Decidability Human-aware Roboticsyzhan442/teaching/CSE355/... · abstract machines •...

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Decidability

• 2017/10/31

• Chapter 4.1 in Sipser

Ø Announcement:q Slides for this lecture are here:

http://www.public.asu.edu/~yzhan442/teaching/CSE355/Lectures/decidability.pdf

q Happy Hollaween!

q Delayed grading for

o HW4 group part

o Project #1

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Performance analysis• Overall average: 78.5% out of a total of 26.5 graded

• Midterm 1 average: 59.3%

• HW average: 86.2%

• Quiz average: 96.0%

78.5% x 108 = 84.8

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Performance analysis

>81 >92 >102

28

15

6

13

6

22

>71>61

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Last chapter

FA (DFA & NFA) expressRegular Expressions (RE)CFL

Recursively enumerable

• The church-turing thesiso Turing machines

q Definition of TM

q Computation of TM

q Design of TM

o Variants of TMs

o Multitape TM

o Nondeterminstic TM

o Enumerators

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Theory of computation

• Automata Theoryabstract machines

• Computability Theoryfundamental capabilities

and limitations of abstract

machines

• Complexity Theorywhy certain problems are

harder than others

Computation

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What are algorithms?Ø A process that can be determined in finite number of operations. Why

finite?

• Alonzo Church’s lambda-calculus

• Alan Turing’s machines (Turing machine)

MAccept

Reject

lambda-calculus

decider

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What are algorithms?Ø A process that can be determined in finite number of operations. Why

finite?

• Alonzo Church’s lambda-calculus

• Alan Turing’s machines (Turing machine)

MAccept

Reject

lambda-calculus

These two

representations

are equivalent!

decider

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machines


harder than others

Computation

decider

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machines


harder than others

Computation

decider

Hilbert’s tenth problem

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Why do we care?

Computation

decider


Ø Identify problems that are too hardo change the problem

Ø Understand limitations of computers

D

D1

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Outline for today• Decidability

o Regular languages

q Acceptance of DFA

q Acceptance of NFA

q Generation of RE

q Test emptiness of RL

q Test equivalence of DFAs

o Context-free languages

q Acceptance of CFG

q Test emptiness of CFG

Ø Goals:

o Learn about several solvable problems about RLs

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Acceptance of DFA• Does a DFA accept a given string (the acceptance problem)?

•

• problem is solvable algorithmically == the corresponding language is decidable

MAccept

Reject

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• Does a DFA accept a given string (the acceptance problem)?

Acceptance of DFA

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Implementationlevel?

MAccept

Reject

Acceptance of DFA

(Q,⌃, �, q0, F )

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Encoding of B is • Check whether input string is a valid finite automaton

Simulate B on w• TM keeps track of the current state of B and current position on w. • Updates are done through δ• When having processed the last input symbol, check whether the state is

accepting.

Acceptance of DFA

(Q,⌃, �, q0, F )

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o Regular languages

q Acceptance of DFA

q Acceptance of NFA

q Generation of RE




q Acceptance of CFG


Ø Goals:


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• Does a NFA accept a given string?

•

• M on input <B, w>:

• Simulate B nondeterminstically using NTM

• If the simulation accepts, accept; else, reject.

Acceptance of NFA

Incorrect since the machine may not stop

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•

•

Acceptance of NFA

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•

• <B,w>

MAccept

Reject

N

Acceptance of NFA

<C,w>ConvertNFAtoDFA

ConvertingNFAtoDFAmustbedoneinfinitestepsalways!

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o Regular languages

q Acceptance of DFA

q Acceptance of NFA

q Generation of RE




q Acceptance of CFG


Ø Goals:


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Generation of RE• Does an RE generate a string w?

•

• M on input <R, w>:

1. Try a possible instantiation of the regular operations so that the

strings are in shortlex order (how do we do this?)

2. Compare with w. If match, accept; else, continue to 1.

3. If the length of string generated is longer than w, reject

Works but clumsy

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• Does an RE generate a string w?

•

• M on input <B, w>:

• Simulate B nondeterminstically

• If the simulation accepts, accept; else, reject.

Generation of RE

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o Regular languages

q Acceptance of DFA

q Acceptance of NFA

q Generation of RE




q Acceptance of CFG


Ø Goals:


• Reading assignment for the

next class:

o Sipser Sec. 4.1 - Quiz

link will be sent out;

due date is before the

beginning of the next

class

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Last time• Decidability

o Regular languages

q Acceptance of DFA

q Acceptance of NFA

q Generation of RE




q Acceptance of CFG


Ø Goals:


MAccept

Reject

lambda-calculus

decider

solvability == decider

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o Regular languages

q Acceptance of DFA

q Acceptance of NFA

q Generation of RE




q Acceptance of CFG


Ø Goals:

o Learn about several solvable problems

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Test emptiness of RL• Is the language of a DFA empty?

