CSCI-2200FOUNDATIONS OF COMPUTER SCIENCE

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Announcements• Homework 7 is due April 12, 2019 at 11:59pm.

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Warm UpConstruct a DFA that accepts all strings with an even number of 0s and an odd number of 1s. The alphabet is {0,1}.

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Last Time - NFAs• NFA generalize FA by adding nondeterminism• Allows several alternative computations for the same input string.• Alternative computations execute in parallel on different copies of the

machine.• Three changes:• Allow δ(q, a) to specify more than one successor state

• Add λ-transitions: transitions made for “free” without consuming any input.

• Do not need to specify a transition for every symbol in the alphabet

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a

a

qiqj

qk

λqi qj

Example NFA

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Theorem:

Languages acceptedby NFAs

RegularLanguages=

NFAs and FAs have the same computational power

Languages acceptedby FAs

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Languages acceptedby NFAs

RegularLanguages

Languages acceptedby NFAs

RegularLanguages

We can show:

⊆

⊇

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Languages acceptedby NFAs

RegularLanguages

Proof-Step 1

Proof: Every DFA is trivially an NFA

Any language accepted by a DFAis also accepted by an NFA

L

⊇

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Languages acceptedby NFAs

RegularLanguages

Proof-Step 2

Proof: Any NFA can be converted to anequivalent DFA

Any language accepted by an NFAis also accepted by a DFA

L

⊆

CONTEXT-FREE LANGUAGES

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Regular Languages

}{ nnba }{ RwwContext-Free Languages

Context-Free Languages• A context free language is a language that can be

recognized by a new type of computational model –a pushdown automaton

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stack

automaton

Inputstring Output

accept or reject

Pushdown Automata (PDA)• Similar to an FA, but with different transition specification

• If a is on input tape and b is on top of stack then:• 1. read a from tape• 2. pop b from stack• 3. push c on stack

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!, # → %qi qj

Pushdown Automata• Similar to an NFA, a PDA does not need a transition for

every symbol in the tape alphabet.• If there is no transition, the machine hangs and the

string is rejected.

• A string is accepted by a PDA if the machine is an accept state and all input has been consumed.

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Context Free Grammars

• A grammar gives rules for expressing a language.• A context-free language can be described by a context-free grammar• Theorem: A language is context-free if and only if some

(nondeterministic) pushdown automaton recognizes it.

• Context-free grammars can be used to describe:• Programming languages• Used by compilers

• The English language• Used in Natural Language Processing, Grammar checkers

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Example of a Context-Free Grammar17

sentence → noun_ phrase predicate

noun_ phrase → article noun

predicate → verb

article → a

article → the

noun → cat

noun → dog

verb → runs

verb →walks

terminal

variable

start variable

production rule

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sentence → noun_ phrase predicate

noun_ phrase → article noun

predicate → verb

article → a

article → the

noun → cat

noun → dog

verb → runs

verb →walks

Formal definition of a CFG

• A context-free grammar (CFG) is a 4-tuple G = (V,Σ,R,S) where• V is a finite set called the variables• Σ is a finite set, disjoint from V, called the terminals• R is a finite set of productions (rules)• S ∈ V is the start variable

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l®®

SaSbS Variables are capital letters.

Terminals are lower-case letters

Language of a Grammar• The sequence of substitutions used to obtain a particular string

is called a derivation.• We write

• The language of a grammar, denoted L(G), is the set of all strings generated by derivations.

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l®®

SaSbS

S ⇒ w*

Another Example Grammar21

l®®®

AaAbAAbS

Convenient Notation• We can combine rules that have the same variable on the

left-hand side.• The “|” symbol acts as an “or”

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Designing a CFGDesign a context-free grammar G such that

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=)(GL }*},{:{ bawwwR Î