Random subgroups of a free group
|
|
- Rosanna Booth
- 5 years ago
- Views:
Transcription
1 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
2 Introduction Any group is isomorphic to quotient group of some free group. Study of lgeric properties of free groups y comintoril methods Grphicl representtion of sugroups : Stllings grphs Comintoril interprettion of prmeters or properties like the rnk, mlnormlity, Whitehed minimlity,... Quntittive study of finitely generted sugroups of free group nd nlysis of relted lgorithms
3 Introduction Any group is isomorphic to quotient group of some free group. Study of lgeric properties of free groups y comintoril methods Grphicl representtion of sugroups : Stllings grphs Comintoril interprettion of prmeters or properties like the rnk, mlnormlity, Whitehed minimlity,... Quntittive study of finitely generted sugroups of free group nd nlysis of relted lgorithms
4 I. Free Group
5 Free group : definition A group F is free if there is suset A of F such tht ny element of F cn e uniquely written s finite product of elements of A nd their inverses. The crdinlity of A is the rnk of the free group. Aprt from the existence of inverses no other reltion exists etween the genertors of free group. Bsic properties The sugroups of free group re free (Nielsen-Schreier Theorem). A free group with finite rnk contins sugroups with ny countle rnk.
6 Free group : definition A group F is free if there is suset A of F such tht ny element of F cn e uniquely written s finite product of elements of A nd their inverses. The crdinlity of A is the rnk of the free group. Aprt from the existence of inverses no other reltion exists etween the genertors of free group. Bsic properties The sugroups of free group re free (Nielsen-Schreier Theorem). A free group with finite rnk contins sugroups with ny countle rnk.
7 Free group : definition A group F is free if there is suset A of F such tht ny element of F cn e uniquely written s finite product of elements of A nd their inverses. The crdinlity of A is the rnk of the free group. Aprt from the existence of inverses no other reltion exists etween the genertors of free group. Bsic properties The sugroups of free group re free (Nielsen-Schreier Theorem). A free group with finite rnk contins sugroups with ny countle rnk.
8 Free groups nd reduced words Let A e finite lphet nd F = F(A) e the free group over A. The elements of F(A) re uniquely represented y the reduced words over A A 1 where A 1 = { 1 A}, A word is reduced if it does not contin fctors of the form 1 Exmples : is reduced, 1 1 cc 1 is not reduced Reduction of word : replce in ny order ll occurrences of 1 y the empty wordǫ. Exmple : 1 1 cc 1 = 1 cc 1 = cc 1
9 Free groups nd reduced words Let A e finite lphet nd F = F(A) e the free group over A. The elements of F(A) re uniquely represented y the reduced words over A A 1 where A 1 = { 1 A}, A word is reduced if it does not contin fctors of the form 1 Exmples : is reduced, 1 1 cc 1 is not reduced Reduction of word : replce in ny order ll occurrences of 1 y the empty wordǫ. Exmple : 1 1 cc 1 = 1 cc 1 = cc 1
10 Free groups nd reduced words Let A e finite lphet nd F = F(A) e the free group over A. The elements of F(A) re uniquely represented y the reduced words over A A 1 where A 1 = { 1 A}, A word is reduced if it does not contin fctors of the form 1 Exmples : is reduced, 1 1 cc 1 is not reduced Reduction of word : replce in ny order ll occurrences of 1 y the empty wordǫ. Exmple : 1 1 cc 1 = 1 cc 1 = cc 1
11 Free groups nd reduced words Let A e finite lphet nd F = F(A) e the free group over A. The elements of F(A) re uniquely represented y the reduced words over A A 1 where A 1 = { 1 A}, A word is reduced if it does not contin fctors of the form 1 Exmples : is reduced, 1 1 cc 1 is not reduced Reduction of word : replce in ny order ll occurrences of 1 y the empty wordǫ. Exmple : 1 1 cc 1 = 1 cc 1 = cc 1
12 Free groups nd reduced words Let A e finite lphet nd F = F(A) e the free group over A. The elements of F(A) re uniquely represented y the reduced words over A A 1 where A 1 = { 1 A}, A word is reduced if it does not contin fctors of the form 1 Exmples : is reduced, 1 1 cc 1 is not reduced Reduction of word : replce in ny order ll occurrences of 1 y the empty wordǫ. Exmple : 1 1 cc 1 = 1 cc 1 = cc 1
13 Finitely generted sugroups We re interested in finitely generted free sugroups, i.e., otined from finite set of genertors. Finitely generted free sugroups cn e represented in unique wy y finite grph clled its Stllings grph (Stllings 1983). This description is very useful, some properties of the sugroup cn e directly otined from its grph representtion. A 1st gol To study lgeric properties of finitely generted sugroups of free group with comintoril methods.
