aarjaneiro/planetmath huggingface dataset:
deterministic Turing machine

deterministic Turing machine

The formal definition of a

deterministic Turing machine

is a tuple:

T

=

(

Q

,

Σ

,

Γ

,

δ

,

q

0

,

B

,

F

)

Where

Q

is a

finite set

of states,

Γ

is the finite set of tape symbols,

B

∈

Γ

is the blank symbol,

Σ

⊆

Γ

is the set of input symbols,

δ

:

Q

×

Γ

→

Q

×

Γ

×

{

L

,

R

}

is the move

function

,

q

0

∈

Q

is the

start state

and

F

⊆

Q

is the set of final states. (Note that, despite the name,

δ

is a

partial function

which will be undefined for some pairs

(

q

,

A

)

.)

A

Turing machine

is conceived to be a box with a tape and a tape head. The tape consists of an

infinite

number of cells stretching in both directions, with the tape head always located over exactly one of these cells. Each cell has symbol from

Γ

written on it.

At the beginning of its

computation

,

T

is in state

q

0

and a finite string

S

(the input string) over the

alphabet

Σ

is written on the tape, with the tape located over the first letter of

S

. Each cell before the beginning or after the end of

S

is blank. For each move, if the current state is

q

and the value of the cell under the tape head is

A

then suppose

δ

⁢

(

q

,

A

)

=

(

q

′

,

A

′

,

D

)

. The value of the cell under the tape head is changed to

A

′

, the state changes to

q

′

, and the tape head moves to the left if

D

=

L

and to the right if

D

=

R

. If

δ

⁢

(

q

,

A

)

is undefined then the

machine

halts.

For any

S

∈

Γ

+

, we say

T

accepts

S

if

T

halts in a state

q

∈

F

when

S

is the input string, that

T

rejects

S

if

T

halts in a state

q

∉

F

, and otherwise that

T

does not halt on

S

.

This definition can easily be extended to multiple tapes with various rules.

δ

should be a function from

Q

×

Γ

n

→

Γ

m

×

{

L

,

R

}

n

where

n

is the number of tapes,

m

is the number of input tapes, and

L

is interpreted to mean not moving (rather than moving to the left) for

one-way

tapes.

Title

deterministic Turing machine

Canonical name

DeterministicTuringMachine

Date of creation

2013-03-22 13:01:22

Last modified on

2013-03-22 13:01:22

Owner

Henry (455)

Last modified by

Henry (455)

Numerical id

8

Author

Henry (455)

Entry type

Definition

Classification

msc 68Q05

Classification

msc 68Q10

Related topic

TuringMachine2

Generated on Sat Feb 10 12:39:37 2018 by

LaTeXML
Levin reduction

Levin reduction

If

R

1

and

R

2

are

search problems

and

𝒞

is a

complexity class

then a

𝒞

Levin reduction

of

R

1

to

R

2

consists of three functions

g

1

,

g

2

,

g

3

∈

𝒞

which satisfy:

•

g

1

is a

𝒞

Karp reduction

of

L

⁢

(

R

1

)

to

L

⁢

(

R

2

)

•

If

R

1

⁢

(

x

,

y

)

then

R

2

⁢

(

f

⁢

(

x

)

,

g

⁢

(

x

,

y

)

)

•

If

R

2

⁢

(

f

⁢

(

x

)

,

z

)

then

R

1

⁢

(

x

,

h

⁢

(

x

,

z

)

)

Note that a

𝒞

Cook reduction can be constructed by calculating

f

⁢

(

x

)

, using the oracle to find

z

, and then calculating

h

⁢

(

x

,

z

)

.

𝒫

Levin reductions are just called Levin reductions.

Title

Levin reduction

Canonical name

LevinReduction

Date of creation

2013-03-22 13:01:44

Last modified on

2013-03-22 13:01:44

Owner

Henry (455)

Last modified by

Henry (455)

Numerical id

5

Author

Henry (455)

Entry type

Definition

Classification

msc 68Q05

Classification

msc 68Q10

Generated on Sat Feb 10 12:39:59 2018 by

LaTeXML
non-deterministic finite automaton

non-deterministic finite automaton

A

non-deterministic finite automaton

(or NDFA) can be formally defined as a 5-tuple

(

S

,

Σ

,

δ

,

q

0

,

F

)

, where

1.

S

is a non-empty

finite set

of

states

,

2.

