8 Automata and formal languages. 8.1 Formal languages


 Estella Short
 11 months ago
 Views:
Transcription
1 8 Automt nd forml lnguges This exposition ws developed y Clemens Gröpl nd Knut Reinert. It is sed on the following references, ll of which re recommended reding: 1. Uwe Schöning: Theoretische Informtik  kurz gefsst. 3. Auflge. Spektrum Akdemischer Verlg, Heidelerg, ISBN PROSITE user mnul 3. Sigrist C.J., Cerutti L., Hulo N., Gttiker A., Flquet L., Pgni M., Biroch A., Bucher P.. PROSITE: documented dtse using ptterns nd profiles s motif descriptors. Brief Bioinform. 3: (2002). We will present sic fcts out: forml lnguges, regulr nd contexfree grmmrs, deterministic finite utomt, nondeterministic finite utomt, pushdown utomt. 8.1 Forml lnguges An lphet Σ is nonempty set of symols (lso clled letters). In the following, Σ will lwys denote finite lphet. A word over n lphet Σ is sequence of elements of Σ. This includes the empty word, which contins no letters nd is denoted y ε. For ny lphet Σ, the set Σ is defined to e the set of ll words over the lphet Σ. The set Σ + := Σ \ {ε} contins ll nonempty words Σ. nd E. g., if Σ = {, }, then Σ = {ε,,,,,,,,,...} Σ + = {,,,,,,,,...}. The length of word x is denoted y x. For words x, y Σ we denote their conctention y xy: If x = x 1,... x m nd y = y 1,..., y n (where x i, y j Σ) then xy = x 1,..., x m, y 1,..., y n. For x Σ let x n := xx... x. (Tht is, n conctented copies of x). } {{ } n
2 Finite utomt nd regulr grmmrs, y Clemens Gröpl, Jnury 9, 2014, 12: Thus x n = n x, xy = x + y, nd ε = 0. A (forml) lnguge A over n lphet Σ is simply set of words over Σ, i. e., suset of Σ. The empty lnguge := {} contins no words. (Note: The empty lnguge must not to e confused with the lnguge {ε}, which contins only the empty word.) The complement of lnguge A (over n lphet Σ) is the lnguge Ā := Σ \ A. For lnguges A, B we define their product s using the conctention opertion defined ove. AB := {xy x A, y B}, E. g., if A = {, h}, B = {ε, t, ttu} then AB = {, h, t, ht, ttu, httu}. The powers of lnguge L re defined y L 0 := {ε} L n := L n 1 L, for n 1. The Kleene str (or Kleene hull) of lnguge L is L := L n. i=0 Funnily, n = for n 1, ut = 0 = {ε}, ccording to these definitions. But (defined this wy) lnguge exponentition works s expected: For every lnguge A nd m, n 0, we hve A m A n = A m+n. Most forml lnguges re infinite ojects. In order to del with them lgorithmiclly, we need finite descriptions for them. There re two pproches to this: Grmmrs descrie rules how to produce words from given lnguge. We cn clssify lnguges ccording to the kinds of rules which re llowed. Automt descrie how to test whether word elongs to given lnguge. We cn clssify lnguges ccording to the computtionl power of the utomt which re llowed. Fortuntely, the two pproches cn e shown to e equivlent in mny cses. 8.2 Grmmrs A grmmr is 4tuple G = (V, Σ, P, S) stisfying the following conditions. V is finite set of vrile symols. For revity, the vrile symols re often simply clled vriles, or nonterminls.
3 8002 Finite utomt nd regulr grmmrs, y Clemens Gröpl, Jnury 9, 2014, 12:48 Σ is finite set of terminl symols, lso clled the terminl lphet. This is the lphet of the lnguge we wnt to descrie. We require tht vrile symols nd terminl symols cn e distinguished, i. e., V Σ =. P is finite set of rules or productions. A rule hs the form where lhs (V Σ) + nd rhs (V Σ). S V is the strt vrile. (left hnd side) (right hnd side), We cn think of the productions of grmmr s wys to trnsform words over the lphet V Σ into other words over V Σ. We cn derive v from u in G in one step if there is production y y in P nd x, z (V Σ) such tht u = xyz nd v = xy z. This is denoted y u G v. If the grmmr is cler from the context, we write just u v. The reflexive nd trnsitive closure G of G is defined s follows. We hve u v if nd only if u = v or v cn G e derived from u in series u w 1 w 2... w n v of steps using the grmmr G. Then L(G) := {w Σ S G w} is the lnguge generted y G. Note: The crucil point is tht we wrote w Σ, not w (V Σ). Vriles re not llowed in generted words which re output. (Note 2: This rises nother question: Productions cn lengthen nd shorten the words. How cn we tell how long it will tke until we hve removed ll vrile symols? Well, tht s nother story.) Exmple 1. The following grmmr genertes ll words over Σ = {, } with eqully mny s nd s. G = ({S}, Σ, P, S), where P := { } S ε, S S, (Cn you prove this?) Exmple 2. The following grmmr genertes wellformed rithmetic expressions over the lphet Σ = {(, ),,, c, +, }.
4 Finite utomt nd regulr grmmrs, y Clemens Gröpl, Jnury 9, 2014, 12: G = ({A, M, K}, Σ, P, A), where P := { } A M, A A + M M K, M M K, K, K, K c, K (A), Chomsky descried four sorts of restrictions on grmmr s rewriting rules. The resulting four clsses of grmmrs form hierrchy known s the Chomsky hierchy. In wht follows we use cpitl letters A, B, W, S,... to denote nonterminl symols, smll letters,, c,... to denote terminl symols nd greek letters α, β, γ,... to represent string of terminl nd nonterminl letters. 1. Regulr grmmrs. Only production rules of the form W W or W re llowed. 2. Contextfree grmmrs. Any production of the form W α is llowed. 3. Contextsensitive grmmrs. Productions of the form α 1 Wα 2 α 1 βα 2 re llowed. 4. Unrestricted grmmrs. Any production rule of the form α 1 Wα 2 γ is llowed. 8.3 Regulr grmmrs For Bioinformtics we will e interested in the regulr nd contextfree grmmrs. definition: Hence more detiled A grmmr is clled regulr if ll productions hve the form l r, where l V nd r Σ ΣV. Tht is, we cn only replce vrile with: terminl (r Σ), or terminl followed y vrile (r ΣV). Exmple. The following regulr grmmr genertes vlid identifier nmes in mny progrmming lnguges. G = ({[lph], [lnum]}, {A,... Z,,..., z,, 0, 1,..., 9}, P, [lph]}, where P := { [lph] A,...,, [lph] A[lnum],..., [lnum], [lnum] A,...,, 0,..., 9, [lnum] A[lnum],..., [lnum], 0[lnum],..., 9[lnum] } Here we used comms to write severl productions shring the sme left hnd side in one line.
5 8004 Finite utomt nd regulr grmmrs, y Clemens Gröpl, Jnury 9, 2014, 12: Contextfree grmmrs Here the definition of contextfree grmmr: Definition 1. A context free grmmr G is 4tuple G = (V, Σ, P, S) with V nd Σ eing lphets with V Σ =. V is the nonterminl lphet. Σ is the terminl lphet. S N is the strt symol. P V (V Σ) is the finite set of ll productions. Consider the contextfree grmmr G = ( {S}, {, }, {S S S }, S ). This CFG produces the lnguge of ll plindromes of the form αα R. For exmple the string cn e generted using the following derivtion: S S S S. The plindrome grmmr cn e redily extended to hndle RNA hirpin loops. For exmple, we could model hirpin loops with three se pirs nd gc or g loop using the following productions. S W 1 u cw 1 g gw 1 c uw 1, W 1 W 2 u cw 2 g gw 2 c uw 2, W 2 W 3 u cw 3 g gw 3 c uw 3, W 3 g gc. (We don t mention the lphets V nd Σ explicitly if they re cler from the context.) Grmmrs generte lnguges. They re mens to quickly specify ll (possily n infinite numer) words in lnguge. Now we will turn the ttention to the utomt tht cn decide whether word is in the lnguge or not. If the word is in the lnguge the utomton ccepts the word. We strt with finite utomt nd proove tht they re le to ccept exctly the words generted y regulr grmmr. 8.5 Deterministic finite utomt A deterministic finite utomton (DFA) is 5tuple M = (Z, Σ, δ, z 0, E) stisfying the following conditions. Z is finite set of sttes the utomton cn e in. Σ is the lphet. The utomton moves long the input from left to right. In ech step, it reds single chrcter from the input. δ : Z Σ Z is the trnsition function. When the chrcter hs een red, M chnges its stte depending on the chrcter nd its current stte. Then it proceeds to the next input position.
