CSE 105 THEORY OF COMPUTATION Fall 2016 - - PowerPoint PPT Presentation

CSE 105

THEORY OF COMPUTATION

Fall 2016 http://cseweb.ucsd.edu/classes/fa16/cse105-abc/

T

• day's learning goals Sipser Ch 1.4, 2.1
• CFG Design
• PDA Design
• Equivalence between CFG and PDA
• Non-context-free languages

CFG Design

• Design a CFG for the language

–

L = {anbm | m > n ≥ 0 }

• G=({S,T},{a,b},R,S) where R is the set of rules

–

S → T | ???

–

T → bT | b What rule is missing? A) S → aS B) S → abS C) S → aSb D) S → aabbb E) I don’t know

CFG Design

• Design a CFG for the language

–

L = {a2nban | n ≥ 0 }

• G=({S,T},{a,b},R,S) where R is the set of rules

–

S → T | ???

–

T → b What rule is missing? A) S → aaSa B) S → aSba C) S → aSb D) S → aSa E) I don’t know

CFG Design

• Design a CFG for the language

–

L = {aibjak | i=j or j=k }

• G=({S,T,U,V},{a,b},R,S) where R is the set of

rules

–

S → UT | TV

–

U → aUb | ε

–

V → bVa | ε

–

T → ??? What rule is missing? A) T → aT a | b B) T → aT | a C) T → aT | ε D) T → aT | bT | ε E) I don’t know

CFG Design

• Design a CFG for the language

–

L = {aibjak | i=j or j=k }

• G=({S,T,U,V},{a,b},R,S) where R is the set of

rules

–

S → UT | TV

–

U → aUb | ε

–

V → bVa | ε

–

T → aT | ε Is this grammar ambiguous? A) No, the grammar is unambiguous B) Yes, but it can be transformed into an equivalent unambiguous grammar C) Yes, and there is no equivalent unambiguous grammar D) I don’t know

PDA Design

• Can we design a PDA for the same language?

–

L = {aibjak | i=j or j=k }

• Can you design a deterministic PDA?

PDAs and CFGs are equivalently expressive

Theorem 2.20: A language is context-free if and

• nly if some nondeterministic PDA recognizes it.

Consequences

• Quick proof that every regular language is context free
• To prove closure of class of CFLs under a given operation, can

choose two modes of proof (via CFGs or PDAs) depending on which is easier

Intersection

• Consider the languages

–

X = {aibjak | i=j and k>1 }

–

Y = {aibjak | j=k and i>1 }

–

W = X U Y

–

Z = X ∩ Y = {aibjak | i = j = k >1} Which languages are context-free? A) X,Y B) X,Y,W C) X,Y,W,Z (all) D) None E) I don’t know

Closure properties of ..

The class of regular languages is closed under

• Union
• Concatenation
• Star
• Complementation
• Intersection
• Difgerence
• Reversal

The class of context-free languages is closed under

• Union
• Concatenation
• Star
• Reversal

The class CFL is not closed under

• Intersection
• Complement
• Difgerence

Non-context-free languages

• {aibjak | i= j =k }
• {aibjak | i<j<k }
• { ww | w in {a,b}*}
• Proof: Pumping Lemma for Context-Free

languages

–

See textbook if interested

–

We will not cover this in homework or tests

Context-free languages Context-free languages Regular languages Regular languages ??? ???

Beyond Push Down Automata

• Can you defjne a more powerful automaton that

is able to recognize L3={aibjak | i=j=k }?

• Can you recognize L3 with a 2PDA (Push Down

Automaton with 2 stacks)?

• Can you recognize L4={aibjakbh | i=j=k=h } with

a 3PDA (Push Down Automaton with 3 stacks?

• Can you recognize L4 with a 2PDA

T uring machines

• Unlimited input
• Unlimited (read/write)

memory

• Unlimited time

T uring machine computation

• Read/write head starts at leftmost position on tape
• Input string written on leftmost squares of tape, rest is

blank

• Computation proceeds according to transition function:
• Given current state of machine, and current symbol being read
• the machine
• transitions to new state
• writes a symbol to its current position (overwriting existing symbol)
• moves the tape head L or R
• Computation ends if and when it enters either the accept
• r the reject state.

Formal defjnition of TM

qreject ≠ qaccept qreject ≠ qaccept

Reminders

• Finish reading Sipser Chapter 2. Start Chapter 3
• HW5 due Monday Oct 31
• Review for Exam 2
• T

uesday Nov 1 TBA

• Exam 2 Wednesday Nov 2 (one week from today)
• New seating chart: see Piazza (tonight)
• New study guide: see Piazza (tonight)
• Haskell 3 on CFG is out. Due Nov 7.