1 From NFA to regular expression

Save this PDF as:
 WORD  PNG  TXT  JPG

Size: px
Start display at page:

Download "1 From NFA to regular expression"

Transcription

1 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 y llowing regulr expressions to e written on the edges of the DFA, nd then showing how one cn remove sttes from this generlized utomt (getting new equivlent utomt with the fewer sttes). In the end of this stte removl process, we will remin with generlized utomt with single il stte nd single ccepting stte, nd it would e then esy to convert it into single regulr expression. 1 From NFA to regulr expression 1.1 NFA A Generlized NFA onsider n NFA N where we llowed to write ny regulr expression on the edges, nd not only just symols. The utomt is llowed to trvel on n edge, if it cn mtches prefix of the unred input, to the regulr expression written on the edge. We will refer to such n utomt s NFA (generlized non-deterministic fe utomt [Don t you just love ll these shortcuts?]). Thus, the NFA on the right, ccepts the string, since A B B E. To simplify the discussion, we would enforce the following conditions: A () B E (1) There re trnsitions going from the il stte to ll other sttes, nd there re no trnsitions into the il stte. (2) There is single ccept stte tht hs only trnsitions coming into it (nd no outgoing trnsitions). (3) The ccept stte is distinct from the il stte. (4) Except for the il nd ccepting sttes, ll other sttes re connected to ll other sttes vi trnsition. In prticulr, ech stte hs trnsition to itself. When you cn not ctully go etween two sttes, NFA hs trnsitions lelled with, which will not mtch ny string of input chrcters. We do not hve to drw these trnsitions explicitly in the stte digrms. 1.2 Top-level outline of conversion We will convert DFA to regulr expression s follows: (A) onvert DFA to NFA, dding new il nd finl sttes. (B) emove ll sttes one-y-one, until we hve only the il nd finl sttes. 1

2 () Output regex is the lel on the (single) trnsition left in the NFA. (The word regex is just shortcut for regulr expression.) Lemm 1.1. A DFA M cn e converted into n equivlent NFA G. Proof: We cn consider M to e n NFA. Next, we dd specil il stte q tht is connected to the old il stte vi ε-trnsition. Next, we dd specil finl stte q fin, such tht ll the finl sttes of M re connected to q fin vi n ε-trnsition. The modified NFA M hs n il stte nd single finl stte, such tht no trnsition enters the il stte, nd no trnsition leves the finl stte, thus M comply with conditions (1 3) ove. Next, we consider ll pir of sttes x, y Q(M ), nd if x y there is no trnsition etween them, we introduce the trnsition. The resulting NFA G from M is now complint lso with condition (4). It is esy now to verify tht G is equivlent to the originl DFA M. We will remove ll the intermedite sttes from the GNFA, leving NFA with only il nd finl sttes, connected y one trnsition with (typiclly complex) lel on it. The equivlent regulr expression is ovious: the lel on the trnsition. some regex q S q F Lemm 1.2. Given NFA N with k = 2 sttes, one cn generte n equivlent regulr expression. Proof: A NFA with only two sttes (tht comply with conditions (1)-(4)) hve the following form. q S some regex q F The NFA hs single trnsition from the il stte to the ccepting stte, nd this trnsition hs the regulr expression ssocited with it. Since the il stte nd the ccepting stte do not hve self loops, we conclude tht N ccepts ll words tht mtches the regulr expression. Nmely, L(N) = L(). 1.3 Detils of ripping out stte We first descrie the construction. Since k > 2, there is t lest one stte in N which is not il or ccepting, nd let q rip denote this stte. We will rip this stte out of N nd fix the NFA, so tht we get NFA with one less stte. Trnsition pths going through q rip might come from ny of vriety of sttes q 1, q 2, etc. They might go from q rip to ny of nother set of sttes r 1, r 2, etc. For ech pir of sttes q i nd r i, we need to convert the trnsition through q rip into direct trnsition from q i to r i. q 1 q 2 q 3 q rip r 1 r 2 r 3 2