6 Finite utomt nd regulr grmmrs, y Clemens Gröpl, Jnury 9, 2014, 12: z 0 is the initil stte of M efore the first chrcter is red. E is the set of end sttes or ccepting sttes. If M is in stte contined in E fter the lst letter hs een red, the input is ccepted. DFAs cn e drwn s digrphs very intuitively. Sttes correspond to vertices. They re drwn s single circles; ccepting sttes re indicted y doule circles. Edges correspond to trnsitions nd re leled with letters from Σ. There is n rc from u to v leled if nd only if there is trnsition δ(u, ) = v. The initil stte is mrked y n ingoing rrow. Exmple. The following utomton ccepts the lnguge (where % denotes modulus, i. e. reminder of division) L = {x {, } (# (x) # (x)) % 3 = 1}. z 0 z 1 z 2 Using the definition of DFA, we hve M = ({z 0, z 1, z 2 }, {, }, δ, z 0, {z 1 }), where δ(z 0, ) = z 1 δ(z 1, ) = z 2 δ(z 2, ) = z 0 δ(z 0, ) = z 2 δ(z 1, ) = z 0 δ(z 2, ) = z 1 A lnguge L is clled regulr if there is regulr grmmr tht produces L \ {ε}. Lengthy remrk: The issue with ε is relly just technicl compliction. We cn lwys modify grmmr G tht genertes lnguge L into grmmr G tht genertes the lnguge L {ε} y the following trick: Let S e the strt vrile of G. Let S e new vrile symol not used y G. Then G is otined y replcing the strt vrile y S nd dding the following productions: S S ε. Whether the resulting grmmr G is lso clled regulr (if G ws regulr) depends on the literture. Schöning uses the following εsonderregelung : If ε L(G) is desired, then the production S ε is dmitted, where S is the strt vrile. However, in this cse S must not pper on the right hnd side of production. 8.6 From DFAs to regulr grmmrs Agin, let M = (Z, Σ, δ, z 0, E) e deterministic finite utomton. It is useful to extend the trnsition function δ : Z Σ Z to mpping δ : Z Σ Z, clled the extended trnsition function. We define δ (z, ε) := z for every stte z Z nd inductively, δ (z, x) := δ(δ (z, x), ) for x Σ, Σ.
7 8006 Finite utomt nd regulr grmmrs, y Clemens Gröpl, Jnury 9, 2014, 12:48 Oserve tht if x = x[1.. n] is n input string, then δ (z 0, x[1.. 0]), δ (z 0, x[1.. 1]),..., δ (z 0, x[1.. n]) is the pth of sttes followed y the DFA. Hence, the lnguge ccepted y M is L(M) := {x Σ δ (z 0, x) E}. We re now redy to prove: Theorem 2. Every lnguge which is ccepted y deterministic finite utomton is regulr. Proof: Let M = (Z, Σ, δ, z 0, E) e DFA nd A := L(M). We will construct regulr grmmr G = (V, Σ, P, S) tht genertes A. We let V := Z nd S := z 0. Every rc δ(u, ) = v ecomes production u v P, nd if v E we lso include production u. Tht is, P := {u v δ(u, ) = v} {u δ(u, ) = v E}. Now we hve x[1.. n] L(M) there re sttes z 1,..., z n Z such tht δ(z i 1, x[i]) = z i for i = 1,..., n, where z 0 is the strt stte nd z n E is n ccepting stte there re vriles z 1,..., z n V such tht z i 1 x[i]z i is production in P, where z 0 is the strt vrile, nd z n 1 x[n] is lso production in P we cn derive S = z 0 x[1]z 1 x[1]x[2]z 2... x[1.. n 1]z n 1 x[1.. n] in G, i. e., S G x x[1.. n] L(G). If ε A, i. e., z 0 E, then we need to pply the εsonderregelung nd modify G ccordingly. 8.7 Nondeterministic finite utomt In DFAs, the pth followed upon given input ws completely determined. Next we will introduce nondeterministic finite utomt (NFAs). These re defined similr to DFAs, ut ech stte cn hve more thn one successor stte for ny given letter, or none t ll. A nondeterministic finite utomton (NFA) is 5tuple M = (Z, Σ, δ, U 0, E) stisfying the following conditions. Z is finite set of sttes. Σ is the lphet. δ : Z Σ P(Z) is the trnsition function. Here P(Z) is the power set of Z, i. e. the set of ll susets of Z.
8 Finite utomt nd regulr grmmrs, y Clemens Gröpl, Jnury 9, 2014, 12: U 0 is the set of initil sttes. E is the set of ccepting sttes. When the utomton reds Σ nd is in stte z, it is free to choose one of severl sucessor sttes in δ(z, ), or its gets stuck if δ(z, ) =. A nondeterministic finite utomton ccepts nd input if there is t lest one ccepting pth. Agin, we cn define n extended trnsition function δ : P(Z) Σ P(Z). We let δ (U, ε) := U for ll susets of sttes U Z nd inductively, δ (U, x) := δ(v, ) for x Σ, Σ. v δ (U,x) Then the lnguge ccepted y M is L(M) := {x Σ δ (U 0, x) E }. The following illustrtes the definition of δ. The lrge ule on the left side is δ (U, x), the lrge ule on the right side is δ (U, x), where is some letter. The stte spce (ft dots) is shown twice for clrity. Time goes from left to right. The smller cones indicte δ(v, ) for ech v δ (U, x). NFAs cn e drwn s digrphs, similr to DFAs. The resulting digrphs re more generl: We cn hve severl rrows pointing to strt sttes. The numer of rcs with given lel leving vertex is no longer required to e exctly 1, it cn e ny numer (including 0). Exmple. The following NFA ccepts ll words over the lphet Σ = {, } which do not strt or end with the letter.
