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CS21 Decidability and Tractability. Lecture 27 March 10, 2010. Outline. “Challenges to the (extended) Church-Turing Thesis” quantum computation. Extended Church-Turing Thesis. the belief that TMs formalize our intuitive notion of an efficient algorithm is:

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cs21 decidability and tractability

CS21 Decidability and Tractability

Lecture 27

March 10, 2010

CS21 Lecture 27

outline
Outline
  • “Challenges to the (extended) Church-Turing Thesis”
    • quantum computation

CS21 Lecture 27

extended church turing thesis
Extended Church-Turing Thesis
  • the belief that TMs formalize our intuitive notion of an efficient algorithm is:
  • quantum computationchallenges this belief

The “extended” Church-Turing Thesis

everything we can compute in time t(n) on a physical computer can be computed on a Turing Machine in time t(n)O(1) (polynomial slowdown)

CS21 Lecture 27

for use later
For use later…
  • Fourier transform:

time domain

frequency domain

can recover r from position

r

time domain

frequency domain

CS21 Lecture 27

a different model
A different model
  • infinite tape of a Turing Machine is an idealized model of computer
  • real computer is a Finite Automaton (!)
    • n bits of memory
    • 2n states

CS21 Lecture 27

model of deterministic computation
Model of deterministic computation

2n possible basic states

state at time t+1

state at time t

one 1 per column

CS21 Lecture 27

model of randomized computation
Model of randomized computation

possible states at time t:

pi = 1 piR+

state at time t+1

state at time t

“stochastic matrix ” sum in each column = 1

CS21 Lecture 27

model of randomized computation8
Model of randomized computation
  • at end of computation, see specific state
  • demand correct result with high probability
  • think of as “measuring” system:

see ith basic state with probability pi

CS21 Lecture 27

model of quantum computation
Model of quantum computation

possible states at time t:

|ci|2 = 1 ciC

state at time t+1

state at time t

“unitary matrix ” preserves L2 norm

CS21 Lecture 27

model of quantum computation10
Model of quantum computation
  • at end of computation, see specific state
  • think of as “measuring” system:

see ith basic state with probability |ci|2

CS21 Lecture 27

one quantum register
One quantum register
  • register with n qubits; shorthand for basic states

shorthand for general state

CS21 Lecture 27

two quantum registers
Two quantum registers
  • registers with n, m qubits: shorthand for 2n+m basic states:

CS21 Lecture 27

two quantum registers13
Two quantum registers

shorthand for general pure state

  • shorthand for any other state (entangled state)
  • |a = i,j ai,j|i|j
  • example:

CS21 Lecture 27

partial measurement
Partial measurement
  • general state:
  • if measure just the 2nd register, see state |j in 2nd register with probability
  • state collapses to:

normalization constant

CS21 Lecture 27

epr paradox
EPR paradox
  • register 1 in LA, register 2 sent to NYC
  • measure register 2
    • probability ½: see |0, state collapses to |0|0
    • probability ½: see |1, state collapses to |1|1
  • measure register 1
    • guaranteed to be same as observed in NYC
    • instantaneous “communication”

CS21 Lecture 27

quantum complexity
Quantum complexity
  • classical computation of function f
  • some functions are easy, some hard
  • need to measure “complexity” of Mf

xth position

f(x)th position

Mf = transition matrix for f

CS21 Lecture 27

quantum complexity17
Quantum complexity
  • one measure: complexity of f =

length of shortest sequence of local operations computing f

  • example local operation:

position x = 0010

logical OR

position x’ = 1010

CS21 Lecture 27

quantum complexity18
Quantum complexity
  • analogous notion of “local operation” for quantum systems
  • in each step
    • split qubits into register of 1 or 2, and rest
    • operate only on small register
  • “efficient” in both settings: # local operations polynomial in # bits n

CS21 Lecture 27

efficiently quantum computable functions
Efficiently quantum computable functions
  • For every f:{0,1}n {0,1}m that is efficiently computable classically
  • the unitary transform Uf:
  • note, when 2nd register = |0:

CS21 Lecture 27

efficiently quantum computable functions20
Efficiently quantum computable functions
  • Fourier Transform
    • N=2n;  such that N = 1; unitary matrixFT =
    • usual FT dimension n; this is dimension N
    • note: FT  |0 = all ones vector

