meljun cortes automata theory 21

8/13/2019 MELJUN CORTES Automata Theory 21

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CSC 3130: Automata theory and formal languages

Polynomial time

Fall 2008MELJUN P. CORTES MBA MPA BSCS ACS

MELJUN CORTES


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Efficient algorithms

The running time of analgorithm depends onthe input

For longer inputs, weallow more time

Efficiency is measuredas a function of inputsize

A TMPCP


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Examples of running time

parsingproblem

running time

0n 1n

algorithm LR(1)

O( n ) O( n ) O( n log n )

short paths

Dijkstra

matching

Edmonds

O( n 3 )

CYK

O( n 2 )

n = input size

running time

problem routing

2O( n )

scheduling

2O( n log n ) 2O( n )

theorem proving


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Input representation

Since we measure efficiency in terms of inputsize , how the input is represented will make adifference

For us, any reasonable representation will beokayThe number 17

17

10001 (17 in base two)11111111111111111

OKOKNO

This graph

0000,0010,0001,00101 2

3 4OK

(2,3),(3,4) OK


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Measuring running time

What does it mean when we say:

One step in

all mean different things!

This algorithm runs in 1000 steps

java RAM machine Turing Machineif (x > 0)

y = 5*y + x;write r3; d (q 3, a) = (q 7, b, R)


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Example

L = {0 n 1n : n > 0}

in java:

M(string x) {n = x.len;if n % 2 == 0 reject;else

for (i = 0; i


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Efficiency and the Church-Turing thesis

The Church-Turing thesis says all these modelsare equivalent in power

but not in running time!

java

RAM machine

Turing Machine

multitape TM

UNIVAC


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The Cobham-Edmonds thesis

However, there is an extension to the Church-Turing thesis that says

For any realistic models of computation M 1 and

M 2:

So any task that takes time T on M 1 can be donein time (say) T 2 or T 3 on M 2

M 1 can be simulated on M 2 with at mostpolynomial slowdown


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Efficient simulation

The running time of a program depends on themodel of computation

but in the grand scheme, this is irrelevant

javaRAM machinemultitape TMordinary TMfastslow

Every reasonable model of computation can besimulated efficiently on every other


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Example of efficient simulation

Recall simulating multiple tapes on a single tape

M 0 1 0

0 1

1 0 0

G = {0, 1, }

S 0 1 0 10 # # 0 #1 0

G = {0, 1, , 0, 1, , #}

#


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Running time of simulation

Each move of the multiple tape TM might requiretraversing the whole single tape

after t steps

O( s ) steps of single tape TM

s = rightmost cell ever visited

s 3 t + 4

1 step of 3-tape TM

t steps of 3-tape O( ts ) = O ( t 2 ) single tape steps

multi-tape TM single tape TMquadraticslowdown


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Simulation slowdown

Cobham-Edmonds Thesis:

multi-tapeTM java

single tape TMRAM machine

O( t ) O( t )

O( t 2 )

O( t 2 )

O( t )O( t )

M 1 can be simulated on M 2 with at mostpolynomial slowdown


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Running time of nondeterministic TM

What about nondeterministic TMs?

For ordinary TMs, the running time of M on inputx is the number of transitions M makes before ithalts

But a nondeterministic TM can run for a differenttime on different computation paths


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Example

Definition of running time for nondeterministic TM

q acc

q 0 1/1R

0/0Rq 1

10001

what is the running time?

q rej

running time =

computation path: any possible sequence of transitions

max length of any computation path

5


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Simulation of nondeterministic TM

nondet TMmulti-tape

TM

1 00

1 00

2 21

input tape x

1

simulation tape z

address tape aFor all k > 0For all possible strings a of length kCopy x to z.Simulate N on input z using a as choices

If a specifies an invalid choice orsimulation loops/rejects, abandon simulation.

If N enters its accept state, accept and halt.If N rejected on all a s of length k, reject and halt.

representspossible choicesat each step

each a describesa possiblecomputation

path

N M


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Simulation slowdown fornondeterminism

For all k > 0For all possible strings a of length kCopy x to z.Simulate N on input z using a as choices

If a specifies an invalid choice or

simulation loops/rejects, abandonsimulation.If N enters its accept state, accept and

halt.If N rejected on all a s of length k, reject and

halt.simulation will halt when k = trunning time of N is t

running time of simulation = ( running time for specific a ) ( number of a s of length t )

= O( t ) 2O( t ) = 2 O( t )


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Simulation slowdown

multi-tapeTM java

single tape TMRAM machine

O( t ) O( t )O( t 2 )

O( t 2 )O( t )

O( t )

nondeterministic TM

2O( t )

Do nondeterministic TM violate the Cobham-Edmonds thesis


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Nondeterminism and the CE thesis

Cobham-Edmonds Thesis says:

But is nondetermistic computation realistic?

Any two realistic models of computationcan be simulated with polynomial slowdown


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Example

Recall the scheduling problem

Scheduling with nondeterminism:

CSC 3230 CSC 2110

CSC 3160CSC 3130

Can you schedule final examsso that there are no conflicts ?

Exams vertices

Slots colors

Conflicts edges

Y R B

schedule(int n, Edges edges) { for i := 1 to n:

choose { c[i] := Y; }or { c[i] := R; }

or { c[i] := B; }for all e in edges:

if c[e.left] == c[e.right]reject;

accept;}


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Nondeterministic simulation

If we can do better, this would improve all known combinatorial optimization algorithms!

nondeterministic TM multi-tapeTM

2O( t ) slowdown

Is this the best we can do?