9 8008 Finite utomt nd regulr grmmrs, y Clemens Gröpl, Jnury 9, 2014, 12:48 z 0 z 1 z 2 z DFAs nd NFAs re equivlent Although NFAs re generliztion of DFAs, they ccept the sme lnguges: Theorem 3 (Rin, Scott). For every nondeterministic finite utomton M there is deterministic finite utomton M such tht L(M) = L(M ). Proof. Let M = (Z, Σ, δ, U 0, E) e n NFA. The sic ide of the proof is to view the susets of sttes of Z s single sttes of n DFA M whose stte spce is P(Z). Then the rest of the definition of M is strightforwrd. The power set utomton is defined s M := (P(Z), Σ, δ, U 0, E ), where P(Z) is the stte spce The trnsition function δ : P(Z) Σ P(Z) is defined y δ (U, ) := δ(v, ) = δ (U, ) for U P(Z). U 0 P(Z) is the new strt stte (note tht in M it ws the set of strt sttes). E := {U Z U E } is the new set of end sttes. v U Using the definitions of M nd M, we hve: x[1.. n] L(M) there re ccepting pths in M, δ (U 0, x[1.. n]) E there re susets U 1, U 2,..., U n Z such tht δ (U i 1, x[i]) = U i nd U n E there is n ccepting pth in M, δ (U 0, x[1.. n]) = U n E x[1.. n] L(M ). Remrks:
10 Finite utomt nd regulr grmmrs, y Clemens Gröpl, Jnury 9, 2014, 12: In the power set construction, we cn sfely leve out sttes which cnnot e reched from the strt stte U 0. Tht is, we cn generte the reched stte sets on the fly. 2. The exponentil low up of the numer of sttes (from Z to 2 Z ) cnnot e voided in generl. For exmple, the lnguge L := {x {, } x k nd the klst letter of x is n } hs n NFA with k + 1 sttes, ut it is not hrd to show tht no DFA for L cn hve less tht 2 k sttes. 8.9 From regulr grmmrs to NFAs We hve just seen how to trnsform nondeterministic finite utomton into deterministic finite utomton. We hve seen efore how to trnsform deterministic finite utomton into regulr grmmr. Next we will see how to trnsform regulr grmmr into nondeterministic finite utomton. This concludes the proof tht the regulr lnguges re precisely those which re ccepted y finite utomt (of oth kinds). Theorem 4. Every regulr lnguge is ccepted y nondeterministic finite utomton. Proof. Let G = (V, Σ, P, S) e regulr grmmr nd A := L(G). We will construct n NFA M = (Z, Σ, δ, {z 0 }, E) such tht L(M) = A. Note tht in every derivtion in regulr grmmr, the intermedite words contin exctly one vrile, nd the vrile must e t the end. This vrile will ecome stte of the NFA. We need one more extr stte, which M enters when the vrile is eliminted in the lst step. Thus we let the stte set e Z := V {X}. The only possile initil stte is z 0 := S, the strt vrile of G. The set of end sttes is E := {X} if ε A. If ε A, then we let E := {X, S}. Next we trnslte productions into trnsitions. We define δ : Z Σ P(Z) y Tht is, δ(u, ) v δ(u, ) X iff u v P iff u P δ(u, ) = {v u v P} {X u P}. Note tht the end stte X hs no successor sttes. And if S is n end stte, then y the εsonderregelung there is no wy to get ck to S, s it does not pper on the right side of production. Now we hve for n 1: x[1.. n] L(G) there re vriles z 1,..., z n V such tht we cn derive z 0 = S G x[1]z 1 G x[1]x[2]z 2 G... G x[1.. n 1]z n 1 G x[1.. n] in G, tht is, z i 1 x[i]z i is production in P, where z 0 = S is the strt vrile, nd z n 1 x[n] is lso production in P there re sttes z 1,..., z n Z {X} such tht δ(z i 1, x[i]) z i for i = 1,..., n, where z 0 is the strt stte, nd z n = X is only end stte which is fesile for word of length 1 x[1.. n] L(M) Moreover, y construction we hve ε L(M) ε A.
11 8010 Finite utomt nd regulr grmmrs, y Clemens Gröpl, Jnury 9, 2014, 12: Regulr expressions Regulr expressions re perhps the most populr wy to descrie regulr lnguges in forml terms. Introductory Exmples. At UNIX shell, you might type: > ls s*[re]?t*x script.ux script.tex slides.tex Here we hve used the wildcrd chrcters * nd?. The PROSITE dtse contins common motifs of protein fmilies, some of which re descried y ( restricted form of) regulr expressions. The following two lines re from the PROSITE entry with ccession numer PS00518, which contins pttern for zinc finger C3HC4 domin: ID PA ZINC_FINGER_C3HC4; PATTERN. CxHx[LIVMFY]Cx(2)C[LIVMYA]. The precise syntx used for regulr expressions vries. Here is one. following rules. A regulr expression γ nd the lnguge L(γ) it represents re defined inductively y the 1. is regulr expression. It denotes the empty lnguge. L( ) =. 2. ε is regulr expression. Its lnguge consists of the empty word. L(ε) = {ε}. 3. For ech single letter, Σ, is regulr expression. L() = {}. 4. Let α nd β e regulr expressions. Then the following re lso regulr expressions: (Closure properties) (i) αβ with L(αβ) := L(α)L(β) product (ii) (α β) with L((α β)) := L(α) L(β) union (iii) (α) with L((α) ) := (L(α)) str hull Some oservtions. 1. For every regulr expression α, it holds αε = εα = α nd α = α =. 2. Let x = x[1.. n] Σ. Then x is regulr expression nd L(x) = {x}, y rules 2., 3., nd 4.(i). 3. Let A = {x 1, x 2,..., x k } Σ e finite lnguge. Then (... ((x 1 x 2 ) x 3 )... x k ) is regulr expression nd L((... ((x 1 x 2 ) x 3 )... x k )) = A, y rule 4.(ii) nd the previous oservtion. We shll rther write this expression s (x 1 x 2 x 3... x k ). Exmple 1. The lnguge L = {x {, } x contins s sustring } cn e represented y the following regulr expression: ( ) ( ).
12 Finite utomt nd regulr grmmrs, y Clemens Gröpl, Jnury 9, 2014, 12: Exmple 2. The lnguge L = {x {, } x does not contin s sustring } cn e represented y the following regulr expression: This cn e seen s follows. A DFA for L is ( ) ( ε) z 1 z 2 z 3 We hve δ(z 1, x) = z 1 iff x L(( ) ). After tht we cn mke one more trnsition from z 1 to z 2 reding n Finite utomt nd regulr expressions re equivlent Theorem 5 (Kleene). A lnguge is regulr if nd only if it is descried y regulr expression. (the following proof is not needed for the exmintion. You cn red it t your convenience). Proof. We show oth inclusions in turn. ( ) Let γ e regulr expression. We show tht L(γ) is ccepted y nondeterministic finite utomton. We follow the cses in the inductive definition of regulr expressions. Cses 1., 2., nd 3. re trivil: Clerly there re NFAs for L( ) =, for L(ε) = {ε}, nd for L() = {} (where Σ). Cse 4.(ii): γ hs the form γ = (α β). We cn ssume y induction tht we lredy hve two NFAs M 1 = (Z 1, Σ, δ 1, U 1, E 1 ) nd M 2 = (Z 2, Σ, δ 2, U 2, E 2 ) stisfying L(α) = L(M 1 ), L(β) = L(M 2 ), nd Z 1 Z 2 =. Then we cn simply tke the union of M 1 nd M 2. Tht is, we uild the NFA M := (Z 1 Z 2, Σ, δ 1 δ 2, U 1 U 2, E 1 E 2 ). Here δ 1 δ 2 susumes ll the trnsitions present in M 1 nd M 2. There re no trnsitions etween the M 1 nd M 2 prts in M. Formlly, (δ 1 δ 2 )(U) = δ 1 (U Z 1 ) δ 2 (U Z 2 ). Then L(M) = L(α) L(β) = L(γ). Cse 4.(i): γ hs the form γ = αβ. Here we plug two NFAs M 1 nd M 2 for L(α) nd L(β) in series to otin M. (We cn ssume tht the stte sets re disjoint, Z 1 Z 2 =.) M hs the strt stte set U 1, if ε L(α). If ε L(α), then M hs the strt stte set U 1 U 2. In oth cses, the end stte set of M is E 2.
