1. LING 388 Language and Computers Lecture 6 9/18/03 Sandiway FONG

2. Administrivia Poll: 2 nd Call for votes Poll: 2 nd Call for votes  How many would want an extra (optional) introductory computer lab session?  Send email to Charles (clin@u.arizona.edu)

3. Administrivia Reminder Reminder  Next Tuesday  Computer Lab Class Meet in SBS 224Meet in SBS 224  Next Thursday  Computer Lab Class continues…  Homework 2 will be handed out

4. Review Last Lecture: Last Lecture:  Formal Grammars  production rules  X -> Y 1 …Y n n>=0 X  V N, Y i  (V N u V T ) 1<=i<=nX  V N, Y i  (V N u V T ) 1<=i<=n  rule application and sentential forms S =>* ,   (V N u V T )*S =>* ,   (V N u V T )* –=>* means “derives through zero or more rule applications”

5. Regular Grammars Regular Grammars are a class of formal grammars with a special restriction Regular Grammars are a class of formal grammars with a special restriction  aka Chomsky hierarchy type-3 grammars  Production rules are constrained to have format: X -> Y tX -> t (left recursive)X -> Y tX -> t (left recursive)  or X -> t YX -> t (right recursive)X -> t YX -> t (right recursive)  where X,Y  V N, t  V T i.e. all rules contain only a single non-terminal, and (possibly) a single terminal) on the right hand side  Note:  can’t have both left and right recursive rules at the same time

6. Regular Grammars (Remarkable) Property (Remarkable) Property  Regular grammars are formally equivalent to FSA (and regular expressions)

7. Regular Grammars Example: Example:  Grammar G ab  S -> aB  B -> aB  B -> bC  B -> b  C -> bC  C -> b  Note: G ab is right recursive  Starting non-terminal: S  V N = {S, B, C}(set of non-terminals)  V T = {a, b}(set of terminals)

8. Regular Grammars Question: Question:  What does grammar G ab generate? Answer: Answer:  Language: L(G ab ) = {a + b + } “one or more a s followed by one or more b s”

9. Regular Grammars Example of Derivation: Example of Derivation:  S  aB  aaB (using rule B -> aB)  … can iterate this to get aa…aB – i.e. any number of a s  From aaB, we can get:  aab(using rule B-> b) OR  aabC(using rule B-> bC)  aabbC(using rule C-> bC) … can iterate this to get aabb…bC – i.e. any number of b s… can iterate this to get aabb…bC – i.e. any number of b s  From aabbC, we can get: aabbbb(using rule C-> b)aabbbb(using rule C-> b)

10. Regular Grammars Consider a FSA equivalent to G ab that also describes L ab = {a + b + } Consider a FSA equivalent to G ab that also describes L ab = {a + b + }

11. Equivalent FSA SB C a a b b

12. Regular Grammars FSA Cases (right- recursive): Cases (right- recursive):  A -> a  A -> aB  A -> aA Note: Note:  A left-recursive grammar can be transformed to a weakly-equivalent right-recursive one a a A B A a A

13. Equivalent FSA Basically, we have that… Basically, we have that…  Each non-terminal corresponds to a state  Each terminal corresponds to a transition between two states  A recursive rule is implemented by a loop

14. Limitations Production Rule Limitation: Production Rule Limitation:  can’t have both left and right recursive rules at the same time  e.g. can’t have rules A -> aBA -> aB A -> BaA -> Ba in the same grammar  Question: Why?  Answer:  Would change the expressive power of the grammar formalism No longer preserve the equivalence with FSA and regular expressionsNo longer preserve the equivalence with FSA and regular expressions

15. Limitations Example: Example:  L = {a n b n | n>=0 } is not a regular language  However, the following grammar (with both left- recursive and right-recursive rules) will generate L :  A ->   A -> aB  B -> Ab  B -> b  Starting non-terminal: A

16. Limitations Sample Derivations : Sample Derivations :  A  aB  aAb  ab (using rule A -> )  or  aaBb aabb (using rule B -> b)aabb (using rule B -> b)  or aaAbbaaAbb aaaBbbaaaBbb aaabbb(using rule B -> b)aaabbb(using rule B -> b)

17. Definite Clauses Grammars (DCG) Introduction Introduction  DCGs are built-in to Prolog  Prolog was originally designed for natural language processing  Notation designed for a transparent encoding of grammar production rules in the database  Can handle general grammars  … not just regular grammars  Next Thursday’s Computer Lab

18. Definite Clause Grammars (DCG) Syntax : Syntax :  We have been writing production rules like  X -> Y t  In Prolog notation, the equivalent DCG rule looks something like  x --> y, [t]. Notes: Notes:  --> replaces ->  Terminal symbols are distinguished from non-terminal symbols by enclosing them in square brackets, e.g. [t]  All terminal and non-terminal symbols begin with a lowercase letter, i.e. are names  General case: can be structures  Commas separate symbols on the right hand side of the rule  A DCG rule ends with a period

19. Definite Clause Grammars (DCG) Example grammar: Example grammar:  Grammar G ab L(G ab ) = {a + b + }L(G ab ) = {a + b + }  Production Rules: S -> aBs --> [a],b.S -> aBs --> [a],b. B -> aBb --> [a],b.B -> aBb --> [a],b. B -> bCb --> [b],c.B -> bCb --> [b],c. B -> bb --> [b].B -> bb --> [b]. C -> bCc --> [b],c.C -> bCc --> [b],c. C -> bc --> [b].C -> bc --> [b].

20. Definite Clause Grammars (DCG) Querying the grammar: Querying the grammar:  Each DCG rule like x --> [a,b]. defines a clause for x/2  In x(X,Y)  X represents the input list to be parsed  Y represents the remainder of the input list after rule x has been parsed  In Prolog parlance, the pair X,Y is known as a difference list  Example queries:  ?- s([a,a,b,b,b],[]).  Yes.  ?- s([a,b,a],[]).  No.