CS 3261 Computability Course Summary

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CS 3261 Computability Course Summary

Zeph Grunschlag

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Announcements

Last hw due nowLook out for a final exam practice problems coming out over the weekend I will hold final review session on Tuesday 12/11, 3-5 pm, 833 Mudd. Pick-up final hw’s. I will hold daily OH’s next week and Monday 12/17, 12:00-1:30 except Thursday, 12/13 Final exam: Tuesday 12/18, 9-12, 833 Mudd

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Computability ConceptsAIM: Reduce Computer Science to

its bare theoretical essentials.

APPROACH: Algorithmic Problems

Formal Languages

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Formal LanguagesFundamental insight of Theoretical CS:By understanding how formal languages can

be computed, will understand how any algorithmic problem can be solved.

Algorithmic input/output problems involve creating procedures for procuring outputs from given inputs. Can be turned into a formal languages by re-writing as yes/no questions.

EG: “Find the shortest path…” becomes“Is there a path shorter than…”

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Computability ConceptsAIM: Reduce Computer Science to its bare

theoretical essentials.Algorithmic Problems Formal LanguagesComputers Graph based machine models

Questions to investigate: 1) What sorts of problems can be solved by each

computer model? 2) What languages does each model accept?3) What are the practical limits on what a computer

can do?

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Abstract Machine ModelsDFA’s

DFA’s model computers with strictly bounded memory.

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3

2

a

b

b

a

a,b

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Abstract Machine ModelsDFA’s

Q: What’s the accepted language?

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3

2

a

b

b

a

a,b

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Abstract Machine ModelsDFA’s

A: a*b+

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3

2

a

b

b

a

a,b

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Abstract Machine ModelsNFA’s

Nondeterminism is a powerful concept. Often 1st view of a problem is nondeterministic.

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3

2

a

a

b

a

a,b

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Abstract Machine ModelsNFA’s

Q: What’s the accepted language?

1

3

2

a

a

b

a

a,b

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Abstract Machine ModelsNFA’s

A: a+b*

1

3

2

a

a

b

a

a,b

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Abstract Machine ModelsPDA’s

By allowing a pushdown stack, increase flexibility and accept more languages.

1 2

3

a,X

b,X

0

$ $

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Abstract Machine ModelsPDA’s

Q: What’s the accepted language?

1 2

3

a,X

b,X

0

$ $

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Abstract Machine ModelsPDA’s

A: {an bn | n 0}

1 2

3

a,X

b,X

0

$ $

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Abstract Machine ModelsTM’s

By allowing a read-write tape, amazingly get most general possible computer model!

1 2

XR

0

1$,RL acc

34XL

$L

1L

$R

1X,R

1R

XR

XL

5 1L

1|XL

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Abstract Machine ModelsTM’s

Q: What’s the accepted language?

1 2

XR

0

1$,RL acc

34XL

$L

1L

$R

1X,R

1R

XR

XL

5 1L

1|XL

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Abstract Machine ModelsTM’s

A: Unary powers of 2.

1 2

XR

0

1$,RL acc

34XL

$L

1L

$R

1X,R

1R

XR

XL

5 1L

1|XL

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I/O VersionsEach class of languages has its own I/O

version. Regular: Finite State Transducers More powerful models exist (e.g. with

’s)

Context free: (didn’t study any) “Compilers”: Input is a string, output is

a parse-tree (or even executable code)

Turing Machines: I/O TM’s

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Robust Formal Language Classes

Turns out these models are very robust Many equivalent ways to generate same classes: Regular languages

FA’s, NFA’s, Regular Expressions, Right-Linear Grammars Context Free Languages

PDA’s, Context Free Grammars Recognizable languages –Church-Turing thesis

TM’s, k-tape machines, k-track machines NTM’s, Queue Machines, 2-Stack PDA’s, RAM’s, Unrestricted Grammars

Complexity classes P and NP For NP: Poly. NTM’s, Poly. Verifiers, Poly. Proofs

We learned algorithms for converting between most of the different views

Language classes closed under natural operations.

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System DesignOften computer system creation involves designing

a formal language to describe system communication. Components receive communication streams and have to effect actions based on these. Computability theory can help drive design at a high level.

EG: Might come up with a communication stream that’s seems like regular language. Could then show that it isn’t using pumping lemma. With this knowledge, final design tweaks original to obtain a regular language and therefore DFA based ultra-fast and super-reliable system components!

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Negative Examples

As above, it is very important to be able to tell when particular languages cannot be accepted by a certain model of computation. We have several tools at our disposal: Irregularity: pumping lemma (PL) Non-Context-Freeness: CFPL Undecidability: Reductions from undecidable languages Intractability: Poly-time reduction from NP-hard languages

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System Design

Learned useful concepts that can help modularization when designing systems. Often can express a language as a union, intersection, negation, concatenation or Kleene-* of simpler languages. More complex language may be put together by using simple components along with “off the shelf” reconstruction techniques:

Language Design

Class Negate Concat.

Kleene-*

DFA’sCartesianProduct

Accept Non-

accepts

NFA’s ParallelCartesia

n Product

Serial Loop

PDA’s Parallel Serial Loop

Deciders Run in parallel

Accept Non-

accepts

Break string

up

Recursive

algorithm

Recognizers Run in parallel ” ”

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Language Class Hierarchy

AllREC = accepted by TM

DEC = decided by TMContext Free

Deterministic Context Free

Regular = accepted by FA’s

Finite languages

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KnownComplexity Hierarchy

Get the following RAM hierarchy diagram: REC

DECP

TIME(n)

CFLTIME(n3)

REG

EQREX

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UnknownComplexity Hierarchy

DecidableNP

NP but not NP-hard

P

Finite languages

Does anythingexist here?

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Conjectured HierarchyInside of DEC most conjecture:

DEC

NP P co-NPNPcomplete

PRIMESAT

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Follow-ups to Computability

Related Electives: Analysis of Algorithms 4231 (fall) Computational Complexity 4236 (spring) Cryptography –generic course no. 4995

(this Spring with Michael Rabin!!!)

Courses Requiring Computability: Programming Languages and

Translators 4115 (every semester) Compilers 4117 (this Spring with Al

Aho!!!) Portions of several other courses

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Final Remarks

With the horrors at the beginning of the semester….

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Final Remarks

Thanks for putting in the effort and

helping make this my best semester

thus far!