13 8012 Finite utomt nd regulr grmmrs, y Clemens Gröpl, Jnury 9, 2014, 12:48 Moreover we dd rcs from M 1 to M 2 s follows: For every rc δ 1 (u, ) v in M 1 tht enters stte in E 1, we dd couple of rcs δ(u, ) s 2, for ll s 2 U 2, in M. Hence in M we cn rech stte in U 2 if nd only if the chrcters which re red long the wy form word in L(M 1 ). After tht the rest of the input is processed using M 2. This implies L(M) = L(M 1 )L(M 2 ) = L(α)L(β) = L(αβ) = L(γ). Cse 4.(iii): γ hs the form γ = (α). Agin let M 1 s ove e n NFA for L(α). Then we construct n NFA M which ccepts L((α) ) s follows. M hs the sme sttes, initil sttes, nd end sttes s M 1. The difference is tht we dd trnsitions similr to Cse 4.(ii). Nmely, for ech trnsition δ 1 (u, ) v, where v E 1, we dd couple of rcs δ(u, ) s 1 for ll s 1 U 1. This mens tht whenever we would rech n end stte of M 1, we cn s well strt reding nother word of L(M 1 ) from one of its initil sttes. Using the union construction, we cn dd ε to the ccepted lnguge using Cse 4.(i), if necessry. Note tht L((M) ) = L(ε L(M 1 ) + ), s needed. ( ) Now let M = (Z, Σ, δ, z 1, E) e deterministic finite utomton. We show tht L(M) is descried y regulr expression. We cn ssume w.l.o.g. tht the vertex set of M is numered Z = {z 1, z 2,..., z n } nd z 1 is the initil stte. The key to the proof is the next definition. R k i,j := {x Σ δ (z i, x) = z j nd δ (z i, x[1.. l]) {z 1,..., z k } for ll 1 l < x }. A word x is in R k i,j iff when we strt reding x eginning in stte z i, then we end up in stte x j, nd moreover, ll intermedite sttes (i. e. those sttes where we hve een fter reding proper, nonempty prefix x[1.. l] of x), hve indices t most k. R k is so importnt ecuse it llows use to solve the prolem using 3prmeter recursion nd dynmic i,j progrmming. The initil cses k = 0 re esy: No intermedite sttes re llowed t ll, thus we hve R 0 i,j = { Σ δ(z i, ) = z j } for i j { Σ δ(z i, ) = z i } {ɛ} for i = j. In oth cses, R 0 is finite lnguge nd we cn write down regulr expression for it. i,j Now for the recursive cse k 1. The difference etween R k+1 nd R k i,j i,j is tht z k+1 ecomes llowed s n intermedite stte. Thus we cn prtition ny word x R k+1 \ R k i,j i,j etween its visits of the stte z k+1. We strt t z i. Either we go to z j without visiting z k+1. Or we visit z k+1 t lest once. The sustring etween two consecutive visits of z k+i is word from the lnguge R k k+1,k+1. Then we go from z k+1 to z j without nother visit t z k+1. Thus it holds: R k+1 i,j = R k i,j Rk i,k+1 (Rk k+1,k+1 ) R k k+1,j. This trnsltes lmost literlly into regulr expression. expressions γ k i,j such tht Rk i,j = L(γk ). Then i,j Assume y induction tht we hve regulr is regulr expression such tht L(γ k+1 i,j ) = R k+1 i,j. γ k+1 i,j := (γ k i,j γk i,k+1 (γk k+1,k+1 ) γ k k+1,j )
14 Finite utomt nd regulr grmmrs, y Clemens Gröpl, Jnury 9, 2014, 12: where To summrize, R k i,j := {x Σ δ (z i, x) = z j nd δ (z i, x[1.. l]) {z 1,..., z k } for ll 1 l < x }, R 0 i,j = { Σ δ(z i, ) = z j } for i j { Σ δ(z i, ) = z i } {ɛ} for i = j. R k+1 i,j = R k i,j Rk i,k+1 (Rk k+1,k+1 ) R k k+1,j. When we hve reched k = n, then the restriction which is imposed y the upper index in R k i,j ecomes void. Note tht L(M) = {R n 1,j z j E}. Therefore we let γ := (γ n 1,i 1... γ n 1,i l ), where E = {i 1,..., i l } is n enumertion of the end sttes. Then L(γ) = L(M) Exmple for Kleene s trnsformtion We hve seen DFA for the lnguge efore. L = {x {, } (# (x) # (x)) 1 mod 3} The recurrences from the theorem yield γ 0 i,j γ 1 i,j ε 0 ε 1 ε 1 (ε ) ( ) 2 ε 2 ( ) (ε ) z 0 z 1 z 2 γ 2 i,j (ε () ) () ( () ( )) 1 () () () ( ) 2 ( ( )() ) ( )() ((ε ) ( )() ( )) Since z 2 is the only end stte, regulr expression for L is γ 3 0,1 = γ2 0,2 (γ2 2,2 ) γ 2 2,1 = ( () ( ))(((ε ) ( )() ( ))) ( )().
15 8014 Finite utomt nd regulr grmmrs, y Clemens Gröpl, Jnury 9, 2014, 12:48 Finlly we return to contextfree grmmrs nd introduce the utomton tht ccepts contextfree lnguge, the pushdown utomton (PDA). We will first define it nd then show t n exmple how we cn decide whether string is in the lnguge generted y given CFG. Recll our smll hirpin generting CFG: S W 1 u cw 1 g gw 1 c uw 1, W 1 W 2 u cw 2 g gw 2 c uw 2, W 2 W 3 u cw 3 g gw 3 c uw 3, W 3 g gc. There is n elegnt representtion for derivtions of sequence in CFG clled the prse tree. The root of the tree is the nonterminl S. The leves re the terminl symols, nd the inner nodes re nonterminls. For exmple if we extend the ove productions with S SS we cn get the following: S S W 1 S W 1 W 2 W 2 W 3 W 3 c g g c u g g g u g c c c Using pushdown utomton we cn prse sequence left to right ccording. A (nondeterministic) PDA is formlly defined s 6tuple: M = (Z, Σ, Γ, δ, z 0, S) where Z is finite set of sttes, Σ is finite input lphet, Γ is finite stck lphet δ : Z Σ {ɛ} Γ P(Z Γ ) is the trnsition function, where P(S) denotes the power set of S. z 0 is the strt stte, S is the initil stck symol S Γ is the lowest stck symol. There is of course lso deterministic version, however the nondeterministic PDA llows for simple construction when CFG is given. If M is in stte z nd reds the input nd if A is the top stck symol, then M cn go to stte z nd replce A y other stck symols. This implies, tht A cn e deleted, replced, or ugmented.
16 Finite utomt nd regulr grmmrs, y Clemens Gröpl, Jnury 9, 2014, 12: After reding the input, the utomton ccepts the word if the stck is empty. Given CFG G = (V, Σ, P, S) we define the corresponding PDA s M = ({z}, Σ, V Σ, δ, z, S). Using the production set P we cn define δ. For ech rule A α P we define δ such tht (z, α) δ(z, ɛ, A) nd (z, ɛ) δ(z,, ). Lets look t our exmple: S W 1 u cw 1 g gw 1 c uw 1, W 1 W 2 u cw 2 g gw 2 c uw 2, W 2 W 3 u cw 3 g gw 3 c uw 3, W 3 g gc. Hence M = ({z}, {, c, g, u}, {, c, g, u, W 1, W 2, W 3, S}, δ, z, S) with δ s explined (lckord). Lets see how the utomton prses word in our hirpin lnguge. Given our CFG, the utomton s stck is initilized with the strt symol S. Then the following steps re iterted until no symols remin. If the stck is empty when no input symols remin, then the sequence hs een successfully prsed. 1. Pop symol off the stck. 2. If the popped symol is nonterminl: Peek hed in the input nd choose vlid production for the symol. (For deterministic PDAs, there is t most one choice. For nondeterministic PDAs, ll possile choices need to e evluted individully.) If there is no vlid trnsision, terminte nd reject. Push the right side of the production on the stck, rightmost symols first. 3. If the popped symol is terminl: Compre it to the current symol of the input. If it mtches, move the utomton to the right on the input. If not, reject nd terminte. Lets try this with the string gccgcggc. Exmple. (The current symol is written using cpitl letter): Input string Stck Opertion Gccgcggc S Pop S. Produce S>gW1c Gccgcggc gw1c Pop g. Accept g. Move right on input. gccgcggc W1c Pop W1. Produce W1>cW2g gccgcggc cw2gc Pop c. Accept c. Move right on input. gccgcggc W2gc Pop W2. Produce W2>cW3g gccgcggc cw3ggc Pop c. Accept c. Move right on input. gccgcggc W3ggc Pop W3. Produce W3>gc gccgcggc gcggc Pop g. Accept g. Move right on input (severl cceptnces) gccgcggc c Pop c. Accept c. Move right on input. gccgcggc$  Stck empty, input string empty. Accept!
