Complexity Classes
Karl Lieberherr
Source
• From
• http://people.cs.umass.edu/~immerman/desc
riptive_complexity.html
Polynomial-time
Hierarchy
complete
complete
SO
coNP=SO A
NP=SO E
NP intersect coNP
complete
P
Existential second-order logic
3-colorability can be expressed quite informally as:
∃ a coloring (“the coloring is a 3-coloring of the graph”)
A little more formally as:
∃R∃G∃B (“Every point is in exactly one of the sets R, G, or B, and
no two points that are connected by an edge are both in R, or
both in G, or both in B”)
This formula can be expressed formally in existential secondorder logic (∃SO)
So 3-colorability can be expressed in ∃SO.
Capturing NP with logic
Fagin’s Theorem (1974): NP = ∃SO
Example: 3-colorability
Surprising, since characterizing a complexity
class in terms of logic, where there is no
notion of machine, computation, polynomial,
or time.
NP and coNP
• NP is the set of languages that have short
proofs.
• coNP is the set of languages that have short
refutations.
• Note that coNP is not the complement of NP.
NP intersect coNP is non-empty.
Problems believed to be in NP
intersect coNP but not in P
• Graph Isomorphism
• several others