![Page 1: Resource-Bounded Computation Previously: can something be done? Now: how efficiently can it be done? Goal: conserve computational resources: Time, space,](https://reader038.vdocument.in/reader038/viewer/2022110304/5518b4ec550346c31f8b50c6/html5/thumbnails/1.jpg)
Resource-Bounded ComputationPreviously: can something be done?
Now: how efficiently can it be done?
Goal: conserve computational resources:
Time, space, other resources?
Def: L is decidable within time O(t(n)) if some TM M that decides L always halts on all wÎå* within O(t(|w|)) steps / time.
Def: L is decidable within space O(s(n)) if some TM M that decides L always halts on all wÎå* while never using more than O(s(|w|)) space / tape cells.
![Page 2: Resource-Bounded Computation Previously: can something be done? Now: how efficiently can it be done? Goal: conserve computational resources: Time, space,](https://reader038.vdocument.in/reader038/viewer/2022110304/5518b4ec550346c31f8b50c6/html5/thumbnails/2.jpg)
Complexity Classes
Def: DTIME(t(n))={L | L is decidable within
time O(t(n)) by some deterministic TM}
Def: NTIME(t(n))={L | L is decidable within
time O(t(n)) by some non-deterministic TM}
Def: DSPACE(s(n))={L | L decidable within
space O(s(n)) by some deterministic TM}
Def: NSPACE(s(n))={L | L decidable within
space O(s(n)) by some non-deterministic TM}
Prope
rties
!
![Page 3: Resource-Bounded Computation Previously: can something be done? Now: how efficiently can it be done? Goal: conserve computational resources: Time, space,](https://reader038.vdocument.in/reader038/viewer/2022110304/5518b4ec550346c31f8b50c6/html5/thumbnails/3.jpg)
Time is Tape DependentTheorem: The time depends on the # of TM tapes.
Idea: more tapes can enable higher efficiency.
Ex: {0n1n | n>0} is in DTIME(n2) for 1-tape
TM’s, and is in DTIME(n) for 2-tape TM’s.
Note: For multi-tape TM’s, input tape space does not
“count” in the total space s(n). This enables
analyzing sub-linear space complexities.
![Page 4: Resource-Bounded Computation Previously: can something be done? Now: how efficiently can it be done? Goal: conserve computational resources: Time, space,](https://reader038.vdocument.in/reader038/viewer/2022110304/5518b4ec550346c31f8b50c6/html5/thumbnails/4.jpg)
Space is Tape IndependentTheorem: The space does not depend on the # tapes.
Proof: 1 0 1 10
0 1 0
1
1
1 1 1 001
0
1 0 1 10
0 1 0
1
1
1 1 1 001
0
Note: This does not asymptotically increase the overall space (but can increase the total time).
Theorem: A 1-tape TM can simulate a t(n)-time-bounded k-tape TM in time O(k×t2(n)).
Idea: Tapes can be “interlaced” space-efficiently:
![Page 5: Resource-Bounded Computation Previously: can something be done? Now: how efficiently can it be done? Goal: conserve computational resources: Time, space,](https://reader038.vdocument.in/reader038/viewer/2022110304/5518b4ec550346c31f8b50c6/html5/thumbnails/5.jpg)
Space-Time RelationsTheorem: If t(n) < t’(n) "n>1 then:
DTIME(t(n)) Í DTIME(t’(n))
NTIME(t(n)) Í NTIME(t’(n))
Theorem: If s(n) < s’(n) "n>1 then:
DSPACE(s(n)) Í DSPACE(s’(n))
NSPACE(s(n)) Í NSPACE(s’(n))
Example: NTIME(n) Í NTIME(n2)
Example : DSPACE(log n) Í DSPACE(n)
![Page 6: Resource-Bounded Computation Previously: can something be done? Now: how efficiently can it be done? Goal: conserve computational resources: Time, space,](https://reader038.vdocument.in/reader038/viewer/2022110304/5518b4ec550346c31f8b50c6/html5/thumbnails/6.jpg)
Examples of Space & Time UsageLet L1={0n1n | n>0}:
For 1-tape TM’s:
L1 Î DTIME(n2)
L1 Î DSPACE(n)
L1 Î DTIME(n log n)
For 2-tape TM’s:
L1 Î DTIME(n)
L1 Î DSPACE(log n)
![Page 7: Resource-Bounded Computation Previously: can something be done? Now: how efficiently can it be done? Goal: conserve computational resources: Time, space,](https://reader038.vdocument.in/reader038/viewer/2022110304/5518b4ec550346c31f8b50c6/html5/thumbnails/7.jpg)
Examples of Space & Time UsageLet L2=S*
L2 Î DTIME(n) Theorem: every regular language is in DTIME(n)
L2 Î DSPACE(1)
Theorem: every regular language is in DSPACE(1)
L2 Î DTIME(1)
Let L3={w$w | w in S*}
L3 Î DTIME(n2) L3 Î DSPACE(n)
L3 Î DSPACE(log n)
![Page 8: Resource-Bounded Computation Previously: can something be done? Now: how efficiently can it be done? Goal: conserve computational resources: Time, space,](https://reader038.vdocument.in/reader038/viewer/2022110304/5518b4ec550346c31f8b50c6/html5/thumbnails/8.jpg)
Special Time Classes
Def: P = È DTIME(nk) "k>1
P º deterministic polynomial time
Note: P is robust / model-independent
Def: NP = È NTIME(nk) "k>1
NP º non-deterministic polynomial time
Theorem: P Í NP
Conjecture: P = NP ? Million $ question!
![Page 9: Resource-Bounded Computation Previously: can something be done? Now: how efficiently can it be done? Goal: conserve computational resources: Time, space,](https://reader038.vdocument.in/reader038/viewer/2022110304/5518b4ec550346c31f8b50c6/html5/thumbnails/9.jpg)
Other Special Space Classes
Def: PSPACE =È DSPACE(nk) "k>1
PSPACE º deterministic polynomial space
Def: NPSPACE = È NSPACE(nk) "k>1
NPSPACE º non-deterministic polynomial space
Theorem: PSPACE Í NPSPACE (obvious)
Theorem: PSPACE = NPSPACE (not obvious)
![Page 10: Resource-Bounded Computation Previously: can something be done? Now: how efficiently can it be done? Goal: conserve computational resources: Time, space,](https://reader038.vdocument.in/reader038/viewer/2022110304/5518b4ec550346c31f8b50c6/html5/thumbnails/10.jpg)
Other Special Space Classes
Def: EXPTIME = È DTIME(2nk) "k>1
EXPTIME º exponential time
Def: EXPSPACE = È DSPACE(2nk) "k>1
EXPSPACE º exponential space
Def: L = LOGSPACE = DSPACE(log n)
Def: NL = NLOGSPACE = NSPACE(log n)
![Page 11: Resource-Bounded Computation Previously: can something be done? Now: how efficiently can it be done? Goal: conserve computational resources: Time, space,](https://reader038.vdocument.in/reader038/viewer/2022110304/5518b4ec550346c31f8b50c6/html5/thumbnails/11.jpg)
Space/Time RelationshipsTheorem: DTIME(f(n)) Í DSPACE(f(n))
Theorem: DTIME(f(n)) Í DSPACE(f(n) / log(f(n)))
Theorem: NTIME(f(n)) Í DTIME(cf(n) ) for some c depending on the language.