3 1.3.1 eworking connections for specific triple of sttes To understnd how this works, let us focus on the connections etween q rip nd two other specific sttes q in nd q out. Notice tht q in nd q out might e the sme stte, ut they oth hve to e different from q rip. The stte q rip hs self loop with regulr expression rip ssocited with it. So, consider frgment of n ccepting trce tht goes through q rip. It trnsition into q rip from stte q in with regulr expression in nd trvels out of q rip into stte q out on n edge with the ssocited regulr expression eing out. This trce, corresponds to the regulr expression in followed y 0 or more times of trveling on the self loop ( rip is used ech time we trverse the loop), nd then trnsition out to q out using the regulr expression out. As such, we cn introduce direct trnsition from q in to q out with the regulr expression = in ( rip ) out. rip lerly, ny frgment of trce trveling q in q in q rip q out cn e replced y the direct q rip q out trnsition q in q out. So, let us do this replcement for ny two such stges, we connect them directly vi new trnsition, so tht they no longer need to trvel through q rip. in ( rip ) out lerly, if we do tht for ll such pirs, the new utomt ccepts the sme lnguge, ut no longer need to use q rip. As such, we cn just remove q rip from the resulting utomt. And let M denote the resulting utomt. The utomt M is not quite legl, yet. Indeed, we will hve now prllel trnsitions ecuse of the ove process (we might even hve prllel self loops). But this is esy to fix: We replce two such prllel trnsitions q 1 i qj nd q 2 i qj, y single trnsition q i qj. As such, for the triple q in, q rip, q out, if the originl lel on the direct trnsition from q in to q out ws originlly dir, then the output lel for the new trnsition (tht skips q rip ) will e in out dir + in ( rip ) out. (1) lerly the new trnsition, is equivlent to the two trnsitions it replces. If we repet this process for ll the prllel trnsitions, we get new NFA M which hs k 1 sttes, nd furthermore it ccepts exctly the sme lnguge s N. 1.4 Proof of correctness of the ripping process Lemm 1.3. Given NFA N with k > 2 sttes, one cn generte n equivlent NFA M with k 1 sttes. Proof: Since k > 2, N contins lest one stte in N which is not ccepting, nd let q rip denote this stte. We will rip this stte out of N nd fix the NFA, so tht we get NFA with one less stte. For every pir of sttes q in nd q out, oth distinct from q rip, we replce the trnsitions tht go through q rip with direct trnsitions from q in to q out, s descried in the previous section. 3

4 orrectness. onsider n ccepting trce T for N for word w. If T does not use the stte q rip thn the sme trce exctly is n ccepting trce for M. So, ssume tht it uses q rip, in prticulr, the trce looks like T =... q i S i qrip 0 or more times {}}{ S i+1 S j 1 qrip... qrip S j 1 qj.... Where S i S i+1..., S j is sustring of w. lerly, S i in, where in is the regulr expression ssocited with the trnsition q i q rip. Similrly, S j 1 out, where out is the regulr expression ssocited with the trnsition q rip q j. Finlly, S i+1 S i+2 S j 1 ( rip ), where rip is the regulr expression ssocited with the self loop of q rip. Now, clerly, the string S i S i+1... S j mtches the regulr expression in ( out ) out. in prticulr, we cn replce this portion of the trce of T y S i S i+1... S j 1 S j T =... q i qj.... This trnsition is using the new trnsition etween q i nd q j introduced y our construction. epeting this replcement process in T till ll the ppernces of q rip re removed, results in n ccepting trce T of M. Nmely, we proved tht ny string ccepted y N is lso ccepted y M. We need lso to prove the other direction. Nmely, given n ccepting trce for M, we cn rewrite it into n equivlent trce of N which is ccepting. This is esy, nd done in similr wy to wht we did ove. Indeed, if portion of the trce uses new trnsition of M (tht does not pper in N), we cn plce it y frgment of trnsitions going through q rip. In light of the ove proof, it is esy nd we omit the strightforwrd ut tedious detils. Theorem 1.4. Any DFA cn e trnslted into n equivlent regulr expression. Proof: Indeed, convert the DFA into NFA N. As long s N hs more thn two sttes, reduce its numer of sttes y removing one of its sttes using Lemm 1.3. epet this process till N hs only two sttes. Now, we convert this NFA into n equivlent regulr expression using Lemm Exmples 2.1 Exmple: From NFA to regex in 8 esy figures 1: The originl NFA. A B 2: Normlizing it. A B A, + 4