17 8016 Finite utomt nd regulr grmmrs, y Clemens Gröpl, Jnury 9, 2014, 12: Summry Forml lnguges re sic mens in computer science to formlly descrie ojects tht follow certin rules, tht is tht cn e generted using grmmr. Two fundmentl views on forml lnguges re i) to view them s generted y grmmr, or ii) to view them s ccepted y n utomton. The Chomsky hierrchy plces different restrictions on the grmmrs. This limits the possiilities you hve, ut mkes the decision whether word is in such lnguge esier. The nondeterministic utomt (DFA nd PDA) re s powerful s te deterministic counterprts in deciding whether word is in lnguge or not. However, it is esier to define nondeterministic utomton. The importnce for ioinfomtics lies in the stochstic versions of the grmmrs which re used to trin cceptors for iologicl sequence ojects (i.e. Genes using Hidden Mrkov models, or RNA using stochstic CFGs).
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 informationGrammar. Languages. Content 5/10/16. Automata and Languages. Regular Languages. Regular Languages
5//6 Grmmr Automt nd Lnguges Regulr Grmmr Contextfree Grmmr Contextsensitive Grmmr Prof. Mohmed Hmd Softwre Engineering L. The University of Aizu Jpn Regulr Lnguges Context Free Lnguges Context Sensitive
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 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 informationOverview HC9. Parsing: TopDown & LL(1) ContextFree Grammars (1) Introduction. CFGs (3) ContextFree Grammars (2) Vertalerbouw HC 9: Ch.
Overview H9 Vertlerouw H 9: Prsing: opdown & LL(1) do 3 mei 2001 56 heo Ruys h. 8  Prsing 8.1 ontextfree Grmmrs 8.2 opdown Prsing 8.3 LL(1) Grmmrs See lso [ho, Sethi & Ullmn 1986] for more thorough
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 informationAutomata and Languages
Automt nd Lnguges Prof. Mohmed Hmd Softwre Engineering Lb. The University of Aizu Jpn Grmmr Regulr Grmmr Contextfree Grmmr Contextsensitive Grmmr Regulr Lnguges Context Free Lnguges Context Sensitive
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 informationChapter 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 informationNONDETERMINISTIC FSA
Tw o types of nondeterminism: NONDETERMINISTIC FS () Multiple strtsttes; strtsttes S Q. The lnguge L(M) ={x:x tkes M from some strtstte to some finlstte nd ll of x is proessed}. The string x = is
More informationNondeterministic Biautomata and Their Descriptional Complexity
Nondeterministic Biutomt nd Their Descriptionl Complexity Mrkus Holzer nd Sestin Jkoi Institut für Informtik JustusLieigUniversität Arndtstr. 2, 35392 Gießen, Germny 23. Theorietg Automten und Formle
More informationMA123, Chapter 10: Formulas for integrals: integrals, antiderivatives, and the Fundamental Theorem of Calculus (pp.
MA123, Chpter 1: Formuls for integrls: integrls, ntiderivtives, nd the Fundmentl Theorem of Clculus (pp. 27233, Gootmn) Chpter Gols: Assignments: Understnd the sttement of the Fundmentl Theorem of Clculus.
More informationexpression simply by forming an OR of the ANDs of all input variables for which the output is
2.4 Logic Minimiztion nd Krnugh Mps As we found ove, given truth tle, it is lwys possile to write down correct logic expression simply y forming n OR of the ANDs of ll input vriles for which the output
More informationComplementing Büchi Automata with a Subsettuple Construction
DEPARTEMENT D INFORMATIQUE DEPARTEMENT FÜR INFORMATIK Bd de Pérolles 90 CH1700 Friourg www.unifr.ch/informtics WORKING PAPER Complementing Büchi Automt with Susettuple Construction J. Allred & U. UltesNitsche
More informationPrefixFree RegularExpression Matching
PrefixFree RegulrExpression Mthing YoSu Hn, Yjun Wng nd Derik Wood Deprtment of Computer Siene HKUST PrefixFree RegulrExpression Mthing p.1/15 Pttern Mthing Given pttern P nd text T, find ll sustrings
More information5: The Definite Integral
5: The Definite Integrl 5.: Estimting with Finite Sums Consider moving oject its velocity (meters per second) t ny time (seconds) is given y v t = t+. Cn we use this informtion to determine the distnce
More informationarxiv: v2 [cs.lo] 26 Dec 2016
On Negotition s Concurrency Primitive II: Deterministic Cyclic Negotitions Jvier Esprz 1 nd Jörg Desel 2 1 Fkultät für Informtik, Technische Universität München, Germny 2 Fkultät für Mthemtik und Informtik,
More informationGeneral idea LR(0) SLR LR(1) LALR To best exploit JavaCUP, should understand the theoretical basis (LR parsing);
Bottom up prsing Generl ide LR(0) SLR LR(1) LLR To best exploit JvCUP, should understnd the theoreticl bsis (LR prsing); 1 Topdown vs Bottomup Bottomup more powerful thn topdown; Cn process more powerful
More information63. Representation of functions as power series Consider a power series. ( 1) n x 2n for all 1 < x < 1
3 9. SEQUENCES AND SERIES 63. Representtion of functions s power series Consider power series x 2 + x 4 x 6 + x 8 + = ( ) n x 2n It is geometric series with q = x 2 nd therefore it converges for ll 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 informationINF1383 Bancos de Dados
3//0 INF383 ncos de Ddos Prof. Sérgio Lifschitz DI PUCRio Eng. Computção, Sistems de Informção e Ciênci d Computção LGER RELCIONL lguns slides sedos ou modificdos dos originis em Elmsri nd Nvthe, Fundmentls
More informationPolynomial Approximations for the Natural Logarithm and Arctangent Functions. Math 230
Polynomil Approimtions for the Nturl Logrithm nd Arctngent Functions Mth 23 You recll from first semester clculus how one cn use the derivtive to find n eqution for the tngent line to function t given
More informationNondeterministic Finite Automata
Nondeterministi Finite utomt The Power of Guessing Tuesdy, Otoer 4, 2 Reding: Sipser.2 (first prt); Stoughton 3.3 3.5 S235 Lnguges nd utomt eprtment of omputer Siene Wellesley ollege Finite utomton (F)
More informationOn the Relative Succinctness of Nondeterministic Büchi and cobüchi Word Automata
On the Reltive Succinctness of Nondeterministic Büchi nd cobüchi Word Automt Benjmin Aminof, Orn Kupfermn, nd Omer Lev Herew University, School of Engineering nd Computer Science, Jeruslem 91904, Isrel
More informationUniversitaireWiskundeCompetitie. Problem 2005/4A We have k=1. Show that for every q Q satisfying 0 < q < 1, there exists a finite subset K N so that
Problemen/UWC NAW 5/7 nr juni 006 47 Problemen/UWC UniversitireWiskundeCompetitie Edition 005/4 For Session 005/4 we received submissions from Peter Vndendriessche, Vldislv Frnk, Arne Smeets, Jn vn de
More informationNUMERICAL INTEGRATION. The inverse process to differentiation in calculus is integration. Mathematically, integration is represented by.