Theorem: DSPACE(f(n)) Í DTIME(cf(n) )for some c, depending on the language.
Theorem [Savitch]: NSPACE(f(n)) Í DSPACE(f 2(n))
Corollary: PSPACE = NPSPACE
Theorem: NSPACE(nr) Í DSPACE(nr+e) " r>0, e>0
![Page 12: Resource-Bounded Computation Previously: can something be done? Now: how efficiently can it be done? Goal: conserve computational resources: Time, space,](https://reader038.vdocument.in/reader038/viewer/2022110304/5518b4ec550346c31f8b50c6/html5/thumbnails/12.jpg)
PS
PAC
E-c
ompl
ete
QB
F
The Extended Chomsky Hierarchy
Finite {a,b}Regular a*Det. CF anbn
Context-free wwRP anbncn
NP
PSPACE
EXPSPACE R
ecog
niza
ble
N
ot R
ecog
niza
ble
HH
Decidable Presburger arithmetic
NP
-com
plet
e S
AT
Not
fin
itel
y de
scri
babl
e ?
2S*
EXPTIME
EX
PT
IME
-com
plet
e G
o
EX
PS
PAC
E-c
ompl
ete
=R
ETuring
degrees Context sensitive LBA
![Page 13: Resource-Bounded Computation Previously: can something be done? Now: how efficiently can it be done? Goal: conserve computational resources: Time, space,](https://reader038.vdocument.in/reader038/viewer/2022110304/5518b4ec550346c31f8b50c6/html5/thumbnails/13.jpg)
![Page 14: Resource-Bounded Computation Previously: can something be done? Now: how efficiently can it be done? Goal: conserve computational resources: Time, space,](https://reader038.vdocument.in/reader038/viewer/2022110304/5518b4ec550346c31f8b50c6/html5/thumbnails/14.jpg)
Time Complexity Hierarchy
Theorem: for any t(n)>0 there exists adecidable language LÏDTIME(t(n)).
Þ No time complexity class contains all the decidable languages, and the time hierarchy is ¥!
Þ There are decidable languages that take arbitrarily long times to decide!
Note: t(n) must be computable & everywhere definedProof: (by diagonalization)
Fix lexicographic orders for TM’s: M1, M2, M3, . . .Interpret TM inputs iÎΣ* as encodings of integers:
a=1, b=2, aa=3, ab=4, ba=5, bb=6, aaa=7, …
Juris Hartmanis Richard Stearns
![Page 15: Resource-Bounded Computation Previously: can something be done? Now: how efficiently can it be done? Goal: conserve computational resources: Time, space,](https://reader038.vdocument.in/reader038/viewer/2022110304/5518b4ec550346c31f8b50c6/html5/thumbnails/15.jpg)
Time Complexity Hierarchy (proof)Define L={i | Mi does not accept i within t(i) time}
Note: L is decidable (by simulation)
Q: is LÎDTIME(t(n)) ?
Assume (towards contradiction) LÎDTIME(t(n))
i.e., $ a fixedKÎN such that Turing machine MK
decides L within time bound t(n)
i
MK Þ decides / accepts L
If Mi accepts i within t(i) time thenelse
RejectAccept
![Page 16: Resource-Bounded Computation Previously: can something be done? Now: how efficiently can it be done? Goal: conserve computational resources: Time, space,](https://reader038.vdocument.in/reader038/viewer/2022110304/5518b4ec550346c31f8b50c6/html5/thumbnails/16.jpg)
Time Complexity Hierarchy (proof)
Consider whether KÎL:
KÎL Þ MK must accept K within t(K) time
Þ MK must reject K Þ KÏL
KÏL Þ MK must reject K within t(K) time
Þ MK must accept K Þ KÎL
So (KÎL) Û (KÏL), a contradiction!
Þ Assumption is false Þ LÏDTIME(t(n))
K
MK Þ decides / accepts L
If MK accepts K within t(K) time thenelse
RejectAccept
![Page 17: Resource-Bounded Computation Previously: can something be done? Now: how efficiently can it be done? Goal: conserve computational resources: Time, space,](https://reader038.vdocument.in/reader038/viewer/2022110304/5518b4ec550346c31f8b50c6/html5/thumbnails/17.jpg)
i = 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 …wiÎΣ* = a b aa ab ba bb aaa aab aba abb baa bab bba bbbaaaa…
M1(i) Þ √ ´ √ √ ´ ´ √ √ √ ´ ´ ´ ´ √ ´ …
M2(i) Þ ´ ´ √ ´ ´ √ ´ ´ √ ´ √ √ ´ ´ ´ …
M3(i) Þ √ √ √ ´ √ ´ ´ ´ √ √ √ √ ´ √ ´ …
M4(i) Þ √ ´ √ ´ √ √ ´ √ √ ´ ´ √ ´ ´ ´ …
M5(i) Þ ´ √ √ ´ ´ √ √ ´ √ √ √ √ ´ √ ´ …
. . .
Time Hierarchy (another proof)Consider all t(n)-time-bounded TM’s on all inputs:
M’(i) Þ is t(n) time-bounded.But M’ computes a different function than any Mj
Þ Contradiction!
´ ´√ √ √ . . .
![Page 18: Resource-Bounded Computation Previously: can something be done? Now: how efficiently can it be done? Goal: conserve computational resources: Time, space,](https://reader038.vdocument.in/reader038/viewer/2022110304/5518b4ec550346c31f8b50c6/html5/thumbnails/18.jpg)
“Lexicographic order.”
