Automt 1 LTL Property Ptterns Quite often the requirements of system follow some simple ptterns. Sometimes we wnt to specify tht property should only hold in certin context, clled

### Chapter 4 Regular Grammar and Regular Sets. (Solutions / Hints)

C K Ngpl Forml Lnguges nd utomt Theory Chpter 4 Regulr Grmmr nd Regulr ets (olutions / Hints) ol. (),,,,,,,,,,,,,,,,,,,,,,,,,, (),, (c) c c, c c, c, c, c c, c, c, c, c, c, c, c c,c, c, c, c, c, c, c, c,

### CDM Memoryless Machines

CDM Memoryless Mchines Zero Spce Klus Sutner Crnegie Mellon Universlity 2 Finite Stte Mchines Fll 27 3 DFA Decision Problems Where Are We? 3 Turing Mchines 4 We hve description of bstrct computbility in

### Math 8 Winter 2015 Applications of Integration

Mth 8 Winter 205 Applictions of Integrtion Here re few importnt pplictions of integrtion. The pplictions you my see on n exm in this course include only the Net Chnge Theorem (which is relly just the Fundmentl

### Agenda. Agenda. Regular Expressions. Examples of Regular Expressions. Regular Expressions (crash course) Computational Linguistics 1

Agend CMSC/LING 723, LBSC 744 Kristy Hollingshed Seitz Institute for Advnced Computer Studies University of Mrylnd HW0 questions? Due Thursdy before clss! When in doubt, keep it simple... Lecture 2: 6

### 1.3 Regular Expressions

56 1.3 Regulr xpressions These hve n importnt role in describing ptterns in serching for strings in mny pplictions (e.g. wk, grep, Perl,...) All regulr expressions of lphbet re 1.Ønd re regulr expressions,

### 7 Automata and formal languages. 7.1 Formal languages

7 Automt nd forml lnguges This exposition ws developed by Clemens Gröpl nd Knut Reinert. It is bsed on the following references, ll of which re recommended reding: 1. Uwe Schöning: Theoretische Informtik

### Automata Theory 101. Introduction. Outline. Introduction Finite Automata Regular Expressions ω-automata. Ralf Huuck.

Outline Automt Theory 101 Rlf Huuck Introduction Finite Automt Regulr Expressions ω-automt Session 1 2006 Rlf Huuck 1 Session 1 2006 Rlf Huuck 2 Acknowledgement Some slides re sed on Wolfgng Thoms excellent

### 1 Online Learning and Regret Minimization

2.997 Decision-Mking in Lrge-Scle Systems My 10 MIT, Spring 2004 Hndout #29 Lecture Note 24 1 Online Lerning nd Regret Minimiztion In this lecture, we consider the problem of sequentil decision mking in

### Acceptance Sampling by Attributes

Introduction Acceptnce Smpling by Attributes Acceptnce smpling is concerned with inspection nd decision mking regrding products. Three spects of smpling re importnt: o Involves rndom smpling of n entire

### PART 1 MULTIPLE CHOICE Circle the appropriate response to each of the questions below. Each question has a value of 1 point.

PART MULTIPLE CHOICE Circle the pproprite response to ech of the questions below. Ech question hs vlue of point.. If in sequence the second level difference is constnt, thn the sequence is:. rithmetic

### Matrices and Determinants

Nme Chpter 8 Mtrices nd Determinnts Section 8.1 Mtrices nd Systems of Equtions Objective: In this lesson you lerned how to use mtrices, Gussin elimintion, nd Guss-Jordn elimintion to solve systems of liner

### 16 Newton s Laws #3: Components, Friction, Ramps, Pulleys, and Strings

Chpter 16 Newton s Lws #3: Components, riction, Rmps, Pulleys, nd Strings 16 Newton s Lws #3: Components, riction, Rmps, Pulleys, nd Strings When, in the cse of tilted coordinte system, you brek up the

### Formal Languages and Automata Theory. D. Goswami and K. V. Krishna

Forml Lnguges nd Automt Theory D. Goswmi nd K. V. Krishn Novemer 5, 2010 Contents 1 Mthemticl Preliminries 3 2 Forml Lnguges 4 2.1 Strings............................... 5 2.2 Lnguges.............................