5 3: emove stte A. A B A + 4: edrwn without old edges. B A + 5: emoving B. 6: edrwn. B + A + + A 7: emoving. ( +)( +) + 8: edrwn. ( +)( +) A + A Thus, this utomt is equivlent to the regulr expression ( + )( + ). 5

Homework 3 Solutions

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}.

More information

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

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)

More information

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

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.

More information

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

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

More information

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

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

More information

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

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

More information

Chapter 2 Finite Automata

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

More information

1 Nondeterministic Finite Automata

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

More information

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

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

More information

Non-deterministic Finite Automata

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

More information

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

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

More information

3 Regular expressions

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

More information

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

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

More information

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

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,

More information

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

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.

More information

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

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

More information

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

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

More information

State Minimization for DFAs

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

More information

NON-DETERMINISTIC FSA

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

More information

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

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.............................

More information

Farey Fractions. Rickard Fernström. U.U.D.M. Project Report 2017:24. Department of Mathematics Uppsala University

Farey Fractions. Rickard Fernström. U.U.D.M. Project Report 2017:24. Department of Mathematics Uppsala University U.U.D.M. Project Report 07:4 Frey Frctions Rickrd Fernström Exmensrete i mtemtik, 5 hp Hledre: Andres Strömergsson Exmintor: Jörgen Östensson Juni 07 Deprtment of Mthemtics Uppsl University Frey Frctions

More information

Automata and Languages

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

More information

input tape head moves current state

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

More information

Nondeterministic Biautomata and Their Descriptional Complexity

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

More information

How Deterministic are Good-For-Games Automata?

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

More information

Recursively Enumerable and Recursive. Languages

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

More information

On Determinisation of History-Deterministic Automata.

On Determinisation of History-Deterministic Automata. On Deterministion of History-Deterministic Automt. Denis Kupererg Mich l Skrzypczk University of Wrsw YR-ICALP 2014 Copenhgen Introduction Deterministic utomt re centrl tool in utomt theory: Polynomil

More information

Prefix-Free Regular-Expression Matching

Prefix-Free Regular-Expression Matching Prefix-Free Regulr-Expression Mthing Yo-Su Hn, Yjun Wng nd Derik Wood Deprtment of Computer Siene HKUST Prefix-Free Regulr-Expression Mthing p.1/15 Pttern Mthing Given pttern P nd text T, find ll sustrings

More information

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

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

More information

General Algorithms for Testing the Ambiguity of Finite Automata

General Algorithms for Testing the Ambiguity of Finite Automata Generl Algorithms for Testing the Amiguity of Finite Automt Cyril Alluzen 1,, Mehryr Mohri 2,1, nd Ashish Rstogi 1, 1 Google Reserch, 76 Ninth Avenue, New York, NY 10011. 2 Cournt Institute of Mthemticl

More information

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

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

More information

Analytically, vectors will be represented by lowercase bold-face Latin letters, e.g. a, r, q.

Analytically, vectors will be represented by lowercase bold-face Latin letters, e.g. a, r, q. 1.1 Vector Alger 1.1.1 Sclrs A physicl quntity which is completely descried y single rel numer is clled sclr. Physiclly, it is something which hs mgnitude, nd is completely descried y this mgnitude. Exmples

More information

Convex Sets and Functions

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

More information

Lecture 9: LTL and Büchi Automata

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

More information

5.1 Definitions and Examples 5.2 Deterministic Pushdown Automata

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

More information

Nondeterministic Finite Automata

Nondeterministic Finite Automata Nondeterministi Finite utomt The Power of Guessing Tuesdy, Otoer 4, 2 Reding: Sipser.2 (first prt); Stoughton 3.3 3.5 S235 Lnguges nd utomt eprtment of omputer Siene Wellesley ollege Finite utomton (F)

More information

Prefix-Free Subsets of Regular Languages and Descriptional Complexity

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

More information

MATH 101A: ALGEBRA I PART B: RINGS AND MODULES 35

MATH 101A: ALGEBRA I PART B: RINGS AND MODULES 35 MATH 101A: ALGEBRA I PART B: RINGS AND MODULES 35 9. Modules over PID This week we re proving the fundmentl theorem for finitely generted modules over PID, nmely tht they re ll direct sums of cyclic modules.

