Computing with finite semigroups: part I

Size: px
Start display at page:

Download "Computing with finite semigroups: part I"

Transcription

1 Computing with finite semigroups: prt I J. D. Mitchell School of Mthemtics nd Sttistics, University of St Andrews Novemer 20th, 2015 J. D. Mitchell (St Andrews) Novemer 20th, / 34

2 Wht is this tlk out? Given: finite semigroup S; nd question out S. Aim: to descrie how to nswer your question using computer descrie the stte of the rt. Why? perform low-level clcultions such s multipliction, inversion,... suggests new theoreticl results otin counter-exmples gin more detiled understnding perform more intricte clcultions. J. D. Mitchell (St Andrews) Novemer 20th, / 34

3 Insert semigroup into computer... numer 1 Cyley tles J. D. Mitchell (St Andrews) Novemer 20th, / 34

4 Insert semigroup into computer... numer 1 Cyley tles Resons not to: Too mny! semigroups up to isomorphism nd nti-isomorphism with 10 elements (Distler-Kelsey 13); Complexity! O( S 3 ) just to verify ssocitivity; Hrd to input! A semigroup with 1000 elements hs 1 million entries in the Cyley tle; Requires nerly complete knowledge! J. D. Mitchell (St Andrews) Novemer 20th, / 34

5 Insert semigroup into computer... numer 2 Presenttions Words in genertors nd reltions:, 2 =, =, 2 =, 3 =, 2 =. Resons not to: Reltively difficult to find! given semigroup S it cn e difficult to find presenttion for S; Undecidility! lmost every meningful question is undecidle, i.e. word prolem, isomorphism prolem,... J. D. Mitchell (St Andrews) Novemer 20th, / 34

6 Insert semigroup into computer... numer 3 Genertors Specify genertors of prticulr type. Definition A trnsformtion is function f from {1,..., n} to itself written: ( ) 1 2 n f =. 1f 2f nf A trnsformtion semigroup is just semigroup consisting of set of trnsformtions under composition of functions. Theorem (Cyley s theorem) Every semigroup is isomorphic to permuttion trnsformtion semigroup. J. D. Mitchell (St Andrews) Novemer 20th, / 34

7 Fundmentl tsks Input: genertors A (trnsformtions, prtil perms, mtrices, inry reltions, prtitions,... ) for semigroup S. Output: the size of S memership in S fctorise elements over the genertors the numer of idempotents (x 2 = x) the mximl su(semi)groups the idel structurl of S (i.e. Green s reltions) is S group? n inverse semigroup? regulr semigroup? the utomorphism group of S the congruences of S... J. D. Mitchell (St Andrews) Novemer 20th, / 34

8 An lgorithm S cting on itself y right multipliction Input: set A of genertors (trnsformtions, prtil perms, mtrices, inry reltions, prtitions,... ) for semigroup S. Output: the elements X of S. Assumes: we cn multiply nd check equlity. Supposing the genertors re distinct. 1: X := A 2: for x X do 3: for A do 4: if x X then 5: ppend x to X 6: return X J. D. Mitchell (St Andrews) Novemer 20th, / 34

9 An exmple Let S e the semigroup generted y the trnsformtions ( ) ( ) = nd = The grph of the ctions of nd :, forth J. D. Mitchell (St Andrews) Novemer 20th, / 34

10 The elements nd the right Cyley grph Edges of the form: x y xy , 2 2 =, =, 2 =, 2 =, 3 =, 2 =, 3 =, =, 2 = ck forth J. D. Mitchell (St Andrews) Novemer 20th, / 34

11 The left Cyley grph Edges of the form x y yx..., 2 2, 2 =, =, 2 =, 2 =, 3 =, 2 =, 3 =, =, 2 = ck forth J. D. Mitchell (St Andrews) Novemer 20th, / 34

12 R-clsses 2, 2 The R-clsses re the strongly connected components of the right Cyley grph. J. D. Mitchell (St Andrews) Novemer 20th, / 34

13 L -clsses, 2 2, The L -clsses re the strongly connected components of the left Cyley grph. J. D. Mitchell (St Andrews) Novemer 20th, / 34

14 The Green s structure The D-clsses re the strongly connected components of the union of the left nd right Cyley grphs. The prtil order of the D-clsses is the trnsitive reflexive closure of the quotient of the union of the left nd right Cyley grphs y its strongly connected components. J. D. Mitchell (St Andrews) Novemer 20th, / 34

15 Semigroupe V. Froidure nd J.-E. Pin, Algorithms for computing finite semigroups, in Foundtions of Computtionl Mthemtics, F. Cucker et M. Shu (eds), Berlin, 1997, pp , Springer. J.-E. Pin, Algorithmic spects of finite semigroup theory, tutoril, jep/pdf/exposes/standrews.pdf J.-E. Pin, Semigroupe, C progrmme, ville t jep/logiciels/semigroupe2.0/semigroupe2.html The Semigroups pckge for GAP version 3.0 (not yet relesed) J. D. Mitchell (St Andrews) Novemer 20th, / 34

16 GAP nd Semigroupe J. D. Mitchell (St Andrews) Novemer 20th, / 34

17 Pros nd Cons Pros: only requires: equlity tester multipliction then we cn run the lgorithm! Does not use the representtion of the semigroup! Cons: hs complexity O( S A ) it cn e costly to multiply elements it cn e costly to check if we ve seen n element efore ll the elements re stored, which uses lots of memory Does not use the representtion of the semigroup! J. D. Mitchell (St Andrews) Novemer 20th, / 34

18 The limittions of exhustive enumertion n # trnsformtions memory unit its ytes ytes k k k m m g g t t.... n n n n n n 16 its Storing the elements of semigroup is imprcticl. J. D. Mitchell (St Andrews) Novemer 20th, / 34