14 Finitely generted sugroups We re interested in finitely generted free sugroups, i.e., otined from finite set of genertors. Finitely generted free sugroups cn e represented in unique wy y finite grph clled its Stllings grph (Stllings 1983). This description is very useful, some properties of the sugroup cn e directly otined from its grph representtion. A 1st gol To study lgeric properties of finitely generted sugroups of free group with comintoril methods.
15 Finitely generted sugroups We re interested in finitely generted free sugroups, i.e., otined from finite set of genertors. Finitely generted free sugroups cn e represented in unique wy y finite grph clled its Stllings grph (Stllings 1983). This description is very useful, some properties of the sugroup cn e directly otined from its grph representtion. A 1st gol To study lgeric properties of finitely generted sugroups of free group with comintoril methods.
16 Stllings foldings Let Y = { 1 1, 2 1, }. Gol Build directed grph representing the free sugroup generted y Y First step Build directed cycle leled with 1 1 the first element of Y 1 1 i
17 Stllings foldings Let Y = { 1 1, 2 1, }. Gol Build directed grph representing the free sugroup generted y Y First step Build directed cycle leled with 1 1 the first element of Y 1 1 i
18 Stllings foldings Second step Build from the sme vertex i directed cycle leled with 2 1 the second element of Y i
19 Stllings foldings Third step Build from the sme vertex i directed cycle leled with the third nd lst element of Y i 1 1
20 Stllings foldings Forml inverses Reverse ll edges leled y 1 re nd replce their lel y. i
21 Stllings foldings Foldings to otin determinism nd codeterminism Apply s mny times s possile the following rules of merging (or folding) : The result does not depend on the order in which the trnsformtions re performed.
22 Stllings foldings - 1st folding i i
23 Stllings foldings - 2nd folding i i
24 Stllings foldings - 3rd folding A i B B A i
25 Stllings foldings - 4th folding i i
26 Stllings foldings - Lst folding nd Stllings grph i The Stllings grph representing the free sugroup generted y Y = { 1 1, 2 1, }. i
27 Stllings grphs : definition The grph (with distinguished vertex i) otined is Stllings grph. Stllings grph It is deterministic nd co-deterministic : ech letter cts like prtil injection on the set of sttes. it is connected ll ut the distinguished stte i hve degree t lest two Unicity of the representtion A Stllings grph represents in unique wy finitely generted sugroup of the free group generted y the lphet of the lels.
28 Stllings grphs : definition The grph (with distinguished vertex i) otined is Stllings grph. Stllings grph It is deterministic nd co-deterministic : ech letter cts like prtil injection on the set of sttes. it is connected ll ut the distinguished stte i hve degree t lest two Unicity of the representtion A Stllings grph represents in unique wy finitely generted sugroup of the free group generted y the lphet of the lels.
29 Stllings grphs : definition The grph (with distinguished vertex i) otined is Stllings grph. Stllings grph It is deterministic nd co-deterministic : ech letter cts like prtil injection on the set of sttes. it is connected ll ut the distinguished stte i hve degree t lest two Unicity of the representtion A Stllings grph represents in unique wy finitely generted sugroup of the free group generted y the lphet of the lels.
30 Stllings grphs exmples of use One cn check whether (reduced) word elongs the sugroup or not. Check if there exists cycle leled y the word eginning in i One cn compute sis nd the rnk of the sugroup rnk = E ( V 1) To otin sis, choose spnning tree of the Stllings grph. Ech edge e tht is not in the tree corresponds to genertor of the se : the lel of cycle eginning in i using e nd edges in the spnning tree. One cn check whether the sugroup hs finite index or not. All letters ct like permuttions on the set of vertices
31 Stllings grphs exmples of use One cn check whether (reduced) word elongs the sugroup or not. Check if there exists cycle leled y the word eginning in i One cn compute sis nd the rnk of the sugroup rnk = E ( V 1) To otin sis, choose spnning tree of the Stllings grph. Ech edge e tht is not in the tree corresponds to genertor of the se : the lel of cycle eginning in i using e nd edges in the spnning tree. One cn check whether the sugroup hs finite index or not. All letters ct like permuttions on the set of vertices
32 Stllings grphs exmples of use One cn check whether (reduced) word elongs the sugroup or not. Check if there exists cycle leled y the word eginning in i One cn compute sis nd the rnk of the sugroup rnk = E ( V 1) To otin sis, choose spnning tree of the Stllings grph. Ech edge e tht is not in the tree corresponds to genertor of the se : the lel of cycle eginning in i using e nd edges in the spnning tree. One cn check whether the sugroup hs finite index or not. All letters ct like permuttions on the set of vertices
33 Exmple for the rnk The Stllings grph of the sugroup genrted y Y = { 1 1, 2 1, } : Therefore{ 2 1, 1 1 } is sis of the sugroup nd the rnk is 2. i
34 Stllings grphs lgorithmic point of view Stlling foldings cn e computed in O(n log n) where n is the totl length of the genertors. The lgorithm due Touikn (2006) mkes use of Union nd Find. The intersection (resp. union) of two sugroups cn e computed in time nd spce O(n 1 n 2 ) where n 1 (resp. n 2 ) is the size (here the numer of vertices) of the first (resp. second) Stllings grph.