Σ

is the

alphabet

(defining what set of input strings the

automaton

operates on),

3.

δ

:

S

×

Σ

→

𝒫

⁢

(

S

)

is a

function

called the

transition function

,

4.

q

0

∈

Q

is the starting state, and

5.

F

⊆

S

is a set of final (or accepting) states.

Some authors also relax the fourth condition by permitting

multiple

starting states (see remark below).

Note how this definition differs from that of a

deterministic finite automaton

(DFA) only by the definition of the transition function

δ

.

Operation

of the NDFA begins at

q

0

, and movement from state to state is governed by the transition function

δ

.

The transition function takes the first symbol of the (remaining) input string and the current state as its input, and after the transition this first symbol is removed only if the transition is defined for a symbol in

Σ

instead of

λ

. Conceptually, all possible transitions from a current state are followed

simultaneously

(hence the non-determinism). Once every possible transition has been executed, the NDFA is halted. If any of the states reached upon halting are in

F

for some input string,

and

the entire input string is consumed to reach that state, then the NDFA accepts that string.

An NDFA can be represented visually as a

directed graph

called the

state diagram

. Circular vertices denote states, and the set of directed edges, labelled by symbols in

Σ

∪

λ

, denotes

T

. The starting state

q

0

is usually denoted by an arrow pointing to it that points from no other vertex. States in

F

are usually denoted by double circles.

NDFAs accept

regular languages

, and can be used to test whether any string in

Σ

*

is in the

language

it represents. Given an NDFA

M

, the language accepted by

M

is denoted by

L

⁢

(

M

)

.

Consider the following regular language over the alphabet

Σ

:=

{

𝚊

,

𝚋

}

(represented by the

regular expression

aa*b

):

<

⁢

S

>

::=

a

⁢

A

<

⁢

A

>

::=

λ

B

|

𝚊

A

<

⁢

B

>

::=

b

This language can be represented by the NDFA with state diagram:

Generated on Sat Feb 10 12:40:15 2018 by

LaTeXML
non-deterministic Turing machine

non-deterministic Turing machine

The definition of a

non-deterministic Turing machine

is the same as the definition of a

deterministic Turing machine

except that

δ

is a

relation

, not a

function

. Hence, for any particular state and symbol, there may be multiple possible legal moves.

If

S

∈

Γ

+

we say

T

accepts

S

if, when

S

is the input, there is some

finite sequence

of legal moves such that

δ

is undefined on the state and symbol pair which results from the last move in the sequence and such that the

final state

is an element of

F

. If

T

does not accept

S

then it rejects

S

.

An alternative definition of a non-deterministic Turing machine is as a deterministic Turing machine with an extra

one-way

, read-only tape, the guess tape. Then we say

T

accepts

S

if there is any string

c

⁢

(

S

)

such that, when

c

⁢

(

S

)

is placed on the guess tape,

T

accepts

S

. We call

c

⁢

(

S

)

a

certificate

for

S

, and otherwise that it rejects

S

. In some cases the guess tape is allowed to be two-way; this generates different time and space

complexity classes

than the one-way case (the one-way case is

equivalent

to the original definition).

Title

non-deterministic Turing machine

Canonical name

NondeterministicTuringMachine

Date of creation

2013-03-22 13:01:25

Last modified on

2013-03-22 13:01:25

Owner

Henry (455)

Last modified by

Henry (455)

Numerical id

6

Author

Henry (455)

Entry type

Definition

Classification

msc 68Q05

Related topic

TuringMachine2

Defines

certificate

Generated on Sat Feb 10 12:40:17 2018 by

LaTeXML
oracle

oracle

An

oracle

is a way to allow

Turing machines

access to

information

they cannot necessarily

calculate

. It makes it possible to consider whether solving one problem would make it possible to solve another problem, and therefore to ask whether one problem is harder than another.

If

T

is a Turing machine and

π

is a problem (either a search problem or a

decision problem

) with appropriate

alphabet

then

T

π

denotes the

machine

which runs

T

using

π

as an oracle.

A Turing machine with an oracle has three special states

?

,

y

and

n

. Whenever

T

π

enters state

?

it

queries

the oracle about

x

, the series of non-blank symbols to the right of the tape head. If

π

is a decision problem then the machine immediately changes to state

y

if in

x

∈

π

and

n

otherwise. If

π

is a search problem then it switches to state

n

if there is no

z

such that

π

⁢

(

x

,

z

)

and to state

y

is there is such a

z

. In the latter case the

section

of the tape containing

x

is changed to contain

z

.