NUMERICAL INTEGRATION 1 Introduction The inverse process to differentition in clculus is integrtion. Mthemticlly, integrtion is represented by f(x) dx which stnds for the integrl of the function f(x) with
More informationAutomata for Analyzing and Querying Compressed Documents Barbara FILA, LIFO, Orl eans (Fr.) Siva ANANTHARAMAN, LIFO, Orl eans (Fr.) Rapport No
Automt for Anlyzing nd Querying Compressed Documents Brr FILA, LIFO, Orléns (Fr.) Siv ANANTHARAMAN, LIFO, Orléns (Fr.) Rpport N o 200603 Automt for Anlyzing nd Querying Compressed Documents Brr Fil, Siv
More informationFractions arise to express PART of a UNIT 1 What part of an HOUR is thirty minutes? Fifteen minutes? tw elve minutes? (The UNIT here is HOUR.
6 FRACTIONS sics MATH 0 F Frctions rise to express PART of UNIT Wht prt of n HOUR is thirty minutes? Fifteen minutes? tw elve minutes? (The UNIT here is HOUR.) Wht frction of the children re hppy? (The
More informationDesign and Analysis of Distributed Interacting Systems
Design nd Anlysis of Distriuted Intercting Systems Lecture 6 LTL Model Checking Prof. Dr. Joel Greenyer My 16, 2013 Some Book References (1) C. Bier, J.P. Ktoen: Principles of Model Checking. The MIT
More informationConvex Sets and Functions
B Convex Sets nd Functions Definition B1 Let L, +, ) be rel liner spce nd let C be subset of L The set C is convex if, for ll x,y C nd ll [, 1], we hve 1 )x+y C In other words, every point on the line
More informationContinuous Random Variables
STAT/MATH 395 A  PROBABILITY II UW Winter Qurter 217 Néhémy Lim Continuous Rndom Vribles Nottion. The indictor function of set S is relvlued function defined by : { 1 if x S 1 S (x) if x S Suppose tht
More information8. Complex Numbers. We can combine the real numbers with this new imaginary number to form the complex numbers.
8. Complex Numers The rel numer system is dequte for solving mny mthemticl prolems. But it is necessry to extend the rel numer system to solve numer of importnt prolems. Complex numers do not chnge the
More informationArithmetic & 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 informationSCHOOL OF ENGINEERING & BUILT ENVIRONMENT. Mathematics
SCHOOL OF ENGINEERING & BUIL ENVIRONMEN Mthemtics An Introduction to Mtrices Definition of Mtri Size of Mtri Rows nd Columns of Mtri Mtri Addition Sclr Multipliction of Mtri Mtri Multipliction 7 rnspose
More informationWaveguide Guide: A and V. Ross L. Spencer
Wveguide Guide: A nd V Ross L. Spencer I relly think tht wveguide fields re esier to understnd using the potentils A nd V thn they re using the electric nd mgnetic fields. Since Griffiths doesn t do it
More informationExam 2 Solutions ECE 221 Electric Circuits
Nme: PSU Student ID Numer: Exm 2 Solutions ECE 221 Electric Circuits Novemer 12, 2008 Dr. Jmes McNmes Keep your exm flt during the entire exm If you hve to leve the exm temporrily, close the exm nd leve
More informationarxiv: v1 [cs.fl] 30 Nov 2016
Some Subclsses of Liner Lnguges bsed on Nondeterministic Liner Automt Benjmín Bedregl Deprtmento de Informátic e Mtemátic Aplicd, Universidde Federl do Rio Grnde do Norte bedregl@dimp.ufrn.br rxiv:1611.10276v1
More informationThe final exam will take place on Friday May 11th from 8am 11am in Evans room 60.
Mth 104: finl informtion The finl exm will tke plce on Fridy My 11th from 8m 11m in Evns room 60. The exm will cover ll prts of the course with equl weighting. It will cover Chpters 1 5, 7 15, 17 21, 23
More informationCOMPOSITIONALITY AND REACHABILITY WITH CONDITIONS ON PATH LENGTHS
compositionlity Interntionl Journl of Foundtions of Computer Science c World Scientific Pulishing Compny COMPOSITIONALITY AND REACHABILITY WITH CONDITIONS ON PATH LENGTHS INGO FELSCHER Lehrstuhl Informtik
More informationContextFree Language Induction by Evolution of Deterministic PushDown Automata Using Genetic Programming
ContextFree Lnguge Induction y Evolution of Deterministic PushDown Automt Using Genetic Progrmming Afr Zomorodin Computer Science Deprtment Stnford University P. O. Box 7171 Stnford, CA 94309 fr@cs.stnford.edu
More informationAutomatabased Pattern Mining from Imperfect Traces
Automtsed Pttern Mining from Imperfect Trces Giles Reger University of Mnchester Oxford Rod, M13 9PL Mnchester, UK regerg@cs.mn.c.uk Howrd Brringer University of Mnchester Oxford Rod, M13 9PL Mnchester,
More information4.1. Probability Density Functions
STT 1 4.14. 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 informationEntropy and Ergodic Theory Notes 10: Large Deviations I
Entropy nd Ergodic Theory Notes 10: Lrge Devitions I 1 A chnge of convention This is our first lecture on pplictions of entropy in probbility theory. In probbility theory, the convention is tht ll logrithms
More information8 factors of x. For our second example, let s raise a power to a power:
CH 5 THE FIVE LAWS OF EXPONENTS EXPONENTS WITH VARIABLES It s no time for chnge in tctics, in order to give us deeper understnding of eponents. For ech of the folloing five emples, e ill stretch nd squish,
More informationAutomatabased Pattern Mining from Imperfect Traces
Automtsed Pttern Mining from Imperfect Trces Giles Reger University of Mnchester Oxford Rod, M13 9PL Mnchester, UK regerg@cs.mn.c.uk Howrd Brringer University of Mnchester Oxford Rod, M13 9PL Mnchester,
More informationLECTURE NOTE #12 PROF. ALAN YUILLE
LECTURE NOTE #12 PROF. ALAN YUILLE 1. Clustering, Kmens, nd EM Tsk: set of unlbeled dt D = {x 1,..., x n } Decompose into clsses w 1,..., w M where M is unknown. Lern clss models p(x w)) Discovery of
More informationParallel Projection Theorem (Midpoint Connector Theorem):
rllel rojection Theorem (Midpoint onnector Theorem): The segment joining the midpoints of two sides of tringle is prllel to the third side nd hs length onehlf the third side. onversely, If line isects
More informationThe Bernoulli Numbers John C. Baez, December 23, x k. x e x 1 = n 0. B k n = n 2 (n + 1) 2
The Bernoulli Numbers John C. Bez, December 23, 2003 The numbers re defined by the eqution e 1 n 0 k. They re clled the Bernoulli numbers becuse they were first studied by Johnn Fulhber in book published
More informationWeek 7 Riemann Stieltjes Integration: Lectures 1921
Week 7 Riemnn Stieltjes Integrtion: Lectures 1921 Lecture 19 Throughout this section α will denote monotoniclly incresing function on n intervl [, b]. Let f be bounded function on [, b]. Let P = { = 0
More informationQUADRATIC EQUATIONS OBJECTIVE PROBLEMS
QUADRATIC EQUATIONS OBJECTIVE PROBLEMS +. The solution of the eqution will e (), () 0,, 5, 5. The roots of the given eqution ( p q) ( q r) ( r p) 0 + + re p q r p (), r p p q, q r p q (), (d), q r p q.
