• Professor Jeanne Ferrante

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http://www.jflap.org/jflaptmp/

CSE 105 Theory of

Computation

Today’s Agenda – Context-Free Grammars – Ambiguity

Reminders and announcements: – Exam 1 grades will be available…

• Very unlikely: Friday More Likely: Monday

– Reading Quiz 4: Due MONDAY Apr 25, 11:59 pm – HW 4: Due WEDNESDAY, Apr 27, 11:59 pm

• Discussion helpful for homework

– Office Hours next week will be updated • Solutions for exam problems

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MODELING GRAMMATICAL UNDERSTANDING

Grammar intuition

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Vote your first instinct!

Which is the correct order? A. a beautiful blue sailing boat B. a blue beautiful sailing boat C. a sailing beautiful blue boat D. a blue sailing beautiful boat

Credits: Quiz questions taken from: http://web2.uvcs.uvic.ca/elc/studyzone/410/grammar/adjord.htm 4

English Adjective Categories & Order

1. General opinion (good, brilliant, awful)

2. Specific opinion (intelligent, tasty, comfortable)

3. Size (small, large, tiny) 4. Shape (square, round, flat, thin )

5. Age (new, ancient)

6. Colour (blue, pink, reddish)

7. Nationality (French, eastern, Martian)

8. Material (steel, wooden, paper)

9. Purpose (sailing, carving)

Sources: https://learnenglish.britishcouncil.org/en/english-grammar/adjectives/order-adjectives http://web2.uvcs.uvic.ca/elc/studyzone/410/grammar/adjord.htm 5

Did you know you knew that? • Could you have written down those rules?

– Then how did you apply them? – Did anyone ever explicitly teach you how to correctly

structure multi-adjective clauses using that list? • Linguists, Psychologists, Neurologists, and Anthropologists

have long been amazed and puzzled by the human capacity for acquiring language

• Context-free grammars are one way that scholars have tried to model how language, and our brain’s models of language, might be structured – Key concept: representing and generating infinite

languages with a complex structure, using just a few simple rules

• Formalizing these rules allows software tools to use them

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Why use grammars in computer science?

• Precise specification of language structure – For example, begin – end hierarchy

• Good basis of automatic tools – For example, parser generators

• Provides feedback to language designers – For example, whether new construct easy/hard

to add

• Adaptable – Easy to add new language constructs

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CONTEXT-FREE GRAMMARS CFGS

They generate languages!

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Context-Free Grammars

• Finite set of Variables: {S} • Designated Start variable: S • Finite set of Terminals: {0} • Finite set of Rules (Productions):

S 0S (1) S 0 (2)

Example: S 0S 00S 000 1 1 2

Example Grammar G

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What is the language of this CFG?

S 0S | 0 (Note: 2 rules, separated by I ) A. {0} B. {0, 00, 000, …} C. {ε , 0, 00, 000, …} D. None of the above

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Formal Definition of CFG A context-free grammar is a 4-tuple (V, Σ , R, S), where 1. V is a finite set of variables 2. Σ is a finite set of terminals, disjoint from

V 3. R is a finite set of rules

A u where A є V and u є (V U Σ )*

4. S є V is the designated start variable Rules capture patterns of words in language

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Derivations in CFG Given a CFG G, with start variable S, and

terminals Σ • u A v u w v if A w is a rule in G and

u, v, and w are strings of variables and terminals of G.

• u * v if u = v, or there is a sequence u1 … uk

such that u = u1 u2 ….. uk-1 uk = v

• L(G) = { w in Σ* │ S * w } – All strings derived when start with S and apply 0 or

more rules to get w

• G is equivalent to G’ if L(G) = L(G’)

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G generates a LANGUAGE L(G) • L(G) includes every string of terminals

generated from the start variable S – Might include the empty string ε

• L(G) does not include strings that cannot be generated from the start variable S

• L(G) includes ONLY strings of terminals, not strings of terminals and variables

• A language L is context-free if there is a CFG G with L(G) = L

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What is the language of this CFG?