35 II. Distriutions on Sugroups
36 A grph-sed distriution on sugroups A rndom sugroup is given y choosing uniformly t rndom Stllings grph of size n Studied y Bssino, Nicud, Weil (2008, 2013, 2015) Wht does the Stllings grph of such rndom sugroup look like? FIGURE: A rndom sugroup with 200 vertices for the grph-sed distriution (The lphet is of size 2).
37 The clssicl word-sed distriution on sugroups A rndom sugroup is given y choosing rndomly nd uniformly k genertors of length t most n, where k is fixed Studied y Gromov (1987), Arzhntsev nd Ol shnskiǐ (1996), Jitsukw (2002),... Wht does the Stllings grph of such rndom sugroup look like? FIGURE: A rndom sugroup for the word-sed distriution with 5 words of lengths t most 40 (The lphet is of size 2.)
38 A word-sed distriution (few genertors) Fix the numer k of genertors nd the mximl length n of ech genertor. Consider the uniform distriution over the k-tuples of reduced words of length t most n. Let R n the numer of reduced words of length n, R n = 2r(2r 1) n 1 The length of word in rndom k-tuple is ner to n.
39 A word-sed distriution (few genertors) Fix the numer k of genertors nd the mximl length n of ech genertor. Consider the uniform distriution over the k-tuples of reduced words of length t most n. Let R n the numer of reduced words of length n, R n = 2r(2r 1) n 1 The length of word in rndom k-tuple is ner to n.
40 A word-sed distriution (few genertors) Length, prefixes nd suffixes Let 0 < α < 1. A reduced word in R n hs length greter thnαn with proility tht tends towrd 1 when n tends towrd+. Let 0 < β < α/2. A k-uple of reduced words of R n is such tht the prefixes of lengthβn of ll words nd their inverses re pirwise distinct with proility tht tends towrd 1 when n tends towrd+. Consequence Ech of the k reduced words hs n outer loop of length t lest n(α 2β) with proility tht tends to 1 when n tends to+.
41 A grph-sed distriution : Proilistic results Theorem (Bssino, Nicud, Weil 2008) The proility for rndom r-tuple of prtil injections of size n to form Stllings grph tends towrd 1 when n tends towrd +. Stllings grph It is deterministic nd co-deterministic : ech letter cts like prtil injection on the set of sttes. The proof it is connected ll ut the distinguished stte i hve degree t lest two is study of prtil injections siclly uses the sddle-point method
42 A grph-sed distriution : Proilistic results Theorem (Bssino, Nicud, Weil 2008) The proility for rndom r-tuple of prtil injections of size n to form Stllings grph tends towrd 1 when n tends towrd +. Stllings grph It is deterministic nd co-deterministic : ech letter cts like prtil injection on the set of sttes. The proof it is connected ll ut the distinguished stte i hve degree t lest two is study of prtil injections siclly uses the sddle-point method
43 A grph-sed distriution : Proilistic results Theorem (Bssino, Nicud, Weil 2008) The proility for rndom r-tuple of prtil injections of size n to form Stllings grph tends towrd 1 when n tends towrd +. Stllings grph It is deterministic nd co-deterministic : ech letter cts like prtil injection on the set of sttes. The proof it is connected ll ut the distinguished stte i hve degree t lest two is study of prtil injections siclly uses the sddle-point method
44 A grph-sed distriution : Prtil injections A prtil injection cn e seen s set of cycles nd of non-empty sequences. Set(Cycle or non-empty Sequences) With the symolic method : I(z) = n 0 I n n! zn = exp ( log With the sddle point method : I n n! e 2 π e2 1 1 z + z ) = 1 1 z 1 z ez/(1 z) 1 2 n n 1 4
45 A grph-sed distriution : Prtil injections A prtil injection cn e seen s set of cycles nd of non-empty sequences. Set(Cycle or non-empty Sequences) With the symolic method : I(z) = n 0 I n n! zn = exp ( log With the sddle point method : I n n! e 2 π e2 1 1 z + z ) = 1 1 z 1 z ez/(1 z) 1 2 n n 1 4
46 Connectedness Theorem The proility for r prtil injections of size n to form connected grph is ( ) p n = 1 2r 1 n r 1 + o n r 1 Proof Let J(z) = n>0 j nz n = n>0 Ir nz n /n!. Then 1+J(z) = exp(c(z)) nd C(z) = log(1+j(z)). From Bender theorem (1974) it is enough to check tht j n = o(j n 1 ) nd tht for some s 1, n s k=s j kj n k = O(j n s ), to otin tht c n = j n ( 1 2r n r 1 + o ( 1 n r 1 ))