19 Bck to semigroups... Suppose we wnt to compute the trnsformtion semigroup S generted y: ( ) = (2 3), = (1 2 3)(4 5), c = We wnt to use lgorithms from computtionl group theory. We do not wnt to find or store the elements of S. J. D. Mitchell (St Andrews) Novemer 20th, / 34

20 Schreier s Lemm for semigroups Suppose tht S = A cts on the right on set Ω. If Σ Ω, then we denote y S Σ the group of permuttions of Σ induced y elements of the stiliser of Σ in S. If s S is such tht Σ s = Σ, then s induces permuttion of Σ, denote y s Σ. Proposition (Linton-Pfeiffer-Roertson-Ruškuc 98) Let {Σ 1,..., Σ n } e s.c.c. of the ction of S on P(Ω). Then: (i) for every i > 1, there exist u i, v i S such tht Σ 1 u i = Σ i, Σ i v i = Σ 1, (u i v i ) Σ1 = id Σ1 nd (v i u i ) Σi = id Σi (ii) S Σ1 = (u i v j ) Σ1 : 1 i, j n, A, Σ i = Σ j. forth J. D. Mitchell (St Andrews) Novemer 20th, / 34

21 Stilisers Let S e the semigroup generted y: α 1 1, 2, 3, 4, 5 α 2 1, 2, 3 α 3 1, 3 α 4 1, 2 α 5 2, 3 α 6 3 α 7 2 α 8 1 = (2 3), = (1 2 3)(4 5), c = α 1 α 2 ( ) α 4 α 3 α 5, c c, c c,, c α 7 α 6,, c J. D. Mitchell (St Andrews) Novemer 20th, / 34 c α 8, c

22 Stilisers Let S e the semigroup generted y: α 1 1, 2, 3, 4, 5 α 2 1, 2, 3 α 3 1, 3 α 6 3 = (2 3), = (1 2 3)(4 5), c = S {1,2,3,4,5} = (2 3), (1 2 3)(4 5) S {1,2,3} = (2 3), (1 2 3) S {1,3} = (1 3), c S {3} = id, c α 1 c α 2, α 3 ( ) α 4, c α 5 c α 6 c α 7,, c α 8 J. D. Mitchell (St Andrews) Novemer 20th, / 34, c

23 Relting the ction nd the R-clsses Proposition Let S e trnsformtion semigroup, let x S, nd let R e the R-clss of x in S. Then: (i) { im(y) : y R } is s.c.c. of the ction of S (ii) { y R : im(y) = im(x) } is group isomorphic to the stiliser S im(x) (iii) if im(y) elongs to the s.c.c. of im(x), then S im(x) = Sim(y). An R-clss R cn e represented y triple consisting of the representtive x the s.c.c. of im(x) the stiliser S im(x). forth J. D. Mitchell (St Andrews) Novemer 20th, / 34

24 The structure of n R-clss Proposition Let S e trnsformtion semigroup, let x S, nd let R e the R-clss of x in S. Then: (i) { im(y) : y R } is s.c.c. of the ction of S (ii) { y R : im(y) = im(x) } is group isomorphic to the stiliser S im(x) (iii) if im(y) elongs to the s.c.c. of im(x), then S im(x) = Sim(y). The R-clss R c 2 the representtive of c 2 cn e represented y the triple: c 2 = ( ) the s.c.c. {{1, 3}, {1, 2}, {2, 3}} of im(c 2 ) the stiliser S im(c 2 ) = S {1,3} = (1 3) J. D. Mitchell (St Andrews) Novemer 20th, / 34

25 The structure of n R-clss Proposition (i) { im(y) : y R } is s.c.c. of the ction of S (ii) { y R : im(y) = im(x) } is group isomorphic to the stiliser S im(x) (iii) if im(y) elongs to the s.c.c. of im(x), then S im(x) = Sim(y). {1, 3} {1, 2} {2, 3} R c 2 c 2 y y J. D. Mitchell (St Andrews) Novemer 20th, / 34

26 Finding the R-clsses... Input: set A of trnsformtions generting semigroup S. Output: the R-clsses of S. 1: find the ction of S on {1,..., n} the orit lgorithm 2: find the s.c.c.s of the ction stndrd grph lgorithms 3: R := {1} R-clss reps 4: for x R do 5: for A do 6: if (x, y) R for ny y R then see the next slide 7: ppend x to R 8: return R. J. D. Mitchell (St Andrews) Novemer 20th, / 34

27 Vlidity Suppose tht S =,. If s S, then write Then = 1 R s = min. length of word in nd equl to s. = 1 R if nd only if (, ) R... Suppose R = {r 1 =, r 2,..., r k } contins representtives of R-clsses of elements s S with s < N for some N (nd mye more elements). if s S nd s = N, then s = t or s = t for some t S with t = N 1. (t, r i ) R for some i, nd so (s, r i ) = (t, r i ) R (R is left congruence) The previous lgorithm is vlid! J. D. Mitchell (St Andrews) Novemer 20th, / 34

28 Testing memership in n R-clss - I If x, y S, then xry implies tht ker (x) = ker (y). For exmple, c 2 = since ( ) ( ) = ( ) R c 2 ker (c 2 ) = {{1}, {2, 3, 4, 5}} {{1, 2, 4, 5}, {3}} = ker (c 2 ). forth J. D. Mitchell (St Andrews) Novemer 20th, / 34

29 Testing memership in n R-clss - II Is x = ( ) R c 2? {1, 3} {1, 2} {2, 3} c 2 c 2 c 2 2 R c 2 S {1,3} = (1 3) c 2 (1 3) c 2 (1 3) c 2 (1 3) 2 Every element of R c 2 is of the form: c 2 g i where g S {1,3}. ck forth J. D. Mitchell (St Andrews) Novemer 20th, / 34