More information

Triangles The following examples explore aspects of triangles:

Triangles The following examples explore aspects of triangles: Tringles The following exmples explore spects of tringles: xmple 1: ltitude of right ngled tringle + xmple : tringle ltitude of the symmetricl ltitude of n isosceles x x - 4 +x xmple 3: ltitude of the

More information

The University of Nottingham

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

More information

USA Mathematical Talent Search Round 1 Solutions Year 21 Academic Year

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.

More information

4 VECTORS. 4.0 Introduction. Objectives. Activity 1

4 VECTORS. 4.0 Introduction. Objectives. Activity 1 4 VECTRS Chpter 4 Vectors jectives fter studying this chpter you should understnd the difference etween vectors nd sclrs; e le to find the mgnitude nd direction of vector; e le to dd vectors, nd multiply

More information

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

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

More information

CS12N: The Coming Revolution in Computer Architecture Laboratory 2 Preparation

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

More information

Problem Set 9. Figure 1: Diagram. This picture is a rough sketch of the 4 parabolas that give us the area that we need to find. The equations are:

Problem Set 9. Figure 1: Diagram. This picture is a rough sketch of the 4 parabolas that give us the area that we need to find. The equations are: (x + y ) = y + (x + y ) = x + Problem Set 9 Discussion: Nov., Nov. 8, Nov. (on probbility nd binomil coefficients) The nme fter the problem is the designted writer of the solution of tht problem. (No one

More information

8 factors of x. For our second example, let s raise a power to a power:

8 factors of x. For our second example, let s raise a power to a power: CH 5 THE FIVE LAWS OF EXPONENTS EXPONENTS WITH VARIABLES It s no time for chnge in tctics, in order to give us deeper understnding of eponents. For ech of the folloing five emples, e ill stretch nd squish,

More information

CSCI FOUNDATIONS OF COMPUTER SCIENCE

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

More information

Parallel Projection Theorem (Midpoint Connector Theorem):

Parallel Projection Theorem (Midpoint Connector Theorem): rllel rojection Theorem (Midpoint onnector Theorem): The segment joining the midpoints of two sides of tringle is prllel to the third side nd hs length one-hlf the third side. onversely, If line isects

More information

MTH 505: Number Theory Spring 2017

MTH 505: Number Theory Spring 2017 MTH 505: Numer Theory Spring 207 Homework 2 Drew Armstrong The Froenius Coin Prolem. Consider the eqution x ` y c where,, c, x, y re nturl numers. We cn think of $ nd $ s two denomintions of coins nd $c

More information

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

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

More information

Project 6: Minigoals Towards Simplifying and Rewriting Expressions

Project 6: Minigoals Towards Simplifying and Rewriting Expressions MAT 51 Wldis Projet 6: Minigols Towrds Simplifying nd Rewriting Expressions The distriutive property nd like terms You hve proly lerned in previous lsses out dding like terms ut one prolem with the wy

More information

General Algorithms for Testing the Ambiguity of Finite Automata and the Double-Tape Ambiguity of Finite-State Transducers

General Algorithms for Testing the Ambiguity of Finite Automata and the Double-Tape Ambiguity of Finite-State Transducers Interntionl Journl of Foundtions of Computer Science c World Scientific Pulishing Compny Generl Algorithms for Testing the Amiguity of Finite Automt nd the Doule-Tpe Amiguity of Finite-Stte Trnsducers