47 Connectedness Theorem The proility for r prtil injections of size n to form connected grph is ( ) p n = 1 2r 1 n r 1 + o n r 1 Proof Let J(z) = n>0 j nz n = n>0 Ir nz n /n!. Then 1+J(z) = exp(c(z)) nd C(z) = log(1+j(z)). From Bender theorem (1974) it is enough to check tht j n = o(j n 1 ) nd tht for some s 1, n s k=s j kj n k = O(j n s ), to otin tht c n = j n ( 1 2r n r 1 + o ( 1 n r 1 ))
48 Vertices with zero or one outgoing or ingoing edge If x is vertex with 0 or 1 edge, then x must e isolted for r 1 injections nd n endpoint for the remining injection. The proility it is isolted for n injection is I n 1 I n, which is smller thn 1 n. Let I n,k e the numer of size-n injections hving k sequences, nd let I(z, u) e the ivrite generting function defined y : ( ( )) zu 1 I(z, u) = exp 1 z + log = 1 ( ) zu 1 z 1 z exp 1 z Using the sddle point theorem we otin tht the expected numer of sequences is 1 n nd tht the proility tht given vertex is n endpoint is in O( 1 n ).
49 Trimness Therefore A given vertex hs degree 0 or 1 with proility O(n r+1/2 ), there is such vertex with proility O(n r+3/2 ) with proility t lesto(n 1/2 ) the grph hs no such vertex.
50 IV. How to compre the two distriutions
51 Méthod A property P is generic for(x n ) when the proility for n element of X n to stisfy P tends towrd 1 when n tends towrd. A property P is negligile for(x n ) when the proility for n element of X n to stisfy P tends towrd O when n tends towrd. In the following, we present generic or negligile lgeric properties for ech distriution.
52 Méthod A property P is generic for(x n ) when the proility for n element of X n to stisfy P tends towrd 1 when n tends towrd. A property P is negligile for(x n ) when the proility for n element of X n to stisfy P tends towrd O when n tends towrd. In the following, we present generic or negligile lgeric properties for ech distriution.
53 Experimentl results FIGURE: On the left, rndom sugroup for the word-sed distriution with 5 words of lengths t most 40. On the right, rndom sugroup with 200 vertices for the grph-sed distriution (The lphet is of size 2).
54 Rnk One cn compute the rnk of finitely generted sugroup from its Stllings grph rnk = E ( V 1) In the word sed distriution (k words of mximl length n), the verge rnk is k In the grph sed distriution the verge rnk is ( A 1)n A n+1.
55 Rnk One cn compute the rnk of finitely generted sugroup from its Stllings grph rnk = E ( V 1) In the word sed distriution (k words of mximl length n), the verge rnk is k In the grph sed distriution the verge rnk is ( A 1)n A n+1.
56 Mlnormlity A sugroup H of G is norml when for ny g G, g 1 Hg = H. A sugroup is mlnorml when for ny g / H, g 1 Hg H = 1. Theorem (comintoril chrcteriztion) A sugroup of free group is non-mlnorml if nd only, in its Stllings grph, if there exists two vertices x y nd non-empty reduced word u, such tht u is the lel of loop on x nd of loop on y.
57 Mlnormlity A sugroup H of G is norml when for ny g G, g 1 Hg = H. A sugroup is mlnorml when for ny g / H, g 1 Hg H = 1. Theorem (comintoril chrcteriztion) A sugroup of free group is non-mlnorml if nd only, in its Stllings grph, if there exists two vertices x y nd non-empty reduced word u, such tht u is the lel of loop on x nd of loop on y.
58 Mlnormlity Theorem For the word-sed distriution, mlnormlity is generic, ut it is negligile for the grph-sed. Proof The proof in the word-sed distriution is due to Jitsukw (2002). Bsiclly loops re long enough to e distinct with high proility. The proility tht prtil injection contins t most one cycle nd tht the length of this cycle is 1 is e n.
59 Mlnormlity Theorem For the word-sed distriution, mlnormlity is generic, ut it is negligile for the grph-sed. Proof The proof in the word-sed distriution is due to Jitsukw (2002). Bsiclly loops re long enough to e distinct with high proility. The proility tht prtil injection contins t most one cycle nd tht the length of this cycle is 1 is e n.
