got the Gödel Prize in 1995! ⟶ solved many problems in the aftermath
Th: Immermann-Szelepcsenyi:coNSPACE(f(n)) ⊆ NSPACE(f(n))
for $f(n)≥ \log (n)$
Reminder: GAP is a.k.a REACH
First step: coGAP ∈ NL
is there NO path between $s$ and $t$?
We will guess the set of reachable states, but we can’t remember them (too large: doesn’t fit in Logspace) ⟶ instead, one remembers their number
d = 0; N = 1 repeat pN = N # previous value of N N = 0 for all nodes n in G: new = false # is it a new one ? cpt = 0 for all nodes n' in G: guess if s ⟶_≤d n' # the NL algo for GAP if yes: cpt += 1 if n' ⟶_≤1 n: -> if n == t: reject N += 1 new = true if cpt < pN: reject # the machine has forgotten some paths s ⟶_≤d n' d = d+1 # N is the new value until N = pN -> now accept
The run which always guesses right gives the correct computation.
What about $t$? ⟶ the $→$ at the beginning of the lines
Now when it comes to our $f$:
Let $L ∈ coNSPACE(f(n))$
⟹ there exists a non-deterministic TM $ℳ$ recognizing $L$
Given $x ∈ A^\ast$, let’s build a non-deterministic (in space $f(n)$) TM $ℳ’$ which decides “$ℳ$ rejects $x$”.
⟹ Now the graph is the configurations of $ℳ$ on $x$.
$ℳ’$ runs algo for coGAP on this graph, with $s$ being the initial configuration on $x$, $t$ being a final one.
But one has to add those instructions:
(at the beginning) size seen so far = size of |x| (after the initial repeat) configuration of size ≤ size seen so far + O(1) # to go from one node to another, without testing all the possible configurations (it would require that f is computable)
NB: $f ≥ \log n$ because of the pointer in the input ⟶ not bounded by $f$, bu the need to store it in the configurations.
Corollary: for $f = \log$:coNL ⊆ NL
And as $co(coNL) = NL$, we have the other inclusion:
QBF: Quantified Boolean Formulae
Ex: $∃x. ∀y. (x ∧ ∃z.(y ∧ z))$
QBF ∈ PSPACE
eval(φ): if φ = ∃x. ψ: eval(ψ[x←true]) or eval(ψ[x←false]) if φ = ∀x. ψ: eval(ψ[x←true]) and eval(ψ[x←false]) if φ = ψ ∨ χ: eval(ψ) or eval(χ) if φ = ψ ∧ χ: eval(ψ) and eval(χ) if φ = true: return true else: return false
Stack of size $\vert φ \vert$, each level being of size $\vert φ \vert$ ⟶ overall space memory = $\vert φ \vert^2$
$QBF$ is $PSPACE$-complete
Proof: we will reuse the same ideas than when we showed that $NP \ni L ≼ SAT$
$L$ recognized by $ℳ$ in time $t(n)$, space $f(n)$.
We will assume, wlog, that the machine writes on the input tape, in space $≤ f(n)$.
As the space is now in $f(n)$, the time is exponential in $f(n)$, so we use the quantifiers of the QBF formula to resort to dichotomy on the time.
The problem is that the formulas “going from $x$ to $y$ in $2^d$ steps” use exponential space ⟶ trick: we use universal quantifiers not to copy them twice at each recursive step.