CS 236 Language and Computation. Alphabet. Definition. I.2.1. Formal Languages (10.1)

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1 C 236 Lnguge nd Computtion Course Notes Prt I: Grmmrs for Defining yntx (II) Chpter I.2: yntx nd Grmmrs (10, 12.1) Anton etzer (Bsed on ook drft y J. V. Tucker nd K. tephenson) Dept. of Computer cience, wnse University csetzer/lectures/ lngugecomputtion/10/index.html Decemer 8, 2010 I.2.1. Forml Lnguges (10.1) I.2.2. Grmmrs nd Derivtions (10.2) I Grmmrs ( ) I Exmples of Grmmrs nd trings ( ) I Derivtions ( ) I Lnguge Genertion ( ) I Designing yntx using Grmmrs ( ) I.2.3. The Chomsky Hierrchy (12.1) I.2.4. Modulrity nd BNF nottion (10.3) I A simple modulr grmmr ( ) I The import construct ( ) I BNF Nottion ( ) C 236 ect. I.2.1 1/ 128 I.2.1. Forml Lnguges (10.1) I.2.1. Forml Lnguges (10.1) I.2.2. Grmmrs nd Derivtions (10.2) I Grmmrs ( ) I Exmples of Grmmrs nd trings ( ) I Derivtions ( ) I Lnguge Genertion ( ) I Designing yntx using Grmmrs ( ) I.2.3. The Chomsky Hierrchy (12.1) I.2.4. Modulrity nd BNF nottion (10.3) I A simple modulr grmmr ( ) I The import construct ( ) I BNF Nottion ( ) Alphet C 236 ect. I.2.1 2/ 128 I.2.1. Forml Lnguges (10.1) An lphet is finite non-empty set T. We shll consider the elements of T to e symols. Exmples The lphet of deciml digits is T Digit = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9} The lower-cse lphet of the English lnguge is T English Lowercse Alphet = {,, c, d, e, f, g, h, i, j, k, l, m, n, o, p, q, r, s, t, u, v, w, x, y, z} C 236 ect. I / 128 C 236 ect. I / 128

2 Alphet I.2.1. Forml Lnguges (10.1) tring I.2.1. Forml Lnguges (10.1) The lower-cse Welsh lphet is T Welsh Lowercse Alphet = {,, c, ch, d, dd, e, f, ff, g, ng, h, i, l, ll, m, n, o, p, ph, r, rh, s, t, th, u, w, y} (This lphet hs een checked crefully). Notice: j, k, q, v, x, z re not elements of this lphet. Therefore Txi written s Tcsi in Welsh. The Unicode lphet hs over 100,000 chrcters. The Ascii lphet hs 128 chrcters, of which 33 re non-printing control chrcters. A string or word over n lphet T is finite sequence elements of T. The set of ll strings orr words over the lphet T is T := {t 1 t 2 t n n 0, t 1, t 2,..., t n T } Note tht one exmple is the empty string, which is represented y ɛ. Mny lnguges, e.g. Hskell, identify trings with lists of chrcters. C 236 ect. I / 128 I.2.1. Forml Lnguges (10.1) tring in Hskell C 236 ect. I / 128 I.2.1. Forml Lnguges (10.1) Exmples of trings :: tring = "Hello" :: tring = [ H, e, l, l, o ] csetzer@cspcnton:~> ghci... Prelude> :lod testtring.hs... *Min> "Hello" *Min> "Hello" 1984 nd 2000 re strings over the lphet T Digit of deciml digits. forwrds nd sdrwkc re words over the lower-cse English lphet T English Lowercse Alphet. C 236 ect. I / 128 C 236 ect. I / 128

3 Length of tring The length of string is given y the function _ : T N w is the numer of symols from the lphet it contins. If we identify strings with List(T ), it is the length of the corresponding list. Exmples 2000 = 4, forwrds = 8, ɛ = 0 T + is the set of non-empty strings: T + = T \ {ɛ} = {t 1 t 2 t n n 1, t 1,..., t n T } Conctention I.2.1. Forml Lnguges (10.1) 1. The conctention of two strings u = u 1 u m nd v = v 1 v n is the string uv = u 1,..., u m, v 1,..., v n 2. The conctention function is the function. : T T T For instnce, if u = 1984 nd v = 2000 then uv = We hve uv = u + v Conctention I.2.1. Forml Lnguges (10.1) C 236 ect. I / 128 I.2.1. Forml Lnguges (10.1) Forml Lnguge We define w n := ww }{{ w}. n times For instnce forwrds 3 = forwrdsforwrdsforwrds A forml lnguge L over n lphet T is suset L of the set of strings over T, i.e. L T We usully sy lnguge for forml lnguge. C 236 ect. I / 128 C 236 ect. I / 128

4 Exmple 1 I.2.1. Forml Lnguges (10.1) Exmple 1 (Cont) I.2.1. Forml Lnguges (10.1) Let We cn enumerte T, T, T, = {, } in the following wy = {ɛ,,,,,,,,,,,,,,,...} o we write first strings of length 0, then of length 1, then of length 2 etc. There is only one string of length 0, nmely ɛ. The strings of length n + 1 re otined y dding in front of ech string of length n nd then do the sme with. Exercise Write progrm which enumertes the strings of length n for given n. Here re few exmples of lnguges over T, : 1. L =. 2. L = {, }. 3. L = {ɛ}. 4. L = {}. 5. L = {, }. 6. L = { n n is even}. 7. L = { n n n 0}. 8. L = { n n+1 n 0}. 9. L = {() n n 0}. C 236 ect. I / 128 I.2.1. Forml Lnguges (10.1) Exmple 2: URLs C 236 ect. I / 128 I.2.1. Forml Lnguges (10.1) Exmple 3: Infix Numers Let T = {,,..., z, A, B,..., Z, 0, 1,..., 9,,, /,.m, :} Let L = {w T w is n http ddress } L contins simple ddresses such s http : // or http : // ut complex exmples like re not in L, since for exmple? nd + re not in T. Let T Infix Arithmetic = {0, 1, +,., )} Expressions of type nturl numers re strings such s (0 + 1) + 1, (1 + 0).0, etc. C 236 ect. I / 128 C 236 ect. I / 128