30 Testing memership in n R-clss - III x = ( ) x R c 2 if nd only if x = c 2 g 2 for some g S {1,3} = (1 3) if nd only if x = c 2 g for some g S {1,3} = (1 3) {1, 3} {1, 2} {2, 3} R c 2 c 2 x x J. D. Mitchell (St Andrews) Novemer 20th, / 34

31 Testing memership in n R-clss - IV x R c 2 if nd only if x = c 2 g for some g S {1,3} = (1 3) if nd only if (c 2 ) 1 x = g S {1,3} = (1 3) = S {1,3} (c 2 ) 1 x (c 2 ) 1 x J. D. Mitchell (St Andrews) Novemer 20th, / 34 forth

32 ( ) = (2 3), = (1 2 3)(4 5), c = r r 2 c r 3 c c r 4 c r 5 c r 6 cc r 4 r 7 c c r 8 cc r 7 r 9 (c) , r 10 c r 11 c 2 c r 10 r 12 (c) r 1 r 2 r 6 c c c c c, r 3 c r 12 r 9, c r 5 = id R = (1 ( 3)(4 5) R ) c = r 8 c r 11 ck forth J. D. Mitchell (St Andrews) Novemer 20th, / 34

33 Complexity In the worst cse the ove lgorithm hs the sme complexity s the Froidure-Pin Algorithm O( S A ) where S = A. The worst cse is relised when S is J -trivil. In the est cse the complexity is the sme s tht of the Schreier-Sims Algorithm. The est cse is relised when S hppens to e group (ut mye doesn t know it). If S = T n, i.e. S hs lots of lrge sugroups nd R-clsses, the complexity is O(2 n ) compred with O(n n ) for the Froidure-Pin Algorithm. J. D. Mitchell (St Andrews) Novemer 20th, / 34

34 More theory It is possile to generlize the technique descried ove to ritrry susemigroups of regulr semigroup. Exmples include: semigroups of mtrices over finite fields susemigroups of the prtition monoid semigroups nd inverse semigroups of prtil permuttions susemigroups of regulr Rees 0-mtrix semigroups.... The theory is descried in: J. Est, A. Egri-Ngy, J. D. Mitchell, nd Y. Péresse, Computing finite semigroups, 45 pges. J. D. Mitchell (St Andrews) Novemer 20th, / 34

378 Relations Solutions for Chapter 16. Section 16.1 Exercises. 3. Let A = {0,1,2,3,4,5}. Write out the relation R that expresses on A.

378 Relations Solutions for Chapter 16. Section 16.1 Exercises. 3. Let A = {0,1,2,3,4,5}. Write out the relation R that expresses on A. 378 Reltions 16.7 Solutions for Chpter 16 Section 16.1 Exercises 1. Let A = {0,1,2,3,4,5}. Write out the reltion R tht expresses > on A. Then illustrte it with digrm. 2 1 R = { (5,4),(5,3),(5,2),(5,1),(5,0),(4,3),(4,2),(4,1),

More information

Lecture 3: Equivalence Relations

Lecture 3: Equivalence Relations Mthcmp Crsh Course Instructor: Pdric Brtlett Lecture 3: Equivlence Reltions Week 1 Mthcmp 2014 In our lst three tlks of this clss, we shift the focus of our tlks from proof techniques to proof concepts

More information

Random subgroups of a free group

Random subgroups of a free group Rndom sugroups of free group Frédérique Bssino LIPN - Lortoire d Informtique de Pris Nord, Université Pris 13 - CNRS Joint work with Armndo Mrtino, Cyril Nicud, Enric Ventur et Pscl Weil LIX My, 2015 Introduction

More information

Coalgebra, Lecture 15: Equations for Deterministic Automata

Coalgebra, Lecture 15: Equations for Deterministic Automata Colger, Lecture 15: Equtions for Deterministic Automt Julin Slmnc (nd Jurrin Rot) Decemer 19, 2016 In this lecture, we will study the concept of equtions for deterministic utomt. The notes re self contined

More information

1. For each of the following theorems, give a two or three sentence sketch of how the proof goes or why it is not true.

1. For each of the following theorems, give a two or three sentence sketch of how the proof goes or why it is not true. York University CSE 2 Unit 3. DFA Clsses Converting etween DFA, NFA, Regulr Expressions, nd Extended Regulr Expressions Instructor: Jeff Edmonds Don t chet y looking t these nswers premturely.. For ech

More information

Minimal DFA. minimal DFA for L starting from any other

Minimal DFA. minimal DFA for L starting from any other Miniml DFA Among the mny DFAs ccepting the sme regulr lnguge L, there is exctly one (up to renming of sttes) which hs the smllest possile numer of sttes. Moreover, it is possile to otin tht miniml DFA

More information

A negative answer to a question of Wilke on varieties of!-languages

A negative answer to a question of Wilke on varieties of!-languages A negtive nswer to question of Wilke on vrieties of!-lnguges Jen-Eric Pin () Astrct. In recent pper, Wilke sked whether the oolen comintions of!-lnguges of the form! L, for L in given +-vriety of lnguges,

More information

Generating finite transformation semigroups: SgpWin

Generating finite transformation semigroups: SgpWin Generting finite trnsformtion semigroups: SgpWin Donld B. McAlister ( don@mth.niu.edu ) Deprtment of Mthemticl Sciences Northern Illinois University nd C.A.U.L. Septemer 5, 2006 Donld B. McAlister ( don@mth.niu.edu

More information

The Value 1 Problem for Probabilistic Automata

The Value 1 Problem for Probabilistic Automata The Vlue 1 Prolem for Proilistic Automt Bruxelles Nthnël Fijlkow LIAFA, Université Denis Diderot - Pris 7, Frnce Institute of Informtics, Wrsw University, Polnd nth@lif.univ-pris-diderot.fr June 20th,