### Speech Recognition Lecture 2: Finite Automata and Finite-State Transducers

Speech Recognition Lecture 2: Finite Automt nd Finite-Stte Trnsducers Eugene Weinstein Google, NYU Cournt Institute eugenew@cs.nyu.edu Slide Credit: Mehryr Mohri Preliminries Finite lphet, empty string.

### Improper Integrals. The First Fundamental Theorem of Calculus, as we ve discussed in class, goes as follows:

Improper Integrls The First Fundmentl Theorem of Clculus, s we ve discussed in clss, goes s follows: If f is continuous on the intervl [, ] nd F is function for which F t = ft, then ftdt = F F. An integrl

### Homework Solution - Set 5 Due: Friday 10/03/08

CE 96 Introduction to the Theory of Computtion ll 2008 Homework olution - et 5 Due: ridy 10/0/08 1. Textook, Pge 86, Exercise 1.21. () 1 2 Add new strt stte nd finl stte. Mke originl finl stte non-finl.

### PHYSICS 211 MIDTERM I 21 April 2004

PHYSICS MIDERM I April 004 Exm is closed book, closed notes. Use only your formul sheet. Write ll work nd nswers in exm booklets. he bcks of pges will not be grded unless you so request on the front of

### NON-DETERMINISTIC FSA

Tw o types of non-determinism: NON-DETERMINISTIC FS () Multiple strt-sttes; strt-sttes S Q. The lnguge L(M) ={x:x tkes M from some strt-stte to some finl-stte nd ll of x is proessed}. The string x = is

### Math 100 Review Sheet

Mth 100 Review Sheet Joseph H. Silvermn December 2010 This outline of Mth 100 is summry of the mteril covered in the course. It is designed to be study id, but it is only n outline nd should be used s

### Where did dynamic programming come from?

Where did dynmic progrmming come from? String lgorithms Dvid Kuchk cs302 Spring 2012 Richrd ellmn On the irth of Dynmic Progrmming Sturt Dreyfus http://www.eng.tu.c.il/~mi/cd/ or50/1526-5463-2002-50-01-0048.pdf

### USA Mathematical Talent Search Round 1 Solutions Year 21 Academic Year

1/1/21. Fill in the circles in the picture t right with the digits 1-8, one digit in ech circle with no digit repeted, so tht no two circles tht re connected by line segment contin consecutive digits.

### Time Optimal Control of the Brockett Integrator

Milno (Itly) August 8 - September, 011 Time Optiml Control of the Brockett Integrtor S. Sinh Deprtment of Mthemtics, IIT Bomby, Mumbi, Indi (emil : sunnysphs4891@gmil.com) Abstrct: The Brockett integrtor

### General idea LR(0) SLR LR(1) LALR To best exploit JavaCUP, should understand the theoretical basis (LR parsing);

Bottom up prsing Generl ide LR(0) SLR LR(1) LLR To best exploit JvCUP, should understnd the theoreticl bsis (LR prsing); 1 Top-down vs Bottom-up Bottom-up more powerful thn top-down; Cn process more powerful

### 1-Way Multihead Quantum Finite State Automata

Applied Mthemtics, 26, 7, 5-22 Published Online My 26 in SciRes. http://www.scirp.org/journl/m http://dx.doi.org/.4236/m.26.7988 -Wy Multihed Quntum Finite Stte Automt Debyn Gnguly, Kingshu Chtterjee,

### Lecture 7 notes Nodal Analysis

Lecture 7 notes Nodl Anlysis Generl Network Anlysis In mny cses you hve multiple unknowns in circuit, sy the voltges cross multiple resistors. Network nlysis is systemtic wy to generte multiple equtions

