CS21 Decidability and Tractability
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March 1, 2017 CS21 Lecture 22 1 CS21 Decidability and Tractability Lecture 22 March 1, 2017
description
CS21 Decidability and Tractability. Lecture 22 February 28, 2014. Outline. the class NP NP-complete problems: subset sum NP-complete problems: NAE-3-SAT, max-cut the class co-NP the class NP Å coNP. SUBSET-SUM is NP-complete. Theorem : the following language is NP-complete: - PowerPoint PPT Presentation
Transcript of CS21 Decidability and Tractability
March 1, 2017 CS21 Lecture 22 1
CS21 Decidability and Tractability
Lecture 22March 1, 2017
Outline
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March 1, 2017 CS21 Lecture 22 3
Quantifier characterization of coNP
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Proof interpretation of coNP
“proof” “proof verifier”“short” proof
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coNP
• what complexity class do the following languages belong in?– COMPOSITES = {x : integer x is a composite}– PRIMES = {x : integer x is a prime number}– GRAPH-NONISOMORPHISM = {(G, H) : G
and H are graphs that are not isomorphic}– EXPANSION = {(G = (V,E), > 0): every
subset S V of size at most |V|/2 has at least |S| neighbors}
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coNP
• Picture of the way we believe things are:
decidable languages
NPP
EXPcoNP
NP coNP
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NP coNP
• Might guess NP coNP = P by analogy with RE (since RE coRE = DECIDABLE)
• Not believed to be true.• A problem in NP coNP not believed to
be in P:L = {(x, k): integer x has a prime factor p < k}
(decision version of factoring)
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NP coNP
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PRIMES in NP
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Summary
• Picture of the way we believe things are:
decidable languages
NPP
EXP coNP
NP coNP
(decision version of ) FACTORING
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Space complexity
Definition: the space complexity of a TM M is a function
f:NN → NNwhere f(n) is the maximum number of tape cells M scans on any input of length n.
• “M uses space f(n),” “M is a f(n) space TM”
March 1, 2017 CS21 Lecture 22 12
Space complexity
Definition: SPACE(t(n)) = {L : there exists a TM M that decides L in space O(t(n))}
PSPACE = k ≥ 1 SPACE(nk)
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PSPACE
• NP PSPACE, coNP PSPACE (proof?)• PSPACE EXP (proof?)• containments believed to be proper
PSPACE
NPP
EXP coNP
decidable languages
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PSPACE
• A PSPACE-complete problem:• Quantified Satisfiability:
QSAT = {φ : φ is a 3-CNF, and x1x2x3x4x5…xn φ(x1, x2, x3, … xn) }
• example: φ = (x1x2x3)(x2x3)
x1x2x3φ? YES: x1=T; if x2=T, set x3=F; if x2=F, set x3=T
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PSPACE
• A PSPACE-complete problem:• Quantified Satisfiability:
QSAT = {φ : φ is a 3-CNF, and x1x2x3x4x5…xn φ(x1, x2, x3, … xn) }
• example: φ = (x1x2x3)(x2)
x1x2x3φ? NO: x1=T; if x2=T…; x1=F; if x2=T…
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QSAT is PSPACE-completeTheorem: QSAT is PSPACE-complete.• Proof:
– in PSPACE: x1x2x3 … Qxn φ(x1, x2, …, xn)?– “x1”: for both x1 = 0, x1 = 1, recursively solve
x2x3 … Qxn φ(x1, x2, …, xn)?• if at least one “yes”, return “yes”; else return “no”
– “x1”: for both x1 = 0, x1 = 1, recursively solve x2x3 … Qxn φ(x1, x2, …, xn)?
• if at least one “no”, return “no”; else return “yes”– base case: evaluating a 3-CNF expression– poly(n) recursion depth– poly(n) bits of state at each level