In either case, the oracle allows the machine to behave as if it could solve

π

, since each time it accesses the oracle, the result is the same as a machine which solves

π

.

Alternate but

equivalent

definitions use a special tape to contain the query and response.

Clearly if

L

≠

L

′

then

L

⁢

(

T

L

)

may not be equal to

L

⁢

(

T

L

′

)

.

By definition, if

T

is a Turing machine appropriate for a

complexity class

𝒞

then

T

L

is a

𝒞

Cook reduction

of

L

⁢

(

T

L

)

to

L

.

If

𝒞

is a complexity class then

T

𝒞

is a complexity class with

T

𝒞

=

{

L

∣

L

=

L

⁢

(

T

C

)

∧

C

∈

𝒞

}

. If

𝒟

is another complexity class then

𝒟

𝒞

=

{

L

∣

L

∈

T

𝒞

∧

L

⁢

(

T

∅

)

∈

𝒟

}

.

Title

oracle

Canonical name

Oracle

Date of creation

2013-03-22 13:01:36

Last modified on

2013-03-22 13:01:36

Owner

Henry (455)

Last modified by

Henry (455)

Numerical id

6

Author

Henry (455)

Entry type

Definition

Classification

msc 68Q05

Defines

query

Generated on Sat Feb 10 12:40:19 2018 by

LaTeXML
self-reducible

self-reducible

A

search problem

R

is

𝒞

self-reducible

if there is a

𝒞

Cook reduction

of

R

to

L

⁢

(

R

)

. That is, if the

decision problem

for

L

⁢

(

R

)

is

𝒞

then so is the search problem for

R

.

If

R

is polynomially self-reducible then it is called self-reducible.

Note that

L

⁢

(

R

)

is trivially Cook reducible to

R

.

Title

self-reducible

Canonical name

Selfreducible

Date of creation

2013-03-22 13:01:47

Last modified on

2013-03-22 13:01:47

Owner

Henry (455)

Last modified by

Henry (455)

Numerical id

5

Author

Henry (455)

Entry type

Definition

Classification

msc 68Q05

Classification

msc 68Q10

Generated on Sat Feb 10 12:40:31 2018 by

LaTeXML
universal Turing machine

universal Turing machine

A

universal Turing machine

U

is a

Turing machine

with a single binary

one-way

read-only input tape, on which it expects to find the

encoding

of an arbitrary Turing machine

M

. The set of all Turing machine encodings must
be prefix-free, so that no special end-marker or ‘blank’ is needed to recognize
a code’s end. Having transferred the description of

M

onto its worktape,

U

then proceeds to simulate the behaviour of

M

on the remaining
contents of the input tape. If

M

halts, then

U

cleans up its worktape,
leaving it with just the output of

M

, and halts too.

If we denote by

M

⁢

(

)

the

partial function

computed by

machine

M

,
and by

<

M

>

the encoding of machine

M

as a binary string,
then we have

U

⁢

(

<

M

>

⁢

x

)

=

M

⁢

(

x

)

.

There are two kinds of universal Turing machine, depending on whether the
input tape

alphabet

of the simulated machine is

{

0

,

1

,

#

}

or just

{

0

,

1

}

.
The first kind is a

plain

Universal Turing machine;
while the second is a

prefix

Universal Turing machine,
which has the nice property
that the set of inputs on which it halts is prefix free.

The letter

U

is commonly used to denote a fixed

universal

machine, whose
type is either mentioned explicitly or assumed clear from

context

.

Title

universal Turing machine

Canonical name

UniversalTuringMachine

Date of creation

2013-03-22 13:43:46

Last modified on

2013-03-22 13:43:46

Owner

tromp (1913)

Last modified by

tromp (1913)

Numerical id

7

Author

tromp (1913)

Entry type

Definition

Classification

msc 68Q05

Related topic

ArtificialInteglligence

Related topic

StrongAIThesis

Generated on Sat Feb 10 12:40:51 2018 by

LaTeXML
Cook reduction

Cook reduction

Given two (search or decision) problems

π

1

and

π

2

and a

complexity class

𝒞

, a

𝒞

Cook reduction

of

π

1

to

π

2

is a

Turing machine

appropriate for

𝒞

which solves

π

1

using

π

2

as an oracle (the Cook reduction itself is not in

𝒞

, since it is a Turing machine, not a problem, but it should be the class of bounded Turing machines corresponding to

𝒞

). The most common type are

𝒫

Cook reductions, which are often just called Cook reductions.