More information

Formal Languages and Automata

Formal Languages and Automata Moile Computing nd Softwre Engineering p. 1/5 Forml Lnguges nd Automt Chpter 2 Finite Automt Chun-Ming Liu cmliu@csie.ntut.edu.tw Deprtment of Computer Science nd Informtion Engineering Ntionl Tipei University

More information

Finite Automata Theory and Formal Languages TMV027/DIT321 LP4 2018

Finite Automata Theory and Formal Languages TMV027/DIT321 LP4 2018 Finite Automt Theory nd Forml Lnguges TMV027/DIT321 LP4 2018 Lecture 10 An Bove April 23rd 2018 Recp: Regulr Lnguges We cn convert between FA nd RE; Hence both FA nd RE ccept/generte regulr lnguges; More

More information

Three termination problems. Patrick Dehornoy. Laboratoire de Mathématiques Nicolas Oresme, Université de Caen

Three termination problems. Patrick Dehornoy. Laboratoire de Mathématiques Nicolas Oresme, Université de Caen Three termintion prolems Ptrick Dehornoy Lortoire de Mthémtiques Nicols Oresme, Université de Cen Three termintion prolems Ptrick Dehornoy Lortoire Preuves, Progrmmes, Systèmes Université Pris-Diderot

More information

set is not closed under matrix [ multiplication, ] and does not form a group.

set is not closed under matrix [ multiplication, ] and does not form a group. Prolem 2.3: Which of the following collections of 2 2 mtrices with rel entries form groups under [ mtrix ] multipliction? i) Those of the form for which c d 2 Answer: The set of such mtrices is not closed

More information

Formal languages, automata, and theory of computation

Formal languages, automata, and theory of computation Mälrdlen University TEN1 DVA337 2015 School of Innovtion, Design nd Engineering Forml lnguges, utomt, nd theory of computtion Thursdy, Novemer 5, 14:10-18:30 Techer: Dniel Hedin, phone 021-107052 The exm

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

First Midterm Examination

First Midterm Examination 24-25 Fll Semester First Midterm Exmintion ) Give the stte digrm of DFA tht recognizes the lnguge A over lphet Σ = {, } where A = {w w contins or } 2) The following DFA recognizes the lnguge B over lphet

More information

Algebraic systems Semi groups and monoids Groups. Subgroups and homomorphisms Cosets Lagrange s theorem. Ring & Fields (Definitions and examples)

Algebraic systems Semi groups and monoids Groups. Subgroups and homomorphisms Cosets Lagrange s theorem. Ring & Fields (Definitions and examples) Prepred y Dr. A.R.VIJAYALAKSHMI Algeric systems Semi groups nd monoids Groups Sugroups nd homomorphms Cosets Lgrnge s theorem Ring & Fields (Definitions nd exmples Stndrd Nottions. N :Set of ll nturl numers.

More information

CM10196 Topic 4: Functions and Relations

CM10196 Topic 4: Functions and Relations CM096 Topic 4: Functions nd Reltions Guy McCusker W. Functions nd reltions Perhps the most widely used notion in ll of mthemtics is tht of function. Informlly, function is n opertion which tkes n input

More information

CS415 Compilers. Lexical Analysis and. These slides are based on slides copyrighted by Keith Cooper, Ken Kennedy & Linda Torczon at Rice University

CS415 Compilers. Lexical Analysis and. These slides are based on slides copyrighted by Keith Cooper, Ken Kennedy & Linda Torczon at Rice University CS415 Compilers Lexicl Anlysis nd These slides re sed on slides copyrighted y Keith Cooper, Ken Kennedy & Lind Torczon t Rice University First Progrmming Project Instruction Scheduling Project hs een posted

More information

Free groups, Lecture 2, part 1

Free groups, Lecture 2, part 1 Free groups, Lecture 2, prt 1 Olg Khrlmpovich NYC, Sep. 2 1 / 22 Theorem Every sugroup H F of free group F is free. Given finite numer of genertors of H we cn compute its sis. 2 / 22 Schreir s grph The

More information

Chapter 14. Matrix Representations of Linear Transformations

Chapter 14. Matrix Representations of Linear Transformations Chpter 4 Mtrix Representtions of Liner Trnsformtions When considering the Het Stte Evolution, we found tht we could describe this process using multipliction by mtrix. This ws nice becuse computers cn

More information

Regular Language. Nonregular Languages The Pumping Lemma. The pumping lemma. Regular Language. The pumping lemma. Infinitely long words 3/17/15

Regular Language. Nonregular Languages The Pumping Lemma. The pumping lemma. Regular Language. The pumping lemma. Infinitely long words 3/17/15 Regulr Lnguge Nonregulr Lnguges The Pumping Lemm Models of Comput=on Chpter 10 Recll, tht ny lnguge tht cn e descried y regulr expression is clled regulr lnguge In this lecture we will prove tht not ll

More information

LECTURE 2: ARTIN SYMBOL, ARTIN MAP, ARTIN RECIPROCITY LAW AND FINITENESS OF GENERALIZED IDEAL CLASS GROUP

LECTURE 2: ARTIN SYMBOL, ARTIN MAP, ARTIN RECIPROCITY LAW AND FINITENESS OF GENERALIZED IDEAL CLASS GROUP Clss Field Theory Study Seminr Jnury 25 2017 LECTURE 2: ARTIN SYMBOL, ARTIN MAP, ARTIN RECIPROCITY LAW AND FINITENESS OF GENERALIZED IDEAL CLASS GROUP YIFAN WU Plese send typos nd comments to wuyifn@umich.edu