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Millenium prize problems

Recall how in 1900, Hilbert gave 23 problemsthat guided mathematics in the 20 th century

In 2000, the Clay Mathematical Institute gave 7problems for the 21 st century

1 P versus NP2 The Hodge conjecture3 The Poincar conjecture4 The Riemann hypothesis5 Yang Mills existence and mass gap6 Navier Stokes existence andsmoothness7 The Birch and Swinnerton-Dyer

conjecture

$1,000,000

Hilberts 8 th problemPerelman 2006 (refused money)

computer science


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The P versus NP question

Among other things, this asks: Is nondeterminism a realistic feature of computation? Can the choose construct be efficiently implemented? Can we efficiently optimize any well -posed problem?

nondeterministic TM ordinary TM

Can nondeterministic TM be simulated onordinary TM with polynomial slowdown?

poly( t )

Most people think not,but nobody knows for sure!


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The class P

P is the class of all languagesthat can be decided on anordinary TM whose running

time is some polynomialin the length of the input

By the CE thesis, we can replaceordinary TM by any realistic model of computation

multi-tape TM java RAM


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Examples of languages in P

parsingproblem

running time

0n 1n

algorithm LR(1)

O( n ) O( n ) O( n log n )

short paths

Dijkstra

matching

Edmonds

O( n 3 )

CYK

O( n 2 )

n = input size

L 01 = {0 n 1n : n > 0}

L G = { x : x is generated by G}

PATH = {(G, a , b , L ): G is a graph with apath of length L from a to b }

G is some CFG

MATCH = {G, a , b , L : G is a graph with aperfect matching } context-free

P (efficient)

decidable

L 01

L GPATH MATCH


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Languages believed to be outside P

running timeof best-known algorithm

problem routing2O( n )

scheduling

2O( n ) 2O( n )thm-proving

We do not know if theseproblems have faster algorithms,but we suspect not

P (efficient)

decidable

L GPATH

MATCH

ROUTESCHED

PROVE?

To explain why, first we needto understand what theseproblems have in common


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More problems

1 2

3 4

Graph G

A clique is a subset of verticesthat are all interconnected

{1, 4}, {2, 3, 4}, {1} are cliques

An independent set is a subset ofvertices so that no pair is connected

{1, 2}, {1, 3}, {4} are independent setsthere is no independent set of size 3

A vertex cover is a set of verticesthat touches (covers) all edges

{2, 4}, {3, 4}, {1, 2, 3} are vertex covers


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Boolean formula satisfiability

A boolean formula is an expression made up ofvariables, ands, ors, and negations, like

The formula is satisfiable if one can assignvalues to the variables so the expressionevaluates to true

( x 1 x 2 ) ( x 2 x 3 x 4 ) ( x 1 )

x 1 = F x 2 = F x 3 = T x 4 = T Above formula is satisfiable because this assignmentmakes it true:


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Status of these problems

CLIQUE = {(G, k ): G is a graph with a clique of k vertices }

IS = {(G, k ): G is a graph with an independent set of k vertices }

VC = {(G, k ): G is a graph with a vertex cover of k vertices }

SAT = { f : f is a satisfiable Boolean formula }

running time

of best-known algorithm

problem CLIQUE

2O( n )

IS

2O( n )

SAT

2O( n )

VC

2O( n )

What do these problems have in common ?


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Checking solutions efficiently

We dont know how to solve them efficiently

But if someone told us the solution, we would beable to check it very quickly

1

2

3

4

5

6

78

9

10

11

12

13

14

15

Is (G, 5) in CLIQUE ?

1,5,9,12,14

Example:


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Cliques via nondeterminism

Checking solutions efficiently is equivalent todesigning efficient nondeterministic algorithms

Is (G, k ) in CLIQUE ?Example:clique(Graph G, int k) { C = {}; % potential clique

for i := 1 to G.n: % choose cliquechoose { C := union(C, {i}); }

or {}if size(C) != k reject; % check size is kfor i := 1 to G.n: % check all edges

for j := 1 to G.n: % are inif i in C and j in C

if G.isedge(i,j) == falsereject;

accept;}


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Example: Formula satisfiability

( x 1 x 2 ) ( x 2 x 3 x 4 ) ( x 1 ) f =

Checking solution: Nondeterministic algorithm:

FFTTsubstitutex 1 = F x 2 = F x 3 = T x 4 = T

evaluate formula

( F T ) ( F T F ) ( T ) f =can be done in linear time

sat(Formula f) { x = new bool[f.n];for i := 1 to n:

choose { x[i] := true; }or { x[i] := false; }

if f.eval(x) == true accept;else reject;

}


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The class NP

The class NP :

L can be solved on a nondeterministic TM inpolynomial time iff its solutions can bechecked intime polynomial in the input length

NP is the class of all languages that can bedecided on a nondeterministic TM whoserunning time is some polynomial in the lengthof the input


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P versus NP

because an ordinaryTM is only weaker

than anondeterministic one

Conceptually, finding solutions can only beharder than checking them

P (efficient)

decidable

L GPATH

MATCH

CLIQUE

SAT ISNP (efficiently checkable)

VC

P is contained inNP


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P versus NP

The answer to the question

is not known. But one reason it is believed to benegative is because, intuitively, searching isharder than verifying

For example, solving homework problems(searching for solutions) is harder than grading (verifying the solution is correct)

Is P equal toNP?

$1,000,000


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Searching versus verifying

Mathematician:Given a mathematical claim, come up with a proof for it.

Scientist:Given a collection of data on some phenomena, nd a

theory explaining it.

Engineer:Given a set of constraints (on cost, physical laws, etc.)

come up with a design (of an engine, bridge, etc) whichmeets them.

Detective:Given the crime scene, nd whos done it.

meljun cortes automata theory 21

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