60 Finite presenttion The ide is to quotient the free group y norml finitely generted sugroup. Let E e n ritrry suset, nd N(E) e its norml closure, tht is the smllest norml sugroup contining E. Equivlently ech word x of E ecomes reltor x = 1. In the word-sed distriution genericlly the quotient sugroup is infinite (Jitsukw, 2002). But in the grph-sed distriution, the quotient group is genericlly trivil.
61 Finite presenttion The ide is to quotient the free group y norml finitely generted sugroup. Let E e n ritrry suset, nd N(E) e its norml closure, tht is the smllest norml sugroup contining E. Equivlently ech word x of E ecomes reltor x = 1. In the word-sed distriution genericlly the quotient sugroup is infinite (Jitsukw, 2002). But in the grph-sed distriution, the quotient group is genericlly trivil.
62 Finite presenttion The ide is to quotient the free group y norml finitely generted sugroup. Let E e n ritrry suset, nd N(E) e its norml closure, tht is the smllest norml sugroup contining E. Equivlently ech word x of E ecomes reltor x = 1. In the word-sed distriution genericlly the quotient sugroup is infinite (Jitsukw, 2002). But in the grph-sed distriution, the quotient group is genericlly trivil.
63 Finite presenttion The ide is to quotient the free group y norml finitely generted sugroup. Let E e n ritrry suset, nd N(E) e its norml closure, tht is the smllest norml sugroup contining E. Equivlently ech word x of E ecomes reltor x = 1. In the word-sed distriution genericlly the quotient sugroup is infinite (Jitsukw, 2002). But in the grph-sed distriution, the quotient group is genericlly trivil.
64 Finite presenttion Theorem Genericlly the gcd of the lengths of the cycles of prtil injection of size n is 1. Theorem Genericlly the gcd of the lengths of the cycles of permuttion of size n is 1. Permuttion prt of n injection Genericlly the permuttion prt of size n injection is greter thn n 1/3 nd the gcd of the length of the cycles is 1.
65 Finite presenttion Theorem Genericlly the gcd of the lengths of the cycles of prtil injection of size n is 1. Theorem Genericlly the gcd of the lengths of the cycles of permuttion of size n is 1. Permuttion prt of n injection Genericlly the permuttion prt of size n injection is greter thn n 1/3 nd the gcd of the length of the cycles is 1.
66 Finite presenttion Theorem Genericlly the gcd of the lengths of the cycles of prtil injection of size n is 1. Theorem Genericlly the gcd of the lengths of the cycles of permuttion of size n is 1. Permuttion prt of n injection Genericlly the permuttion prt of size n injection is greter thn n 1/3 nd the gcd of the length of the cycles is 1.
67 Thnk you for your ttention!
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 informationFirst 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 informationConvert 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 informationMinimal 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 informationCS 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)
More informationComputing with finite semigroups: part I
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, 2015 1 / 34 Wht is this tlk
More informationLinear 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 informationCoalgebra, 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 informationLecture 08: Feb. 08, 2019
4CS4-6:Theory of Computtion(Closure on Reg. Lngs., regex to NDFA, DFA to regex) Prof. K.R. Chowdhry Lecture 08: Fe. 08, 2019 : Professor of CS Disclimer: These notes hve not een sujected to the usul scrutiny
More informationSurface 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 informationFinite-State Automata: Recap
Finite-Stte Automt: Recp Deepk D Souz Deprtment of Computer Science nd Automtion Indin Institute of Science, Bnglore. 09 August 2016 Outline 1 Introduction 2 Forml Definitions nd Nottion 3 Closure under
More informationGenerating 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 informationSymbolic enumeration methods for unlabelled structures
Go & Šjn, Comintoril Enumertion Notes 4 Symolic enumertion methods for unlelled structures Definition A comintoril clss is finite or denumerle set on which size function is defined, stisfying the following
More informationHomework 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 informationRegular 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 informationCS415 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 informationSpeech Recognition Lecture 2: Finite Automata and Finite-State Transducers. Mehryar Mohri Courant Institute and Google Research
Speech Recognition Lecture 2: Finite Automt nd Finite-Stte Trnsducers Mehryr Mohri Cournt Institute nd Google Reserch mohri@cims.nyu.com Preliminries Finite lphet Σ, empty string. Set of ll strings over
More informationFirst 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 informationCHAPTER 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
More informationBases 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 informationInfinitely presented graphical small cancellation groups