5 I.2.1. Forml Lnguges (10.1) Exmple 4: Prefix Numers Tokens I.2.1. Forml Lnguges (10.1) Let T Prefix Arithmetic = {zero, succ, dd, mult,,, (, )} Expressions of type nturl numers re expressions such s succ(succ(zero)), dd(zero, succ(succ(zero))) Let L Prefix Arithmetic = {w TPrefix Arithmetic w is n rithmetic expression} Most prser genertors (or compiler-compiler) hve two phses: In phse one the incoming strem of chrcters is grouped into simple tokens. This cn e words like zero, succ ove. Or even longer words considered s one entity, such s <Identifier>. In phse two the text is prsed using grmmr which refers to the tokens generted in the first phse. Token genertion will mke use of regulr expressions, prsing in phse two will use restricted context free grmmrs. Notions regulr expression nd context free grmmr will e introduced lter. Tokens will ply the role of the input lphet for the context free grmmr. Token genertion is the process of generting text in the input lphet of the context free grmmr (tokens) from the input lphet of the lnguge (e.g. Ascii or Unicode chrcters). C 236 ect. I / 128 I.2.1. Forml Lnguges (10.1) Desigining yntx Using Forml Lnguges C 236 ect. I / 128 I.2.1. Forml Lnguges (10.1) Recognition Prolem The syntx of lnguge is defined in two steps: 1. Choose n lphet T. 2. Define the lnguge L T. Let L T e forml lnguge over T. The recognition prolem for T is: Given ny w T decide whether or not w L. C 236 ect. I / 128 C 236 ect. I / 128

6 I.2.2. Grmmrs nd Derivtions (10.2) I.2.2. Grmmrs nd Derivtions (10.2) I Grmmrs ( ) I.2.1. Forml Lnguges (10.1) Exmple: Prse Tree for n English entence I.2.2. Grmmrs nd Derivtions (10.2) I Grmmrs ( ) I Exmples of Grmmrs nd trings ( ) I Derivtions ( ) I Lnguge Genertion ( ) I Designing yntx using Grmmrs ( ) NounPhrse entence VerPhrse I.2.3. The Chomsky Hierrchy (12.1) I.2.4. Modulrity nd BNF nottion (10.3) I A simple modulr grmmr ( ) I The import construct ( ) I BNF Nottion ( ) I g Ver o NounPhrse h o m e A simplified prse tree for the English entence I go home. Terminls C 236 ect. I / 128 I.2.2. Grmmrs nd Derivtions (10.2) I Grmmrs ( ) C 236 ect. I / 128 I.2.2. Grmmrs nd Derivtions (10.2) I Grmmrs ( ) Non-terminls entence entence NounPhrse VerPhrse NounPhrse VerPhrse I Ver NounPhrse I Ver NounPhrse g o h o m e g o h o m e I,, g, o etc. re the terminls of the grmmr. They re elements of the lphet of the lnguge. entence, NounPhrse, VerPhrse etc. re non-terminls. They re intermedite steps in the grmmr. Elements derivle from them form sulnguge, often grmmticl entity. C 236 ect. I / 128 C 236 ect. I c/ 128

7 trt ymol I.2.2. Grmmrs nd Derivtions (10.2) I Grmmrs ( ) Productions I.2.2. Grmmrs nd Derivtions (10.2) I Grmmrs ( ) entence entence NounPhrse VerPhrse NounPhrse VerPhrse I Ver NounPhrse I Ver NounPhrse g o h o m e g o h o m e entence is the strt symol. The lnguge derived is the set of words formed from non-terminls derivle from them. We need s well rules clled which re used to crry out derivtions. E.g.: entence Nounphrse Verphrse Productions C 236 ect. I d/ 128 I.2.2. Grmmrs nd Derivtions (10.2) I Grmmrs ( ) C 236 ect. I e/ 128 I.2.2. Grmmrs nd Derivtions (10.2) I Grmmrs ( ) Grmmrs ( ) NounPhrse I entence g Ver NounPhrse I VerPhrse Ver NounPhrse o VerPhrse NounPhrse h o m e A Grmmr G = (T, N,, P) consists of 1. finite set T clled the lphet or set of terminl symols, 2. finite set N of non-terminl symols or vrile symols such tht T N =, 3. specil non-terminl symol N clled the strt symol, 4. finite set P of sustitution or rewrite rules, clled, ech of which hs the form u v where 4.1 The left hnd string u (T N) + (esp. u is non-empty), 4.2 The right hnd string v (T N). C 236 ect. I f/ 128 C 236 ect. I / 128

8 I.2.2. Grmmrs nd Derivtions (10.2) I Grmmrs ( ) I.2.2. Grmmrs nd Derivtions (10.2) I Grmmrs ( ) Remrk Presenttion of Grmmrs We present grmmr s 4-tuple G = (T, N,, P) nd lso use displyed version: Note tht oth left nd right hnd strings cn contin oth terminls nd non-terminls. Terminls is nother word for elements of the lphet of grmmr. grmmr terminls nonterminls G T N strt symol P C 236 ect. I / 128 I.2.2. Grmmrs nd Derivtions (10.2) I Grmmrs ( ) Grmmr for English entence C 236 ect. I / 128 I.2.2. Grmmrs nd Derivtions (10.2) I Grmmrs ( ) Derivtions (Informl) grmmr terminls nonterminls strt symol impleenglishgrmmrexmple,,c,...,z,a,b,...,z, entence, Nounphrse, Verphrse,Ver entence A grmmr G defines forml lnguge L(G) T. The elements of L(G) re the set of strings we cn otin s follows: trt with strt symol. elect production such tht the left hnd string is sustring of wht you derived so fr. Replce this sustring y the right hnd string of the production. entence Nounphrse Verphrse Verphrse Ver Nounphrse Ver g o Nounphrse I Nounphrse h o m e Once you hve otined string consisting of non-terminls, possily stop. We will first give some exmples nd then forml definition of L(G). C 236 ect. I / 128 C 236 ect. I / 128