![Page 19: Resource-Bounded Computation Previously: can something be done? Now: how efficiently can it be done? Goal: conserve computational resources: Time, space,](https://reader038.vdocument.in/reader038/viewer/2022110304/5518b4ec550346c31f8b50c6/html5/thumbnails/19.jpg)
![Page 20: Resource-Bounded Computation Previously: can something be done? Now: how efficiently can it be done? Goal: conserve computational resources: Time, space,](https://reader038.vdocument.in/reader038/viewer/2022110304/5518b4ec550346c31f8b50c6/html5/thumbnails/20.jpg)
Juris Hartmanis Richard Stearns
Space Complexity HierarchyTheorem: for any s(n)>0 there exists a
decidable language LÏDSPACE(s(n)).Þ No space complexity class contains all the
decidable languages, and the space hierarchy is ¥!Þ There are decidable languages that take arbitrarily
much space to decide!Note: s(n) must be computable & everywhere defined
Proof: (by diagonalization)
Fix lexicographic orders for TM’s: M1, M2, M3, . . .
Interpret TM inputs iÎΣ* as encodings of integers:
a=1, b=2, aa=3, ab=4, ba=5, bb=6, aaa=7, …
![Page 21: Resource-Bounded Computation Previously: can something be done? Now: how efficiently can it be done? Goal: conserve computational resources: Time, space,](https://reader038.vdocument.in/reader038/viewer/2022110304/5518b4ec550346c31f8b50c6/html5/thumbnails/21.jpg)
Space Complexity Hierarchy (proof)Define L={i | Mi does not accept i within t(i) space}
Note: L is decidable (by simulation; ¥-loops?)
Q: is LÎDSPACE(s(n)) ?
Assume (towards contradiction) LÎDSPACE(s(n))
i.e., $ a fixedKÎN such that Turing machine MK
decides L within space bound s(n)
i
MK Þ decides / accepts L
If Mi accepts i within t(i) space thenelse
RejectAccept
![Page 22: Resource-Bounded Computation Previously: can something be done? Now: how efficiently can it be done? Goal: conserve computational resources: Time, space,](https://reader038.vdocument.in/reader038/viewer/2022110304/5518b4ec550346c31f8b50c6/html5/thumbnails/22.jpg)
Space Complexity Hierarchy (proof)
Consider whether KÎL:
KÎL Þ MK must accept K within s(K) space
Þ MK must reject K Þ KÏL
KÏL Þ MK must reject K within s(K) space
Þ MK must accept K Þ KÎL
So (KÎL) Û (KÏL), a contradiction!
Þ Assumption is false Þ LÏDSPACE(s(n))
K
MK Þ decides / accepts L
If MK accepts K within s(K) space thenelse
RejectAccept
![Page 23: Resource-Bounded Computation Previously: can something be done? Now: how efficiently can it be done? Goal: conserve computational resources: Time, space,](https://reader038.vdocument.in/reader038/viewer/2022110304/5518b4ec550346c31f8b50c6/html5/thumbnails/23.jpg)
i = 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 …wiÎΣ* = a b aa ab ba bb aaa aab aba abb baa bab bba bbbaaaa…
M1(i) Þ √ ´ √ √ ´ ´ √ √ √ ´ ´ ´ ´ √ ´ …
M2(i) Þ ´ ´ √ ´ ´ √ ´ ´ √ ´ √ √ ´ ´ ´ …
M3(i) Þ √ √ √ ´ √ ´ ´ ´ √ √ √ √ ´ √ ´ …
M4(i) Þ √ ´ √ ´ √ √ ´ √ √ ´ ´ √ ´ ´ ´ …
M5(i) Þ ´ √ √ ´ ´ √ √ ´ √ √ √ √ ´ √ ´ …
. . .
Space Hierarchy (another proof)Consider all s(n)-space-bounded TM’s on all inputs:
M’(i) Þ is s(n) space-bounded.But M’ computes a different function than any Mj
Þ Contradiction!
´ ´√ √ √ . . .
![Page 24: Resource-Bounded Computation Previously: can something be done? Now: how efficiently can it be done? Goal: conserve computational resources: Time, space,](https://reader038.vdocument.in/reader038/viewer/2022110304/5518b4ec550346c31f8b50c6/html5/thumbnails/24.jpg)
Savitch’s TheoremTheorem: NSPACE(f(n)) Í DSPACE (f2(n))Proof: Simulation: idea is to aggressively conserve
and reuse space while sacrificing (lots of) time.Consider a sequence of TM states in one branch of
an NSPACE(f(n))-bounded computation:
Computation time / length is bounded by cf(n) (why?)We need to simulate this branch and all others too!Q: How can we space-efficiently simulate these?A: Use divide-and-conquer with heavy space-reuse!
Walter Savitch
![Page 25: Resource-Bounded Computation Previously: can something be done? Now: how efficiently can it be done? Goal: conserve computational resources: Time, space,](https://reader038.vdocument.in/reader038/viewer/2022110304/5518b4ec550346c31f8b50c6/html5/thumbnails/25.jpg)
Savitch’s TheoremPick a midpoint state along target path:
Verify it is a valid intermediate state by recursively solving both subproblems.
Iterate for all possible midpoint states!The recursion stack depth is at most log(cf(n))=O(f(n))Each recursion stack frame size is O(f(n)).Þ total space needed is O(f(n)*f(n))=O(f2(n))Note: total time is exponential (but that’s OK).Þ non-determinism can be eliminated by squaring
the space: NSPACE(f(n)) Í DSPACE (f2(n))
Walter Savitch
![Page 26: Resource-Bounded Computation Previously: can something be done? Now: how efficiently can it be done? Goal: conserve computational resources: Time, space,](https://reader038.vdocument.in/reader038/viewer/2022110304/5518b4ec550346c31f8b50c6/html5/thumbnails/26.jpg)
Savitch’s TheoremCorollary: NPSPACE = PSPACE
Proof: NPSPACE = È NSPACE(nk) k>1
Í È DSPACE(n2k) k>1
= È DSPACE(nk) k>1
= PSPACE
i.e., polynomial space is invariant with respect to non-determinism!
Q: What about polynomial time?A: Still open! (P=NP)
Walter Savitch
![Page 27: Resource-Bounded Computation Previously: can something be done? Now: how efficiently can it be done? Goal: conserve computational resources: Time, space,](https://reader038.vdocument.in/reader038/viewer/2022110304/5518b4ec550346c31f8b50c6/html5/thumbnails/27.jpg)
Space & ComplementationTheorem: Deterministic space is closed under
complementation, i.e., DSPACE(S(n)) = co-DSPACE(S(n))
= {S*-L | L Î DSPACE(S(n)) }Proof: Simulate & negate.
Theorem [Immerman, 1987]: Nondeterministic space is closed under complementation, i.e. NSPACE(S(n)) = co-NSPACE(S(n))
Proof idea: Similar strategy to Savitch’s theorem.
No similar result is known for any of the standardtime complexity classes!
Q: Is NP = co-NP?A: Still open!