Infinitely presented grphicl smll cncelltion groups Dominik Gruer Université de Neuchâtel Stevens Group Theory Interntionl Weinr Decemer 10, 2015 Dominik Gruer (Neuchâtel) 2/30 Motivtion Grphicl smll cncelltion
More informationChapter 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 informationCSCI 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 informationConverting Regular Expressions to Discrete Finite Automata: A Tutorial
Converting Regulr Expressions to Discrete Finite Automt: A Tutoril Dvid Christinsen 2013-01-03 This is tutoril on how to convert regulr expressions to nondeterministic finite utomt (NFA) nd how to convert
More informationCS 275 Automata and Formal Language Theory
CS 275 utomt nd Forml Lnguge Theory Course Notes Prt II: The Recognition Prolem (II) Chpter II.5.: Properties of Context Free Grmmrs (14) nton Setzer (Bsed on ook drft y J. V. Tucker nd K. Stephenson)
More informationLexical 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
More informationJava II Finite Automata I
Jv II Finite Automt I Bernd Kiefer Bernd.Kiefer@dfki.de Deutsches Forschungszentrum für künstliche Intelligenz Finite Automt I p.1/13 Processing Regulr Expressions We lredy lerned out Jv s regulr expression
More informationChapter 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 informationa,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 informationNFAs continued, Closure Properties of Regular Languages
Algorithms & Models of Computtion CS/ECE 374, Fll 2017 NFAs continued, Closure Properties of Regulr Lnguges Lecture 5 Tuesdy, Septemer 12, 2017 Sriel Hr-Peled (UIUC) CS374 1 Fll 2017 1 / 31 Regulr Lnguges,
More informationHarvard 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
More informationFingerprint idea. Assume:
Fingerprint ide Assume: We cn compute fingerprint f(p) of P in O(m) time. If f(p) f(t[s.. s+m 1]), then P T[s.. s+m 1] We cn compre fingerprints in O(1) We cn compute f = f(t[s+1.. s+m]) from f(t[s.. s+m
More informationNFAs continued, Closure Properties of Regular Languages
lgorithms & Models of omputtion S/EE 374, Spring 209 NFs continued, losure Properties of Regulr Lnguges Lecture 5 Tuesdy, Jnury 29, 209 Regulr Lnguges, DFs, NFs Lnguges ccepted y DFs, NFs, nd regulr expressions
More informationGNFA 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 informationGrammar. 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 informationp-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 informationCMPSCI 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 informationCS103B 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 informationFarey 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 informationAutomata 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 informationSpeech 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 informationTypes of Finite Automata. CMSC 330: Organization of Programming Languages. Comparing DFAs and NFAs. Comparing DFAs and NFAs (cont.) Finite Automata 2
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 informationFormal 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 informationFinite 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 informationClosure Properties of Regular Languages
Closure Properties of Regulr Lnguges Regulr lnguges re closed under mny set opertions. Let L 1 nd L 2 e regulr lnguges. (1) L 1 L 2 (the union) is regulr. (2) L 1 L 2 (the conctention) is regulr. (3) L
More informationCS 301. Lecture 04 Regular Expressions. Stephen Checkoway. January 29, 2018
CS 301 Lecture 04 Regulr Expressions Stephen Checkowy Jnury 29, 2018 1 / 35 Review from lst time NFA N = (Q, Σ, δ, q 0, F ) where δ Q Σ P (Q) mps stte nd n lphet symol (or ) to set of sttes We run n NFA
More informationCMSC 330: Organization of Programming Languages
CMSC 330: Orgniztion of Progrmming Lnguges Finite Automt 2 CMSC 330 1 Types of Finite Automt Deterministic Finite Automt (DFA) Exctly one sequence of steps for ech string All exmples so fr Nondeterministic
More informationFinite Automata-cont d
Automt Theory nd Forml Lnguges Professor Leslie Lnder Lecture # 6 Finite Automt-cont d The Pumping Lemm WEB SITE: http://ingwe.inghmton.edu/ ~lnder/cs573.html Septemer 18, 2000 Exmple 1 Consider L = {ww
More informationLecture 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 informationTypes 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 information1 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 information12.1 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 informationThoery of Automata CS402
Thoery of Automt C402 Theory of Automt Tle of contents: Lecture N0. 1... 4 ummry... 4 Wht does utomt men?... 4 Introduction to lnguges... 4 Alphets... 4 trings... 4 Defining Lnguges... 5 Lecture N0. 2...
More informationAUTOMATA AND LANGUAGES. Definition 1.5: Finite Automaton
25. Finite Automt AUTOMATA AND LANGUAGES A system of computtion tht only hs finite numer of possile sttes cn e modeled using finite utomton A finite utomton is often illustrted s stte digrm d d d. d q
More informationReview 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 informationTable of contents: Lecture N Summary... 3 What does automata mean?... 3 Introduction to languages... 3 Alphabets... 3 Strings...