9 I.2.2. Grmmrs nd Derivtions (10.2) I Exmples of Grmmrs nd trings ( ) Exmple 0: English Grmmr Exmple 1 I.2.2. Grmmrs nd Derivtions (10.2) I Exmples of Grmmrs nd trings ( ) Consider the grmmr G [01] 1 = ({0, 1}, {},, { 1, 0, 1}) The following derives the sentence I go home : entence NounPhrse VerPhrse I VerPhrse I Ver NounPhrse I g o NounPhrse I g o h o m e displyed s follows grmmr G [01] 1 terminls 0, 1 nonterminls strt symol 1, 0, 1 C 236 ect. I / 128 I.2.2. Grmmrs nd Derivtions (10.2) I Exmples of Grmmrs nd trings ( ) Exmple 1 (Genertion of trings) We ssign numers to the production rules: Rule 1 1 Rule 2 0 Rule 3 1 A derivtion of 1001 L(G [01] 1 ) is s follows: 1 Rule 3 10 Rule Rule Rule 1 A derivtion of 011 L(G [01] 1 ) is s follows: 0 Rule 2 01 Rule Rule 1 C 236 ect. I / 128 I.2.2. Grmmrs nd Derivtions (10.2) I Exmples of Grmmrs nd trings ( ) Derivtions of trings in G {01} 1 The following shows ll strings derivle in this grmmr. Note tht this is not derivtion tree, its the tree of ll possile derivtions in this lnguge C 236 ect. I / 128 C 236 ect. I / 128

10 L(G {01} 1 ) I.2.2. Grmmrs nd Derivtions (10.2) I Exmples of Grmmrs nd trings ( ) I.2.2. Grmmrs nd Derivtions (10.2) I Exmples of Grmmrs nd trings ( ) Exmple 2 (Grmmrs) Consider the grmmr We cn see tht the elements of L(G {01} 1 ) re the strings in {0, 1} which end with 1: Remrk L(G {01} 1 ) = {w1 w {0, 1} } G {01} 1 is n exmple of regulr grmmr (this notion will e introduced lter). G n n = ({, }, {},, {, ) displyed s follows grmmr terminls nonterminls strt symol G n n,, C 236 ect. I / 128 I.2.2. Grmmrs nd Derivtions (10.2) I Exmples of Grmmrs nd trings ( ) Derivtions of strings in G n n L(G n n ) C 236 ect. I / 128 I.2.2. Grmmrs nd Derivtions (10.2) I Exmples of Grmmrs nd trings ( ) We cn see tht the elements of L(G n n ) re the strings of the form n n : L(G n n ) = { n n n 1} Remrk is n exmple of context-free grmmr (this notion will e introduced lter). G n n Note gin tht this is not derivtion tree, ut tree determining ll posssile derivle strings in this lnguge. C 236 ect. I / 128 C 236 ect. I / 128

11 Exmple 3 (Grmmrs) Consider the grmmr G n n c n = ({,, c}, {, B, C},, { BC, BC, CB BC, B, B, C c, cc cc}) displyed s follows grmmr terminls nonterminls strt symol G n n c n,, c, B, C BC, BC, CB BC, B, B, C c, cc cc. I.2.2. Grmmrs nd Derivtions (10.2) I Exmples of Grmmrs nd trings ( ) Derivtion of ccc Exmple Derivtion I.2.2. Grmmrs nd Derivtions (10.2) I Exmples of Grmmrs nd trings ( ) A derivtion of cc in this grmmr is s follows: BC BCBC BBCC BCC CC cc cc C 236 ect. I / 128 I.2.2. Grmmrs nd Derivtions (10.2) I Exmples of Grmmrs nd trings ( ) No Prse Trees for Context ensitive Grmmrs A derivtion of ccc will now e shown on the lck/whiteord. Note tht for context sensitive grmmrs nd unrestricted grmmrs we don t get prse trees. They exist in generl only for context free nd regulr grmmrs. C 236 ect. I / 128 C 236 ect. I / 128

12 L(G n n c n ) I.2.2. Grmmrs nd Derivtions (10.2) I Exmples of Grmmrs nd trings ( ) I.2.2. Grmmrs nd Derivtions (10.2) I Derivtions ( ) Derivtions ( ) We cn see tht the elements of L(G n n c n ) re the strings of the form n n c n : L(G n n c n ) = { n n c n n 1} Remrk is often in ooks nd wepges presented s n exmple of context-sensitive grmmr (this notion will e introduced lter). G n n c n It is not, ut cn e trnsformed into context sensitve one, see ect. I.2.3. Exmple Let G = (T, N,, P) e grmmr, w (T N) + e non-empty word, w (T N) e possily empty word. We sy tht w is derived from w w G w in one step, with nottion iff there exists production u v P such tht u occurs in w, w is the result of replcing one occurrence of u in w y v. C 236 ect. I / 128 I.2.2. Grmmrs nd Derivtions (10.2) I Derivtions ( ) (Cont.) C 236 ect. I / 128 I.2.2. Grmmrs nd Derivtions (10.2) I Derivtions ( ) (Cont.) (Cont) 2. We sy s well w is immeditely generted from w or w is one-step generted from w, or for w G w. w w o w hs the form sut for some s, t (T N), nd w = svt where u v is production. Informlly w is the result of replcing in w sustring, which is equl to the right hnd string of production y the left hnd string of tht production. 3. Let G = (T, N,, P) e grmmr, w, w (T N) e possily empty words. We sy tht w derived from w, with nottion w G w iff is either w = w or there exists sequence of words w0,..., w n (T N) s.t. w 0 = w, w n = w for 1 i n 1 we hve w i G w i+1. C 236 ect. I / 128 C 236 ect. I / 128