Neil Immerman
![Page 28: Resource-Bounded Computation Previously: can something be done? Now: how efficiently can it be done? Goal: conserve computational resources: Time, space,](https://reader038.vdocument.in/reader038/viewer/2022110304/5518b4ec550346c31f8b50c6/html5/thumbnails/28.jpg)
From:
![Page 29: Resource-Bounded Computation Previously: can something be done? Now: how efficiently can it be done? Goal: conserve computational resources: Time, space,](https://reader038.vdocument.in/reader038/viewer/2022110304/5518b4ec550346c31f8b50c6/html5/thumbnails/29.jpg)
![Page 30: Resource-Bounded Computation Previously: can something be done? Now: how efficiently can it be done? Goal: conserve computational resources: Time, space,](https://reader038.vdocument.in/reader038/viewer/2022110304/5518b4ec550346c31f8b50c6/html5/thumbnails/30.jpg)
Neil Immerman
![Page 31: Resource-Bounded Computation Previously: can something be done? Now: how efficiently can it be done? Goal: conserve computational resources: Time, space,](https://reader038.vdocument.in/reader038/viewer/2022110304/5518b4ec550346c31f8b50c6/html5/thumbnails/31.jpg)
PS
PAC
E-c
ompl
ete
QB
F
The Extended Chomsky Hierarchy
Finite {a,b}Regular a*Det. CF anbn
Context-free wwRP anbncn
NP
Rec
ogni
zabl
e
Not
Rec
ogni
zabl
e
HH
Decidable Presburger arithmetic
NP
-com
plet
e S
AT
Not
fin
itel
y de
scri
babl
e ?
2S*
EXPTIME
EX
PT
IME
-com
plet
e G
o
EX
PS
PAC
E-c
ompl
ete
=R
ETuring
degrees Context sensitive LBA
EXPSPACE=co-EXPSPACE
PSPACE=NPSPACE=co-NPSPACE
![Page 32: Resource-Bounded Computation Previously: can something be done? Now: how efficiently can it be done? Goal: conserve computational resources: Time, space,](https://reader038.vdocument.in/reader038/viewer/2022110304/5518b4ec550346c31f8b50c6/html5/thumbnails/32.jpg)
Q: Can we enumerate TM’s for all languages in P?
Q: Can we enumerate TM’s for all languages in
NP, PSPACE? EXPTIME? EXPSPACE?
Note: not necessarily in a lexicographic order.
Enumeration of Resource-Bounded TMs
Extra credit
![Page 33: Resource-Bounded Computation Previously: can something be done? Now: how efficiently can it be done? Goal: conserve computational resources: Time, space,](https://reader038.vdocument.in/reader038/viewer/2022110304/5518b4ec550346c31f8b50c6/html5/thumbnails/33.jpg)
Denseness of Space HierarchyQ: How much additional space does it
take to recognize more languages?A: Very little more!
Theorem: Given two space bounds s1 and s2 such that
Lim s1(n) / s2(n)=0 as n®¥, i.e., s1(n) = o(s2(n)), $ a decidable language L such that LÎDSPACE(s2(n)) but LÏDSPACE(s1(n)).
Proof idea: Diagonalize efficiently.
Note: s2(n) must be computable within s2(n) space.
Þ Space hierarchy is infinite and very dense!
Juris Hartmanis Richard Stearns
![Page 34: Resource-Bounded Computation Previously: can something be done? Now: how efficiently can it be done? Goal: conserve computational resources: Time, space,](https://reader038.vdocument.in/reader038/viewer/2022110304/5518b4ec550346c31f8b50c6/html5/thumbnails/34.jpg)
Space hierarchy is infinite
and very dense!
Examples:
DSPACE(log n) Ì DSPACE(log2 n)
DSPACE(n) Ì DSPACE(n log n)
DSPACE(n2) Ì DSPACE(n2.001)
DSPACE(nx) Ì DSPACE(ny) "1<x<y
Corollary: LOGSPACE ¹ PSPACE
Corollary: PSPACE ¹ EXPSPACE
Denseness of Space Hierarchy
Juris Hartmanis Richard Stearns
![Page 35: Resource-Bounded Computation Previously: can something be done? Now: how efficiently can it be done? Goal: conserve computational resources: Time, space,](https://reader038.vdocument.in/reader038/viewer/2022110304/5518b4ec550346c31f8b50c6/html5/thumbnails/35.jpg)
Q: How much additional time does it take to recognize more languages?
A: At most a logarithmic factor more!
Theorem: Given two time bounds t1 and t2 such that
t1(n)×log(t1(n)) = o(t2(n)), $ a decidable language L
such that LÎDTIME(t2(n)) but LÏDTIME(t1(n)).
Proof idea: Diagonalize efficiently.
Note: t2(n) must be computable within t2(n) time.
Þ Time hierarchy is infinite and pretty dense!
Denseness of Time Hierarchy
Juris Hartmanis Richard Stearns
![Page 36: Resource-Bounded Computation Previously: can something be done? Now: how efficiently can it be done? Goal: conserve computational resources: Time, space,](https://reader038.vdocument.in/reader038/viewer/2022110304/5518b4ec550346c31f8b50c6/html5/thumbnails/36.jpg)
Time hierarchy is infinite
and pretty dense!
Examples:
DTIME(n) Ì DTIME(n log2 n)
DTIME(n2) Ì DTIME(n2.001)
DTIME(2n) Ì DTIME(n22n)
DTIME(nx) Ì DTIME(ny) "1<x<y
Corollary: LOGTIME ¹ P
Corollary: P ¹ EXPTIME
Denseness of Time Hierarchy
Juris Hartmanis Richard Stearns
![Page 37: Resource-Bounded Computation Previously: can something be done? Now: how efficiently can it be done? Goal: conserve computational resources: Time, space,](https://reader038.vdocument.in/reader038/viewer/2022110304/5518b4ec550346c31f8b50c6/html5/thumbnails/37.jpg)
Complexity Classes RelationshipsTheorems: LOGTIME Í L Í NL Í P Í NP Í PSPACE
Í EXPTIME Í NEXPTIME Í EXPSPACE Í . . .
Theorems: L ¹ PSPACE ¹ EXPSPACE ...¹
Theorems: LOGTIME ¹ P ¹ EXPTIME ...¹
Conjectures: L¹NL, NL¹P, P¹NP, NP¹PSPACE,
PSPACE¹EXPTIME, EXPTIME¹NEXPTIME,
NEXPTIME¹EXPSPACE, . . .
Theorem: At least two of the above conjectures are true!