Tle of contents: Lecture N0.... 3 ummry... 3 Wht does utomt men?... 3 Introduction to lnguges... 3 Alphets... 3 trings... 3 Defining Lnguges... 4 Lecture N0. 2... 7 ummry... 7 Kleene tr Closure... 7 Recursive
More informationCS 330 Formal Methods and Models Dana Richards, George Mason University, Spring 2016 Quiz Solutions
CS 330 Forml Methods nd Models Dn Richrds, George Mson University, Spring 2016 Quiz Solutions Quiz 1, Propositionl Logic Dte: Ferury 9 1. (4pts) ((p q) (q r)) (p r), prove tutology using truth tles. p
More informationThe area under the graph of f and above the x-axis between a and b is denoted by. f(x) dx. π O
1 Section 5. The Definite Integrl Suppose tht function f is continuous nd positive over n intervl [, ]. y = f(x) x The re under the grph of f nd ove the x-xis etween nd is denoted y f(x) dx nd clled the
More informationThree 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 informationCS375: 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
More informationAssignment 1 Automata, Languages, and Computability. 1 Finite State Automata and Regular Languages
Deprtment of Computer Science, Austrlin Ntionl University COMP2600 Forml Methods for Softwre Engineering Semester 2, 206 Assignment Automt, Lnguges, nd Computility Smple Solutions Finite Stte Automt nd
More informationWe will see what is meant by standard form very shortly
THEOREM: For fesible liner progrm in its stndrd form, the optimum vlue of the objective over its nonempty fesible region is () either unbounded or (b) is chievble t lest t one extreme point of the fesible
More informationChapter 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 informationRegular 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 informationNondeterminism and Nodeterministic Automata
Nondeterminism nd Nodeterministic Automt 61 Nondeterminism nd Nondeterministic Automt The computtionl mchine models tht we lerned in the clss re deterministic in the sense tht the next move is uniquely
More informationA 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 informationWorked out examples Finite Automata
Worked out exmples Finite Automt Exmple Design Finite Stte Automton which reds inry string nd ccepts only those tht end with. Since we re in the topic of Non Deterministic Finite Automt (NFA), we will
More informationCSCI 340: Computational Models. Transition Graphs. Department of Computer Science
CSCI 340: Computtionl Models Trnsition Grphs Chpter 6 Deprtment of Computer Science Relxing Restrints on Inputs We cn uild n FA tht ccepts only the word! 5 sttes ecuse n FA cn only process one letter t
More information4.1. Probability Density Functions
STT 1 4.1-4. 4.1. Proility Density Functions Ojectives. Continuous rndom vrile - vers - discrete rndom vrile. Proility density function. Uniform distriution nd its properties. Expected vlue nd vrince of
More informationConnected-components. Summary of lecture 9. Algorithms and Data Structures Disjoint sets. Example: connected components in graphs
Prm University, Mth. Deprtment Summry of lecture 9 Algorithms nd Dt Structures Disjoint sets Summry of this lecture: (CLR.1-3) Dt Structures for Disjoint sets: Union opertion Find opertion Mrco Pellegrini
More informationHomework 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 informationDesigning 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 informationOn the Hanna Neumann Conjecture
On the Hnn Neumnn Conjecture Toshiki Jitsukw Bill Khn Alexei G. Mysnikov Astrct The Hnn Neumnn conjecture sttes tht if F is free group, then for ll nontrivil finitely generted sugroups H, K F, rnk(h K)
More information5. (±±) Λ = fw j w is string of even lengthg [ 00 = f11,00g 7. (11 [ 00)± Λ = fw j w egins with either 11 or 00g 8. (0 [ ffl)1 Λ = 01 Λ [ 1 Λ 9.
Regulr Expressions, Pumping Lemm, Right Liner Grmmrs Ling 106 Mrch 25, 2002 1 Regulr Expressions A regulr expression descries or genertes lnguge: it is kind of shorthnd for listing the memers of lnguge.