13 I.2.2. Grmmrs nd Derivtions (10.2) I Derivtions ( ) (Cont.) Exmple I.2.2. Grmmrs nd Derivtions (10.2) I Derivtions ( ) In exmple 3 ove we hd the following derivtion: (Cont) 4. We sy s well w is generted from w or for w G w. w w BC BCBC BBCC BCC CC cc cc Therefore we hve for instnce cc BCC BCBC BCC BCBC BCBC Remrk C 236 ect. I / 128 I.2.2. Grmmrs nd Derivtions (10.2) I Derivtions ( ) C 236 ect. I / 128 I.2.2. Grmmrs nd Derivtions (10.2) I Lnguge Genertion ( ) Lnguge Genertion ( ) Remrk The ove definitions introduced reltions G (T N) + (T N) G (T N) (T N) G is the reflexive nd trnsitive closure of G Let G = (T, N,, P) e grmmr. The lnguge generted y the grmmr G, denoted y L(G) T is defined s the set of terminl strings generted from the strt symol, i.e. Remrk L(G) := {w T G w} If lnguge is generted y grmmr, then there re infinitely mny grmmrs which generte the sme lnguge. C 236 ect. I / 128 C 236 ect. I / 128

14 I.2.2. Grmmrs nd Derivtions (10.2) I Lnguge Genertion ( ) Equivlence of Grmmrs Exmple I.2.2. Grmmrs nd Derivtions (10.2) I Lnguge Genertion ( ) We sy tht two grmmrs G 1 nd G 2 re equivlent iff L(G 1 ) = L(G 2 ) We show tht L(G n n ) = { n n n 1} One cn esily show tht t n 0(t = n n t = n+1 n+1 ) follows y induction on length of the derivtion t. : show first the ssertion for t = n n y induction on n. Then the ssertion for t = n+1 n+1 follows s well. C 236 ect. I / 128 I.2.2. Grmmrs nd Derivtions (10.2) I Designing yntx using Grmmrs ( ) Designing yntx using Grmmrs ( ) C 236 ect. I / 128 I.2.2. Grmmrs nd Derivtions (10.2) I Designing yntx using Grmmrs ( ) Designing yntx using Grmmrs For progrmming lnguges L it is usully difficult to find grmmr G s.t. L = L(G). One prolem is the fct tht vriles usully need to e defined efore eing used. We will lter see tht lnguges with such property cn usully not e defined y context-free grmmr. Without ny restrictions on the grmmr it is possile, ut lnguges with unrestricted grmmrs re difficult to prse. Insted one defines first grmmr which reflects the sic structure of the lnguge. Then one selects those strings which fulfil the dditionl criteri. In order to define lnguge L one usully proceeds s follows: tep 1 Choose n lphet s.t. L T. tep 2 Choose simple grmmr G with lphet T s.t. L L(G) T The grmmr should give good explntion of how the lnguge is formed. tep 3 Define n lgorithm which determines for t L(G) whether t L or not. C 236 ect. I / 128 C 236 ect. I / 128

15 I.2.3. The Chomsky Hierrchy (12.1) I.2.3. The Chomsky Hierrchy (12.1) I.2.1. Forml Lnguges (10.1) I.2.2. Grmmrs nd Derivtions (10.2) I Grmmrs ( ) I Exmples of Grmmrs nd trings ( ) I Derivtions ( ) I Lnguge Genertion ( ) I Designing yntx using Grmmrs ( ) I.2.3. The Chomsky Hierrchy (12.1) I.2.4. Modulrity nd BNF nottion (10.3) I A simple modulr grmmr ( ) I The import construct ( ) I BNF Nottion ( ) C 236 ect. I / 128 I.2.3. The Chomsky Hierrchy (12.1) Unrestricted Grmmrs Chomsky Hierrchy The Chomsky hierrchy is the clssifiction of grmmrs y mens of 4 properties of its production rules: Unrestricted grmmrs. The limit of grmmrs. Context-sensitive grmmrs. It s usully n ccident if grmmr of lnguge is context-sensitive. C nd C++ hve some context-sensitive spects (delt with y selecting correct strings fter the prsing). Context-free grmmrs. Esy to understnd nd supported y prse genertors. In lnguge design one ims t lnguges hving n underlying context-free grmmr. Regulr grmmrs. imple to prse. Used for dividing the input strem of chrcters into tokens. C 236 ect. I / 128 I.2.3. The Chomsky Hierrchy (12.1) Context-ensitive Grmmrs Let in the following four definitions G = (T, N,, P) e grmmr. Any grmmr G is of Type 0 or unrestricted, so ny production u v for u (T N) +, v (T N) re llowed. Any grmmr G is of Type 1 or context-sensitive, if ll its hve the form uav uwv where A N is nonterminl, which rewrites to non-empty string w (T N) +, ut only where A is in the context of strings u, v (T N). Furthermore production A ɛ is llowed, ut only if A does not occur in the right hnd side of ny production. C 236 ect. I / 128 C 236 ect. I / 128

