BU CS 332 Theory of Computation Lecture 3: Reading: Nondeterminism - - PowerPoint PPT Presentation

BU CS 332 – Theory of Computation

Lecture 3:

• Nondeterminism
• Equivalence of NFAs and

DFAs

• More on closure properties

Reading: Sipser Ch 1.1‐1.2

Mark Bun January 29, 2020

Non Nondeterminism

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1 1 1 0,1 A Nondeterministic Finite Automaton (NFA) accepts if there is a way to make it reach an accept state.

Non Nondeterminism

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1 1 1 0,1 Example: Does this NFA accept the string 1100?

Some special transitions

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0, 1 1

Example

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1

Example

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0,1 0, 1 0,1 1

Formal Definition of a NFA

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is the set of states is the alphabet 

• is the transition function

0 

is the start state is the set of accept states

An NFA is a 5‐tuple 

accepts a string if there exists a path from

0 to

an accept state that can be followed by reading .

Example

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1 

𝟒



𝟏 𝟏, 𝟐 𝟑 𝟒 𝟓 𝟓



𝟑 𝟐 𝟑 𝟒 𝟓 𝟏

Example

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0,1 0, 1 0,1 1

𝟐 𝟑 𝟒 𝟏



𝟏 𝟏, 𝟐 𝟑 𝟒 𝟒



𝟏



𝟑



𝟏



𝟐

Nondeterminism

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Ways to think about nondeterminism

• (restricted)

parallel computation

• tree of possible

computations

• guessing and

verifying the “right” choice Deterministic Computation Nondeterministic Computation accept or reject accept reject

Why study NFAs?

• Not really a realistic model of computation: Real

computing devices can’t really try many possibilities in parallel But:

• Useful tool for understanding power of DFAs/regular

languages

• NFAs can be simpler than DFAs
• Lets us study “nondeterminism” as a resource

(cf. P vs. NP)

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NFAs can be simpler than DFAs

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An NFA that recognizes the language : 1 1 0,1 0,1 A DFA that recognizes the language :

Sometimes DFAs mu must st be larger

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• Theorem. Every DFA for the language

must have at least 3 states. Proof:

Equivalence of NFAs and DFAs

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Equivalence of NFAs and DFAs

Every DFA is an NFA, so NFAs are at least as powerful as DFAs Theorem: For every NFA , there is a DFA such that Corollary: A language is regular if and only if it is recognized by an NFA

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Equivalence of NFAs and DFAs (Proof)

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Let  be an NFA Intuition: Run all threads of in parallel, maintaining the set of states where all threads are.

accept reject

Goal: Construct DFA  

0

 recognizing Formally: “The Subset Construction”

NFA ‐> DFA Example

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1 a b

Subset Construction (Formally)

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  for all and .     

0

 Input: NFA Output: DFA  

0



NFA ‐> DFA Example

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0,1 ε 2 3 1 1

Subset Construction (Formally)

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 

∈

for all and .     

0

   contains some accept state of Input: NFA Output: DFA  

0



Proving the Construction Works

Claim: For every string , running

• n

leads to state There exists a computation

• f
• n input

ending at Proof idea: By induction on

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Historical Note

Subset Construction introduced in Rabin & Scott’s 1959 paper “Finite Automata and their Decision Problems”

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1976 ACM Turing Award citation For their joint paper "Finite Automata and Their Decision Problem," which introduced the idea of nondeterministic machines, which has proved to be an enormously valuable concept. Their (Scott & Rabin) classic paper has been a continuous source

• f inspiration for subsequent work in this

field.

Is this construction the best we can do?

Subset construction converts an state NFA into a ‐state DFA Could there be a construction that always produces, say, an

‐state DFA?

Theorem: For every , there is a language such that

• 1. There is an

‐state NFA recognizing .

• 2. There is no DFA recognizing with fewer than

states. Conclusion: For finite automata, nondeterminism provides an exponential savings over determinism (in the worst case).

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