More information

Complementing Büchi Automata with a Subset-tuple Construction

Complementing Büchi Automata with a Subset-tuple Construction DEPARTEMENT D INFORMATIQUE DEPARTEMENT FÜR INFORMATIK Bd de Pérolles 90 CH-1700 Friourg www.unifr.ch/informtics WORKING PAPER Complementing Büchi Automt with Suset-tuple Construction J. Allred & U. Ultes-Nitsche

More information

8 Automata and formal languages. 8.1 Formal languages

8 Automata and formal languages. 8.1 Formal languages 8 Automt nd forml lnguges This exposition ws developed y Clemens Gröpl nd Knut Reinert. It is sed on the following references, ll of which re recommended reding: 1. Uwe Schöning: Theoretische Informtik

More information

Quadratic reciprocity

Quadratic reciprocity Qudrtic recirocity Frncisc Bozgn Los Angeles Mth Circle Octoer 8, 01 1 Qudrtic Recirocity nd Legendre Symol In the eginning of this lecture, we recll some sic knowledge out modulr rithmetic: Definition

More information

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

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.

More information

arxiv: v2 [cs.lo] 26 Dec 2016

arxiv: v2 [cs.lo] 26 Dec 2016 On Negotition s Concurrency Primitive II: Deterministic Cyclic Negotitions Jvier Esprz 1 nd Jörg Desel 2 1 Fkultät für Informtik, Technische Universität München, Germny 2 Fkultät für Mthemtik und Informtik,

More information

20 MATHEMATICS POLYNOMIALS

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

More information

Chapter 4 Contravariance, Covariance, and Spacetime Diagrams

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

More information

Resources. Introduction: Binding. Resource Types. Resource Sharing. The type of a resource denotes its ability to perform different operations

Resources. Introduction: Binding. Resource Types. Resource Sharing. The type of a resource denotes its ability to perform different operations Introduction: Binding Prt of 4-lecture introduction Scheduling Resource inding Are nd performnce estimtion Control unit synthesis This lecture covers Resources nd resource types Resource shring nd inding

More information

Regular Expressions and NFAs without ε-transitions

Regular Expressions and NFAs without ε-transitions Regulr Expressions nd NFAs without ε-trnsitions Georg chnitger Institut für Informtik, Johnn Wolfgng Goethe-Universität, Robert Myer trße 11 15, 60054 Frnkfurt m Min, Germny georg@thi.informtik.uni-frnkfurt.de

More information

Where did dynamic programming come from?

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

More information

State Complexity of Union and Intersection of Binary Suffix-Free Languages

State Complexity of Union and Intersection of Binary Suffix-Free Languages Stte Complexity of Union nd Intersetion of Binry Suffix-Free Lnguges Glin Jirásková nd Pvol Olejár Slovk Ademy of Sienes nd Šfárik University, Košie 0000 1111 0000 1111 Glin Jirásková nd Pvol Olejár Binry

More information

Lecture V. Introduction to Space Groups Charles H. Lake

Lecture V. Introduction to Space Groups Charles H. Lake Lecture V. Introduction to Spce Groups 2003. Chrles H. Lke Outline:. Introduction B. Trnsltionl symmetry C. Nomenclture nd symols used with spce groups D. The spce groups E. Derivtion nd discussion of

More information

Introduction to Electrical & Electronic Engineering ENGG1203

Introduction to Electrical & Electronic Engineering ENGG1203 Introduction to Electricl & Electronic Engineering ENGG23 2 nd Semester, 27-8 Dr. Hden Kwok-H So Deprtment of Electricl nd Electronic Engineering Astrction DIGITAL LOGIC 2 Digitl Astrction n Astrct ll

More information

Separating Regular Languages with First-Order Logic

Separating Regular Languages with First-Order Logic Seprting Regulr Lnguges with First-Order Logic Thoms Plce Mrc Zeitoun LBRI, Bordeux University, Frnce firstnme.lstnme@lri.fr Astrct Given two lnguges, seprtor is third lnguge tht contins the first one

More information

Automata-based Pattern Mining from Imperfect Traces

Automata-based Pattern Mining from Imperfect Traces Automt-sed Pttern Mining from Imperfect Trces Giles Reger University of Mnchester Oxford Rod, M13 9PL Mnchester, UK regerg@cs.mn.c.uk Howrd Brringer University of Mnchester Oxford Rod, M13 9PL Mnchester,

More information

On Decentralized Observability of Discrete Event Systems

On Decentralized Observability of Discrete Event Systems 2011 50th IEEE Conference on Decision nd Control nd Europen Control Conference (CDC-ECC) Orlndo, FL, USA, Decemer 12-15, 2011 On Decentrlized Oservility of Discrete Event Systems M.P. Csino, A. Giu, C.