16 I.2.3. The Chomsky Hierrchy (12.1) Context-Free Grmmrs Regulr Grmmrs I.2.3. The Chomsky Hierrchy (12.1) 1. A grmmr G is left-liner, iff ll its hve the form Any grmmr G is of Type 2 or context-free, if ll its hve the form A w where A N is nonterminl, which rewrites to string w (T N). C 236 ect. I / 128 I.2.3. The Chomsky Hierrchy (12.1) A Hierrchy of Lnguges A B or A or A ɛ 2. A grmmr G is right-liner, iff ll its hve the form A B or A or A ɛ 3. A grmmr G is of Type 3 or regulr, iff it is left-liner or right-liner In the ove we hve A, B N re nonterminl nd T. Note tht in regulr grmmr either ll must e left-liner or ll must e right-liner, so no mixing of the left-liner nd right-liner is llowed. C 236 ect. I / 128 I.2.3. The Chomsky Hierrchy (12.1) Hierrchy of Lnguges A lnguge L T is unrestricted, context-sensitive, context-free, or regulr, iff there exists grmmr G of the relevnt type such tht L(G) = L. Remrk For ny L we hve L regulr L context-free L context-sensitive L unrestricted. We hve tht every regulr grmmr is context-free. every context-sensitive grmmr is n unrestricted grmmr. However not every context-free grmmr is context sensitive, since context-sensitive lnguges llow only A ɛ if A does not occur t the right hnd side of production. (Otherwise ll unrestricted lnguges would e context-sensitive). However one cn construct from context-free grmmr context-free grmmr of the sme lnguge, which hs only A ɛ, if A does not occur on the right hnd side of production. This grmmr is therefore context-sensitive s well. C 236 ect. I / 128 C 236 ect. I / 128

17 I.2.3. The Chomsky Hierrchy (12.1) I.2.3. The Chomsky Hierrchy (12.1) Hierrchy of Lnguges Exmples of Equivlent Grmmrs We give grmmrs of ech type for defining the lnguge regulr context-free context sensitive unrestricted L 2n := { i i is even } C 236 ect. I / 128 I.2.3. The Chomsky Hierrchy (12.1) Unrestricted Grmmr for L 2n C 236 ect. I / 128 I.2.3. The Chomsky Hierrchy (12.1) Context-ensitive Grmmr for L 2n grmmr G unrestricted,2n grmmr G context sensitive,2n terminls terminls nonterminls nonterminls, T strt symol strt symol ɛ ɛ T T T T C 236 ect. I / 128 C 236 ect. I / 128

18 I.2.3. The Chomsky Hierrchy (12.1) I.2.3. The Chomsky Hierrchy (12.1) Context-Free Grmmr for L 2n Regulr Grmmr for L 2n grmmr G context free,2n grmmr G regulr,2n terminls terminls nonterminls nonterminls, A strt symol strt symol ɛ ɛ A A C 236 ect. I / 128 I.2.3. The Chomsky Hierrchy (12.1) Exmple 1 (Grmmrs of the Levels of the Chomsky Hierrchy) Exmple 2 C 236 ect. I / 128 I.2.3. The Chomsky Hierrchy (12.1) grmmr G grmmr G terminls, terminls nonterminls nonterminls strt symol strt symol,, ɛ, L(G) =? G is of which type? L(G) =? G is of which type? C 236 ect. I / 128 C 236 ect. I / 128

19 Exmple 3 I.2.3. The Chomsky Hierrchy (12.1) Exmple 4 (Grmmrs) grmmr terminls nonterminls strt symol L(G) =? G is of which type? G,, Consider the grmmr grmmr terminls nonterminls strt symol G n n c n,, c, B, C BC, BC, CB BC, B, B, C c, cc cc. Exmple 4 is not context sensitive grmmr. Why? C 236 ect. I / 128 Exmple 5 (Grmmrs) Exmple Derivtion I.2.3. The Chomsky Hierrchy (12.1) Here is vrint which is context sensitive: grmmr terminls nonterminls strt symol G n n c n,, c, B, C, H BC, BC, CB HB, HB HC, HC BC, B, B, C c, cc cc. A derivtion of cc in Exmple 5 is s follows: BC BCBC BHBC BHCC BBCC BCC CC cc cc C 236 ect. I / 128

20 I.2.3. The Chomsky Hierrchy (12.1) I.2.4. Modulrity nd BNF nottion (10.3) Exmples I.2.1. Forml Lnguges (10.1) regulr { n n 1} context context-free sensitive { n n n 1} { n n c n n 1} unrestricted I.2.2. Grmmrs nd Derivtions (10.2) I Grmmrs ( ) I Exmples of Grmmrs nd trings ( ) I Derivtions ( ) I Lnguge Genertion ( ) I Designing yntx using Grmmrs ( ) I.2.3. The Chomsky Hierrchy (12.1) I.2.4. Modulrity nd BNF nottion (10.3) I A simple modulr grmmr ( ) I The import construct ( ) I BNF Nottion ( ) C 236 ect. I / 128 I.2.4. Modulrity nd BNF nottion (10.3) I A simple modulr grmmr ( ) pecifying yntx using Grmmrs: Modulrity nd BNF Nottion (10.3) A imple Modulr Grmmr for Progrmming Lnguge ( ) We introduce grmmr for simple while lnguge for computing nturl numers. It will hve the following components: identifiers, nturl numers, rithmetic expressions, Boolen expressions, progrms. C 236 ect. I / 128 Grmmr for Identifiers grmmr terminls nonterminls strt symol G Identifier,,..., z, A, B,..., Z Letter, Id Id Id Letter Id Letter Id Letter Letter Letter z Letter A Letter B Letter Z C 236 ect. I / 128