More information

CSE : Exam 3-ANSWERS, Spring 2011 Time: 50 minutes

CSE : Exam 3-ANSWERS, Spring 2011 Time: 50 minutes CSE 260-002: Exm 3-ANSWERS, Spring 20 ime: 50 minutes Nme: his exm hs 4 pges nd 0 prolems totling 00 points. his exm is closed ook nd closed notes.. Wrshll s lgorithm for trnsitive closure computtion is

More information

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

CISC 4090 Theory of Computation

CISC 4090 Theory of Computation 9/6/28 Stereotypicl computer CISC 49 Theory of Computtion Finite stte mchines & Regulr lnguges Professor Dniel Leeds dleeds@fordhm.edu JMH 332 Centrl processing unit (CPU) performs ll the instructions

More information

Recitation 3: More Applications of the Derivative

Recitation 3: More Applications of the Derivative Mth 1c TA: Pdric Brtlett Recittion 3: More Applictions of the Derivtive Week 3 Cltech 2012 1 Rndom Question Question 1 A grph consists of the following: A set V of vertices. A set E of edges where ech

More information

Bases for Vector Spaces

Bases for Vector Spaces Bses for Vector Spces 2-26-25 A set is independent if, roughly speking, there is no redundncy in the set: You cn t uild ny vector in the set s liner comintion of the others A set spns if you cn uild everything

More information

Lecture 3. In this lecture, we will discuss algorithms for solving systems of linear equations.

Lecture 3. In this lecture, we will discuss algorithms for solving systems of linear equations. Lecture 3 3 Solving liner equtions In this lecture we will discuss lgorithms for solving systems of liner equtions Multiplictive identity Let us restrict ourselves to considering squre mtrices since one

More information

Math Lecture 23

Math Lecture 23 Mth 8 - Lecture 3 Dyln Zwick Fll 3 In our lst lecture we delt with solutions to the system: x = Ax where A is n n n mtrix with n distinct eigenvlues. As promised, tody we will del with the question of

More information

Rectangular group congruences on an epigroup

Rectangular group congruences on an epigroup cholrs Journl of Engineering nd Technology (JET) ch J Eng Tech, 015; 3(9):73-736 cholrs Acdemic nd cientific Pulisher (An Interntionl Pulisher for Acdemic nd cientific Resources) wwwsspulishercom IN 31-435X

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

Review: set theoretic definition of the numbers. Natural numbers:

Review: set theoretic definition of the numbers. Natural numbers: Review: reltions A inry reltion on set A is suset R Ñ A ˆ A, where elements p, q re written s. Exmple: A Z nd R t pmod nqu. A inry reltion on set A is... (R) reflexive if for ll P A; (S) symmetric if implies

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

Linear Systems with Constant Coefficients

Linear Systems with Constant Coefficients Liner Systems with Constnt Coefficients 4-3-05 Here is system of n differentil equtions in n unknowns: x x + + n x n, x x + + n x n, x n n x + + nn x n This is constnt coefficient liner homogeneous system

More information

Intermediate Math Circles Wednesday, November 14, 2018 Finite Automata II. Nickolas Rollick a b b. a b 4

Intermediate Math Circles Wednesday, November 14, 2018 Finite Automata II. Nickolas Rollick a b b. a b 4 Intermedite Mth Circles Wednesdy, Novemer 14, 2018 Finite Automt II Nickols Rollick nrollick@uwterloo.c Regulr Lnguges Lst time, we were introduced to the ide of DFA (deterministic finite utomton), one

More information

Linear Inequalities. Work Sheet 1

Linear Inequalities. Work Sheet 1 Work Sheet 1 Liner Inequlities Rent--Hep, cr rentl compny,chrges $ 15 per week plus $ 0.0 per mile to rent one of their crs. Suppose you re limited y how much money you cn spend for the week : You cn spend

More information

Torsion in Groups of Integral Triangles

Torsion in Groups of Integral Triangles Advnces in Pure Mthemtics, 01,, 116-10 http://dxdoiorg/1046/pm011015 Pulished Online Jnury 01 (http://wwwscirporg/journl/pm) Torsion in Groups of Integrl Tringles Will Murry Deprtment of Mthemtics nd Sttistics,

More information

Chapter 8.2: The Integral

Chapter 8.2: The Integral Chpter 8.: The Integrl You cn think of Clculus s doule-wide triler. In one width of it lives differentil clculus. In the other hlf lives wht is clled integrl clculus. We hve lredy eplored few rooms in

More information

Fast Frequent Free Tree Mining in Graph Databases

Fast Frequent Free Tree Mining in Graph Databases The Chinese University of Hong Kong Fst Frequent Free Tree Mining in Grph Dtses Peixing Zho Jeffrey Xu Yu The Chinese University of Hong Kong Decemer 18 th, 2006 ICDM Workshop MCD06 Synopsis Introduction

More information

Revision Sheet. (a) Give a regular expression for each of the following languages:

Revision Sheet. (a) Give a regular expression for each of the following languages: Theoreticl Computer Science (Bridging Course) Dr. G. D. Tipldi F. Bonirdi Winter Semester 2014/2015 Revision Sheet University of Freiurg Deprtment of Computer Science Question 1 (Finite Automt, 8 + 6 points)

More information

Convert the NFA into DFA

Convert the NFA into DFA Convert the NF into F For ech NF we cn find F ccepting the sme lnguge. The numer of sttes of the F could e exponentil in the numer of sttes of the NF, ut in prctice this worst cse occurs rrely. lgorithm:

More information

p-adic Egyptian Fractions

p-adic Egyptian Fractions p-adic Egyptin Frctions Contents 1 Introduction 1 2 Trditionl Egyptin Frctions nd Greedy Algorithm 2 3 Set-up 3 4 p-greedy Algorithm 5 5 p-egyptin Trditionl 10 6 Conclusion 1 Introduction An Egyptin frction

