Announcements ICS 6B Final on Tuesday in class Comprehensive - PDF document

Announcements ICS 6B � Final on Tuesday in class � Comprehensive Boolean Algebra & Logic ● Similar to quizzes ● A little more focus on last lecture � Let me know if anyone wants some suggested Lecture Notes for Summer Quarter, 2008 problems � Let me know if anyone wants to attend a final Michele Rousseau review on Monday ● I will announce if I am able to get a room Set 10 – Ch. 12.1, 12.2, 12.3 � Quiz #4 regrades due today (Some slides inspired and adapted from Alessandra Pantano) 2 Lecture Set 10 - Chpts 12.1, 12.2, 12.3 Today’s Lecture � Chapter 12�12.1, 12.2, 12.3� ● Languages and Grammars �12.1� Chapter 12: Section 12.1 ● Finite State Machines with Outputs �12.2� ● Finite State Machines with No Output �12.3� Languages and Grammars Lecture Set 10 - Chpts 12.1, 12.2, 12.3 3 Examples Preliminary Definitions: Example 1 � Vocabulary: a not empty finite set of symbols ● V��a,B,C,d� is a vocabulary ● Also called an “alphabet” ● aBaaC and dBaB are sentences ● The set �aBaaC, dBaB� is a language � Sentence: a finite string of symbols Example 2 ● Also called a “word” ● V��English words� � g � ● Any finite sequence of words is a sentence � Language: any collection of sentences ● The set �“The Blue The”, “sky is blue”, “goat two”� is a language Example 3 ● V��letters in the alphabet� A sentence can be the empty string of symbols. ● Any sequence of the letters is a sentence In this case the sentence is denoted by λ�lambda� ● The set �“blue”, “sky”, “the”� is a language Lecture Set 10 - Chpts 12.1, 12.2, 12.3 5 Lecture Set 10 - Chpts 12.1, 12.2, 12.3 6 1

Grammar: Informal Definition Example � A grammar is a way to construct a language from a given � Consider the vocabulary V��a,B,C,d�. vocabulary � Say that B is the start element. Choose a special element S in the vocabulary to begin with 1. � Choose �a,d� to be the terminal elements, and �B,C� �S will be called the “start symbol”� to be the non‐terminal elements. Divide your set of symbols �i.e. your vocabulary� into two 2. categories: the “terminal” and the “non‐terminal” � Allow the following replacements: elements. l t add The non‐terminal elements are the ones that can be 3. ● B�aC �aBCd � aBBcd �… replaced by other symbols. Strings containing non‐ aaCCd �… terminal elements can be replaced by other strings. Assign a set of rules �called “productions”� to replace 4. certain strings by other stings Start from you “start element” S, and apply the rules 5. �“productions”� to create the sentences in the language. 7 8 Lecture Set 10 - Chpts 12.1, 12.2, 12.3 Lecture Set 10 - Chpts 12.1, 12.2, 12.3 Example 1 A more formal definition Consider the vocabulary V�T �N with A phrase structure grammar G��V,T,S,P� consists of: ● T��a, the, fat, rabbit, mathematician, hops, eats� � A vocabulary v ● N��sentence, article, adjective, noun, verb� ● i.e. a not empty set of symbols � Let “sentence” be the start element. � A subset T � V of terminal elements productions � Allow the following replacements ● �i.e. a subset of symbols that cannot be replaced by other symbols� 1. sentence � �article� �adjective� �noun� �verb� � A start element S 2. article � a 3. article � the ● To begin our construction 4. adjective � fat � A finite set of productions 5. noun � mathematician ● A finite set of rules that specify which strings of symbols 6. noun � rabbit can be replaced by other strings 7. verb � hops 8. verb � eats Lecture Set 10 - Chpts 12.1, 12.2, 12.3 9 10 Example 1 (2) Example 1 (3) If we start from S�sentence and apply a sequence of We obtained strings of the following form production �until no further production is S � �article� �adjective� �noun� �verb� applicable�, we produce the following valid � sentences: adjective �article� �fat� �noun� �verb� article verb noun The article can be replaced by “a” or “the” The article can be replaced by a or the ● a fat mathematician hops f h i i h The noun can be replaced by “rabbit” or “mathematician” ● the fat mathematician hops I am only The verb can be replaced by “eats” or “hops” interested ● a fat rabbit hops in the sentences Notice that we can’t produce the string ● the fat rabbit hops that contain “a mathematician eats the rabbit” ● a fat mathematician eats “terminal” symbols ● the fat mathematician eats By applying a sequence of productions? (those that can’t be ● a fat rabbit eats replaced) �� This is not a valid string ● the fat rabbit eats 11 Lecture Set 10 - Chpts 12.1, 12.2, 12.3 12 2

Example 2 Derivations G��V,T,S,P� with � Let G��V,T,S,P� be a phrase‐structure grammar. � Let w 0 and w 1 be strings over V. ● V��a,b,A,B,S� � We say that w 1 is directly derivable from w 0 if w 1 can be ● T��a,b� obtained from w 0 by applying a single production ● S�the start symbol More precisely, we can write w 0 and w 1 as a concatenation of ● P��S �ABa, A �BB, B �ab, AB �b� w 0 � l, z 0 r, w 1 � l, z 1 r �1� �1� �2� �3� �4� �2� �3� �4� And there exists a production z 0 � z 1. Lets see what are all the valid sentences that we can If w 1 is directly derivable from w 0 , we write w 0 � w 1 produce �by applying a sequence of productions to S� � We say that w 1 is derivable from w 0 if w 1 can be obtained ba from w 0 by applying a sequence of productions. �4� �2� �3�, �3�, �3� More precisely, there exists strings u 1, u 2 , …, u k such that S �ABa � BBBa �1� � abababa w 0 � u 1 � u 2 � … � u k � w 1 �3�, �3� Aaba �BBaba * �3� In this case we write w 0 � w 1 �2� The sequence of steps w 0 � u 1 � u 2 � … � u k � w 1 is called a derivation. 13 14 Lecture Set 10 - Chpts 12.1, 12.2, 12.3 For example Language of a Grammar � In example 2, � Let G��V, T, S, P� be a grammar. The language of G ‐ denoted L�G� ‐ is the set of all sequences of ● Bbaba is derivable from Aba, and a derivation is: terminal elements that can be derived from P: Aaba � Aaba � BBaba �3� �2� L�G���w �T * : S � w� * � To construct the language of G, you start from S and ● Another derivation is ABa � BBBa � BBaba AB � BBB � BB b apply all possible derivations that lead to sentences l ll ibl d i ti th t l d t t �2� �3� which only contain the symbols in T. � In Example 1, L�G� consists of exactly 8 sentences. � In Example 2, L�G� consists of exactly 2 sentences ● �“ba” and “abababa”� Lecture Set 10 - Chpts 12.1, 12.2, 12.3 15 Lecture Set 10 - Chpts 12.1, 12.2, 12.3 16 Type 1: Construct the Lang of a Grammar 2 Main Types of Problems: Example 3 1 – Given a grammar G, find L�G� G ��V, T, S, P� with ● V��S, A, a, b� ● T��a, b� 2 – Given a language L, find a grammar G such ● S� the start element that L L�G� that L�L�G� ● P��S�aA S �b A�aa� ● P��S�aA, S �b, A�aa� Find L�G� S �aA �aaa � L(G) = {b, aaa} b Lecture Set 10 - Chpts 12.1, 12.2, 12.3 17 Lecture Set 10 - Chpts 12.1, 12.2, 12.3 18 3