CSCI 1010 Models of Computa3on. Lecture 16 The Chomsky Language Hierarchy

Transcription

1 CSCI 1010 Models of Computa3on Lecture 16 The Chomsky Language Hierarchy

2 Overview Defini3ons of phrase structure, context-free and regular languages Proof showing that languages defined by regular grammars and languages recognized by finite-state machines are the same. Parse trees for CFLs Conver3ng CFGs to Chomsky normal form

3 Chomsky Hierarchy Four language types, one less expressive than the next, each with its own grammar rules. Regular Context-Free Context-Sensi3ve Phrase Structure

4 The Chomsky Hierarchy Phrase structure languages are most expressive and recognized by Turing machines. Context-sensi3ve languages are recognized by linear-bounded automata, TM s with amount of space bounded by O( input ). Context-free languages are recognized by pushdown automata. Regular languages are recognized by FSMs.

5 Phrase Structure Languages Defined by grammars: G = (N, T, R, S) 1. N = non-terminals, T = terminals, V = N T, and start symbol S N 2. Rules R V + V*, R is finite. For (a, b) R, a contains at least one non-terminal. Also, (a, a) R for all a in N If (a, b) R, we write a b and say b is derived from a 4. Let u V + and a be a substring of u. Let a b. If v is obtained by replacing a by b in u, we write u G v (immediate deriva4on of v from u.) 5. If u G x 1 G x 2 G... G x n G v, we write u * G v. Here * G is the transi3ve closure of G. 6. The language defined by G is the set of terminal strings derived from S using the rules R, L(G) = {v T* S * G v}

6 Context-Sensi3ve Languages Context-sensi3ve grammars are phrase structure grammars in which for each rule (a, b) R has a b. Context-sensi3ve languages are generated by context-sensi3ve grammars.

7 Example G1 = (N 1,T 1 R 1,S) is context-sensi3ve where N 1 = {S,B,C}, T 1 = {a, b, c} and R 1 shown below. Context is important! L(G) contains aabbcc, which follows from rules (a), (b), (c), (d), (e), (f), and (g) that produce a terminal string: S asbc aabcbc aabbcc aabbcc aabbcc aabbcc aabbcc. S a n (BC) n is possible using (a) n-1 4mes and (b). If (c) is not used to produce S a n B n C n, substring cb occurs for which there is no rule. Thus, L(G) = {a n b n c n n 1}.

8 Context-Free Languages A context-free grammar is a phrase-structure grammar G = (N, T, R, S) in which rules have a single non-terminal on the leh. Context-free languages generate context-free grammars. Example: N 2 = {S}, T 2 = {ε,a,b}, R 2 = {S asb, S ε}. Then, G 2 = (N 2,T 2,R 2,S) is context-free. L(G 2 )= {a n b n n 0}. To see this, apply S asb n 3mes to give S a n Sb n aher which apply S ε.

9 Context-Free Languages Context-free languages are widely used to parse a large por3on of programming languages. They need to be augmented with seman3c analysis because such languages are not context-free. For example, in the statement name1 = name2; name2 could be either a func3on or variable depending on context. A parse of tree of would be augmented with this type of informa3on.

10 Regular Languages A regular grammar is a context-free grammar G = (N,T,R,S) in which the right-hand side of each rule is either a terminal or a terminal followed by a non-terminal. That is, they are of the form A bc or A a. Regular languages are generated by regular grammars.

11 Regular Languages Example: Let N 4 = {S,A B}, T 4 = {0,1}, R 4 below. The rules given above are equivalent to S 0, S 01B, B 01B, B 0. Thus, the original and new grammars both generate the language L(G 4 ) = (01)*0. We now give an FSM that recognizes L(G 4 ).

12 Recognizing Regular Languages Theorem: The regular languages and those recognized by FSM s are the same. Proof If G is regular, L(G) is recognized by an FSM. Replace each rule A a by the two rules A af and F ε where F is a new non-terminal. Construct a state for each non-terminal. Insert edge from state A to state B with label a for each rule A ab. Make A final if A ε. This FSM accepts w such that S wb where B ε. It recognizes L(G). The FSM is nondeterminis3c.