### 5.5 The Substitution Rule

5.5 The Substitution Rule Given the usefulness of the Fundmentl Theorem, we wnt some helpful methods for finding ntiderivtives. At the moment, if n nti-derivtive is not esily recognizble, then we re in

### Convex Sets and Functions

B Convex Sets nd Functions Definition B1 Let L, +, ) be rel liner spce nd let C be subset of L The set C is convex if, for ll x,y C nd ll [, 1], we hve 1 )x+y C In other words, every point on the line

### 4. GREEDY ALGORITHMS I

4. GREEDY ALGORITHMS I coin chnging intervl scheduling scheduling to minimize lteness optiml cching Lecture slides by Kevin Wyne Copyright 2005 Person-Addison Wesley http://www.cs.princeton.edu/~wyne/kleinberg-trdos

### NUMERICAL INTEGRATION. The inverse process to differentiation in calculus is integration. Mathematically, integration is represented by.

NUMERICAL INTEGRATION 1 Introduction The inverse process to differentition in clculus is integrtion. Mthemticlly, integrtion is represented by f(x) dx which stnds for the integrl of the function f(x) with

### 440-2 Geometry/Topology: Differentiable Manifolds Northwestern University Solutions of Practice Problems for Final Exam

440-2 Geometry/Topology: Differentible Mnifolds Northwestern University Solutions of Prctice Problems for Finl Exm 1) Using the cnonicl covering of RP n by {U α } 0 α n, where U α = {[x 0 : : x n ] RP

### Design and Analysis of Distributed Interacting Systems

Design nd Anlysis of Distriuted Intercting Systems Lecture 6 LTL Model Checking Prof. Dr. Joel Greenyer My 16, 2013 Some Book References (1) C. Bier, J.-P. Ktoen: Principles of Model Checking. The MIT

### MATH34032: Green s Functions, Integral Equations and the Calculus of Variations 1

MATH34032: Green s Functions, Integrl Equtions nd the Clculus of Vritions 1 Section 1 Function spces nd opertors Here we gives some brief detils nd definitions, prticulrly relting to opertors. For further

### DESCRIBING MOTION: KINEMATICS IN ONE DIMENSION

DESCRIBING MOTION: KINEMATICS IN ONE DIMENSION Responses to Questions. A cr speedometer mesures only speed. It does not give ny informtion bout the direction, so it does not mesure velocity.. If the velocity

### Line Integrals. Partitioning the Curve. Estimating the Mass

Line Integrls Suppose we hve curve in the xy plne nd ssocite density δ(p ) = δ(x, y) t ech point on the curve. urves, of course, do not hve density or mss, but it my sometimes be convenient or useful to

### How do we solve these things, especially when they get complicated? How do we know when a system has a solution, and when is it unique?

XII. LINEAR ALGEBRA: SOLVING SYSTEMS OF EQUATIONS Tody we re going to tlk out solving systems of liner equtions. These re prolems tht give couple of equtions with couple of unknowns, like: 6= x + x 7=

### Line and Surface Integrals: An Intuitive Understanding

Line nd Surfce Integrls: An Intuitive Understnding Joseph Breen Introduction Multivrible clculus is ll bout bstrcting the ides of differentition nd integrtion from the fmilir single vrible cse to tht of

### Math 113 Exam 2 Practice

Mth Em Prctice Februry, 8 Em will cover sections 6.5, 7.-7.5 nd 7.8. This sheet hs three sections. The first section will remind you bout techniques nd formuls tht you should know. The second gives number

### 200 points 5 Problems on 4 Pages and 20 Multiple Choice/Short Answer Questions on 5 pages 1 hour, 48 minutes

PHYSICS 132 Smple Finl 200 points 5 Problems on 4 Pges nd 20 Multiple Choice/Short Answer Questions on 5 pges 1 hour, 48 minutes Student Nme: Recittion Instructor (circle one): nme1 nme2 nme3 nme4 Write