21 I.2.4. Modulrity nd BNF nottion (10.3) I A simple modulr grmmr ( ) I.2.4. Modulrity nd BNF nottion (10.3) I A simple modulr grmmr ( ) Grmmr for Numers Import grmmr G Numer terminls 0, 1,..., 9 nonterminls strt symol Numer, Digit Numer Numer Digit Numer Digit Numer Digit 0 Digit 1 Digit 9 The grmmr for rithmetic expressions G Arithmetic Expression will import G Identifier G Numer. This mens the following: The terminls/nonterminls/ of these grmmrs re dded to the terminls/nonterminls/ of G Arithmetic Expression. C 236 ect. I / 128 I.2.4. Modulrity nd BNF nottion (10.3) I A simple modulr grmmr ( ) Grmmr for Arithmetic Expressions Amiguity C 236 ect. I / 128 I.2.4. Modulrity nd BNF nottion (10.3) I A simple modulr grmmr ( ) grmmr import G Arithmetic Expression G Identifier, G Numer terminls +,,, /, (, ) nonterminls strt symol AExp, AOp AExp AExp Id AExp Numer AExp ( AExp ) AExp AExp AOp AExp AOp + AOp AOp AOp / The grmmr G Arithmetic Expression is miguous, concept discussed in ection I.2.5. There we will discuss how to mke this grmmr unmiguous. The sme pplies to the grmmrs G Boolen Expression nd G while (ecuse of the sequencing opertion) introduced on the next two slides. C 236 ect. I / 128 C 236 ect. I / 128

22 Grmmr for Boolen Expressions grmmr import G Boolen Expression G Arithmetic Expression terminls true, flse, not, nd, or, =, < nonterminls strt symol BExp, BOp1, BOp2, RelOp BExp BExp BOp1 BExp BExp BExp BOp2 BExp BExp AExp RelOp AExp BExp true BExp flse BOp1 not BOp2 nd BOp2 or RelOp = RelOp < I.2.4. Modulrity nd BNF nottion (10.3) I BNF Nottion ( ) User Friendly Grmmrs nd BNF Nottion (10.3.3) I.2.4. Modulrity nd BNF nottion (10.3) I A simple modulr grmmr ( ) Grmmr for While Progrms grmmr import G while G Arithmetic Expression, G Boolen Expression terminls skip, if, then, else, fi, while, do, od, :=, ; nonterminls strt symol Exmple 1 Progrm Progrm Progrm skip Progrm Id := AExp Progrm Progrm ; Progrm Progrm if BExp then Progrm else Progrm fi Progrm while BExp do Progrm od C 236 ect. I / 128 I.2.4. Modulrity nd BNF nottion (10.3) I BNF Nottion ( ) The Bckus-Nur-Form or BNF is the presenttion of grmmr using the following conventions: 1. The terminl symols re often written in old font. 2. The non-terminl symols re fmilir terms for syntctic components enclosed in ngle rckets, e.g. <sttement>, <expresssion>, <identifier>. 3. The strt symol is the non-terminl tht is presented first. 4. The symol ::= replces nd extends y listing the possile for non-terminl; lterntive possiilities for the right hnd sides of prticulr production re seprted y. For instnce the rules re condensed into one rule BExp ::= true BExp ::= flse BExp ::= BOp1 BExp BExp ::= BExp BOp2 BExp BExp ::= AExp RelOp AExp BExp ::= true flse BOp1 BExp BExp BOp2 BExp AExp RelOp AExp C 236 ect. I / 128 C 236 ect. I / 128

23 I.2.4. Modulrity nd BNF nottion (10.3) I BNF Nottion ( ) I.2.4. Modulrity nd BNF nottion (10.3) I BNF Nottion ( ) Exmple 2 Extended BNF nf rules Letter Letter ::= LowerCse UpperCse LowerCse ::= c d e f g h i j k l m n o p q r s t u v w x y z UpperCse ::= A B C D E F G H I J K L M N O P Q R T U V W X Y Z Extended BNF or EBNF dds to BNF the following conventions: 1. An optionl occurrence of some portion of production rule choise is enclosed in squre rckets [ ]. [u] mens zero or one occurrence of u where u (T N) An ritrry numer of occurences of some portion of production rule choice is enclosed in rces { }. {u} mens zero or more occurrences of u where u (T N) +. Exmple C 236 ect. I / 128 I.2.4. Modulrity nd BNF nottion (10.3) I BNF Nottion ( ) Exmple C 236 ect. I / 128 I.2.4. Modulrity nd BNF nottion (10.3) I BNF Nottion ( ) In norml BNF, numers cn e defined s follows nf Numer rules Numer Digits Digit ::= Digit NonZeroDigit Digits ::= Digit Digit Digits ::= 0 NonZeroDigit NonZeroDigit ::= In EBNF, we cn define them s follows: enf Numer rules Numer Digit ::= 0 NonZeroDigit { Digit } ::= 0 NonZeroDigit NonZeroDigit ::= C 236 ect. I / 128 C 236 ect. I / 128

24 I.2.4. Modulrity nd BNF nottion (10.3) I BNF Nottion ( ) Theorem I Lemm Let L T e lnguge. L is definle in EBNF iff it is definle in BNF. Proof: Exercise. I.2.1. Forml Lnguges (10.1) I.2.2. Grmmrs nd Derivtions (10.2) I Grmmrs ( ) I Exmples of Grmmrs nd trings ( ) I Derivtions ( ) I Lnguge Genertion ( ) I Designing yntx using Grmmrs ( ) I.2.3. The Chomsky Hierrchy (12.1) I.2.4. Modulrity nd BNF nottion (10.3) I A simple modulr grmmr ( ) I The import construct ( ) I BNF Nottion ( ) C 236 ect. I / 128 Derivtion Trees or Prse Trees C 236 ect. I / 128 Exmple Consider the grmmr grmmr G Context free Grmmrs (revited s CFG in the following) llow to pply to non-terminl t position without needing the context. Therefore we cn expnd the non-terminls independently of ech other. This llows us to define derivtion trees (lso clled prse trees). terminls nonterminls strt symol, C 236 ect. I / 128 C 236 ect. I / 128