CS21 Lecture 27

shor s factoring algorithm
Shor’s factoring algorithm
  • well-known: factoring equivalent to order finding
    • input: y, N
    • output : smallest r>0 such that

yr = 1 mod N

CS21 Lecture 27

factoring step 1
Factoring: step 1

input: y, N

  • start state: |0|0
  • apply FT on register 1: ( |i)  |0
  • apply Uf for function f(i) = yi mod N

“quantum parallelization”

CS21 Lecture 27

factoring step 123
Factoring: step 1
  • given y, N; f(i) = yi mod N; have

in each vector, period = r, the order of y mod N

offset depends on 2nd register

CS21 Lecture 27

factoring step 2
Factoring: step 2
  • measure register 2
  • state collapses to:

Key: period = r (the number we are seeking)

CS21 Lecture 27

factoring step 3
Factoring: step 3
  • Apply FT to register 1

large in positions b such that rb close to N

  • measure register 1
  • obtain b
  • determine r from b (classically, basic number theory)

CS21 Lecture 27

quantum computation
Quantum computation
  • if can build quantum computers, they will be capable of factoring in polynomial time
    • big “if”
  • do not believe factoring possible in polynomial time classically
    • but factoring in P if P = NP
  • serious challenge to extended Church-Turing Thesis

CS21 Lecture 27

course summary continued
Course Summary continued…

(Skipped in class)

CS21 Lecture 27

summary
Summary

Part II: Turing Machines and decidability

CS21 Lecture 27

turing machines
Turing Machines
  • New capabilities:
    • infinite tape
    • can read OR write to tape
    • read/write head can move left and right

input tape

1

1

0

0

1

1

0

0

0

0

1

1

finite control

read/write head

q0

CS21 Lecture 27

deciding and recognizing
Deciding and Recognizing
  • accept
  • reject
  • loop forever
  • TM M:
    • L(M) is the language it recognizes
    • if M rejects every x  L(M) it decides L
    • set of languages recognized by some TM is called Turing-recognizable or recursively enumerable (RE)
    • set of languages decided by some TM is called Turing-decidable or decidable or recursive

input

machine

CS21 Lecture 27

church turing thesis
Church-Turing Thesis
  • the belief that TMs formalize our intuitive notion of an algorithm is:
  • Note: this is a belief, not a theorem.

The Church-Turing Thesis

everything we can compute on a physical computer

can be computed on a Turing Machine

CS21 Lecture 27

the halting problem
The Halting Problem

box (M, x): does M halt on x?

inputs

Y

Turing Machines

n

Y

The existence of H which tells us yes/no for each box allows us to construct a TM H’ that cannot be in the table.

n

n

Y

n

H’ :

n

Y

n

Y

Y

n

Y

CS21 Lecture 27

decidable re core
Decidable, RE, coRE…

co-HALT

some language

some problems (e.g HALT) have no algorithms

{anbn : n ≥ 0 }

co-RE

decidable

all languages

regular languages

context free languages

RE

{anbncn : n ≥ 0 }

HALT

CS21 Lecture 27

definition of reduction
Definition of reduction
  • More refined notion of reduction:
    • “many-one” reduction (commonly)
    • “mapping” reduction (book)

A

B

f

yes

yes

reduction from language A to language B

f

no

no

CS21 Lecture 27

using reductions
Using reductions
  • Used reductions to prove lots of problems were:
    • undecidable (reduce from undecidable)
    • non-RE (reduce from non-RE)
      • or show undecidable, and coRE
    • non-coRE (reduce from non-coRE)
      • or show undecidable, and RE

Rice’s Theorem: Every nontrivial TM property is undecidable.

CS21 Lecture 27

the recursion theorem
The Recursion Theorem

Theorem: Let T be a TM that computes fn:

t: Σ* x Σ* → Σ*

There is a TM R that computes the fn:

r: Σ* → Σ*

defined as r(w) = t(w, <R>).

  • In the course of computation, a Turing Machine can output its own description.

CS21 Lecture 27

incompleteness theorem
Incompleteness Theorem

Theorem: Peano Arithmetic is not complete.

(same holds for any reasonable proof system for number theory)

Proof outline:

  • the set of theorems of PA is RE
  • the set of true sentences (= Th(N)) is not RE

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summary38
Summary

Part III: Complexity

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complexity
Complexity
  • Complexity Theory = study of what is computationally feasible (or tractable) with limited resources:
    • running time
    • storage space
    • number of random bits
    • degree of parallelism
    • rounds of interaction
    • others…

main focus

not in this course

CS21 Lecture 27

time and space complexity
Time and Space Complexity

Definition: the time complexity of a TM M is a function f:N→ N, where f(n) is the maximum number of steps M uses on any input of length n.