### Calculus - Activity 1 Rate of change of a function at a point.

Nme: Clss: p 77 Mths Helper Plus Resource Set. Copright 00 Bruce A. Vughn, Techers Choice Softwre Clculus - Activit Rte of chnge of function t point. ) Strt Mths Helper Plus, then lod the file: Clculus

### { } = E! & \$ " k r t +k +1

Chpter 4: Dynmic Progrmming Objectives of this chpter: Overview of collection of clssicl solution methods for MDPs known s dynmic progrmming (DP) Show how DP cn be used to compute vlue functions, nd hence,

### DETERMINANTS. All Mathematical truths are relative and conditional. C.P. STEINMETZ

All Mthemticl truths re reltive nd conditionl. C.P. STEINMETZ 4. Introduction DETERMINANTS In the previous chpter, we hve studied bout mtrices nd lgebr of mtrices. We hve lso lernt tht system of lgebric

### Physics 9 Fall 2011 Homework 2 - Solutions Friday September 2, 2011

Physics 9 Fll 0 Homework - s Fridy September, 0 Mke sure your nme is on your homework, nd plese box your finl nswer. Becuse we will be giving prtil credit, be sure to ttempt ll the problems, even if you

### New Expansion and Infinite Series

Interntionl Mthemticl Forum, Vol. 9, 204, no. 22, 06-073 HIKARI Ltd, www.m-hikri.com http://dx.doi.org/0.2988/imf.204.4502 New Expnsion nd Infinite Series Diyun Zhng College of Computer Nnjing University

### Arithmetic & Algebra. NCTM National Conference, 2017

NCTM Ntionl Conference, 2017 Arithmetic & Algebr Hether Dlls, UCLA Mthemtics & The Curtis Center Roger Howe, Yle Mthemtics & Texs A & M School of Eduction Relted Common Core Stndrds First instnce of vrible

### STURM-LIOUVILLE BOUNDARY VALUE PROBLEMS

STURM-LIOUVILLE BOUNDARY VALUE PROBLEMS Throughout, we let [, b] be bounded intervl in R. C 2 ([, b]) denotes the spce of functions with derivtives of second order continuous up to the endpoints. Cc 2

### Software Engineering using Formal Methods

Softwre Engineering using Forml Methods Propositionl nd (Liner) Temporl Logic Wolfgng Ahrendt 13th Septemer 2016 SEFM: Liner Temporl Logic /GU 160913 1 / 60 Recpitultion: FormlistionFormlistion: Syntx,

### STEP FUNCTIONS, DELTA FUNCTIONS, AND THE VARIATION OF PARAMETERS FORMULA. 0 if t < 0, 1 if t > 0.

STEP FUNCTIONS, DELTA FUNCTIONS, AND THE VARIATION OF PARAMETERS FORMULA STEPHEN SCHECTER. The unit step function nd piecewise continuous functions The Heviside unit step function u(t) is given by if t

### approaches as n becomes larger and larger. Since e > 1, the graph of the natural exponential function is as below

. Eponentil nd rithmic functions.1 Eponentil Functions A function of the form f() =, > 0, 1 is clled n eponentil function. Its domin is the set of ll rel f ( 1) numbers. For n eponentil function f we hve.

GRADE Division WORKSHEETS Division division is shring nd grouping Division cn men shring or grouping. There re cndies shred mong kids. How mny re in ech shre? = 3 There re 6 pples nd go into ech bsket.