More information

Good-for-Games Automata versus Deterministic Automata.

Good-for-Games Automata versus Deterministic Automata. Good-for-Gmes Automt versus Deterministic Automt. Denis Kuperberg 1,2 Mich l Skrzypczk 1 1 University of Wrsw 2 IRIT/ONERA (Toulouse) Séminire MoVe 12/02/2015 LIF, Luminy Introduction Deterministic utomt

More information

Lecture 7 notes Nodal Analysis

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

More information

MATH 573 FINAL EXAM. May 30, 2007

MATH 573 FINAL EXAM. May 30, 2007 MATH 573 FINAL EXAM My 30, 007 NAME: Solutions 1. This exm is due Wednesdy, June 6 efore the 1:30 pm. After 1:30 pm I will NOT ccept the exm.. This exm hs 1 pges including this cover. There re 10 prolems.

More information

On the degree of regularity of generalized van der Waerden triples

On the degree of regularity of generalized van der Waerden triples On the degree of regulrity of generlized vn der Werden triples Jcob Fox Msschusetts Institute of Technology, Cmbridge, MA 02139, USA Rdoš Rdoičić Deprtment of Mthemtics, Rutgers, The Stte University of

More information

C/CS/Phys C191 Bell Inequalities, No Cloning, Teleportation 9/13/07 Fall 2007 Lecture 6

C/CS/Phys C191 Bell Inequalities, No Cloning, Teleportation 9/13/07 Fall 2007 Lecture 6 C/CS/Phys C9 Bell Inequlities, o Cloning, Teleporttion 9/3/7 Fll 7 Lecture 6 Redings Benenti, Csti, nd Strini: o Cloning Ch.4. Teleporttion Ch. 4.5 Bell inequlities See lecture notes from H. Muchi, Cltech,

More information

UniversitaireWiskundeCompetitie. Problem 2005/4-A We have k=1. Show that for every q Q satisfying 0 < q < 1, there exists a finite subset K N so that

UniversitaireWiskundeCompetitie. Problem 2005/4-A We have k=1. Show that for every q Q satisfying 0 < q < 1, there exists a finite subset K N so that Problemen/UWC NAW 5/7 nr juni 006 47 Problemen/UWC UniversitireWiskundeCompetitie Edition 005/4 For Session 005/4 we received submissions from Peter Vndendriessche, Vldislv Frnk, Arne Smeets, Jn vn de

More information

Automata-based Pattern Mining from Imperfect Traces

Automata-based Pattern Mining from Imperfect Traces Automt-sed Pttern Mining from Imperfect Trces Giles Reger University of Mnchester Oxford Rod, M13 9PL Mnchester, UK regerg@cs.mn.c.uk Howrd Brringer University of Mnchester Oxford Rod, M13 9PL Mnchester,

More information

expression simply by forming an OR of the ANDs of all input variables for which the output is

expression simply by forming an OR of the ANDs of all input variables for which the output is 2.4 Logic Minimiztion nd Krnugh Mps As we found ove, given truth tle, it is lwys possile to write down correct logic expression simply y forming n OR of the ANDs of ll input vriles for which the output

More information

5.4 The Quarter-Wave Transformer

5.4 The Quarter-Wave Transformer 3/4/7 _4 The Qurter Wve Trnsformer /.4 The Qurter-Wve Trnsformer Redg Assignment: pp. 73-76, 4-43 By now you ve noticed tht qurter-wve length of trnsmission le ( = λ 4, β = π ) ppers often microwve engeerg

More information

The Caucal Hierarchy of Infinite Graphs in Terms of Logic and Higher-order Pushdown Automata