More information

Exercises with (Some) Solutions

Exercises with (Some) Solutions Exercises with (Some) Solutions Techer: Luc Tesei Mster of Science in Computer Science - University of Cmerino Contents 1 Strong Bisimultion nd HML 2 2 Wek Bisimultion 31 3 Complete Lttices nd Fix Points

More information

MTH 5102 Linear Algebra Practice Exam 1 - Solutions Feb. 9, 2016

MTH 5102 Linear Algebra Practice Exam 1 - Solutions Feb. 9, 2016 Nme (Lst nme, First nme): MTH 502 Liner Algebr Prctice Exm - Solutions Feb 9, 206 Exm Instructions: You hve hour & 0 minutes to complete the exm There re totl of 6 problems You must show your work Prtil

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

1. For each of the following theorems, give a two or three sentence sketch of how the proof goes or why it is not true.

1. For each of the following theorems, give a two or three sentence sketch of how the proof goes or why it is not true. York University CSE 2 Unit 3. DFA Clsses Converting etween DFA, NFA, Regulr Expressions, nd Extended Regulr Expressions Instructor: Jeff Edmonds Don t chet y looking t these nswers premturely.. For ech

More information

The practical version

The practical version Roerto s Notes on Integrl Clculus Chpter 4: Definite integrls nd the FTC Section 7 The Fundmentl Theorem of Clculus: The prcticl version Wht you need to know lredy: The theoreticl version of the FTC. Wht

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

Fundamentals of Computer Science

Fundamentals of Computer Science Fundmentls of Computer Science Chpter 3: NFA nd DFA equivlence Regulr expressions Henrik Björklund Umeå University Jnury 23, 2014 NFA nd DFA equivlence As we shll see, it turns out tht NFA nd DFA re equivlent,

More information

DFA minimisation using the Myhill-Nerode theorem

DFA minimisation using the Myhill-Nerode theorem DFA minimistion using the Myhill-Nerode theorem Johnn Högerg Lrs Lrsson Astrct The Myhill-Nerode theorem is n importnt chrcteristion of regulr lnguges, nd it lso hs mny prcticl implictions. In this chpter,

More information

dx dt dy = G(t, x, y), dt where the functions are defined on I Ω, and are locally Lipschitz w.r.t. variable (x, y) Ω.

dx dt dy = G(t, x, y), dt where the functions are defined on I Ω, and are locally Lipschitz w.r.t. variable (x, y) Ω. Chpter 8 Stility theory We discuss properties of solutions of first order two dimensionl system, nd stility theory for specil clss of liner systems. We denote the independent vrile y t in plce of x, nd

More information

Math 4310 Solutions to homework 1 Due 9/1/16

Math 4310 Solutions to homework 1 Due 9/1/16 Mth 4310 Solutions to homework 1 Due 9/1/16 1. Use the Eucliden lgorithm to find the following gretest common divisors. () gcd(252, 180) = 36 (b) gcd(513, 187) = 1 (c) gcd(7684, 4148) = 68 252 = 180 1

More information

arxiv: v1 [math.ra] 1 Nov 2014

arxiv: v1 [math.ra] 1 Nov 2014 CLASSIFICATION OF COMPLEX CYCLIC LEIBNIZ ALGEBRAS DANIEL SCOFIELD AND S MCKAY SULLIVAN rxiv:14110170v1 [mthra] 1 Nov 2014 Abstrct Since Leibniz lgebrs were introduced by Lody s generliztion of Lie lgebrs,

More information

Math 520 Final Exam Topic Outline Sections 1 3 (Xiao/Dumas/Liaw) Spring 2008

Math 520 Final Exam Topic Outline Sections 1 3 (Xiao/Dumas/Liaw) Spring 2008 Mth 520 Finl Exm Topic Outline Sections 1 3 (Xio/Dums/Liw) Spring 2008 The finl exm will be held on Tuesdy, My 13, 2-5pm in 117 McMilln Wht will be covered The finl exm will cover the mteril from ll of

More information

The University of Nottingham SCHOOL OF COMPUTER SCIENCE A LEVEL 2 MODULE, SPRING SEMESTER LANGUAGES AND COMPUTATION ANSWERS

The University of Nottingham SCHOOL OF COMPUTER SCIENCE A LEVEL 2 MODULE, SPRING SEMESTER LANGUAGES AND COMPUTATION ANSWERS The University of Nottinghm SCHOOL OF COMPUTER SCIENCE LEVEL 2 MODULE, SPRING SEMESTER 2016 2017 LNGUGES ND COMPUTTION NSWERS Time llowed TWO hours Cndidtes my complete the front cover of their nswer ook

More information

CS103 Handout 32 Fall 2016 November 11, 2016 Problem Set 7

CS103 Handout 32 Fall 2016 November 11, 2016 Problem Set 7 CS103 Hndout 32 Fll 2016 Novemer 11, 2016 Prolem Set 7 Wht cn you do with regulr expressions? Wht re the limits of regulr lnguges? On this prolem set, you'll find out! As lwys, plese feel free to drop

More information

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?

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=

More information

HW3, Math 307. CSUF. Spring 2007.