![Page 38: Resource-Bounded Computation Previously: can something be done? Now: how efficiently can it be done? Goal: conserve computational resources: Time, space,](https://reader038.vdocument.in/reader038/viewer/2022110304/5518b4ec550346c31f8b50c6/html5/thumbnails/38.jpg)
Theorem: At least two
of the following
conjectures are true:
L¹LNL¹PP¹NP
NP¹PSPACE
PSPACE¹EXPTIME
EXPTIME¹NEXPTIME
NEXPTIME¹EXPSPACE
Open problems!
Theorem: P ¹ SPACE(n) Open: P Ì SPACE(n) ? Open: SPACE(n) Ì P ?Open: NSPACE(n) ¹ DSPACE(n) ?
![Page 39: Resource-Bounded Computation Previously: can something be done? Now: how efficiently can it be done? Goal: conserve computational resources: Time, space,](https://reader038.vdocument.in/reader038/viewer/2022110304/5518b4ec550346c31f8b50c6/html5/thumbnails/39.jpg)
……………
……………
PS
PAC
E-c
ompl
ete
QB
F
The Extended Chomsky Hierarchy Reloaded
Context-free wwRP anbncn
NP
Rec
ogni
zabl
e
Not
Rec
ogni
zabl
e
HH
Decidable Presburger arithmetic
NP
-com
plet
e S
AT
Not
fin
itel
y de
scri
babl
e ?
2S*
EXPTIME
EX
PT
IME
-com
plet
e G
o
EX
PS
PAC
E-c
ompl
ete
=R
E
Context sensitive LBA
EXPSPACE=co-EXPSPACE
PSPACE=NPSPACE=co-NPSPACE
Dense infinite time & space complexity hierarchies……………
……………
……………
……………
……………
……………
……………
……………Regular a*
…… … ……
…… … …………………
Turingdegrees
Other infinite complexity & descriptive hierarchies
……………Det. CF anbn
……………Finite {a,b}
![Page 40: Resource-Bounded Computation Previously: can something be done? Now: how efficiently can it be done? Goal: conserve computational resources: Time, space,](https://reader038.vdocument.in/reader038/viewer/2022110304/5518b4ec550346c31f8b50c6/html5/thumbnails/40.jpg)
Gap Theorems$ arbitrarily large space & time complexity gaps!
Theorem [Borodin]: For any computable function g(n), $ t(n) such that DTIME(t(n)) = DTIME(g(t(n)).
Ex: DTIME(t(n)) = DTIME(22t(n)) for some t(n)
Theorem [Borodin]: For any computable function g(n), $ s(n) such that DSPACE(s(n)) = DSPACE(g(s(n)).
Ex: DSPACE(s(n)) = DSPACE(S(n)s(n)) for some s(n)
Proof idea: Diagonalize over TMs & construct a gap that avoids all TM complexities from falling into it.
Corollary: $ f(n) such that DTIME(f(n)) = DSPACE(f(n)).
Note: does not contradict the space and time hierarchy theorems, since t(n), s(n), f(n) may not be computable.
Allan Borodin
![Page 41: Resource-Bounded Computation Previously: can something be done? Now: how efficiently can it be done? Goal: conserve computational resources: Time, space,](https://reader038.vdocument.in/reader038/viewer/2022110304/5518b4ec550346c31f8b50c6/html5/thumbnails/41.jpg)
The First Complexity GapThe first space “gap” is between O(1) and O(log log n)
Theorem: LÎDSPACE(o(log log n)) Þ
LÎDSPACE(O(1)) Þ L is regular!
All space classes below O(log log n) collapes to O(1).
Allan Borodin
![Page 42: Resource-Bounded Computation Previously: can something be done? Now: how efficiently can it be done? Goal: conserve computational resources: Time, space,](https://reader038.vdocument.in/reader038/viewer/2022110304/5518b4ec550346c31f8b50c6/html5/thumbnails/42.jpg)
Speedup TheoremThere are languages for which there are no asymptotic
space or time lower bounds for deciding them!
Theorem [Blum]: For any computable function g(n), $ a
language L such that if TM M accepts L within t(n) time,
$ another TM M’ that accepts L within g(t(n)) time.
Corollary [Blum]: There is a problem such that if any
algorithm solves it in time t(n), $ other algorithms that solve it, in times O(log t(n)), O(log(log t(n))),O(log(log(log t(n)))), ...
Þ Some problems don’t have an “inherent” complexity!
Note: does not contradict the time hierarchy theorem!
Manuel Blum
![Page 43: Resource-Bounded Computation Previously: can something be done? Now: how efficiently can it be done? Goal: conserve computational resources: Time, space,](https://reader038.vdocument.in/reader038/viewer/2022110304/5518b4ec550346c31f8b50c6/html5/thumbnails/43.jpg)
From:
Manuel Blum
![Page 44: Resource-Bounded Computation Previously: can something be done? Now: how efficiently can it be done? Goal: conserve computational resources: Time, space,](https://reader038.vdocument.in/reader038/viewer/2022110304/5518b4ec550346c31f8b50c6/html5/thumbnails/44.jpg)
![Page 45: Resource-Bounded Computation Previously: can something be done? Now: how efficiently can it be done? Goal: conserve computational resources: Time, space,](https://reader038.vdocument.in/reader038/viewer/2022110304/5518b4ec550346c31f8b50c6/html5/thumbnails/45.jpg)
![Page 46: Resource-Bounded Computation Previously: can something be done? Now: how efficiently can it be done? Goal: conserve computational resources: Time, space,](https://reader038.vdocument.in/reader038/viewer/2022110304/5518b4ec550346c31f8b50c6/html5/thumbnails/46.jpg)
Abstract Complexity TheoryComplexity theory can be machine-independent!
Instead of referring to TM’s, we state simple axioms
that any complexity measure F must satisfy.
Example: the Blum axioms:
1) F(M,w) is finite iff M(w) halts; and
2) The predicate “F(M,w)=n” is decidable.
Theorem [Blum]: Any complexity measure satisfying these axioms gives rise to hierarchy, gap, & speedup theorems.
Corollary: Space & time measures satisfy these axioms.