If a Cook reduction exists then

π

2

is in some sense “at least as hard” as

π

1

, since a

machine

which solves

π

2

could be used to construct one which solves

π

1

. When

𝒞

is

closed under

appropriate

operations

, if

π

2

∈

𝒞

and

π

1

is

𝒞

-Cook reducible to

π

2

then

π

1

∈

𝒞

.

A

𝒞

Karp reduction

is a special kind of

𝒞

Cook reduction for

decision problems

L

1

and

L

2

. It is a

function

g

∈

𝒞

such that:

x

∈

L

1

↔

g

⁢

(

x

)

∈

L

2

Again,

𝒫

Karp reductions are just called Karp reductions.

A Karp reduction provides a Cook reduction, since a Turing machine could decide

L

1

by calculating

g

⁢

(

x

)

on any input and determining whether

g

⁢

(

x

)

∈

L

2

. Note that it is a stronger condition than a Cook reduction. For

instance

, this machine requires only one use of the oracle.

Title

Cook reduction

Canonical name

CookReduction

Date of creation

2013-03-22 13:01:41

Last modified on

2013-03-22 13:01:41

Owner

Henry (455)

Last modified by

Henry (455)

Numerical id

7

Author

Henry (455)

Entry type

Definition

Classification

msc 68Q10

Classification

msc 68Q05

Defines

Karp reduction

Generated on Sat Feb 10 12:41:01 2018 by

LaTeXML
Chomsky hierarchy

Chomsky hierarchy

The

Chomsky hierarchy

or

Chomsky-Schützenberger hierarchy

is a way of classifying

formal grammars

into four types, with the lower numbered types being more general.

Recall that a formal grammar

G

=

(

Σ

,

N

,

P

,

σ

)

consists of an

alphabet

Σ

, an alphabet

N

of non-terminal symbols properly included in

Σ

, a non-empty

finite set

P

of productions, and a symbol

σ

∈

N

called the start symbol. The non-empty alphabet

T

:=

Σ

-

N

is the set of terminal symbols. Then

G

is called a

Type-0 grammar

if there are no

restrictions

on the productions. Type-0 grammar is also known as an

unrestricted grammar

, or a

phrase-structure grammar

.

Type-1 grammar

if the productions are of the form

u

⁢

A

⁢

v

→

u

⁢

W

⁢

v

, where

u

,

v

,

W

∈

Σ

*

with

W

≠

λ

, and

A

∈

N

, or

σ

→

λ

, provided that

σ

does not occur on the

right hand side

of any productions in

P

. As

A

is surrounded by words

u

,

v

, a type-1 grammar is also known as a context-sensitive grammar.

Type-2 grammar

if the productions are of the form

A

→

W

, where

A

∈

N

and

W

∈

Σ

*

. Type-2 grammars are also called context-free grammars, because the left hand side of any productions are “free” of

contexts

.

Type-3 grammar

if the productions are of the form

A

→

u

or

A

→

u

⁢

B

, where

A

,

B

∈

N

and

u

∈

T

*

. Owing to the fact that

languages

generated by type-3 grammars can be represented by

regular expressions

, type-3 grammars are also known as regular grammars.

It is clear that every type-

i

grammar

is type-0, and every type-3 grammar is type-2. A type-2 grammar is not necessarily type-1, because it may contain both

σ

→

λ

and

A

→

W

, where

λ

occurs in

W

. Nevertheless, the relevance of the hierarchy has more to do with the languages generated by the grammars. Call a

formal language

a type-

i

language if it is generated by a type-

i

grammar, and denote

ℒ

i

the family of type-

i

languages. Then it can be shown that

ℒ

3

⊂

ℒ

2

⊂

ℒ

1

⊂

ℒ

0

where each

inclusion

is strict.

Below is a table summarizing the four types of grammars, the languages they generate, and the

equivalent

computational devices accepting the languages.