S 0S | 1S | ε (Note: 3 separate rules!) A. Same as regular expression 0*1* B. Same as regular expression 0* U 1* C. Same as regular expression (0 U 1)* D. None of the above

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Are all regular languages context-free? Discuss and try to prove or disprove.

A. Yes B. No C. Don’t know

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Designing CFG’s • If you can divide language you want to

generate into pieces, write a separate grammar for each piece, with start Si , and then add a new start variable S S1 | S2 | … Sn

• If you need to link information, such as the number of 0’s to the same number of 1’s, use rule of form R u R v

• You can construct a CFG from a DFA! – For state qi in DFA use variable Ri in CFG – For start state q0 in DFA make R0 start in CFG – If δ(qi, a) = qj, add rule Ri a Rj in CFG – Add Ri ε if qi is a final state in DFA

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CFG for L = {anbn | n ≥ 0} • What’s the shortest string in language? • If we have built anbn , how do we modify it to

create a longer string in the language?

• What rules should we pick to use this idea?

ε

an+1bn+1 = a anbn b

S ε | aSb

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Demonstrating CFG Power Consider L = { 0n 1n | n ≥ 0}. Which of these CFG’s generate L? A. S QR Q 0Q | ε R 1R | ε B. S RQ R 0R1 | ε Q 0Q1 | ε C. S 0S1 | ε D. S 01S01 | ε E. None of the above

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Design a CFG • L = {an bm cn | n, m ≥ 0} • Shortest string: • Any number of b’s in middle?

• Right order and right number of a’s and

c’s?

ε

B bB | ε

S ε | aSc | B B bB | ε OR S aSc | B B bB | ε

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Derivations and Parse Trees • Leftmost (rightmost) derivation:

Derivation where you always replace the leftmost (rightmost) variable.

• Can put the same information in a parse tree – But ignore the order of the derivation

• Unique leftmost (rightmost) derivation

Parse Tree

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Parse Trees For the following grammar, where Σ = {a, +, x, (, ) } construct the parse tree for the string a+axa

E E+T | T T TxF | F F ( E ) | a

E E + T T T x F F F a a a

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Special Forms of CFG’s “Chomsky Normal Form” • Noam Chomsky, linguist

who has taught at MIT for 55 years

• Every rule of form A → BC │ a or S → ε

“Greibach Normal Form” • Sheila Greibach, UCLA CS

professor • Every rule of form

A → a A1A2…An or S → ε

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Example CFG E E + E | E x E | (E) | a Consider the string a+axa. It has 2 different parse trees: E E E E E E E E E E a + a x a a + a x a

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Ambiguity The existence of 2 distinct parse trees (or 2

leftmost derivations) for the same string means the grammar G is ambiguous.

Ambiguous grammars are a problem (for

example, for compilers) In practice, we often try to rewrite the

grammar to eliminate the ambiguity. See slide 22 for unambiguous grammar for

expressions

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What are the limits of Context-Free Languages? What are their Closure Properties?

Regular

Context-Free

Not Context-Free ??

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True or False: The Context-Free Languages are closed under Union. A. True B. False C. We don’t know yet

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Pushdown Automata: Sneak Peek

NFA

STACK with operations Read top Remove top (Pop) Write top (Push)

Input

a

x

At each step, transition from state q, input a , top of stack x state r, replace x with y on top of stack ( for a, x ≠ ε ) Accept: IF Have read ALL the input, AND in final state of PDA BOTH MUST HOLD! (Don’t require the stack to be empty) 27

JFLAP Diagram of PDA for {0n1n │n > 0}

a,b; c : “When PDA is reading input a, replace symbol b on the top of the stack with symbol c” (This example was from JFLAP, but written a, b c in Sipser) a = ε means don’t read input symbol b = ε means don’t read or pop top of stack c = ε means don’t write on top of stack

Example input: 0011 010

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