AKA “Axiomatic complexity theory [Blum, 1967]
Manuel Blum
![Page 47: Resource-Bounded Computation Previously: can something be done? Now: how efficiently can it be done? Goal: conserve computational resources: Time, space,](https://reader038.vdocument.in/reader038/viewer/2022110304/5518b4ec550346c31f8b50c6/html5/thumbnails/47.jpg)
AlternationAlternation: generalizes non-determinism, where
each state is either “existential” or “universal”Old: existential states
New: universal states • Existential state is accepting iff any
of its child states is accepting (OR)• Universal state is accepting iff all
of its child states are accepting (AND)• Alternating computation is a “tree”.• Final states are accepting• Non-final states are rejecting• Computation accepts iff initial state is accepting
Note: in non-determinism, all states are existential
$
"
"
"
"
"
"
Stockmeyer Chandra
![Page 48: Resource-Bounded Computation Previously: can something be done? Now: how efficiently can it be done? Goal: conserve computational resources: Time, space,](https://reader038.vdocument.in/reader038/viewer/2022110304/5518b4ec550346c31f8b50c6/html5/thumbnails/48.jpg)
AlternationTheorem: a k-state alternating finite automaton
can be converted into an equivalent 2k-state non-deterministic FA.
Proof idea: a generalized powerset construction.
Theorem: a k-state alternating finite automaton can beconverted into an equivalent 22k-state deterministic FA.
Proof: two composed powerset constructions.
Def: alternating Turing machine is an alternating FAwith an unbounded read/write tape.
Theorem: alternation does not increase the language recognition power of Turing machine.
Proof: by simulation.
Stockmeyer Chandra
![Page 49: Resource-Bounded Computation Previously: can something be done? Now: how efficiently can it be done? Goal: conserve computational resources: Time, space,](https://reader038.vdocument.in/reader038/viewer/2022110304/5518b4ec550346c31f8b50c6/html5/thumbnails/49.jpg)
Stockmeyer Chandra
Alternating Complexity ClassesDef: ATIME(t(n))={L | L is decidable in
time O(t(n)) by some alternating TM}
Def: ASPACE(s(n))={L | L decidable inspace O(s(n)) by some alternating TM}
Def: AP = È ATIME(nk) "k>1
AP º alternating polynomial time
Def: APSPACE = È ASPACE(nk) "k>1
APSPACE º alternating polynomial space
Prope
rties
!
![Page 50: Resource-Bounded Computation Previously: can something be done? Now: how efficiently can it be done? Goal: conserve computational resources: Time, space,](https://reader038.vdocument.in/reader038/viewer/2022110304/5518b4ec550346c31f8b50c6/html5/thumbnails/50.jpg)
Alternating Complexity Classes
Def: AEXPTIME = È ATIME(2nk) "k>1
AEXPTIME º alternating exponential time
Def: AEXPSPACE = È ASPACE(2nk) "k>1
AEXPSPACE º alternating exponential space
Def: AL = ALOGSPACE = ASPACE(log n)
AL º alternating logarithmic space
Note: AP, ASPACE, AL are model-independent
Stockmeyer Chandra
![Page 51: Resource-Bounded Computation Previously: can something be done? Now: how efficiently can it be done? Goal: conserve computational resources: Time, space,](https://reader038.vdocument.in/reader038/viewer/2022110304/5518b4ec550346c31f8b50c6/html5/thumbnails/51.jpg)
Stockmeyer Chandra
Alternating Space/Time RelationsTheorem: P Í NP Í AP
Open: NP = AP ?
Open: P = AP ?
Corollary: P=AP Þ P=NP
Theorem: ATIME(f(n)) Í DSPACE(f(n)) Í ATIME(f 2(n))
Theorem: PSPACE = NPSPACE Í APSPACE
Theorem: ASPACE(f(n)) Í DTIME(cf(n))
Theorem: AL = P
Theorem: AP = PSPACE
Theorem: APSPACE = EXPTIME
Theorem: AEXPTIME = EXPSPACE
$
"
![Page 52: Resource-Bounded Computation Previously: can something be done? Now: how efficiently can it be done? Goal: conserve computational resources: Time, space,](https://reader038.vdocument.in/reader038/viewer/2022110304/5518b4ec550346c31f8b50c6/html5/thumbnails/52.jpg)
Quantified Boolean Formula ProblemDef: Given a fully quantified Boolean formula, where each
variable is quantified existentially or universally, does it evaluate to “true”?
Example: Is “"x $ y $ z (x Ù z) Ú y” true?• Also known as quantified satisfiability (QSAT)• Satisfiability (one $ only) is a special case of QBF
Theorem: QBF is PSPACE-complete.Proof idea: combination of [Cook] and [Savitch].Theorem: QBF Î TIME(2n)Proof: recursively evaluate all possibilities.Theorem: QBF Î DSPACE(n)Proof: reuse space during exhaustive evaluations.Theorem: QBF Î ATIME(n)Proof: use alternation to guess and verify formula.
![Page 53: Resource-Bounded Computation Previously: can something be done? Now: how efficiently can it be done? Goal: conserve computational resources: Time, space,](https://reader038.vdocument.in/reader038/viewer/2022110304/5518b4ec550346c31f8b50c6/html5/thumbnails/53.jpg)
QBF and Two-Player Games• SAT solutions can be succinctly (polynomially) specified.• It is not known how to succinctly specify QBF solutions.• QBF naturally models winning strategies
for two-player games:
$ a move for player A
" moves for player B
$ a move for player A
" moves for player B
$ a move for player A...
player A has a winning move!
"
$
$
![Page 54: Resource-Bounded Computation Previously: can something be done? Now: how efficiently can it be done? Goal: conserve computational resources: Time, space,](https://reader038.vdocument.in/reader038/viewer/2022110304/5518b4ec550346c31f8b50c6/html5/thumbnails/54.jpg)
QBF and Two-Player GamesTheorem: Generalized Checkers is EXPTIME-complete.
Theorem: Generalized Chess is EXPTIME-complete.
Theorem: Generalized Go is EXPTIME-complete.
Theorem: Generalized Othello is PSPACE-complete.