HW3, Math 307. CSUF. Spring 2007. HW, Mth 7. CSUF. Spring 7. Nsser M. Abbsi Spring 7 Compiled on November 5, 8 t 8:8m public Contents Section.6, problem Section.6, problem Section.6, problem 5 Section.6, problem 7 6 5 Section.6, problem

More information

Introduction To Matrices MCV 4UI Assignment #1

Introduction To Matrices MCV 4UI Assignment #1 Introduction To Mtrices MCV UI Assignment # INTRODUCTION: A mtrix plurl: mtrices) is rectngulr rry of numbers rrnged in rows nd columns Exmples: ) b) c) [ ] d) Ech number ppering in the rry is sid to be

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

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

Review of Gaussian Quadrature method

Review of Gaussian Quadrature method Review of Gussin Qudrture method Nsser M. Asi Spring 006 compiled on Sundy Decemer 1, 017 t 09:1 PM 1 The prolem To find numericl vlue for the integrl of rel vlued function of rel vrile over specific rnge

More information

Lecture 09: Myhill-Nerode Theorem

Lecture 09: Myhill-Nerode Theorem CS 373: Theory of Computtion Mdhusudn Prthsrthy Lecture 09: Myhill-Nerode Theorem 16 Ferury 2010 In this lecture, we will see tht every lnguge hs unique miniml DFA We will see this fct from two perspectives

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

Chapter 3. Vector Spaces

Chapter 3. Vector Spaces 3.4 Liner Trnsformtions 1 Chpter 3. Vector Spces 3.4 Liner Trnsformtions Note. We hve lredy studied liner trnsformtions from R n into R m. Now we look t liner trnsformtions from one generl vector spce

More information

Jian-yi Shi East China Normal University, Shanghai and Technische Universität Kaiserslautern

Jian-yi Shi East China Normal University, Shanghai and Technische Universität Kaiserslautern IMPRIMITIVE COMPLEX REFLECTION GROUPS G(m, p, n) Jin-yi Shi Est Chin Norml University, Shnghi nd Technische Universität Kiserslutern 1 Typeset by AMS-TEX 2 Jin-yi Shi 1. Preliminries. 1.1. V, n-dim spce/c.

More information

N 0 completions on partial matrices

N 0 completions on partial matrices N 0 completions on prtil mtrices C. Jordán C. Mendes Arújo Jun R. Torregros Instituto de Mtemátic Multidisciplinr / Centro de Mtemátic Universidd Politécnic de Vlenci / Universidde do Minho Cmino de Ver

More information

Lecture 3: Curves in Calculus. Table of contents

Lecture 3: Curves in Calculus. Table of contents Mth 348 Fll 7 Lecture 3: Curves in Clculus Disclimer. As we hve textook, this lecture note is for guidnce nd supplement only. It should not e relied on when prepring for exms. In this lecture we set up

More information

First Midterm Examination

First Midterm Examination Çnky University Deprtment of Computer Engineering 203-204 Fll Semester First Midterm Exmintion ) Design DFA for ll strings over the lphet Σ = {,, c} in which there is no, no nd no cc. 2) Wht lnguge does

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

REPRESENTATION THEORY OF PSL 2 (q)

REPRESENTATION THEORY OF PSL 2 (q) REPRESENTATION THEORY OF PSL (q) YAQIAO LI Following re notes from book [1]. The im is to show the qusirndomness of PSL (q), i.e., the group hs no low dimensionl representtion. 1. Representtion Theory

More information

CMPSCI 250: Introduction to Computation. Lecture #31: What DFA s Can and Can t Do David Mix Barrington 9 April 2014

CMPSCI 250: Introduction to Computation. Lecture #31: What DFA s Can and Can t Do David Mix Barrington 9 April 2014 CMPSCI 250: Introduction to Computtion Lecture #31: Wht DFA s Cn nd Cn t Do Dvid Mix Brrington 9 April 2014 Wht DFA s Cn nd Cn t Do Deterministic Finite Automt Forml Definition of DFA s Exmples of DFA

More information

Chapter Five: Nondeterministic Finite Automata. Formal Language, chapter 5, slide 1

Chapter Five: Nondeterministic Finite Automata. Formal Language, chapter 5, slide 1 Chpter Five: Nondeterministic Finite Automt Forml Lnguge, chpter 5, slide 1 1 A DFA hs exctly one trnsition from every stte on every symol in the lphet. By relxing this requirement we get relted ut more

More information

The Cayley-Hamilton Theorem For Finite Automata. Radu Grosu SUNY at Stony Brook

The Cayley-Hamilton Theorem For Finite Automata. Radu Grosu SUNY at Stony Brook The Cyley-Hmilton Theorem For Finite Automt Rdu Grosu SUNY t Stony Brook How did I get interested in this topic? Convergence of Theories Hyrid Systems Computtion nd Control: - convergence etween control

More information

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

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

More information

Matrix Algebra. Matrix Addition, Scalar Multiplication and Transposition. Linear Algebra I 24

Matrix Algebra. Matrix Addition, Scalar Multiplication and Transposition. Linear Algebra I 24 Mtrix lger Mtrix ddition, Sclr Multipliction nd rnsposition Mtrix lger Section.. Mtrix ddition, Sclr Multipliction nd rnsposition rectngulr rry of numers is clled mtrix ( the plurl is mtrices ) nd the

More information

Parse trees, ambiguity, and Chomsky normal form

Parse trees, ambiguity, and Chomsky normal form Prse trees, miguity, nd Chomsky norml form In this lecture we will discuss few importnt notions connected with contextfree grmmrs, including prse trees, miguity, nd specil form for context-free grmmrs

More information

CHAPTER 1 PROGRAM OF MATRICES

CHAPTER 1 PROGRAM OF MATRICES CHPTER PROGRM OF MTRICES -- INTRODUCTION definition of engineering is the science y which the properties of mtter nd sources of energy in nture re mde useful to mn. Thus n engineer will hve to study the

More information

Hints for Exercise 1 on: Current and Resistance

Hints for Exercise 1 on: Current and Resistance Hints for Exercise 1 on: Current nd Resistnce Review the concepts of: electric current, conventionl current flow direction, current density, crrier drift velocity, crrier numer density, Ohm s lw, electric

More information

Multidimensional. MOD Planes. W. B. Vasantha Kandasamy Ilanthenral K Florentin Smarandache