13 Recognizing Regular Languages

14 Recognizing Regular Languages Proof (cont.) Given FSM M, there is a regular grammar G genera4ng language recognized by M. Let G have one non-terminal q i for each state of M and one rule of the form q i aq j if there is an edge labeled a from q i to q j. Add the rule q i ε if q i is a final state. Ini3al state q 0 is associated with the start symbol S of G. The set of strings {w} that takes M from the ini3al state to a final state is the same set of strings generated by G such that S wb where B ε.

15 Parse Trees for CFLs Example: G 3 = (N 3,T 3,R 3,S) A deriva3on of caacaabcbc and its parse tree. s cmnc camanc ca 2 Ma 2 Nc ca 2 ca 2 Nc ca 2 ca 2 bnbc ca 2 ca 2 bcbc

16 Parse Trees Yield of tree is the string of characters at the leaves. The height of a parse tree is length of its longest path. In a lehmost deriva3on, rules invoked in depth-first leh to right order. Rightmost deriva3on similar.

17 Context-Free Languages (CFLs) Recall: A context-free grammar (CFG) is a phrase-structure grammar G = (N,T,R,S) in which each rule has only a single non-terminal on the leh. CFLs are generated by context-free grammars. Example: Let N 2 = {S}, T 2 = {ε, a, b}, R 2 = {S asb, S ε}. Then, G 2 = (N 2,T 2,R 2,S) is context-free.

18 Chomsky Normal Form A CFG G = (N, T, R, S) is in Chomsky normal form if every rule is of the form A BC or A b, b T, except if ε L(G) in which case S ε is also a rule. Theorem: Every CFL L can be generated by a CFG in Chomsky normal form.

19 Chomsky Normal Form Example Example: G 3 = (N 3,T 3,R 3,S). A Chomsky normal form grammar genera3ng this language uses (c) & (e) and replaces others by: (a) S CD, C c, D ME, E NC, (b) M AF, A a, F MA (d) N BG, B b, G NB

20 Conver3ng to Chomsky Normal Form Theorem: Every CFL L can be generated by a CFG in Chomsky normal form. Proof: If ε L, add S ε. Let L be generated by G. Convert G to G in Chomsky normal form in stages. a) Eliminate from G ε-rules of the form B ε (except for S ε) as follows: for each rule with at least one 1 B in right-hand side, e.g. A αbβbγ (α,β,γ are strings), add all possible rules formed by replacing B by ε in all possible ways e.g. A αβbγ, A αbβγ, A αβγ, giving four rules for one original rule.

21 Conver3ng to Chomsky Normal Form Proof (cont.) b) For rules A αw i β (α,β are strings) with w i T, replace it by A αz i β & add rule Z i w i, where Z i is a new non-terminal. Con3nue un3l all rules have a single terminal on right or a string of non-terminals. This new grammar also generates L.

22 Conver3ng to Chomsky Normal Form Proof (cont.) Rules are now of the form: a) A b for b T, b) S ε, c) A Z 1 Z 2... Z k, for Z i N. Consider rules of type c) with k = 1. Cascading such rules gives deriva3ons A B; delete all rules of type c) with k = 1 and replace them with A B if A B. The same language is generated.

23 Conver3ng to Chomsky Normal Form Proof (cont.) If C D and D b, add C b, dele3ng all rules of the form A B. This generates same language; all remaining rules are of the form S ε, A b or A Z 1 Z 2... Z k with k 2, Z i N. Now replace all rules of the form A Z 1 Z 2... Z k by the rules A Z 1 N 1, N 1 Z 2 N 2,..., N k-3 Z k-2 N k-2, N k-2 Z k-1 Z k where each N i is a new nonterminal. This new grammar is in correct form and generates L. Q.E.D.

24 Example Let G = (N,T,R,E) be grammar with N = {E,T,F}, T = {a,b,+,*,(,)} and let R have following rules: E, T, F denote expressions, terms & factors. Easy to * * show that E (a*b+a)*(a+b) and E a*b+a. This grammar doesn t have ε rules. Use *, (, ), +, as non-terminals for *, (, ), and +.

25 Example Transform as indicated un3l only non-terminals on right. Then, reduce the number of non-terminals on the right to two.

26 Example The grammar is now in Chomsky normal form.

27 Summary Defini3ons of phrase structure, context-free and regular languages Proof showing that languages defined by regular grammars and languages recognized by finite-state machines are the same. Parse trees for CFLs Conver3ng CFGs to Chomsky normal form