Space-Bounded Computation

Komplexitätstheorie

Space-Bounded Computation

• Input tape: read-only, for free

• working tape: read/write, incurs cost

• output tape: write-only, one-way, for free

→ streaming computation Def: A ≼

B iff there is some log-space

computable f: Σ * →Σ * s.t. x ∈ A ⇔ f(x) ∈ B ^.

Call A NL NL ^{-hard if every B} _∈ NL NL satisfies B ≼

A,

NL NL ^-complete if in addition A ∈ NL NL ^holds.

L L ⊆ NL NL ⊆ P P

NP NP

PSPACE=NPSPACE PSPACE=NPSPACE

EXP EXP

Komplexitätstheorie

Reductions

• polynomial-time many-one: A ≼

B

f: Σ * →Σ * P P -computable s.t.: x ∈ A f(x) ∈ B – in general B †

Σ

\B ^(example?)

• log-spacebounded many-one: A ≼

B

f: Σ * →Σ * L L -computable s.t.: x ∈ A f(x) ∈ B

– A ≼

B ⇒ _{A ≼}

pm

B ^(proof?)

• polynomial-time Turing Reduction: A ≼

B

A can be solved in polynomial time

by aid of oracle queries to B: A ∈ P P

– A ≼

B ⇒ A ≼

B ⇒ A ≼

Σ

\B ^(proof?)

• Further reductions: truthtable, parsimonious…

Komplexitätstheorie

Theorem: dirPath ^is NL NL ^-complete

Let A∈

NL NL

M accepts w ⇔ there is a path in G from s to t G=(V,E), V:=all configurations of M of size c·log |w|

(K₁,K₂)∈E :⇔ K₂ is a successor config of K₁

s:=start config of M on w; t:=accept.config (wlog unique)

• M accepts w ⇔ there is a path in G from s to t √

• How large is G? Constructible in logarithmic space?

qed

Komplexitätstheorie

Immerman-Szelepcsényi

L = NL ? =

L = NL ? = coNL coNL ? ? = P ? = P ?

P P

NP NP

L L

NL NL

NL NL

NL NL

P P

P P

(probably do not admit an efficient parallelization)

NL NL

coNL coNL

Theorem (Neil Immerman,

Róbert Szelepcsényi):

NL NL

coNL coNL

Proof: Show

dirGraph

coNL coNL

Theorem: For s(n)≥log n increasing,

NSPACE

NSPACE(

)

coNSPACE( coNSPACE

)

Komplexitätstheorie

Given G=(V,E), s,t∈V={1,…,m}. Goal:

Logspace NTM accepts iff t not reachable from s.

A_i := { v∈V : exists path in G of length ≤i from s to v }, c_i := #A_i , i=0,…,m-1. A₀={s}, c₀=1. Accept iff t∉A_m-1 Definition: NTM computes (partial) f:⊆Σ*→Σ* iff

• ∀ inputs x∈dom(f) there is an accepting computation.

• Every accepting computation outputs f(x).

FNL FNL

NL NL

Given (!) c_i, logspace NTM can even enumerate A_i:

• For each v∈V, ‘guess‘ whether v∈A_i (1) or not (0)

• If guessed 1: output, verify (

NL NL

• In the end, accept iff counter=c !!!

dirGraph _∈ coNL coNL

Komplexitätstheorie

Given G=(V,E), s,t∈V={1,…,m}. Goal:

Logspace NTM accepts iff t not reachable from s.

A_i := { v∈V : exists path in G of length ≤i from s to v }, c_i := #A_i , i=0,…,m-1. A₀={s}, c₀=1. Accept iff t∉A_m-1

Lemma: For each i, A_i∈

NL NL

Given (!) c_i, logspace NTM can even enumerate A_i. Lemma: Given (!) c_i, A_i+1∈

coNL coNL

Enumerate A_i and, if no edge to v found, accept.

Lemma: Given c_i, logspace NTM can compute c_i+1:

For each v, ‘guess‘ whether v∈A_i+1 holds, and verify Proof (Theorem): 1=c₀ → c₁ → c₂ → c₃ → … → c_m-1

dirGraph _∈ coNL coNL