Multidimensional. MOD Planes. W. B. Vasantha Kandasamy Ilanthenral K Florentin Smarandache Multidimensionl MOD Plnes W. B. Vsnth Kndsmy Ilnthenrl K Florentin Smrndche 2015 This book cn be ordered from: EuropNov ASBL Clos du Prnsse, 3E 1000, Bruxelles Belgium E-mil: info@europnov.be URL: http://www.europnov.be/

More information

Model Reduction of Finite State Machines by Contraction

Model Reduction of Finite State Machines by Contraction Model Reduction of Finite Stte Mchines y Contrction Alessndro Giu Dip. di Ingegneri Elettric ed Elettronic, Università di Cgliri, Pizz d Armi, 09123 Cgliri, Itly Phone: +39-070-675-5892 Fx: +39-070-675-5900

More information

To Do. Vectors. Motivation and Outline. Vector Addition. Cartesian Coordinates. Foundations of Computer Graphics (Spring 2010) x y

To Do. Vectors. Motivation and Outline. Vector Addition. Cartesian Coordinates. Foundations of Computer Graphics (Spring 2010) x y Foundtions of Computer Grphics (Spring 2010) CS 184, Lecture 2: Review of Bsic Mth http://inst.eecs.erkeley.edu/~cs184 o Do Complete Assignment 0 Downlod nd compile skeleton for ssignment 1 Red instructions

More information

Regular expressions, Finite Automata, transition graphs are all the same!!

Regular expressions, Finite Automata, transition graphs are all the same!! CSI 3104 /Winter 2011: Introduction to Forml Lnguges Chpter 7: Kleene s Theorem Chpter 7: Kleene s Theorem Regulr expressions, Finite Automt, trnsition grphs re ll the sme!! Dr. Neji Zgui CSI3104-W11 1

More information

Designing finite automata II

Designing finite automata II Designing finite utomt II Prolem: Design DFA A such tht L(A) consists of ll strings of nd which re of length 3n, for n = 0, 1, 2, (1) Determine wht to rememer out the input string Assign stte to ech of

More information

The Regulated and Riemann Integrals

The Regulated and Riemann Integrals Chpter 1 The Regulted nd Riemnn Integrls 1.1 Introduction We will consider severl different pproches to defining the definite integrl f(x) dx of function f(x). These definitions will ll ssign the sme vlue

More information

Myhill-Nerode Theorem

Myhill-Nerode Theorem Overview Myhill-Nerode Theorem Correspondence etween DA s nd MN reltions Cnonicl DA for L Computing cnonicl DFA Myhill-Nerode Theorem Deepk D Souz Deprtment of Computer Science nd Automtion Indin Institute

More information

Name Ima Sample ASU ID

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

More information

Arithmetic & Algebra. NCTM National Conference, 2017

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

More information

CSCI 340: Computational Models. Kleene s Theorem. Department of Computer Science

CSCI 340: Computational Models. Kleene s Theorem. Department of Computer Science CSCI 340: Computtionl Models Kleene s Theorem Chpter 7 Deprtment of Computer Science Unifiction In 1954, Kleene presented (nd proved) theorem which (in our version) sttes tht if lnguge cn e defined y ny

More information

September 13 Homework Solutions

September 13 Homework Solutions College of Engineering nd Computer Science Mechnicl Engineering Deprtment Mechnicl Engineering 5A Seminr in Engineering Anlysis Fll Ticket: 5966 Instructor: Lrry Cretto Septemer Homework Solutions. Are

More information

CS103B Handout 18 Winter 2007 February 28, 2007 Finite Automata

CS103B Handout 18 Winter 2007 February 28, 2007 Finite Automata CS103B ndout 18 Winter 2007 Ferury 28, 2007 Finite Automt Initil text y Mggie Johnson. Introduction Severl childrens gmes fit the following description: Pieces re set up on plying ord; dice re thrown or

More information

1B40 Practical Skills

1B40 Practical Skills B40 Prcticl Skills Comining uncertinties from severl quntities error propgtion We usully encounter situtions where the result of n experiment is given in terms of two (or more) quntities. We then need

More information

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

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

More information

GNFA GNFA GNFA GNFA GNFA

GNFA GNFA GNFA GNFA GNFA DFA RE NFA DFA -NFA REX GNFA Definition GNFA A generlize noneterministic finite utomton (GNFA) is grph whose eges re lele y regulr expressions, with unique strt stte with in-egree, n unique finl stte with

More information

Surface maps into free groups

Surface maps into free groups Surfce mps into free groups lden Wlker Novemer 10, 2014 Free groups wedge X of two circles: Set F = π 1 (X ) =,. We write cpitl letters for inverse, so = 1. e.g. () 1 = Commuttors Let x nd y e loops. The

More information

International workshop on graphs, semigroups, and semigroup acts

International workshop on graphs, semigroups, and semigroup acts Interntionl workshop on grphs, semigroups, nd semigroup cts celebrting the 75th birthdy of Ulrich Knuer October 10 - October 13, 2017 Institute of Mthemtics Technicl University Berlin Strsse des 17. Juni

More information

The size of subsequence automaton

The size of subsequence automaton Theoreticl Computer Science 4 (005) 79 84 www.elsevier.com/locte/tcs Note The size of susequence utomton Zdeněk Troníček,, Ayumi Shinohr,c Deprtment of Computer Science nd Engineering, FEE CTU in Prgue,

More information

Uses of transformations. 3D transformations. Review of vectors. Vectors in 3D. Points vs. vectors. Homogeneous coordinates S S [ H [ S \ H \ S ] H ]

Uses of transformations. 3D transformations. Review of vectors. Vectors in 3D. Points vs. vectors. Homogeneous coordinates S S [ H [ S \ H \ S ] H ] Uses of trnsformtions 3D trnsformtions Modeling: position nd resize prts of complex model; Viewing: define nd position the virtul cmer Animtion: define how objects move/chnge with time y y Sclr (dot) product

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