1 / 24

LING 388 Language and Computers

LING 388 Language and Computers. Lecture 11 10/7 /03 Sandiway FONG. Administrivia. Computer Lab Double Class: Thursday 9th and Tuesday 14th Location: SBS 224 TA Office Hours Change in time and location Now Tuesdays after class 12:15 pm - 1:15 pm SBS 224. Review. Chomsky Hierarchy:

naava
Download Presentation

LING 388 Language and Computers

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. LING 388Language and Computers Lecture 11 10/7/03 Sandiway FONG

  2. Administrivia • Computer Lab • Double Class: • Thursday 9th and Tuesday 14th • Location: • SBS 224 • TA Office Hours • Change in time and location • Now Tuesdays after class 12:15 pm - 1:15 pm • SBS 224

  3. Review • Chomsky Hierarchy: • Type-0 General rewrite rules • Type-1 Context-sensitive rules • anbncn • Implementation: type-2 rules + counter • Type-2 Context-free rules • anbn • Implementation: type-2 rules or type-3 rules + counter • Type-3 Regular grammar rules • a+b+ • Implementation: type-3 rules

  4. Review • Production rule formats: • Type-1: • Next slide… • Type-2: • A -> a • Type-3: • A -> Bc A -> c or • A -> cB A -> c • where … • A e VN, c e VT and a e (VN u VT)*

  5. Context-sensitive Grammars • Type-2 and 3 grammars may only have a single non-terminal on the left. • Type-1 (context-sensitive) grammars extend what’s allowed on the left. • Production rules have the format: • a -> b such that |b| >= |a| • where … • a e (VN u VT)+ Note: a should not be comprised of just terminal symbols • b e (VN u VT)* • Notes: • Length constraint means a rule like A -> l is not permitted

  6. Context-sensitive Grammars • (Almost equivalent) alternative definition: • Production rules have the format: • aAb -> agb • where … • A e VN • a, b, g e (VN u VT)* • Notes: • a and b are “copied” over from the left to the right side unchanged • a..b constitutes the (left and right) contexts for non-terminal A • i.e. A -> g in context a__b • However, • … will be more convenient (for our purposes) to use the first definition

  7. Context-sensitive Grammars • Equivalence (informal) • Can transform a context-sensitive rule of form: • AB -> CD • where A,B,C,D are all different non-terminals into rules respecting the form: • aAb -> agb • Invent non-terminals A’ and B’ • Conversion: • AB -> A’B (left context a empty, right context b = B) • A’B -> A’B’ (left context a = A’, right context b empty) • A’B’ -> CB’ (left context a empty, right context b = B’) • CB’ -> CD (left context a = C, right context b empty) • Note: • All four rules respect aAb -> agb, yet clearly AB =>+ CD

  8. Context-sensitive Grammars • Example: • Gabc is a type-1 grammar such that L(Gabc) = {anbncn | n >=1 } • Gabc has 4 production rules: • S -> aSBc • S -> abc • cB -> Bc • bB -> bb • Notes: • 1st two rules context-free • 3rd/4th rules context-sensitive • c (resp. b) left context for non-terminal B in cB -> Bc (bB -> bb) • Length restriction is respected

  9. Context-sensitive Grammars • Compare Gabc to DCG (based on type-2 rules) shown in Lecture 10: • s --> [a],t(1),[c]. • t(N) --> [a],{M is N+1},t(M),[c]. • t(N) --> u(N). • u(N) --> {N > 1}, [b],{M is N-1},u(M). • u(1) --> [b].

