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Chapter 5 The Witness Reduction Technique: Feasible Closure Properties of #P Greg Goldstein Andrew Learn 18 April 2001

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Outline Introduction Thm: Proper Subtraction and “Closure Completeness” Thm: Complexity Class Collapse Related Results Summary

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Closure Concepts Let be an operation, N x N N Natural Numbers: –Closure under an operation defined as: ( x N) ( y N)[x y N] –Closed under addition, multiplication; not subtraction Functions: –Closure similarly defined for a class of functions: ( f 1 F) ( f 2 F)[h f 1,f 2 F] where h f 1,f 2 (n) = (f 1 (n), f 2 (n)) –e.g. Polynomials closed under addition, subtraction, etc. –We examine another class of functions: #P

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#P -- Counting Class for Accepting Paths Defn: f is a #P function if a NPTM N, s.t. for each input x, f(x) = number of accepting paths of N(x) #P has closure properties –Addition: –Multiplication: f 1 (x) f 2 (x) + f 1 (x) f 2 (x) * N1N1 N1N1 N2N2 N2N2

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#P Subtraction Skepticism What about subtraction? –Can’t have # paths < 0 –Consider Proper Subtraction: a b = max{0, a-b} Ponderable: Is #P closed under proper subtraction?? –Witness Reduction Technique: Reducing the number of accepting paths –If we can do this, class hierarchies come crashing down

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Our Prove-it Roadmap “Closure Completeness” Theorem –Proper subtraction is “hardest” closure property –If #P is closed under proper subtraction, implies: #P closed under all polynomial-time operations Hierarchy collapse follows “Hierarchy Heroics” Theorem –Given one hierarchy collapse from above, prove a lot of others Related Cases –Other “hardest” closure properties –Other classes of functions

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But first, a word from these complexity classes... UP (Unambiguous Polynomial Time): “Unique Paths” –UP = {L | NPTM N such that L=L(N) and x, N(x) has at most one accepting path} x L One accepting path x L Zero accepting paths P (Parity P): “Odd Paths” – P = {L | NPTM N such that x L #acc N (x) 0 (mod 2)} x L Odd number of accepting paths x L Even number of accepting paths

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Complexity classes (continued) PP (Probabilistic Polynomial Time): “Probabilistic Plurality” –Probabilistic machines: NPTM with a twist Binary, fully populated computation tree Path is sequence of coin flips -- one path taken, acceptance not guaranteed –L PP if PPTM M such that x L M(x) accepts with probability 1/2 I.e. number of accepting paths 1/2 of total paths Depth = q(|x|) Total paths = 2 q(|x|), Accepting paths 2 q(|x|)-1

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Handwritten slides here for Theorem 5.5 on Closure Completeness

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“Hierarchy Heroics” Theorem The following statements are equivalent: –UP = PP –UP = NP = coNP = PH = P = PP PP PP PP PP PP ... Given UP = PP from previous theorem, we explore the hierarchy collapse

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Building by Bits... UP = PP = NP = coNP = PH = P = PP PP PP PP PP PP ... –Known facts: UP = PP (from Theorem 1) UP NP PP –Proof: Trivially obvious UP = PP = NP = coNP = PH = P = PP PP PP PP PP PP ... –Known facts: PP is closed under complementation –Proof: PP = coPP Since PP = NP, then NP = coNP

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Hierarchy Proof (continued) UP = PP = NP = coNP = PH = P = PP PP PP PP PP PP ... –Known facts: PH = i = P NP NP NP NP NP NP ... –Proof: Because NP = coNP, we can construct a single NPTM with the power of an oracle. Thus NP = NP NP, and we can collapse any size stack of NP NP NP… NANA NANA One of these must accept; thus we get an oracle-type yes/no answer in polynomial time.

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Hierarchy Proof (continued) UP = PP = NP = coNP = PH = P = PP PP PP PP PP PP ... –Known facts: P UP PH –Proof: Since UP = PH, we get P UP UP It’s also true that UP P UP because UP oracle decides UP. P UP = UP (by double-inclusion) We know UP P PP P (1/0 acc. paths odd/even acc. paths) From Lemma 4.13, we know that PP P P PP Putting these together with what we already know, we get: UP P PP P P PP = P UP = UP UP = P = PP P

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Hierarchy Proof (continued) UP = PP = NP = coNP = PH = P = PP PP PP PP PP PP ... –Proof: On last page we showed PP P = PP Thus we can start with a stack of two PP’s: PP PP = PP P = PP And thus can collapse any size stack of PP PP PP… !

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Related Results Other “hardest” closure properties: Integer Division –Defn: a b = a/b Thm: The following statements are equivalent: 1. #P is closed under integer division 2. #P is closed under every polynomial-time computable operation 3. UP = PP ?

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#P Integer Division... (continued) We’re going to prove the integer division case using the Witness Reduction Technique again, similar to proper subtraction. Conceptually, it works like this: 1. Pick an arbitrary PP language 2. Take the corresponding #P function PP 3. Make a tricky pick of another #P function 4. Using assumed closure property, generate a third #P function, for a machine That accepts the same language, but voila!--is in a different class UP L f g h Assumed closure L

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#P Integer Division... (continued) Proof: –Part 2 trivially implies Part 1 –Part 3 implies Part 2 from previous proof –Show Part 1 implies Part 3: Let L be any PP set; we seek to show L UP Because L PP, NPTM N and integer k 1 s.t. 1. N(x) has exactly 2 |x| k computation paths, each with |x| k choices (steps) 2. x L iff N(x) has at least 2 |x| k -1 accepting paths 3. For each x, N(x) has at least one rejecting path q(|x|) |x| k

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#P Integer Division... (continued) Let f(x) be the #P function for N. Let g(x) = 2 |x| k -1 be another #P function. By our assumption, h(x) = f(x) g(x) is a #P function. x Lx L 2 |x| k - 1 f(x) 2 |x| k -1 2 |x| k -1 f(x) h(x) = f(x)/g(x) = 1h(x) = f(x)/g(x) = 0 Thus the NPTM defined by h is a UP machine for L! L UP UP = PP max # of accepting paths min #

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More Related Results Other complexity classes: OptP and SpanP –OptP Each path considered to have a non-negative output f is an OptP function if f(x) = max of path outputs –SpanP f is a SpanP function if f(x) = number of different path outputs For both classes: –Proper subtraction is a “hardest” closure property –If the class is closed under proper subtraction, some complexity class collapse occurs OptP: NP = coNP SpanP: PP NP = PH = NP

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Summary Definitions –#P = functions that count accepting paths –Witness Reduction = Reducing # of accepting paths “Closure Completeness” Theorem –If #P is closed under proper subtraction: #P is closed under all polynomial-time computable operations UP = PP “Hierarchy Heroics” Theorem –If UP = PP, UP = PP = NP = coNP = PH = P = PP PP PP PP PP PP ... Related Results: –Integer division for #P, Proper subtraction for OptP, SpanP

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