E6. Beyond NP
Gabriele R¨oger
University of Basel
May 15, 2019
Complexity Theory: What we already have seen
Complexity theoryinvestigates which problems are “easy” to solve and which ones are “hard”.
two important problem classes:
P: problems that are solvable inpolynomial time by“normal” computation mechanisms
NP: problems that are solvable inpolynomial time with the help ofnondeterminism
We know that P⊆NP, but we do not know whether P = NP.
Many practically relevant problems are NP-complete:
They belong to NP.
All problems in NP can be polynomially reduced to them.
If there is an efficient algorithm forone NP-complete problem, then there are efficient algorithms forall problems in NP.
coNP
Complexity Class coNP
Definition (coNP)
coNPis the set of all languages Lfor which ¯L∈NP.
Example: The complement ofSATis in coNP.
Hardness and Completeness
Definition (Hardness and Completeness) Let C be a complexity class.
A problemY is called C-hardifX ≤pY for allproblemsX ∈C.
Y is calledC-complete ifY ∈C andY is C-hard.
Example (Tautology)
The following problemTautologyis coNP-complete:
Given: a propositional logic formulaϕ Question: Isϕvalid?
Known Results and Open Questions
Open
NP= coNP? Known
P⊆coNP
IfX is NP-complete then ¯L is coNP-complete.
If NP6= coNP then P6= NP.
If a coNP-complete problem is in NP, then NP = coNP.
If a coNP-complete problem is in P, then P = coNP = NP.
Time and Space Complexity
Time
Definition (Reminder: Accepting a Language in Timef) LetM be a DTM or NTM with input alphabet Σ, L⊆Σ∗ a language andf :N0→N0 a function.
M acceptsL in timef if:
1 for all wordsw ∈L: M accepts w in time f(|w|)
2 for all wordsw ∈/L: M does not accept w
TIME(f): all languages accepted by a DTMin time f. NTIME(f): all languages accepted by aNTM in timef. P =S
k∈NTIME(nk) NP =S
k∈NNTIME(nk)
Space
Analogously: A TM accepts a languageLin space f if every word w ∈Lgets accepted using at most of f(|w|) space besides it input on the tape and no w 6∈Lgets accepted.
SPACE(f): all languages accepted by a DTM in spacef. NSPACE(f): all languages accepted by aNTM in spacef.
Important Complexity Classes Beyond NP
PSPACE =S
k∈NSPACE(nk) NPSPACE =S
k∈NNSPACE(nk) EXPTIME =S
k∈NTIME(2nk) EXPSPACE =S
k∈NSPACE(2nk) Some known results:
PSPACE = NPSPACE (from Savitch’s theorem) PSPACE⊆EXPTIME⊆EXPSPACE
(at least one relationship strict)
P6= EXPTIME, PSPACE6= EXPSPACE P⊆NP⊆PSPACE
Polynomial Hierarchy
Oracle Machines
Anoracle machine is like a Turing machine that has access to an oraclewhich can solve some decision problem in constant time.
Example oracle classes:
PNP={L|Lcan get accepted in polynomial time by a DTM PNP={L|with an oracle that decides some problem in NP}
NPNP ={L|Lcan get accepted in pol. time by a NTM NPNP ={L|with an oracle deciding some problem in NP}
Polynomial Hierarchy
Inductively defined:
∆P0 := ΣP0 := ΠP0 := P
∆Pi+1:= PΣPi ΣPi+1:= NPΣPi ΠPi+1:= coNPΣPi PH :=S
kΣPk
∆P0 = ΣP0 = ΠP0 = P = ∆P1 NP = ΣP1 ΠP1= coNP
PNP= ∆P2 ΣP2 ΠP2
∆P3 ΣP3 ΠP3
...
Polynomial Hierarchy: Results
PH⊆PSPACE (PH= PSPACE is open)? There are complete problems for each level.
If there is a PH-complete problem, then the polynomial hierarchy collapses to some finite level.
If P = NP, the polynomial hierarchy collapses to the first level.
Counting
#P
Complexity class#P
Set of functions f :{0,1} →N0, wheref(n) is the number of accepting paths of a polynomial-time NTM
Example (#SAT)
The following problem#SATis #P-complete:
Given: a propositional logic formulaϕ Question: How many models doesϕhave?
End of Part E
coNP Time and Space Complexity Polynomial Hierarchy Counting End of Part E
What’s Next?
contents of this course:
A. background X
. mathematical foundations and proof techniques B. logic X
. How can knowledge be represented?
. How can reasoning be automated?
C. automata theory and formal languagesX . What is a computation?
D. Turing computability X
. What can be computed at all?
E. complexity theory
. What can be computed efficiently?
F. more computability theory . Other models of computability
What’s Next?
contents of this course:
A. background X
. mathematical foundations and proof techniques B. logic X
. How can knowledge be represented?
. How can reasoning be automated?
C. automata theory and formal languagesX . What is a computation?
D. Turing computability X
. What can be computed at all?
E. complexity theoryX
. What can be computed efficiently?
F. more computability theory . Other models of computability