Title

Chomsky hierarchy

Canonical name

ChomskyHierarchy

Date of creation

2013-03-22 16:27:04

Last modified on

2013-03-22 16:27:04

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

13

Author

CWoo (3771)

Entry type

Definition

Classification

msc 68Q15

Classification

msc 03D55

Synonym

Chomsky-Schützenberger hierarchy

Synonym

Chomsky-Schutzenberger hierarchy

Synonym

unrestricted grammar

Defines

type-0 grammar

Defines

type-0 language

\@unrecurse

Generated on Sat Feb 10 12:41:17 2018 by

LaTeXML
constructible

constructible

A

function

f

:

ℕ

→

ℕ

is

time constructible

if there is a

deterministic Turing machine

T

(with

alphabet

{

0

,

1

,

B

}

) such that when

T

receives as input the a series of

n

ones, it halts after exactly

f

⁢

(

n

)

steps. Similarly

f

is

space constructible

if there is a

similar

Turing machine

which halts after using exactly

f

⁢

(

n

)

cells.

Most ’natural’ functions are both time and space constructible, including

constant functions

, polynomials, and exponentials, for example.

Title

constructible

Canonical name

Constructible

Date of creation

2013-03-22 13:03:20

Last modified on

2013-03-22 13:03:20

Owner

Henry (455)

Last modified by

Henry (455)

Numerical id

7

Author

Henry (455)

Entry type

Definition

Classification

msc 68Q15

Defines

time constructible

Defines

space constructible

Generated on Sat Feb 10 12:41:32 2018 by

LaTeXML
counting complexity class

counting complexity class

If

𝒩

⁢

𝒞

is a

complexity class

associated with non-deterministic machines then

#

⁢

𝒞

=

{

#

⁢

R

∣

R

∈

𝒩

⁢

𝒞

}

is the set of

counting problems

associated with each

search problem

in

𝒩

⁢

𝒞

. In particular,

#

⁢

𝒫

is the class of counting problems associated with

𝒩

⁢

𝒫

search problems.

Title

counting complexity class

Canonical name

CountingComplexityClass

Date of creation

2013-03-22 13:02:31

Last modified on

2013-03-22 13:02:31

Owner

Henry (455)

Last modified by

Henry (455)

Numerical id

4

Author

Henry (455)

Entry type

Definition

Classification

msc 68Q15

Generated on Sat Feb 10 12:41:34 2018 by

LaTeXML
NP-complete

NP-complete

A problem

π

∈

𝒩

⁢

𝒫

is

𝒩

⁢

𝒫

if for any

π

′

∈

𝒩

⁢

𝒫

there is a

Cook reduction

of

π

′

to

π

. Hence if

π

∈

𝒫

then every

𝒩

⁢

𝒫

problem would be in

𝒫

. A slightly stronger definition requires a Karp reduction or Karp reduction of corresponding

decision problems

as appropriate.

A

search problem

R

is

N

⁢

P

hard

if for any

R

′

∈

𝒩

⁢

𝒫

there is a

Levin reduction

of

R

′

to

R

.

Title

NP-complete

Canonical name

NPcomplete

Date of creation

2013-03-22 13:01:49

Last modified on

2013-03-22 13:01:49

Owner

Henry (455)

Last modified by

Henry (455)

Numerical id

4

Author

Henry (455)

Entry type

Definition

Classification

msc 68Q15

Synonym

NP complete

Synonym

NP hard

Defines

NP-hard

Generated on Sat Feb 10 12:41:36 2018 by

LaTeXML
polynomial hierarchy

polynomial hierarchy

The

polynomial hierarchy

is a hierarchy of

complexity classes

generalizing the relationship between

𝒫

and

𝒩

⁢

𝒫

.

We let

Σ

0

p

=

Π

0

p

=

Δ

0

p

=

𝒫

, then

Δ

i

+

1

p

=

𝒫

Σ

i

p

and

Σ

i

+

1

p

=

𝒩

⁢

𝒫

Σ

i

p

. Define

Π

i

p

to be

c

⁢

o

⁢

Σ

i

p

.

For

instance

Σ

1

p

=

𝒩

⁢

𝒫

𝒫

=

𝒩

⁢

𝒫

.

The complexity class

𝒫

⁢

ℋ

=

⋃

i

∈

ℕ

Σ

i

p

.

The polynomial hierarchy is closely related to the

arithmetical hierarchy

; indeed, an alternate

definition

is almost identical to the definition of the arithmetical hierarchy but stricter rules on what

quantifiers

can be used.