![Page 55: Resource-Bounded Computation Previously: can something be done? Now: how efficiently can it be done? Goal: conserve computational resources: Time, space,](https://reader038.vdocument.in/reader038/viewer/2022110304/5518b4ec550346c31f8b50c6/html5/thumbnails/55.jpg)
Meyer
Idea: bound # of “existential” / “universal” states
Old: unbounded existential / universal states
New: at most i existential / universal alternations
Def: a Si-alternating TM has at most i runs of
quantified steps, starting with existential
Def: a Pi-alternating TM has at most i runs of
quantified steps, starting with universal
Note: Pi- and Si- alternation-bounded TMs
are similar to unbounded alternating TMs
Stockmeyer
$
"
"
"
"
"
"
The Polynomial Hierarchy
i=1
i=2
i=3
i=4
i=5
S5-alternating
![Page 56: Resource-Bounded Computation Previously: can something be done? Now: how efficiently can it be done? Goal: conserve computational resources: Time, space,](https://reader038.vdocument.in/reader038/viewer/2022110304/5518b4ec550346c31f8b50c6/html5/thumbnails/56.jpg)
Def: SiTIME(t(n))={L | L is decidable withintime O(t(n)) by some Si-alternating TM}
Def: SiSPACE(s(n))={L | L is decidable within space O(s(n)) by some Si-alternating TM}
Def: PiTIME(t(n))={L | L is decidable within time O(t(n)) by some Pi-alternating TM}
Def: PiSPACE(s(n))={L | L is decidable within space O(s(n)) by some Pi-alternating TM}
Def: SiP = È SiTIME(nk) "k>1
Def: PiP = È PiTIME(nk) "k>1
Stockmeyer
The Polynomial Hierarchy
Meyer
"
"
"
"
"
![Page 57: Resource-Bounded Computation Previously: can something be done? Now: how efficiently can it be done? Goal: conserve computational resources: Time, space,](https://reader038.vdocument.in/reader038/viewer/2022110304/5518b4ec550346c31f8b50c6/html5/thumbnails/57.jpg)
Def: SPH = È SiP "i>1
Def: PPH = È PiP "i>1
Theorem: SPH = PPH
Def: The Polynomial Hierarchy PH = SPH Þ Languages accepted by polynomialtime, unbounded-alternations TMs
Theorem: S0P= P0P= P
Theorem: S1P=NP, P1P= co-NP
Theorem: SiP Í Si+1P, PiP Í Pi+1P
Theorem: SiP Í Pi+1P, PiP Í Si+1P
Stockmeyer
The Polynomial Hierarchy
Meyer
"
"
"
"
"
![Page 58: Resource-Bounded Computation Previously: can something be done? Now: how efficiently can it be done? Goal: conserve computational resources: Time, space,](https://reader038.vdocument.in/reader038/viewer/2022110304/5518b4ec550346c31f8b50c6/html5/thumbnails/58.jpg)
Theorem: SiP Í PSPACE
Theorem: PiP Í PSPACE
Theorem: PH Í PSPACE
Open: PH = PSPACE ?Open: S0P=S1P?Û P=NP ?
Open: P0P=P1P?Û P=co-NP ?
Open: S1P=P1P?Û NP=co-NP ?
Open: SkP = Sk+1P for any k ?
Open: PkP = Pk+1P for any k ?
Open: SkP = PkP for any k ?
Theorem: PH = languages expressible by 2nd-order logic
Infinite number of “P=NP”–type open problems!
StockmeyerMeyer
"
"
"
"
"
The Polynomial Hierarchy
![Page 59: Resource-Bounded Computation Previously: can something be done? Now: how efficiently can it be done? Goal: conserve computational resources: Time, space,](https://reader038.vdocument.in/reader038/viewer/2022110304/5518b4ec550346c31f8b50c6/html5/thumbnails/59.jpg)
Open: Is the polynomial hierarchy infinite ?
Theorem: If any two successive levels conicide (SkP = Sk+1P or SkP = PkP for some k) then the entire polynomial hierarchy collapses to that level (i.e., PH = SkP = PkP).
Corollary: If P = NP then the entire polynomial hierarchy collapses completely (i.e., PH = P = NP).
Theorem: P=NPÛ P=PH
Corollary: To show P≠NP, it suffices to show P≠PH.
Theorem: There exist oracles that separate SkP ≠ Sk+1P.
Theorem: PH contains almost all well-known complexity classes in PSPACE, including P, NP, co-NP, BPP, RP, etc.
StockmeyerMeyer
The Polynomial Hierarchy
![Page 60: Resource-Bounded Computation Previously: can something be done? Now: how efficiently can it be done? Goal: conserve computational resources: Time, space,](https://reader038.vdocument.in/reader038/viewer/2022110304/5518b4ec550346c31f8b50c6/html5/thumbnails/60.jpg)
……………
……………
PS
PAC
E-c
ompl
ete
QB
F
The Extended Chomsky Hierarchy Reloaded
Context-free wwRP anbncn
NP
Rec
ogni
zabl
e
Not
Rec
ogni
zabl
e
HH
Decidable Presburger arithmetic
NP
-com
plet
e S
AT
Not
fin
itel
y de
scri
babl
e ?
2S*
EXPTIME
EX
PT
IME
-com
plet
e G
o
EX
PS
PAC
E-c
ompl
ete
=R
E
Context sensitive LBA
EXPSPACE
PSPACE
Dense infinite time & space complexity hierarchies……………
……………
……………
……………
……………
……………
……………
……………Regular a*
…… … ……
…… … …………………
Turingdegrees
Other infinite complexity & descriptive hierarchies
……………Det. CF anbn
……………Finite {a,b}
…………
…PH
![Page 61: Resource-Bounded Computation Previously: can something be done? Now: how efficiently can it be done? Goal: conserve computational resources: Time, space,](https://reader038.vdocument.in/reader038/viewer/2022110304/5518b4ec550346c31f8b50c6/html5/thumbnails/61.jpg)
Probabilistic Turing MachinesIdea: allow randomness / coin-flips during computation
Old: nondeterministic states New: random states changes via coin-flips• Each coin-flip state has two successor states
Def: Probability of branch B is Pr[B] = 2-k
where k is the # of coin-flips along B.
Def: Probability that M accepts w is sum ofthe probabilities of all accepting branches.
Def: Probability that M rejects w is 1 – (probability that M accepts w).
Def: Probability that M accepts L with probability e if: wÎL Þ probability(M accepts w) ³ 1-e
wÏL Þ probability(M rejects w) ³ 1-e
![Page 62: Resource-Bounded Computation Previously: can something be done? Now: how efficiently can it be done? Goal: conserve computational resources: Time, space,](https://reader038.vdocument.in/reader038/viewer/2022110304/5518b4ec550346c31f8b50c6/html5/thumbnails/62.jpg)
Probabilistic Turing MachinesDef: BPP is the class of languages accepted by
probabilistic polynomial time TMs with error e =1/3.
Note: BPP Bounded-error Probabilistic Polynomial time
Theorem: any error threshold 0< <1/2e can be substituted.Proof idea: run the probabilistic TM multiple times
and take the majority of the outputs.
Theorem [Rabin, 1980]: Primality testing is in BPP.
Theorem [Agrawal et al., 2002]: Primality testing is in P.
Note: BPP is one of the largest practical classesof problems that can be solved effectively.
Theorem: BPP is closed under complement (BPP=co-BPP).Open: BPP Í NP ?Open: NP Í BPP ?
