1. §6 PSPACE and Polynomial Hierarchy
PSPACE-complete problems QBF, 3QBF, GRAPH
(Savitch's Theorem: PSPACE = NPSPACE)
Polynomial Hierarchy: syntactically / semantically
(Baker, Gill & Solovay)

2. QBF and PSPACE NPc coNP ? 3QBF: Given Boolean term Φ(Y1,…Ym), NP ?
does it hold y1 y2 y3 …: Φ(y1,…ym)=1 ?
PSPACE
Recursively evaluate quantifiers:
s(m) = s(m-1) + poly(n)
Theorem: QBF is PSPACE-complete.
Proof: Let A decide LPSPACE in space s:=poly(n).
Encode configurations of A in s bits u=(u1,…us).
edge from u to v if v encodes A's unique config after u.
G=GA,s digraph with u=start config on input x, w unique accepting config.
A accepts x  PG(u,w,s),
: "path of length ≤2m from u to w"
 v: PG(u,v,s-1)  PG(v,w,s-1)
 v s,t: (s=u  t=v)  (s=v  t=w) → PG(s,t,s-1)
Draw w u v
of length ≤2s
SAT: Given Boolean term Φ(Y1,…Yn),
does it hold y1,…, yn{0,1}: Φ(y1,… yn)=1 ?
NPc

3. Two-Player Game on Graphs
3QBF: Given Boolean term Φ(X1,…Xm) in 3CNF,
does it hold x1 x2 x3 …: Φ(x1,…xm)=1 ?
PSPACEc
Fix digraph G with start vertex s. Rules:
Red (start) and blue player alternatingly
mark current vertex, and follow any outgoing edge
to a yet unmarked vertex.
Who cannot move, loses.
Red has a winning strategy if,
however blue reacts, red can follow such that,
however ……… blue looses.
Decision problem: Given (G,s),
does red have a winning strategy?

4. Two-Player Game on Graphs
3QBF: Given Boolean term Φ(X1,…Xm) in 3CNF,
does it hold x1 x2 x3 …: Φ(x1,…xm)=1 ?
PSPACEc n 7
Proof (reduction from 3QBF):
Let Φ = C1  C2  …  Ck
See illustration for  = (x1  x2  x3)  (x2  x3  x7)  …  Ck
Theorem: The following is PSPACE-complete:
Decision problem: Given (G,s),
does red have a winning strategy?

5. Oracle Complexity Theory
Fix ON. An OWHILE+ program AO has test instructions "xjO?"
Definition: Fix some class C of subsets LN.
PC := { LN decided by polytime OWHILE+ AO, OC }
NPC := { LN accep. nondet.poly. OWHILE+ AO, OC }
Examples:
a) MinBF  NPSAT = NPNP  PNPNP
b) PP=P, NPP=NP, PSPACEPSPACE=PSPACE
c) NP  coNP  PNP; „≠“ unless NP=coNP

6. Semantic Polynomial Hierarchy:
Δ3P
Def: Δ0P =Σ0P =Π0P := P
Δk+1P := PΣkP
Σk+1P := NPΣkP
Πk+1P := coNPΣkP
PH := ΣkP
Lemma: a) ΔkP=co-ΔkP
b) ΔkP  ΣkP  ΠkP
c) ΣkP  ΠkP  Δk+1P
d) PH  PSPACE
Σ2PΠ2P
= PΠkP
NPNP
coNPNP
= NPΠkP
=Σ2P
= Π2P
= coNPΠkP
PNP
=Δ2P
NPcoNP NP
coNP
=Σ1P
=Π1P P

7. k+1P = NPkP =NPkP k+1P =coNPΣkP =coNPkP
Syntactic Polynomial Hierarchy
Abbreviate Nn := { yN : ℓ(y)≤n }
Theorem: a) LN belongs to coNP iff
L = { x | yNpoly(n) : x,yV } for some VP
b) L belongs to k+1 iff
L = { x | yNpoly(n) : x,yW } for some Wk or k
c) L belongs to k+1 iff
L = { x | yNpoly(n) : x,yZ } for some Zk or k
d) L belongs to k iff
L = { x | y1Npoly(n) y2Npoly(n) y3… QkykNpoly(n) : x,y1,y2,…ykA } for some AP
n:=ℓ(x)
"" if k odd, "" else
k+1P = NPkP =NPkP k+1P =coNPΣkP =coNPkP

8. Walter Savitch's Theorem
Def: For nondecreasing f:N→N, let TIME(f) := { L {0,1}* decidable by WHILE+ program in time f(n) }
Similarly NTIME(f), SPACE(f), NSPACE(f): nondet. WHILE+
Theorem: For any polynom. p, NSPACE(p)SPACE(p²)
Proof: Let nondet. A accept L in space s:=p(n)
Encode configurations of A in s bits u=(u1,…us).
edge from u to v if v encodes A's possible config after u.
G=GA,s digraph with u=start config on input x, w unique accepting config.
A accepts x  PG(u,w,s),
: "path of length ≤2m from u to w"
 v: PG(u,v,s-1)  PG(v,w,s-1)
Recursive algorithm of depth s stores v in s bits: O(s²)
Draw u w
of length ≤2s
SAT: Given Boolean term Φ(Y1,…Yn),
does it hold y1,…, yn{0,1}: Φ(y1,… yn)=1 ? v