In sensible gramma rs, these strings share some common c - PDF document

Con text-F ree Grammars Notation for recursiv e description of languages. Example: Rol l ! < RO LL > C l ass S tuds < =RO LL > C l ass ! < C LAS S > T ext < =C LAS S > T ext ! C har T ext T ext ! C har C har ! a � � � (other c hars) S tuds ! S tud S tuds S tuds ! � S tud ! < S T U D > T ext < =S T U D > � Generates \do cumen ts" suc h as: <ROLL><CLASS>cs 154</ CLASS > <STUD>Sally</ST UD> <STUD>Fred</STU D> � � � </ROLL> � V ariables (e.g., S tuds ) represen t sets of strings (i.e., languages). ✦ In sensible gramma rs, these strings share some common c haracteristic or roll. � T erminals (e.g., a or < RO LL > ) = sym b ols of whic h strings are comp osed. ✦ \T ags" lik e < RO LL > could b e considered either a single terminal or the concatenation of 6 terminals. � = rules of the form H ead ! Pr o ductions B ody . ✦ H ead is a v ariable. ✦ B ody is a string of zero or more v ariables and/or terminals. � = v ariable that represen ts \the Start Symb ol language." � Notation: G = ( V ; � ; P ; S ) = (v ariables, terminals, pro ductions, start sym b ol). Example A simpler example generates strings of 0's and 1's suc h that eac h blo c k of 0's is follo w ed b y at least as man y 1's. S ! AS j � A ! 0 A 1 j A 1 j 01 � Note v ertical bar separates di�eren t b o dies for the same head. 1

Deriv ations � �A� ) �� � whenev er there is a pro duction A ! � . ✦ Subscript with name of grammar, e.g., ) , if necessary . G ✦ Example: 011 AS ) 0110 A 1 S . * � � ) � means string � can b ecome � in zero or more deriv ation steps. * ✦ Examples: 011 AS ) 011 AS (zero steps); * * 011 AS ) 0110 A 1 S (one step); 011 AS ) 0110011 (three steps). Language of a CF G L ( G ) = set of terminal strings w suc h that * S ) w , where S is the start sym b ol. G Aside: Notation � a; b; : : : are terminals; : : : ; y ; z are strings of terminals. � Greek letters are strings of v ariables and/or terminals, often called sentential forms . � A; B ; : : : are v ariables. � : : : ; Y ; Z are v ariables or terminals. � S is t ypically the start sym b ol. Leftmost/Righ t most Deriv ations � W e ha v e a c hoice of v ariable to replace at eac h step. ✦ Deriv ations ma y app ear di�eren t only b ecause w e mak e the replacemen ts same in a di�eren t order. ✦ T o a v oid suc h di�erences, w e ma y restrict the c hoice. � A deriv ation alw a ys replaces the leftmost leftmost v ariable in a sen ten tial form. ✦ Yields left-sentential forms . � R ightmost de�ned analogously . � ) , ) , etc., used to indicate deriv ations are lm rm leftmost or righ tmost. 2

Example � S ) AS ) A 1 S ) 011 S ) 011 AS ) lm lm lm lm lm 0110 A 1 S ) 0110011 S ) 0110011 lm lm � S ) AS ) AAS ) AA ) A 0 A 1 ) rm rm rm rm rm A 0011 ) A 10011 ) 0110011 rm rm Deriv ation T rees � No des = v ariables, terminals, or � . ✦ V ariables at in terior no des; terminals and � at lea v es. ✦ A leaf can b e � only if it is the only c hild of its paren t. � A no de and its c hildren from the left m ust form the head and b o dy of a pro duction. Example S A S A 1 A S 0 1 0 A 1 � 0 1 Equiv alence of P arse T rees, Leftmost, and Righ tmost Deriv ations The follo wing ab out a grammar G = ( V ; � ; P ; S ) and a terminal string w are all equiv alen t: * 1. S ) w (i.e., w is in L ( G )). * 2. S ) w lm * 3. S ) w rm 4. There is a parse tree for G with ro ot S and (lab els of lea v es, from the left) w . yield � Ob viously (2) and (3) eac h imply (1). 3

P arse T ree Implies LM/RM Deriv ations � Generalize all statemen ts to talk ab out an arbitrary v ariable A in place of S . ✦ Except no w (1) no longer means w is in L ( G ). � Induction on the heigh t of the parse tree. Basis : Heigh t 1: T ree is ro ot A and lea v es w = a ; a ; : : : ; a . 1 2 k � A ! w m ust b e a pro duction, so A ) w and lm A ) w . rm Induction : Heigh t > 1. T ree is ro ot A with c hildren X ; X ; : : : ; X . 1 2 k � Those X 's that are v ariables are ro ots of i shorter trees. ✦ Th us, the IH sa ys that they ha v e LM deriv ations of their yields. � Construct a LM deriv ation of w from A b y starting with A ) X X � � � X , then 1 2 k lm using LM deriv ations from eac h X that is a i v ariable, in order from the left. � RM deriv ation analogous. Deriv ations to P arse T rees Induction on length of the deriv ation. : One step. There is an ob vious parse tree. Basis : More than one step. Induction � Let the �rst step b e A ) X X � � � X . k 1 2 � Subsequen t c hanges can b e reordered so that all c hanges to X and the sen ten tial forms 1 that replace it are done �rst, then those for X , and so on (i.e., w e can rewrite the 2 deriviation as a LM deriv ation). � The deriv ations from those X 's that are i v ariables are all shorter than the giv en deriviation, so the IH applies. � By the IH, there are parse trees for eac h of these deriv ations. � Mak e the ro ots of these trees b e c hildren of a new ro ot lab eled A . Example Consider deriv ation S ) AS ) AAS ) AA ) A 1 A ) A 10 A 1 ) 0110 A 1 ) 0110011 4

� Sub deriv ation from A is: A ) A 1 ) 011 � Sub deriv ation from S is: S ) AS ) A ) 0 A 1 ) 0011 � Eac h has a parse tree; put them together with new ro ot S . 5