languages are not CFL and hence are not This lemma enables us to prove that some recognizable by any PDA.

Transcription

1 Introdction to Comtabilit Theor Lectre7: The Pming Lemma for Contet Free Langages Prof. Amos Israeli 1 The Pming Lemma Let Abe a contet free langage. There eists a nmber sch that for eer w A, if w then wma be diided into fie arts, w= = satisfing: i i 1. for each i, it holds that A.. >. 3.. Note: Withot req. the Theorem is triial. 3 Introdction and Motiation In this lectre we resent the Pming Lemma for Contet Free Langages. This lemma enables s to roe that some langages are not CFL and hence are not recogniable b an PDA. Proof Idea If wis long enogh (to be recisel defined later) it has a large arse tree which has a long enogh ath α from its root to one of its leaes. Under these conditions, some ariable on α shold aear twice. This enables ming of was demonstrated in the net slide: 4

2 Proof Idea i = i = Pming Pming down 5 The Proof Let Gbe a grammar for the langage L. Let bbe the maimm nmber of smbols (ariables and constants) in the right hand side of a rle of G. (Assme b ). In an arse tree, T, for generating w from G, a node of T ma hae no more than bchildren. If the height of Tis hthen w b. h 7 eminder Let Tbe a binar tree. The -thleel of Thas 1= The 1 -thleel of Thas at most = 1 The i-thleel of Thas at most i If T is a b-aritree then its i-thleel has at most b i 6 If the height of Tis hthen w b. Conersel, h If w > b then the height of T is at least h+ 1. Assme that G has V ariables. Then we set + = b V 1. Conclsion: For an w L, if w, then the height of an arse tree of wis at least V + 1. h 8

3 To see how ming works let τ be the arse tree of w with a minimal nmber of The height of the tree τ, is at least V + 1, so it has a ath, α with at least V + nodes, from its root ntil some leaf. The ath α has at least V + 1 ariables and a single terminal. 9 Each occrrence of has a sb-tree rooted at it: Let be the word generated b the er occrrence of and let be the word generated b the lower occrrence of. 11 Since Ghas V ariables and α has at least V + 1 ariables, there eists a ariable,, that reeats itself among the V + 1 lowest ariables of α, as deicted in the following ictre: 1 Since both sb-trees are generated b the same ariable, each of these sb-trees can be relaced b another. This tree is obtained from τ b sbstitting the er sb-tree at the lower occrrence of. 1

4 The word generated is, and since It is generated b a arse tree of G we get A. Additional sbstittions of the er sb-tree at the lower occrrence of, ield the i i conclsion A for each i >. 13 To see that >, assme that this is the sitation. In this case, this tree is a arse tree for wwith less nodes then τ, in contradiction with the choice of τ as a arse tree for w with a minimal nmber of 15 Sbstittion of the lower sb-tree at the er occrrence of ields this ars tree whose generated word is =. Since once again this is a legitimate arse tree we get A. 14 In order to show that recall that we chose 16 so that both its occrrences fall within the bottom V + 1 nodes of the ath α, where α is the longest ath of the tree so the height of the red sb-tree is at most V + 1 and the nmber of its leaes is at most 1 b V + =.

5 Using the Pming Lemma Now we se the ming lemma for CFL to show that the langage L is not CFL. n n n { a b c } = n Assme towards a contradiction that Lis CFL and let be the ming constant. Consider w= a b c. Obiosl w L. 17 Using the Pming Lemma Case : Either or contain two smbols: In this case, the word has more than three blocks of identical letters: In other words: a b c, Q.E.D qod erat demonstrandm(wiktionar) which was to be roed; which was to be demonstrated. Abbreiation: QED 19 Using the Pming Lemma B the ming lemma, there eist a artition w= where >, and for each i i i, it holds that L. Case 1: Both and contain one smbol each: Together the ma hold smbols, so in, the third smbol aears less than the other two. 18 Discssion Some weeks ago we started or qest to find ot What can be comted and what cannot? So far we identified two classes: L-s and CFL-s and fond some eamles which do not belong in neither class

6 Discssion This is what we got so far: L-s E: { a n n }??? CFL-s E: n n { a b n } n n n Non CFL-s E:{ a b c n } 1 eca In this lectre we introdced and roed the Pming Lemma for CFL-s Using this lemma we managed to roe that the { } fairl simle langage L = a n b n c n n, is not CFL. The net ste is to define Tring Machines. 3 Discssion Moreoer: Or most comle eamle, namel, n n n { } the langage L = a b c n is easil recogniable b or eerda comter, so we did not get so far et. Or net attemt to gras the essence of What s Comtable? are Tring Machines. Some srrises are awaiting