Definition: the space complexity of a TM M is a function f:N→ N, where f(n) is the maximum number of tape cells M scans on any input of length n.

CS21 Lecture 27

complexity classes
Complexity Classes

Definition: TIME(t(n)) = {L : there exists a TM M that decides L in space O(t(n))}

P = k ≥ 1 TIME(nk)

EXP = k ≥ 1 TIME(2nk)

Definition: SPACE(t(n)) = {L : there exists a TM M that decides L in space O(t(n))}

PSPACE = k ≥ 1 SPACE(nk)

CS21 Lecture 27

complexity classes42
Complexity Classes

Definition: NTIME(t(n)) = {L : there exists a NTM M that decides L in time O(t(n))}

NP = k ≥ 1 NTIME(nk)

  • Theorem: P  EXP
  • P  NP  PSPACE  EXP
  • Don’t know if any of the containments are proper.

CS21 Lecture 27

alternate definition of np
Alternate definition of NP

Theorem: language L is in NP if and only if it is expressible as:

L = { x | 9y, |y| ≤ |x|k, (x, y)  R }

where R is a language in P.

CS21 Lecture 27

poly time reductions
Poly-time reductions
  • Type of reduction we will use:
    • “many-one” poly-time reduction (commonly)
    • “mapping” poly-time reduction (book)
  • f poly-time computable
  • YES maps to YES
  • NO maps to NO

A

B

f

yes

yes

f

no

no

CS21 Lecture 27

hardness and completeness
Hardness and completeness

Definition: a language L is C-hard if for every language A  C, A poly-time reduces to L; i.e., A ≤PL.

can show L is C-hard by reducing from a known C-hard problem

Definition: a language L is C-complete if L is C-hard andL  C

CS21 Lecture 27

complete problems
Complete problems
  • EXP-complete: ATMB = {<M, x, m> : M is a TM that accepts x within at most m steps}
  • PSPACE-complete: QSAT = {φ : φ is a 3-CNF, and x1x2x3…xn φ(x1, x2, … xn)}
  • NP-complete: 3SAT = {φ : φ is a satisfiable 3-CNF formula}

CS21 Lecture 27

lots of np complete problems
Lots of NP-complete problems
  • Indendent Set
  • Vertex Cover
  • Clique
  • Hamilton Path (directed and undirected)
  • Hamilton Cycle and TSP
  • Subset Sum
  • Scheduling
  • NAE3SAT
  • Max Cut
  • Problem sets: Min Bisection, subgraph isomorphism, (3,3)-SAT, Partition, Knapsack, Max2SAT…

CS21 Lecture 27

other complexity classes
Other complexity classes
  • coNP – complement of NP
    • complete problems: UNSAT, DNF TAUTOLOGY
  • NP intersect coNP
    • contains (decision version of ) FACTORING
  • PSPACE
    • complete problems: QSAT, GEOGRAPHY

CS21 Lecture 27

complexity classes49
Complexity classes

coNP

EXP

all containments believed to be proper

PSPACE

decidable languages

P

NP

CS21 Lecture 27

extended church turing thesis50
Extended Church-Turing Thesis
  • the belief that TMs formalize our intuitive notion of an efficient algorithm is:

The “extended” Church-Turing Thesis

everything we can compute in time t(n) on a physical computer can be computed on a Turing Machine in time t(n)O(1) (polynomial slowdown)

CS21 Lecture 27

challenges to the extended church turing thesis
Challenges to the Extended Church-Turing Thesis
  • Randomized computation – BPP
    • POLYNOMIAL IDENTITY TESTING example of problem in BPP, not known to be in P
  • Quantum computation
    • FACTORING example of problem solvable in quantum polynomial time, not believed to be in P

CS21 Lecture 27

the very last slide
The very last slide
  • Fill out TQFR surveys!
  • Course to consider
    • CS138 (advanced algorithms)
    • CS150 (probability and computation)
    • CS151 (complexity theory)
    • CS153 (current topics in theoretical CS)
  • Good luck
    • on final
    • in CS, at Caltech, beyond…
  • Thank you!

CS21 Lecture 27