The Caucal Hierarchy of Infinite Graphs in Terms of Logic and Higher-order Pushdown Automata The Cucl Hierrchy of Infinite Grphs in Terms of Logic nd Higher-order Pushdown Automt Arnud Cryol 1 nd Stefn Wöhrle 2 1 IRISA Rennes, Frnce rnud.cryol@iris.fr 2 Lehrstuhl für Informtik 7 RWTH Achen, Germny

More information

Automata and semigroups recognizing infinite words

Automata and semigroups recognizing infinite words Automt nd semigroups recognizing infinite words Olivier Crton 1 Dominique Perrin 2 Jen-Éric Pin1 1 LIAFA, CNRS nd Université Denis Diderot Pris 7, Cse 7014, 75205 Pris Cedex 13, Frnce 2 Institut Gsprd

More information

2. VECTORS AND MATRICES IN 3 DIMENSIONS

2. VECTORS AND MATRICES IN 3 DIMENSIONS 2 VECTORS AND MATRICES IN 3 DIMENSIONS 21 Extending the Theory of 2-dimensionl Vectors x A point in 3-dimensionl spce cn e represented y column vector of the form y z z-xis y-xis z x y x-xis Most of the

More information

Math 360: A primitive integral and elementary functions

Math 360: A primitive integral and elementary functions Mth 360: A primitive integrl nd elementry functions D. DeTurck University of Pennsylvni October 16, 2017 D. DeTurck Mth 360 001 2017C: Integrl/functions 1 / 32 Setup for the integrl prtitions Definition:

More information

Section 14.3 Arc Length and Curvature

Section 14.3 Arc Length and Curvature Section 4.3 Arc Length nd Curvture Clculus on Curves in Spce In this section, we ly the foundtions for describing the movement of n object in spce.. Vector Function Bsics In Clc, formul for rc length in

More information

CDM Memoryless Machines

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

More information

Automata for Analyzing and Querying Compressed Documents Barbara FILA, LIFO, Orl eans (Fr.) Siva ANANTHARAMAN, LIFO, Orl eans (Fr.) Rapport No

Automata for Analyzing and Querying Compressed Documents Barbara FILA, LIFO, Orl eans (Fr.) Siva ANANTHARAMAN, LIFO, Orl eans (Fr.) Rapport No Automt for Anlyzing nd Querying Compressed Documents Brr FILA, LIFO, Orléns (Fr.) Siv ANANTHARAMAN, LIFO, Orléns (Fr.) Rpport N o 2006-03 Automt for Anlyzing nd Querying Compressed Documents Brr Fil, Siv

More information

The mth Ratio Convergence Test and Other Unconventional Convergence Tests

The mth Ratio Convergence Test and Other Unconventional Convergence Tests The mth Rtio Convergence Test nd Other Unconventionl Convergence Tests Kyle Blckburn My 14, 2012 Contents 1 Introduction 2 2 Definitions, Lemms, nd Theorems 2 2.1 Defintions.............................

More information

The Minimization Problem. The Minimization Problem. The Minimization Problem. The Minimization Problem. The Minimization Problem

The Minimization Problem. The Minimization Problem. The Minimization Problem. The Minimization Problem. The Minimization Problem Simpler & More Generl Minimiztion for Weighted Finite-Stte Automt Json Eisner Johns Hopkins University My 28, 2003 HLT-NAACL First hlf of tlk is setup - revies pst ork. Second hlf gives outline of the

More information

1 Error Analysis of Simple Rules for Numerical Integration

1 Error Analysis of Simple Rules for Numerical Integration cs41: introduction to numericl nlysis 11/16/10 Lecture 19: Numericl Integrtion II Instructor: Professor Amos Ron Scries: Mrk Cowlishw, Nthnel Fillmore 1 Error Anlysis of Simple Rules for Numericl Integrtion

More information

Polynomial Approximations for the Natural Logarithm and Arctangent Functions. Math 230