![Page 63: Resource-Bounded Computation Previously: can something be done? Now: how efficiently can it be done? Goal: conserve computational resources: Time, space,](https://reader038.vdocument.in/reader038/viewer/2022110304/5518b4ec550346c31f8b50c6/html5/thumbnails/63.jpg)
Probabilistic Turing MachinesTheorem: BPP Í PH
Theorem: P=NP Þ BPP=P
Theorem: NP Í BPP Þ PH Í BPP
Note: the former is unlikely, since this would imply efficient
randomized algorithms for many NP-hard problems.
Def: A pseudorandom number generator (PRNG) is an algorithm for generating number sequences that approximates the properties of random numbers.
Theorem: The existance of strong
PRNGs implies that P=BPP.
“Anyone who considers arithmetical methods of producing random digits
is, of course, in a state of sin.”
John von Neumann
![Page 64: Resource-Bounded Computation Previously: can something be done? Now: how efficiently can it be done? Goal: conserve computational resources: Time, space,](https://reader038.vdocument.in/reader038/viewer/2022110304/5518b4ec550346c31f8b50c6/html5/thumbnails/64.jpg)
……………
……………
PS
PAC
E-c
ompl
ete
QB
F
The Extended Chomsky Hierarchy Reloaded
Context-free wwRP anbncn
NP
Rec
ogni
zabl
e
Not
Rec
ogni
zabl
e
HH
Decidable Presburger arithmetic
NP
-com
plet
e S
AT
Not
fin
itel
y de
scri
babl
e ?
2S*
EXPTIME
EX
PT
IME
-com
plet
e G
o
EX
PS
PAC
E-c
ompl
ete
=R
E
Context sensitive LBA
EXPSPACE
PSPACE
Dense infinite time & space complexity hierarchies……………
……………
……………
……………
……………
……………
……………
……………Regular a*
…… … ……
…… … …………………
Turingdegrees
Other infinite complexity & descriptive hierarchies
……………Det. CF anbn
……………Finite {a,b}
…………
…PH BPP
![Page 65: Resource-Bounded Computation Previously: can something be done? Now: how efficiently can it be done? Goal: conserve computational resources: Time, space,](https://reader038.vdocument.in/reader038/viewer/2022110304/5518b4ec550346c31f8b50c6/html5/thumbnails/65.jpg)
![Page 66: Resource-Bounded Computation Previously: can something be done? Now: how efficiently can it be done? Goal: conserve computational resources: Time, space,](https://reader038.vdocument.in/reader038/viewer/2022110304/5518b4ec550346c31f8b50c6/html5/thumbnails/66.jpg)
![Page 67: Resource-Bounded Computation Previously: can something be done? Now: how efficiently can it be done? Goal: conserve computational resources: Time, space,](https://reader038.vdocument.in/reader038/viewer/2022110304/5518b4ec550346c31f8b50c6/html5/thumbnails/67.jpg)
![Page 68: Resource-Bounded Computation Previously: can something be done? Now: how efficiently can it be done? Goal: conserve computational resources: Time, space,](https://reader038.vdocument.in/reader038/viewer/2022110304/5518b4ec550346c31f8b50c6/html5/thumbnails/68.jpg)
![Page 69: Resource-Bounded Computation Previously: can something be done? Now: how efficiently can it be done? Goal: conserve computational resources: Time, space,](https://reader038.vdocument.in/reader038/viewer/2022110304/5518b4ec550346c31f8b50c6/html5/thumbnails/69.jpg)
![Page 70: Resource-Bounded Computation Previously: can something be done? Now: how efficiently can it be done? Goal: conserve computational resources: Time, space,](https://reader038.vdocument.in/reader038/viewer/2022110304/5518b4ec550346c31f8b50c6/html5/thumbnails/70.jpg)
The “Complexity Zoo”Class inclusion diagram
• Currently 493 named classes!
• Interactive, clickable map
• Shows class subset relations
Legend:
http://www.math.ucdavis.edu/~greg/zoology/diagram.xml Scott Aaronson
![Page 71: Resource-Bounded Computation Previously: can something be done? Now: how efficiently can it be done? Goal: conserve computational resources: Time, space,](https://reader038.vdocument.in/reader038/viewer/2022110304/5518b4ec550346c31f8b50c6/html5/thumbnails/71.jpg)
![Page 72: Resource-Bounded Computation Previously: can something be done? Now: how efficiently can it be done? Goal: conserve computational resources: Time, space,](https://reader038.vdocument.in/reader038/viewer/2022110304/5518b4ec550346c31f8b50c6/html5/thumbnails/72.jpg)
2S*
Recognizable
Decidable
Polynomial space
Exponential space
Deterministicexponential
time
Non-deterministicexponential
time
![Page 73: Resource-Bounded Computation Previously: can something be done? Now: how efficiently can it be done? Goal: conserve computational resources: Time, space,](https://reader038.vdocument.in/reader038/viewer/2022110304/5518b4ec550346c31f8b50c6/html5/thumbnails/73.jpg)
Polynomial space
Deterministicpolynomial time
Non-deterministicpolynomial time
Non-deterministiclinear time
Non-deterministiclinear space
Polynomialtime hierarchy
Interactiveproofs
![Page 74: Resource-Bounded Computation Previously: can something be done? Now: how efficiently can it be done? Goal: conserve computational resources: Time, space,](https://reader038.vdocument.in/reader038/viewer/2022110304/5518b4ec550346c31f8b50c6/html5/thumbnails/74.jpg)
Deterministicpolynomial time
Deterministiclinear time
Non-deterministic
linear time
Poly-logarithmic time
Context-sensitive
Deterministic context-free
Regular
Deterministiclogarithmic space
Non-deterministiclogarithmic space
Empty set
Contextfree
![Page 75: Resource-Bounded Computation Previously: can something be done? Now: how efficiently can it be done? Goal: conserve computational resources: Time, space,](https://reader038.vdocument.in/reader038/viewer/2022110304/5518b4ec550346c31f8b50c6/html5/thumbnails/75.jpg)
……………
……………
PS
PAC
E-c
ompl
ete
QB
F
The Extended Chomsky Hierarchy Reloaded
Context-free wwRP anbncn
NP
Rec
ogni
zabl
e
Not
Rec
ogni
zabl
e
HH
Decidable Presburger arithmetic
NP
-com
plet
e S
AT
Not
fin
itel
y de
scri
babl
e ?
2S*
EXPTIME
EX
PT
IME
-com
plet
e G
o
EX
PS
PAC
E-c
ompl
ete
=R
E
Context sensitive LBA
EXPSPACE
PSPACE
Dense infinite time & space complexity hierarchies……………
……………
……………
……………
……………
……………
……………
……………Regular a*
…… … ……
…… … …………………
Turingdegrees
Other infinite complexity & descriptive hierarchies
……………Det. CF anbn
……………Finite {a,b}
…………
…PH BPP