More informationCompiler Design. Fall Lexical Analysis. Sample Exercises and Solutions. Prof. Pedro C. Diniz
University of Southern Cliforni Computer Science Deprtment Compiler Design Fll Lexicl Anlysis Smple Exercises nd Solutions Prof. Pedro C. Diniz USC / Informtion Sciences Institute 4676 Admirlty Wy, Suite
More informationThe Minimum Label Spanning Tree Problem: Illustrating the Utility of Genetic Algorithms
The Minimum Lel Spnning Tree Prolem: Illustrting the Utility of Genetic Algorithms Yupei Xiong, Univ. of Mrylnd Bruce Golden, Univ. of Mrylnd Edwrd Wsil, Americn Univ. Presented t BAE Systems Distinguished
More informationTutorial Automata and formal Languages
Tutoril Automt nd forml Lnguges Notes for to the tutoril in the summer term 2017 Sestin Küpper, Christine Mik 8. August 2017 1 Introduction: Nottions nd sic Definitions At the eginning of the tutoril we
More informationCM10196 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 information1. 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 informationCHAPTER 1 Regular Languages. Contents. definitions, examples, designing, regular operations. Non-deterministic Finite Automata (NFA)
Finite Automt (FA or DFA) CHAPTER Regulr Lnguges Contents definitions, exmples, designing, regulr opertions Non-deterministic Finite Automt (NFA) definitions, equivlence of NFAs DFAs, closure under regulr
More informationHybrid Control and Switched Systems. Lecture #2 How to describe a hybrid system? Formal models for hybrid system
Hyrid Control nd Switched Systems Lecture #2 How to descrie hyrid system? Forml models for hyrid system João P. Hespnh University of Cliforni t Snt Brr Summry. Forml models for hyrid systems: Finite utomt
More informationParse 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 information2.4 Linear Inequalities and Interval Notation
.4 Liner Inequlities nd Intervl Nottion We wnt to solve equtions tht hve n inequlity symol insted of n equl sign. There re four inequlity symols tht we will look t: Less thn , Less thn or
More informationPavel Rytí. November 22, 2011 Discrete Math Seminar - Simon Fraser University
Geometric representtions of liner codes Pvel Rytí Deprtment of Applied Mthemtics Chrles University in Prgue Advisor: Mrtin Loebl November, 011 Discrete Mth Seminr - Simon Frser University Bckground Liner
More informationɛ-closure, Kleene s Theorem,
DEGefW5wiGH2XgYMEzUKjEmtCDUsRQ4d 1 A nice pper relevnt to this course is titled The Glory of the Pst 2 NICTA Resercher, Adjunct t the Austrlin Ntionl University nd Griffith University ɛ-closure, Kleene
More informationModel 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 informationState space systems analysis (continued) Stability. A. Definitions A system is said to be Asymptotically Stable (AS) when it satisfies
Stte spce systems nlysis (continued) Stbility A. Definitions A system is sid to be Asymptoticlly Stble (AS) when it stisfies ut () = 0, t > 0 lim xt () 0. t A system is AS if nd only if the impulse response
More informationFormal 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 informationRegular 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 informationMath 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 informationSTRUCTURE OF CONCURRENCY Ryszard Janicki. Department of Computing and Software McMaster University Hamilton, ON, L8S 4K1 Canada
STRUCTURE OF CONCURRENCY Ryszrd Jnicki Deprtment of Computing nd Softwre McMster University Hmilton, ON, L8S 4K1 Cnd jnicki@mcmster.c 1 Introduction Wht is concurrency? How it cn e modelled? Wht re the
More information12.1 Nondeterminism Nondeterministic Finite Automata. a a b ε. CS125 Lecture 12 Fall 2014
CS125 Lecture 12 Fll 2014 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 informationCS 330 Formal Methods and Models
CS 330 Forml Methods nd Models Dn Richrds, George Mson University, Spring 2017 Quiz Solutions Quiz 1, Propositionl Logic Dte: Ferury 2 1. Prove ((( p q) q) p) is tutology () (3pts) y truth tle. p q p q
More informationContinuous Random Variables Class 5, Jeremy Orloff and Jonathan Bloom
Lerning Gols Continuous Rndom Vriles Clss 5, 8.05 Jeremy Orloff nd Jonthn Bloom. Know the definition of continuous rndom vrile. 2. Know the definition of the proility density function (pdf) nd cumultive
More informationCS103 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 informationAutomaton groups and complete square complexes
rxiv:1707.00215v1 [mth.gr] 1 Jul 2017 Automton groups nd complete squre complexes Ievgen Bondrenko nd Bohdn Kivv July 4, 2017 Astrct The first exmple of non-residully finite group in the clsses of finitely
More informationChapter 6 Continuous Random Variables and Distributions
Chpter 6 Continuous Rndom Vriles nd Distriutions Mny economic nd usiness mesures such s sles investment consumption nd cost cn hve the continuous numericl vlues so tht they cn not e represented y discrete
More informationBayesian Networks: Approximate Inference
pproches to inference yesin Networks: pproximte Inference xct inference Vrillimintion Join tree lgorithm pproximte inference Simplify the structure of the network to mkxct inferencfficient (vritionl methods,
More informationGenetic Programming. Outline. Evolutionary Strategies. Evolutionary strategies Genetic programming Summary
Outline Genetic Progrmming Evolutionry strtegies Genetic progrmming Summry Bsed on the mteril provided y Professor Michel Negnevitsky Evolutionry Strtegies An pproch simulting nturl evolution ws proposed
More informationRecitation 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