•

• M on input <A>:

• For each w in Σ* in shorlex order:

• Simulate A with w as the input

• If the simulation accepts, reject

• Accept

Incorrect since the machine would not stop

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• Is the language of a DFA empty?

•

• M on input <A>:

• For each w order in shorlex order:

• Simulate A with w in

• If the simulation accepts, reject

• Accept.

Test emptiness of RL

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o Regular languages

q Acceptance of DFA

q Acceptance of NFA

q Generation of RE




q Acceptance of CFG


Ø Goals:


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• Are the languages of DFA A and B the same?

•

• M on input <A, B>:

• Keep track of the current states of A and B

• For each state in the possible current states for A (unchecked), collect all

possible next states from the state

• If there is no a matched state (unchecked) in B’s current states (based

on its transition structure), reject; otherwise, mark the state in A and

the matched state in B as checked and continue

• Accept if no more states can be checked

Test equivalence of RLs

Incorrect since A and B can have different structures!

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•

•

L(A) = L(B) if and only if L(C) is empty


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•


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o Regular languages

q Acceptance of DFA

q Acceptance of NFA

q Generation of RE




q Acceptance of CFG


Ø Goals:


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• Can a string w be generated by a CFG?

•

• How to contruct a Turing machine to decide ACFG?

Acceptance of CFG

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•

Acceptance of CFG

What about APDA?

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•

Acceptance of CFG

What about APDA?

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o Regular languages

q Acceptance of DFA

q Acceptance of NFA

q Generation of RE




q Acceptance of CFG


Ø Goals:


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• Is the language generated by a CFG empty?

•

• How to construct a Turing machine for ECFG

Acceptance of CFG

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• Is the language generated by a CFG empty?

•

Acceptance of CFG

What about EQCFG?

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Put RL and CFL in scope

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Undecidability

• 2017/11/2

• Chapter 4.2 in Sipser

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Undecidability

Computation

decider


Recognizable but undecidable languages

Computation

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Undecidability

Computation

decider


Unrecognizable languages?

Computation

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Undecidability

Computation

decider


Unrecognizable languages? complement of D

Computation

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Undecidability


Unrecognizable languages? complement of D

Computation

Ø A language is co-Turing-recognizable if it is the complement of a Turing-recognizable language.

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Non Turing-recognizable

Claim: There are an infinite number of languages that are not Turing recognizable

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Countably infinite• {1, 2, 3, …} and {2, 4, 6, …}, which one has a bigger size?

one-to-one

onto

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Diagonalization• {1, 2, 3, …} and {2, 4, 6, …}, which one has a bigger size?

• Rational numbers is also countable (countably infinite) m/n

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Diagonalization• {1, 2, 3, …} and {2, 4, 6, …}, which one has a bigger size?

• Rational numbers is also countable (countably infinite) m/n

Is every infinite setcountable?

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Uncountable

Ø If such a mapping exists,For any x in R, there must exist a n such that f(n) = x

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Uncountable


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Uncountable

x != f(n) for all n


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Uncountable

x != f(n) for all n


why doesn’t this work for rational numbers?

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Ø Σ* is countable – so is the number of Turing machines!

Ø The set (power set) of languages for Σ* is uncountable


why?


In fact, it is uncountable!


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Ø Σ* is countable – so is the number of Turing machines!

Ø The set (power set) of languages for Σ* is uncountable


why?


In fact, it is uncountable!


Ø Way more problems cannot be solved (or even recognized) by TMs than those that can!

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any other problems?

Undecidable languages

Let us now return to reality

Ø Findingthemcanhelpusdetermineotherproblems

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?

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is undecidable!

<M>M

Accept

Reject|Loop

D

<M,w>H Accept

Reject

wReject

Accept


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<M>M

Accept

Reject|Loop

D

<M,<M>>H Accept

Reject

<M>Reject

Accept

Undecidable languagesis undecidable!

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<D>D

Accept

Reject|Loop

D

<D,<D>>H Accept

Reject

<D>Reject

Accept


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<D>D

Accept

Reject|Loop

D

<D,<D>>H Accept

Reject

<D>Reject

Accept


Noticeanyissues?Checkouttherecursiontheorem6.1

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M on w where w = <Mi> H on <Mi, w> where w = <Mj>

D on <M>


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o Regular languages

q Acceptance of DFA

q Acceptance of NFA

q Generation of RE




q Acceptance of CFG


• UndecidabilityØ Goals:


o Learn about undecidability and non turing recognizable languages

• Reading assignment for the

next class:

o Sipser Sec. 6.1 - Quiz

link will be sent out;

due date is before the

beginning of the next

class

o Second (and last

project) based on this

o HW6

Decidability Human-aware Roboticsyzhan442/teaching/CSE355/... · abstract machines •...

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