  10. Context-sensitive Grammars • How does Gabc work? • Basic Idea: • Gabc has two stages: • Build an equal number of as, bs and cs • Re-arrange them into the correct linear order • We have 4 production rules: • S -> aSBc Stage 1: Build • S -> abc • cB -> Bc Stage 2: Re-arrange • bB -> bb

  11. Context-sensitive Grammars • Build Stage: • S -> aSBc • S -> abc • Sentential Forms: • S • aSBc Using 1st rule • aaSBcBc Using 1st rule • aaabcBcBc Using 2nd rule • Notes: • same number of as, bs (counting b and B together) and cs at each sentential form

  12. Context-sensitive Grammars • Re-arrangement Stage: • cB -> Bc • bB -> bb • Example Sentential Form: • aaabcBcBc • Question: What can we do? • Answer: • Basic Idea 1: Re-arrange the order of Bs and cs • want to move the Bs to the left • want to move the cs to the right • Question: How do we know when to stop re-arranging? • Answer: • Basic Idea 2: Stop when a B comes into contact with a b

  13. Context-sensitive Grammars • Re-arrangement Stage: • cB -> Bc re-arrange c and B • bB -> bbstopping condition • Example Sentential Form: • aaabcBcBc • Basic Idea 1: Re-arrange the order of Bs and cs • cB -> Bc • “If a c precedes a B, the order is wrong, let’s flip them” • Example of Derivation: • aaabcBcBc • aaabBccBc • aaabBcBcc • aaabBBccc

  14. Context-sensitive Grammars • Re-arrangement Stage: • cB -> Bc re-arrange c and B • bB -> bbstopping condition • Example Sentential Form: • aaabBBccc • We still have non-terminals in the sentential string • so we’re not done yet • Apply stopping condition: • bB -> bb • Example Derivation: • aaabBBccc • aaabbBccc • aaabbbccc (a3b3c3) • Final sentential form contains no non-terminals, so we’re done!

  15. Grammar Rule Implementations • Chomsky Hierarchy: • Type-0 General rewrite rules • Type-1 Context-sensitive rules • anbncn • Implementation: type-2 rules + counter or type-1 rules • Type-2 Context-free rules • anbn • Implementation: type-2 rules or type-3 rules + counter • Type-3 Regular grammar rules • a+b+ • Implementation: type-3 rules

  16. Type-0 Grammars • General rewrite rule system • Rule format: • a -> b • where • a e (VN u VT)+ • b e (VN u VT)*

  17. Equivalent Automata Implementations • Chomsky Hierarchy: • Type-0 General rewrite rules • Implementation: Turing Machine (TM) • Type-1 Context-sensitive rules • anbncn • Implementation: Linear Bounded Automata (LBA) • Type-2 Context-free rules • anbn • Implementation: Non-deterministic Push-Down Automata (NPDA) • Type-3 Regular grammar rules • a+b+ • Implementation: Finite State Automata (FSA)

  18. Equivalent Automata Implementations • Machine Characteristics: • All the machine types have a finite state core • However, they differ with respect to working memory • The difference in expressive power can be traced to limitations on the working memory…

  19. Equivalent Automata Implementations

  20. DCG and Context-sensitive Grammars • DCG formalism is based on type-2 (context-free) grammars • It is powerful enough to encode type-1 (context-sensitive) grammars and beyond… • Example: • in Lecture 10, we exhibited a DCG for anbncn • … based on type-2 rules plus a counter

  21. FSA Regular Expressions Regular Grammars = Type-3 DCG and Chomsky Hierarchy Type-2 Type-1 DCG = Type-0

  22. DCG and Context-sensitive Grammars • However, we cannot write DCG rules for context-sensitive rules directly • Example: • can’t write [c],b --> b, [c]. • for cB -> Bc • Note: • we can write Prolog code that takes a context-sensitive grammar and interprets it • Possible term programming project?

  23. This concludes our tour of the grammar hierarchy for this course • But before we leave …

  24. Extra Credit Homework Question • We now know • {anbncn | n >=1 } is a context-sensitive (not context-free) language • {anbn | n >=1 } is a context-free (not regular) language • How about? • Labcd = {anbncndn | n >=1 } • Write a grammar for Labcd • Notes: • Not a computer lab homework • Extra credit question is entirely optional • A chance to make up some ground if you lost points on Homeworks 1 or 2 • Hand in your solution at the same time as Homework 3

More Related