More informationUniversität Augsburg. Institut für Informatik. OutputDeterminacy and Asynchronous Circuit Synthesis. Victor Khomenko Mark Schaefer Walter Vogler
Ã ÊÇÅÍÆ ËÀÇ¼ Universität Augsburg OutputDetermincy nd Asynchronous Circuit Synthesis Victor Khomenko Mrk Schefer Wlter Vogler Report 200702 Jnury 2007 Institut für Informtik D86135 Augsburg Copyright
More informationBernd Finkbeiner Date: October 25, Automata, Games, and Verification: Lecture 2
Bernd Finkeiner Dte: Octoer 25, 2012 Automt, Gmes, nd Verifiction: Lecture 2 2 Büchi Automt Definition1 AnondeterministicBüchiutomtonAoverlphetΣistuple(S,I,T,F): S : finitesetof sttes I S:susetof initilsttes
More informationUniversity of Bristol  Explore Bristol Research. Peer reviewed version. Link to published version (if available): /j.jda
Strikovsky, T. A., & Vildhøj, H. W. (2015). A suffix tree or not suffix tree? Journl of Discrete Algorithms, 32, 1423. DOI: 10.1016/j.jd.2015.01.005 Peer reviewed version Link to pulished version (if
More information1 Introduction Lrgescle heterogeneous electronic text collections re more ville now thn ever efore nd rnge from pulished documents (e.g. electronic d
The String Tree: A New Dt Structure for String Serch in Externl Memory nd its Applictions. Polo Ferrgin Diprtimento di Informtic Universit di Pis Roerto Grossi Diprtimento di Sistemi e Informtic Universit
More information11.1 Exponential Functions
. Eponentil Functions In this chpter we wnt to look t specific type of function tht hs mny very useful pplictions, the eponentil function. Definition: Eponentil Function An eponentil function is function
More informationThe Fundamental Theorem of Algebra
The Fundmentl Theorem of Alger Jeremy J. Fries In prtil fulfillment of the requirements for the Mster of Arts in Teching with Speciliztion in the Teching of Middle Level Mthemtics in the Deprtment of Mthemtics.
More informationMACsolutions of the nonexistent solutions of mathematical physics
Proceedings of the 4th WSEAS Interntionl Conference on Finite Differences  Finite Elements  Finite Volumes  Boundry Elements MACsolutions of the nonexistent solutions of mthemticl physics IGO NEYGEBAUE
More informationConducting Ellipsoid and Circular Disk
1 Problem Conducting Ellipsoid nd Circulr Disk Kirk T. McDonld Joseph Henry Lbortories, Princeton University, Princeton, NJ 08544 (September 1, 00) Show tht the surfce chrge density σ on conducting ellipsoid,
More informationMapping the delta function and other Radon measures
Mpping the delt function nd other Rdon mesures Notes for Mth583A, Fll 2008 November 25, 2008 Rdon mesures Consider continuous function f on the rel line with sclr vlues. It is sid to hve bounded support
More informationComplexity in Modal Team Logic
ThI Theoretische Informtik Complexity in Modl Tem Logic JulinSteffen Müller Theoretische Informtik 18. Jnur 2012 Theorietg 2012 Theoretische Informtik Inhlt 1 Preliminries 2 Closure properties 3 Model
More informationOn the degree of regularity of generalized van der Waerden triples
On the degree of regulrity of generlized vn der Werden triples Jcob Fox Msschusetts Institute of Technology, Cmbridge, MA 02139, USA Rdoš Rdoičić Deprtment of Mthemtics, Rutgers, The Stte University of
More informationDiscrete Time Process Algebra with Relative Timing
Discrete Time Process Alger with Reltive Timing J.C.M. Beten nd M.A. Reniers Deprtment of Mthemtics nd Computing Science, Eindhoven University of Technology, P.O. Box 513, NL5600 MB Eindhoven, The Netherlnds
More informationNonblocking Supervisory Control of Nondeterministic Systems via Prioritized Synchronization 1
Nonblocking Supervisory Control of Nondeterministic Systems vi Prioritized Synchroniztion 1 Rtnesh Kumr Deprtment of Electricl Engineering University of Kentucky Lexington, KY 405060046 Emil: kumr@engr.uky.edu
More informationLanguages. A language is a set of strings. String: A sequence of letters. Examples: cat, dog, house, Defined over an alphabet:
Languages 1 Languages A language is a set of strings String: A sequence of letters Examples: cat, dog, house, Defined over an alphaet: a,, c,, z 2 Alphaets and Strings We will use small alphaets: Strings
More informationSemantic Analysis. CSCI 3136 Principles of Programming Languages. Faculty of Computer Science Dalhousie University. Winter Reading: Chapter 4
Semnti nlysis SI 16 Priniples of Progrmming Lnguges Fulty of omputer Siene Dlhousie University Winter 2012 Reding: hpter 4 Motivtion Soure progrm (hrter strem) Snner (lexil nlysis) Front end Prse tree
More informationMath 113 Exam 2 Practice
Mth Em Prctice Februry, 8 Em will cover sections 6.5, 7.7.5 nd 7.8. This sheet hs three sections. The first section will remind you bout techniques nd formuls tht you should know. The second gives number
More informationTechnical Appendix: Childhood Family Structure and Schooling Outcomes: Evidence for Germany
Technicl Appendix: Childhood Fmily Structure nd Schooling Outcomes: Evidence for Germny Mrco Frncesconi* Stephen P. Jenkins Thoms Siedler Universy of Essex nd Universy of Essex Universy of Essex Instute
More informationSection 7.1 Area of a Region Between Two Curves
Section 7.1 Are of Region Between Two Curves White Bord Chllenge The circle elow is inscried into squre: Clcultor 0 cm Wht is the shded re? 400 100 85.841cm White Bord Chllenge Find the re of the region
More informationON THE ENTRY SUM OF CYCLOTOMIC ARRAYS. Don Coppersmith IDACCR. John Steinberger UC Davis
INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY 6 (2006), #A26 ON THE ENTRY SUM OF CYCLOTOMIC ARRAYS Don Coppersmith IDACCR John Steinberger UC Dvis Received: 3/29/05, Revised: 0/8/06, Accepted:
More informationClassification: Rules. Prof. Pier Luca Lanzi Laurea in Ingegneria Informatica Politecnico di Milano Polo regionale di Como
Metodologie per Sistemi Intelligenti Clssifiction: Prof. Pier Luc Lnzi Lure in Ingegneri Informtic Politecnico di Milno Polo regionle di Como Rules Lecture outline Why rules? Wht re clssifiction rules?
More informationa a a a a a a a a a a a a a a a a a a a a a a a In this section, we introduce a general formula for computing determinants.
Section 9 The Lplce Expnsion In the lst section, we defined the determinnt of (3 3) mtrix A 12 to be 22 12 21 22 2231 22 12 21. In this section, we introduce generl formul for computing determinnts. Rewriting
More informationNegative Selection Algorithms on Strings with Efficient Training and LinearTime Classification
Negtive Selection Algorithms on Strings with Efficient Trining nd LinerTime Clssifiction Michel Elerfeld, Johnnes Textor Institut für Theoretische Informtik, Universität zu Lüeck, 23538 Lüeck, Germny
More informationManyvalued truth functions, Cerny's conjecture and road. Lemminkaisenkatu 14 A, Turku, Finland and
Mnyvlued truth functions, Cerny's conjecture nd rod coloring Alexndru Mteescu Turku Centre for Computer Science, Lemminkisenktu 4 A, 050 Turku, Finlnd nd Fculty of Mthemtics, University of Buchrest, Romni
More informationPDE Notes. Paul Carnig. January ODE s vs PDE s 1
PDE Notes Pul Crnig Jnury 2014 Contents 1 ODE s vs PDE s 1 2 Section 1.2 Het diffusion Eqution 1 2.1 Fourier s w of Het Conduction............................. 2 2.2 Energy Conservtion.....................................
More information1.3 Regular Expressions
51 1.3 Regular Expressions These have an important role in descriing patterns in searching for strings in many applications (e.g. awk, grep, Perl,...) All regular expressions of alphaet are 1.Øand are
More information1 Exercises about introductory part
1 Exercises out introductory prt 1. Pigeonhole Principle sys: If you hve more pigeons thn pigeonholes, nd ech pigeon ies into some pigeonhole, then there must e t lest one hole tht hs more thn one pigeon.