25 Exmple Derivtion Derivtion Tree We derive in it: C 236 ect. I / 128 Form of the Derivtion Tree C 236 ect. I / 128 Derivtion Tree Nodes re lelled with elements of N T {ɛ}. A node with lel A hs sutree A X 1 X 2... only if A is non-terminl nd there is production where X i T N. A X 1 X 2 X n All leves of the tree together red from left to right form the string derived, nmely. This is clled the frontier of the derivtion tree. We will s well consider derivtion trees not ending in string of terminls, so the frontier is n element of (T N). C 236 ect. I / 128 X n Let G = (T, N,, P) e CFG. A derivtion tree or prse tree for G is finite tree with nodes lelled y elements of N T {ɛ}, s.t. node A hs children with lels X 1,..., X n only if A N nd there is production A X 1 X 2 X n If the node of one of the children of A is ɛ, then this node is the only child of this tree. The frontier of the tree is the set of leves red from left to right in sequence, which is n element (T N). The root of the tree is the node t the to of the derivtion tree. C 236 ect. I / 128

26 Left-Most nd Right-Most Derivtions Exmple Derivtion Tree From derivtion tree we cn otin derivtion in vrious orders. Consider the grmmr grmmr G terminls, A B nonterminls, A, B strt symol AB, A A, A B B, B A B C 236 ect. I / 128 Different Derivtions of C 236 ect. I / 128 Left-Most Derivtion We cn derive in different wys: AB AB B B A left most derivtion AB AB A A A right most derivtion AB AB AB B AB AB AB A AB AB AB B Let G = (T, N,, P) e CFG. A single-step derivtion w w is left-most if rule ws pplied to the left-most non-terminl in w, i.e. w = sat for some A N, s T (consisting only of terminls), t ( T ), nd there exist production A v s.t. w = svt. AB AB AB A C 236 ect. I / 128 C 236 ect. I / 128

27 Left-Most Derivtion Left/Right-Most Derivtion equence Let G = (T, N,, P) e CFG. A single-step derivtion w w is right-most if rule ws pplied to the right-most non-terminl in w, i.e. w = sat for some A N, s ( T ), t T (consisting only of terminls), there exist production A v s.t. w = svt. Let G = (T, N,, P) e CFG 1. A derivtion sequence w 0 w 1 w 2 w n is left-most, if ech dervtion step w i w i+1 is left-most. 2. Right-most derivtion sequences re defined nlogously. C 236 ect. I / 128 Theorem I (Derivtion Trees nd Lnguge Genertion) C 236 ect. I / 128 Proof of Theorem I Theorem Let G = (T, N,, P) e CFG, A T, w, w (T N), Then the following re equivlent (1) There exist derivtion tree with root lelled y A nd frontier w. (2) A w. In cse w T, the derivtion sequence w w cn oth e chosen s left-most nd s right-most derivtion sequence A proof of this theorem cn e found in the dditionl mteril. We illustrte this theorem y n exmple. It will show oth: (red forwrds) how to otin from left-most derivtion derivtion tree, (red ckwrds) how to otin from derivtion tree left-most derivtion. C 236 ect. I / 128 C 236 ect. I / 128

28 Exmple (tep 1) Exmple (tep 2) AB A B A B A B A B C 236 ect. I / 128 Exmple (tep 3) C 236 ect. I / 128 Exmple (tep 4) AB AB AB AB B A B A B A B A B Derivtion tree for is trivil. C 236 ect. I / 128 C 236 ect. I / 128

29 Exmple (tep 5) Exmple (tep 6) AB AB B B AB AB B B A B A B A B A B C 236 ect. I / 128 Theorem I Uniqueness of Derivtion (Trees) Finl derivtion. C 236 ect. I / 128 Proof of Theorem I Theorem Let G = (T, N,, P) e CFG, A N, w T. (1) Assume there re two different derivtion trees with root lelled y A nd frontier w. Then there exist two different left-most nd two different right-most derivtions of A w. (2) Assume there re two different left-most derivtions or two different right-most-derivtions of A w. Then there exist two different derivtion trees of with root lelled y A nd frontier w. The exmple ove demonstrtes how derivtion trees re trnsformed into left-most derivtions in unique wy. A more forml proof cn e found in the dditionl mteril. C 236 ect. I / 128 C 236 ect. I / 128

30 Uniqueness of Derivtion (Trees) Amiguous Grmmrs A direct consequence of the theorem is the following: Theorem Let G = (T, N,, P) e CFG. The following re equivlent: (1) For every w T there exist t most one derivtion tree with lel nd frontier w. (2) For every w T there exist t most one left-most derivtion sequence w. (3) For every w T there exist t most one right-most derivtion sequence w. A CFG G = (T, N,, P) is miguous, if there is string w L(G) hving more thn one derivtion tree (or, equivlently, hving more thn one left-most or right-most derivtion). C 236 ect. I / 128 Exmple 1 C 236 ect. I / 128 Exmple 1 grmmr terminls nonterminls strt symol G, There re two left-most derivtions of : nd And two derivtion trees: C 236 ect. I / 128 C 236 ect. I / 128

