Computability and Complexity 36-1 Computability and Complexity 36-2
The Class #P
A polynomially balanced, polynomial time decidable binary relation is a binary relation U on Σ* such that (a) there is a polynomial time decider for U #P-Completeness (b) if then (u,v)∈L | v |≤ for some polynomial p(| n |) p
Definition Let U be a certain polynomially balanced polynomial time decidable relation. The counting problem associated with U is the following: Given u, how many v are there such that (u,v) ∈ U?
#P is the class of all counting problems associated with a certain polynomially balanced polynomial time decidable relations Computability and Complexity Andrei Bulatov
Computability and Complexity 36-3 Computability and Complexity 36-4
#P-Completeness #SAT
#SAT Definition A polynomial time function f is said to be a parsimonious reduction of a counting problem A to a counting problem B , Instance: A formula Φ in CNF ≤ ∈Σ denoted , if for all instances A p B of x * A, x and f(x) have the same answer Objective: How many satisfying assignments does Φ have?
Definition A language L is said to be #P -complete if, for any counting problem A corresponding to a polynomially balanced Theorem #SAT is #P-complete polynomial time decidable relation, ≤ A p L
Computability and Complexity 36-5 Computability and Complexity 36-6
Proof Reduction
Take a counting problem A from #P . When proving Cook’s theorem we, given a non-deterministic Turing machine NT and an input w for NT , built a CNF such thatΦ There is a polynomially balanced polynomial time decidable relation U NT ,w every satisfying assignment of encoΦ des an accepting computation corresponding to A NT ,w of NT on the input w. That is there exist a polynomial p(n) and a Turing machine T such that • T decides U (on an input (w,c)) in a time p(| w|+| c|) The decider for U is a deterministic Turing machine, that is a particular case of non-deterministic machines • if (w,c) ∈ U then |c| ≤ p(| w|) Φ Consider the formula T (, w,c) We need to do: Φ Since T is deterministic, has at most one sT (, w,c) atisfying assignment Φ Given a string w generate (in polynomial time) a formula w such that Φ the number of satisfying assignments to w = the number of c such that (w,c) ∈ U
1 Computability and Complexity 36-7
The fact that the input for T is (w,c ) is encoded as follows = = Φ Let and w w1w2 Kwn c c1c2 then contains theKcp(n) T (, w,c) clauses wi ∧ ci ∧ Pi 1, ∧ Pi 1, 1≤i≤n n+1≤i≤ p(n)
x where is true if and only if at the time Pi 1, 1 the tape cell number i contains x
wi ∧ ci Φ Φ ∧ Pi 1, ∧ Pi 1, Replace with in whichT (, w,c) T ,w 1≤i≤n n+1≤i≤ p(n) wi is replaced with ∧ Pi 1, 1≤i≤n
0 0 K 0 Then every assignment to Pn+ 1,1 , Pn+ 1,2 , , correspond to a unique pairPp(n), 1 (w,c ) which may or may not belong to U, and gives rise to a unique satisfying Φ ∈ assignment to if and only if T ,w (w,c) U
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