More informationMATH 174A: PROBLEM SET 5. Suggested Solution
MATH 174A: PROBLEM SET 5 Suggested Solution Problem 1. Suppose tht I [, b] is n intervl. Let f 1 b f() d for f C(I; R) (i.e. f is continuous relvlued function on I), nd let L 1 (I) denote the completion
More informationDiophantine Steiner Triples and PythagoreanType Triangles
Forum Geometricorum Volume 10 (2010) 93 97. FORUM GEOM ISSN 15341178 Diophntine Steiner Triples nd PythgorenType Tringles ojn Hvl bstrct. We present connection between Diophntine Steiner triples (integer
More informationMETHODS OF APPROXIMATING THE RIEMANN INTEGRALS AND APPLICATIONS
Journl of Young Scientist Volume III 5 ISSN 448; ISSN CDROM 449; ISSN Online 445; ISSNL 44 8 METHODS OF APPROXIMATING THE RIEMANN INTEGRALS AND APPLICATIONS An ALEXANDRU Scientific Coordintor: Assist
More informationAnonymous Math 361: Homework 5. x i = 1 (1 u i )
Anonymous Mth 36: Homewor 5 Rudin. Let I be the set of ll u (u,..., u ) R with u i for ll i; let Q be the set of ll x (x,..., x ) R with x i, x i. (I is the unit cube; Q is the stndrd simplex in R ). Define
More informationReasoning over Time or Space. CS 188: Artificial Intelligence. Outline. Markov Models. Conditional Independence. Query: P(X 4 )
CS 88: Artificil Intelligence Lecture 7: HMMs nd Prticle Filtering Resoning over Time or Spce Often, we wnt to reson out sequence of oservtions Speech recognition Root locliztion User ttention Medicl monitoring
More informationOrthogonal Polynomials and LeastSquares Approximations to Functions
Chpter Orthogonl Polynomils nd LestSqures Approximtions to Functions **4/5/3 ET. Discrete LestSqures Approximtions Given set of dt points (x,y ), (x,y ),..., (x m,y m ), norml nd useful prctice in mny
More informationThe Lie bialgebra of loops on surfaces of Goldman and Turaev
The Lie bilgebr of loops on surfces of Goldmn nd Turev Consider ny surfce F.Letˆπ be the set of free homotopy clsses of loops on F.LetZ be the vector spce with bsis ˆπ \{null} where null is the homotopy
More informationLazy Security Controllers
Lzy Security Controllers Giulio Crvgn, Griele Cost, Giovnni Prdini Diprtimento di Informtic, Sistemistic e Comuniczione Università degli Studi di MilnoBicocc, Itly Emil: giulio.crvgn@disco.unimi.it Diprtimento
More informationNONDETERMINISTIC KLEENE COALGEBRAS
Logicl Methods in Computer Science Vol. 6 (3:23) 2010, pp. 1 1 39 www.lmcsonline.org Submitted Jn. 3, 2010 Published Sep. 9, 2010 NONDETERMINISTIC KLEENE COALGEBRAS ALEXANDRA SILVA, MARCELLO BONSANGUE
More informationMeasuring Electron Work Function in Metal
n experiment of the Electron topic Mesuring Electron Work Function in Metl Instructor: 梁生 Office: 7318 Emil: shling@bjtu.edu.cn Purposes 1. To understnd the concept of electron work function in metl nd
More informationFachgebiet Rechnersysteme1. 1. Boolean Algebra. 1. Boolean Algebra. Verification Technology. Content. 1.1 Boolean algebra basics (recap)
. Boolen Alger Fchgeiet Rechnersysteme. Boolen Alger Veriiction Technology Content. Boolen lger sics (recp).2 Resoning out Boolen expressions . Boolen Alger 2 The prolem o logic veriiction: Show tht two
More informationOrdinal diagrams. By Gaisi TAKEUTI. (Received April 5, 1957) $(a)\frac{s_{1}s_{2}}{(b)\frac{s_{3}s_{4}}{(c)\frac{s}{s}6\underline{6}}}$ Fig.
cn Journl the Mthemticl Society Jpn Vol 9, No 4, Octor, 1957 Ordinl digrms By Gi TAKEUTI (Received April 5, 1957) In h pper [2] on the constencypro the theory nturl numrs, G Gentzen ssigned to every profigure
More informationFrom the Numerical. to the Theoretical in. Calculus
From the Numericl to the Theoreticl in Clculus Teching Contemporry Mthemtics NCSSM Ferury 67, 003 Doug Kuhlmnn Phillips Acdemy Andover, MA 01810 dkuhlmnn@ndover.edu How nd Why Numericl Integrtion Should
More informationWave Equation on a Two Dimensional Rectangle
Wve Eqution on Two Dimensionl Rectngle In these notes we re concerned with ppliction of the method of seprtion of vriles pplied to the wve eqution in two dimensionl rectngle. Thus we consider u tt = c
More informationLaboratoire de l Informatique du Parallélisme
Lortoire de l Informtique du Prllélisme Ecole Normle Supérieure de Lyon Unité de recherche ssociée u CNRS n 1398 Simultions Between Cellulr Automt on Cyley Grphs Zsuzsnn Rok Decemre 1994 Reserch Report
More informationLIP. Laboratoire de l Informatique du Parallélisme. Ecole Normale Supérieure de Lyon
LIP Lortoire de l Informtique du Prllélisme Eole Normle Supérieure de Lyon Institut IMAG Unité de reherhe ssoiée u CNRS n 1398 Onewy Cellulr Automt on Cyley Grphs Zsuzsnn Rok Mrs 1993 Reserh Report N
More informationE1: CALCULUS  lecture notes
E1: CALCULUS  lecture notes Ştefn Blint Ev Kslik, Simon Epure, Simin Mriş, Aureli Tomoiogă Contents I Introduction 9 1 The notions set, element of set, membership of n element in set re bsic notions of
More informationECON 331 Lecture Notes: Ch 4 and Ch 5
Mtrix Algebr ECON 33 Lecture Notes: Ch 4 nd Ch 5. Gives us shorthnd wy of writing lrge system of equtions.. Allows us to test for the existnce of solutions to simultneous systems. 3. Allows us to solve
More informationAN020. a a a. cos. cos. cos. Orientations and Rotations. Introduction. Orientations
AN020 Orienttions nd Rottions Introduction The fct tht ccelerometers re sensitive to the grvittionl force on the device llows them to be used to determine the ttitude of the sensor with respect to the
More informationNonLinear & Logistic Regression
NonLiner & Logistic Regression If the sttistics re boring, then you've got the wrong numbers. Edwrd R. Tufte (Sttistics Professor, Yle University) Regression Anlyses When do we use these? PART 1: find
More informationHypergraph regularity and quasirandomness
Hypergrph regulrity nd qusirndomness Brendn Ngle Annik Poerschke Vojtěch Rödl Mthis Schcht Astrct Thomson nd Chung, Grhm, nd Wilson were the first to systemticlly study qusirndom grphs nd hypergrphs,
More informationOn the application of explicit spatial filtering to the variables or fluxes of linear equations
Journl of Computtionl Physics 225 (27) 2 27 www.elsevier.com/locte/jcp Short Note On the ppliction of explicit sptil filtering to the vriles or fluxes of liner equtions Christophe Bogey *, Christophe Billy
More informationUsing integration tables
Using integrtion tbles Integrtion tbles re inclue in most mth tetbooks, n vilble on the Internet. Using them is nother wy to evlute integrls. Sometimes the use is strightforwr; sometimes it tkes severl
More informationShortened Array Codes of Large Girth
Revised version, submitted to the IEEE TRANSACTIONS ON INFORMATION THEORY, Februry 1, 2008 1 Shortened Arry Codes of Lrge Girth Olgic Milenkovic, Member, IEEE, Nvin Kshyp, Member, IEEE, nd Dvid Leyb rxiv:cs/0504016v2
More information