31 Exmple 2: Dngling Else Exmple 2: Dngling Else Assume the following grmmr which is cut down version of the grmmr G while introduced in I.2.4 with if then else fi replced y if then nd if then else ): grmmr import G Dngling else G Arithmetic Expression, G Boolen Expression terminls if, then, else, :=, ; nonterminls strt symol Progrm Progrm Progrm Id := AExp Progrm if BExp then Progrm else Progrm Progrm if BExp then Progrm Assume strings 1, 2 deriving from BExp nd string s 1, s 2 deriving from Progrm. The string if 1 then if 2 then s 1 else s 2 hs two derivtion trees (we omit the derivtion trees for i, s i.) C 236 ect. I / 128 First Derivtion Tree C 236 ect. I / 128 econd Derivtion Tree Progrm Progrm if BExp then Progrm if BExp then Progrm else Progrm 1 if BExp then Progrm else Progrm 1 if BExp then Progrm s 2 2 s 1 s 2 2 s 1 C 236 ect. I / 128 C 236 ect. I / 128

32 Different Interprettions of the Progrm Execution following the Derivtion Tree 1 The two different derivtion trees of the progrm In the first the else cse elongs to the second if. It is executed if 1 is true nd 2 is flse. The progrm cn e using intendtion e written s follows: if 1 then if 2 then s 1 else s 2 correspond to two different wys of executing the progrm: if 1 then if 2 then s 1 else s 2 C 236 ect. I / 128 Execution following the Derivtion Tree 2 C 236 ect. I / olutions for olving the Prolem In the second derivtion tree, the else cse elongs to the first if. It is executed if 1 is flse. The progrm cn e using intendtion e written s follows: if 1 then if 2 else s 2 then s 1 There re 2 solutions for solving this prolem. The first solution is to dd to if then nd if then else symol fi (or some other keyword such s end if). lelling the end of the sttement. grmmr import G Unmiguous if G Arithmetic Expression, G Boolen Expression terminls if, then, else, :=, ; nonterminls strt symol Progrm Progrm Progrm Id := AExp Progrm if BExp then Progrm else Progrm fi Progrm if BExp then Progrm fi C 236 ect. I / 128 C 236 ect. I / 128

33 olution 1 olution 2 Now the two interprettions of the originl string would e written in s two different strings: Else elonging to the second if is written s if 1 then if 2 then s 1 else s 2 fi fi Else elong to the first if is written s if 1 then if 2 then s 1 fi else s 2 fi This solution hs een tken for instnce in Algol, in the sh shell (Linux), nd in Ad (where fi is replced y end if ). The 2nd solution is to modify the grmmr so tht the derivtion tree will e possile only for one of the two choices. For this one we modify the grmmr, tht the sttement s 1 in is not mtched y ut only y if 1 then s 1 else s 2 if 1 then s 1 if 1 then s 1 else s 2 This solution hs een tken in most other progrmming lnguges. C 236 ect. I / 128 olution 2 olution 2 C 236 ect. I / 128 For this we split Progrms into two ctegories: Those derived from MtchedIf. In progrm deriving from MtchedIf, ech if is mtched y n else cluse. Those derived from UnmtchedIf. These hve t lest one if with no mtching else cluse. The grmmr will mke sure tht else will lwys e ssocited with the first if to the left, which hs no unmtched else yet. o if then mthf else expression will e prsed s in the first derivtion tree. C 236 ect. I / 128

34 olution 2 Here is the grmmr. We omit sequencing (comining progrms using ; ), since it would result in n miguous grmmr, which needs to e resolved. grmmr import G Dngling Else G Arithmetic Expression, G Boolen Expression terminls if, then, else, :=, ; nonterminls strt symol Progrm Progrm Progrm UnmtchedIf Progrm MtchedIf MtchedIf Id := AExp MtchedIf if BExp then MtchedIf else MtchedIf UnmtchedIf if BExp then Progrm UnmtchedIf if BExp then MtchedIf else UnmtchedIf Exmple: Grmmr for Arithmetic Expressions Rememer the grmmr for rithmetic expressions (using elements of BNF nottion) Unique Derivtion Tree 2nd olution Progrm UnmtchedIf if BExp then Progrm 1 MtchedIf if BExp then MtchedIf else 2 s 1 MtchedIf C 236 ect. I / 128 First Prse tree for AExp s 2 grmmr import G Arithmetic Expression G Identifier, G Numer terminls +,,, /, (, ) nonterminls strt symol AExp, AOp AExp AExp Id Numer AExp ( AExp ) AExp AExp AOp AExp AOp + / AExp AExp AOp AOp AExp Numer + Numer 2 3 AExp Numer 4 C 236 ect. I / 128 C 236 ect. I / 128

35 econd Prse tree for Difference in Evlution AExp AExp Numer AOp + AExp AExp AOp AExp The first prse tree correponds to prsing it s if it were (2 + 3) 4 Evlution will return 20. The second prse tree correponds to prsing it s if it were 2 + (3 4) Evlution will return Numer Numer 3 4 C 236 ect. I / 128 Unmiguous Version C 236 ect. I / 128 Arithmetic Expression Unique Prse Tree for in Gunmiguous AExp grmmr import Arithmetic Expression Gunmiguous G Identifier, G Numer terminls +,,, /, (, ) nonterminls strt symol AExp, Term, Fctor AExp AExp AExp + Term AExp Term Term Term Term Fctor Term/Fctor Fctor Fctor Id Numer ( AExp ) AExp Term Fctor Numer + Term Fctor Numer Term Fctor Numer C 236 ect. I / 128 C 236 ect. I / 128

36 Mking Context Free Grmmrs Unmiguous The following is known out Context Free Grmmrs: There re lnguges defined y context free grmmrs which cnnot e defined y n unmiguous grmmr. Context free grmmrs, for which there exist no equivlent unmigous grmmr, re clled inherently miguous grmmrs. ee Hopcroft/Motwni/Ullmn, 5.4.4, p It is undecdile whether grmmr is miguous. (me ook, 7.4.5, p. 307.) It is undecdile whether grmmr is inherently miguous grmmrs. (me ook, nd 9.5.2, p. 413). C 236 ect. I / 128