### DEFINITION OF ASSOCIATIVE OR DIRECT PRODUCT AND ROTATION OF VECTORS

3 DEFINITION OF ASSOCIATIVE OR DIRECT PRODUCT AND ROTATION OF VECTORS This chpter summrizes few properties of Cli ord Algebr nd describe its usefulness in e ecting vector rottions. 3.1 De nition of Associtive

### Prefix-Free Subsets of Regular Languages and Descriptional Complexity

Prefix-Free Susets of Regulr Lnguges nd Descriptionl Complexity Jozef Jirásek Jurj Šeej DCFS 2015 Prefix-Free Susets of Regulr Lnguges nd Descriptionl Complexity Jozef Jirásek, Jurj Šeej 1/22 Outline Mximl

### How Deterministic are Good-For-Games Automata?

How Deterministic re Good-For-Gmes Automt? Udi Boker 1, Orn Kupfermn 2, nd Mich l Skrzypczk 3 1 Interdisciplinry Center, Herzliy, Isrel 2 The Herew University, Isrel 3 University of Wrsw, Polnd Astrct

### MA123, Chapter 10: Formulas for integrals: integrals, antiderivatives, and the Fundamental Theorem of Calculus (pp.

MA123, Chpter 1: Formuls for integrls: integrls, ntiderivtives, nd the Fundmentl Theorem of Clculus (pp. 27-233, Gootmn) Chpter Gols: Assignments: Understnd the sttement of the Fundmentl Theorem of Clculus.

### CS12N: The Coming Revolution in Computer Architecture Laboratory 2 Preparation

CS2N: The Coming Revolution in Computer Architecture Lortory 2 Preprtion Ojectives:. Understnd the principle of sttic CMOS gte circuits 2. Build simple logic gtes from MOS trnsistors 3. Evlute these gtes

### Overview HC9. Parsing: Top-Down & LL(1) Context-Free Grammars (1) Introduction. CFGs (3) Context-Free Grammars (2) Vertalerbouw HC 9: Ch.

Overview H9 Vertlerouw H 9: Prsing: op-down & LL(1) do 3 mei 2001 56 heo Ruys h. 8 - Prsing 8.1 ontext-free Grmmrs 8.2 op-down Prsing 8.3 LL(1) Grmmrs See lso [ho, Sethi & Ullmn 1986] for more thorough

### Improper Integrals. Introduction. Type 1: Improper Integrals on Infinite Intervals. When we defined the definite integral.

Improper Integrls Introduction When we defined the definite integrl f d we ssumed tht f ws continuous on [, ] where [, ] ws finite, closed intervl There re t lest two wys this definition cn fil to e stisfied:

### arxiv: v1 [cs.fl] 30 Nov 2016

Some Subclsses of Liner Lnguges bsed on Nondeterministic Liner Automt Benjmín Bedregl Deprtmento de Informátic e Mtemátic Aplicd, Universidde Federl do Rio Grnde do Norte bedregl@dimp.ufrn.br rxiv:1611.10276v1

### CS 330 Formal Methods and Models

CS 330 Forml Methods nd Models Dn Richrds, section 003, George Mson University, Fll 2017 Quiz Solutions Quiz 1, Propositionl Logic Dte: Septemer 7 1. Prove (p q) (p q), () (5pts) using truth tles. p q

### MATH STUDENT BOOK. 10th Grade Unit 5

MATH STUDENT BOOK 10th Grde Unit 5 Unit 5 Similr Polygons MATH 1005 Similr Polygons INTRODUCTION 3 1. PRINCIPLES OF ALGEBRA 5 RATIOS AND PROPORTIONS 5 PROPERTIES OF PROPORTIONS 11 SELF TEST 1 16 2. SIMILARITY

### Surface Integrals of Vector Fields

Mth 32B iscussion ession Week 7 Notes Februry 21 nd 23, 2017 In lst week s notes we introduced surfce integrls, integrting sclr-vlued functions over prmetrized surfces. As with our previous integrls, we

### Nondeterministic Biautomata and Their Descriptional Complexity

Nondeterministic Biutomt nd Their Descriptionl Complexity Mrkus Holzer nd Sestin Jkoi Institut für Informtik Justus-Lieig-Universität Arndtstr. 2, 35392 Gießen, Germny 23. Theorietg Automten und Formle