When there is no risk of confusion with the arithmetical hierarchy, the

superscript

p

can be dropped.

Title

polynomial hierarchy

Canonical

name

PolynomialHierarchy

Date of creation

2013-03-22 13:02:11

Last modified on

2013-03-22 13:02:11

Owner

Henry (455)

Last modified by

Henry (455)

Numerical id

6

Author

Henry (455)

Entry type

Definition

Classification

msc 68Q15

Defines

PH

Generated on Sat Feb 10 12:41:37 2018 by

LaTeXML
polynomial hierarchy is a hierarchy

polynomial hierarchy is a hierarchy

The

polynomial hierarchy

is a hierarchy. Specifically:

Σ

i

p

∪

Π

i

p

⊆

Δ

i

+

1

p

⊆

Σ

i

+

1

p

∩

Π

i

+

1

p

.

Proof

To see that

Σ

i

p

∪

Π

i

p

⊆

Δ

i

+

1

p

=

𝒫

Σ

i

p

,
observe that the

machine

which checks its input against its oracle and accepts or rejects when the oracle accepts or rejects (respectively) is easily in

𝒫

, as is the machine which rejects or accepts when the oracle accepts or rejects (respectively). These easily emulate

Σ

i

p

and

Π

i

p

respectively.

Since

𝒫

⊆

𝒩

⁢

𝒫

, it is clear that

Δ

i

p

⊆

Σ

i

p

. Since

𝒫

𝒞

is

closed
under

complementation

for any

complexity class

𝒞

(the associated
machines are

deterministic

and always halt, so the complementary machine just
reverses which states are accepting), if

L

∈

𝒫

Σ

i

p

⊆

Σ

i

p

then so is

L

¯

,
and therefore

L

∈

Π

i

p

.

Unlike the

arithmetical hierarchy

, the polynomial hierarchy is not known to be proper. Indeed, if

𝒫

=

𝒩

⁢

𝒫

then

𝒫

=

𝒫

⁢

ℋ

, so a proof that the hierarchy is proper would be quite significant.

Title

polynomial hierarchy is a hierarchy

Canonical name

PolynomialHierarchyIsAHierarchy

Date of creation

2013-03-22 13:02:14

Last modified on

2013-03-22 13:02:14

Owner

uzeromay (4983)

Last modified by

uzeromay (4983)

Numerical id

5

Author

uzeromay (4983)

Entry type

Result

Classification

msc 68Q15

Generated on Sat Feb 10 12:41:39 2018 by

LaTeXML
time complexity

time complexity

1

Time Complexity

Time complexity

refers to a

function

describing how much time it will take an

algorithm

to execute, based on the

parameters

of its input. The exact value of this function is usually ignored in favour of its order, expressed in the so-called Big-O notation (this is based on the limit of the time complexity function as the values of its parameters increase without bound.)

Complexity classes

are

equivalence classes

of time complexities which are “equal” in Big-O notation. Further, there are meta-complexity classes

1

1

This term has been invented for use in this entry.

of time complexities which have Big-O

expressions

that differ only by some specific parameter. For

instance

,

O

⁢

(

n

2

)

and

O

⁢

(

n

3

)

are both

polynomial time

complexity classes, similarly

O

⁢

(

2

n

)

and

O

⁢

(

3

n

)

are

exponential time

complexity classes.

2

Example

All comparison-based

sorting

has time complexity no better than

O

⁢

(

n

⁢

log

⁡

n

)

, where

n

is the number of

elements

to be sorted. The exact expression for time complexity of a particular sorting algorithm may be something like

T

⁢

(

n

)

=

a

⁢

n

⁢

log

⁡

n

+

b

, with

a

and

b

constants

, which still is order

O

⁢

(

n

⁢

log

⁡

n

)

.

Title

time complexity

Canonical

name

TimeComplexity

Date of creation

2013-03-22 11:47:39

Last modified on

2013-03-22 11:47:39

Owner

akrowne (2)

Last modified by

akrowne (2)

Numerical id

10

Author

akrowne (2)

Entry type

Definition

Classification

msc 68Q15

Defines

polynomial time

Defines

polynomial-time

Defines

exponential time

Defines

exponential-time

Defines

complexity classes

Defines

meta-complexity classes

Generated on Sat Feb 10 12:41:41 2018 by

LaTeXML