Polynomial Approximations for the Natural Logarithm and Arctangent Functions. Math 230 Polynomil Approimtions for the Nturl Logrithm nd Arctngent Functions Mth 23 You recll from first semester clculus how one cn use the derivtive to find n eqution for the tngent line to function t given

More information

SCHOOL OF ENGINEERING & BUILT ENVIRONMENT. Mathematics

SCHOOL OF ENGINEERING & BUILT ENVIRONMENT. Mathematics SCHOOL OF ENGINEERING & BUIL ENVIRONMEN Mthemtics An Introduction to Mtrices Definition of Mtri Size of Mtri Rows nd Columns of Mtri Mtri Addition Sclr Multipliction of Mtri Mtri Multipliction 7 rnspose

More information

Looking for All Palindromes in a String

Looking for All Palindromes in a String Looking or All Plindromes in String Shih Jng Pn nd R C T Lee Deprtment o Computer Science nd Inormtion Engineering, Ntionl Chi-Nn University, Puli, Nntou Hsien,, Tiwn, ROC sjpn@lgdoccsiencnuedutw, rctlee@ncnuedutw

More information

Hopcroft and Karp s algorithm for Non-deterministic Finite Automata

Hopcroft and Karp s algorithm for Non-deterministic Finite Automata Hopcroft nd Krp s lgorithm for Non-deterministic Finite Automt Filippo Bonchi, Dmien Pous To cite this version: Filippo Bonchi, Dmien Pous. Hopcroft nd Krp s lgorithm for Non-deterministic Finite Automt.

More information

Resistors. Consider a uniform cylinder of material with mediocre to poor to pathetic conductivity ( )

Resistors. Consider a uniform cylinder of material with mediocre to poor to pathetic conductivity ( ) 10/25/2005 Resistors.doc 1/7 Resistors Consider uniform cylinder of mteril with mediocre to poor to r. pthetic conductivity ( ) ˆ This cylinder is centered on the -xis, nd hs length. The surfce re of the

More information

The final exam will take place on Friday May 11th from 8am 11am in Evans room 60.

The final exam will take place on Friday May 11th from 8am 11am in Evans room 60. Mth 104: finl informtion The finl exm will tke plce on Fridy My 11th from 8m 11m in Evns room 60. The exm will cover ll prts of the course with equl weighting. It will cover Chpters 1 5, 7 15, 17 21, 23

More information

What else can you do?

What else can you do? Wht else cn you do? ngle sums The size of specil ngle types lernt erlier cn e used to find unknown ngles. tht form stright line dd to 180c. lculte the size of + M, if L is stright line M + L = 180c( stright

More information

ON THE DETERMINIZATION OF WEIGHTED FINITE AUTOMATA

ON THE DETERMINIZATION OF WEIGHTED FINITE AUTOMATA To pper in SIAM Journl on Computing c SIAM 000 ON THE DETERMINIZATION OF WEIGHTED FINITE AUTOMATA ADAM L. BUCHSBAUM, RAFFAELE GIANCARLO, AND JEFFERY R. WESTBROOK Astrct. We study the prolem of constructing

More information

Refined interfaces for compositional verification

Refined interfaces for compositional verification Refined interfces for compositionl verifiction Frédéric Lng INRI Rhône-lpes http://www.inrilpes.fr/vsy Motivtion Enumertive verifiction of concurrent systems Prllel composition of synchronous processes

More information

a a a a a a a a a a a a a a a a a a a a a a a a In this section, we introduce a general formula for computing determinants.

a a a a a a a a a a a a a a a a a a a a a a a a In this section, we introduce a general formula for computing determinants. Section 9 The Lplce Expnsion In the lst section, we defined the determinnt of (3 3) mtrix A 12 to be 22 12 21 22 2231 22 12 21. In this section, we introduce generl formul for computing determinnts. Rewriting

More information

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

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

More information

Tries and suffixes trees

Tries and suffixes trees Trie: A dt-structure for set of words Tries nd suffixes trees Alon Efrt Comuter Science Dertment University of Arizon All words over the lhet Σ={,,..z}. In the slides, let sy tht the lhet is only {,,c,d}

More information

1 Online Learning and Regret Minimization

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

More information