Intro to Theory of Computation LECTURE 26 Last time • Space complexity • The class PSPACE Today • PSPACE-completeness • Hierarchy theorems Sofya Raskhodnikova Review question Let 푓(푛) be a function, where 푓 푛 ≥ 푛. If a language A is decided by a NTM that runs in space 풇(풏) then A is decided by a TM that runs in space O(풇ퟐ(풏)). A. True B. False C. Unknown D. None of the above 12/5/2019 L26.2 Review question Relationships between the following classes are an open question EXPTIME A. P vs NP NP PSPACE = B. P vs PSPACE P NPSPACE C. NP vs EXPTIME D. All of the above E. All of the above and P vs EXPTIME 12/5/2019 L26.3 Hardest problems in PSPACE A language B is PSPACE-complete if 1. B∈ 퐏퐒퐏퐀퐂퐄 2. B is PSPACE-hard,i.e., every language in PSPACE is poly-time reducible to B. P PSPACE B 12/5/2019 L26.4 The TQBF problem • Boolean variables: variables that can take on values T/F (or 1/0) • Boolean operations: ∨, ∧, and ¬ • Boolean formula: expression with Boolean variables and ops • Quantified Boolean formula: Boolean formula with quantifiers (∀, ∃) • Fully Quantified Boolean formula: all variables have quantifiers (∀, ∃) We only consider the form where all quantifiers appear in the beginning. Ex. ∀푥∃푦[ 푥 ∨ 푦 ∧ 푥 ∨ 푦 ] True ∃푦∀푥 [ 푥 ∨ 푦 ∧ 푥 ∨ 푦 ] False • Each fully quantified Boolean formula is either true or false. • The order of quantifiers matters! TQBF = {〈흓〉 ∣ 흓 is a true fully quantified Boolean formula} 12/5/2019 L26.5 TQBF is PSPACE-complete 1. TQBF is in PSPACE 2. TQBF is PSPACE-hard 12/5/2019 L26.6 Prove: T퐐퐁퐅 ∈ PSPACE T = `` On input 〈흓〉, where 흓 is a fully quantified Boolean formula: 1. If 흓 has no quantifiers, it has only constants (and no variables). Evaluate 흓. If true, accept; o.w., reject. 2. If 흓 is of the form ∃풙 흍, recursively call T on 흍 with 풙 = ퟎ and then on 흍 with 풙 = ퟏ. If either call accepts, accept; o.w., reject. 3. If 흓 is of the form ∀풙 흍, recursively call T on 흍 with 풙 = ퟎ and then on 흍 with 풙 = ퟏ. If both calls accept, accept; o.w., reject.’’ • If 푛 is the input length, T uses space 푂 푛 . 12/5/2019 L26.7 Question If TQBF is in P then it implies that A. P = NP EXPTIME B. P = PSPACE C. P = EXPTIME NP PSPACE D. (A) and (B) are true = P NPSPACE E. (A), (B), (C) are true 12/5/2019 L26.8 TQBF as a game • Two players: E (for ∃) and A (for ∀) • Given: a quantified Boolean formula • Players alternate choosing values for variables (in the same order as the quantifiers appear). • E wins if the formula evaluates to 1. • Does E have a winning strategy in the game? (Can she win if both E and A play optimally?) 12/5/2019 L26.9 Winning strategies for games Determining whether there is a winning strategy for a specified game position is hard for many games • Generalized Geography is PSPACE-complete • Generalized Chess is EXPTIME-complete 12/5/2019 L26.10 recognizable Generalized decidable Chess Generalized EXPSPACE Geography EXPTIME TQBF PSPACE=NPSPACE NP coNP P CFL regular Hardest problems in EXPSPACE A language B is EXPSPACE-complete if 1. B∈ 퐄퐗퐏퐒퐏퐀퐂퐄 2. B is EXPSPACE-hard,i.e., every language in EXPSPACE is poly-time reducible to B. P EXPSPACE B The 퐸푄푹푬푿↑ problem • Regular operations: ∪, ∘, ∗ • Regular expressions: use regular operations, ∅, 휺, 횺, members of 횺 • We can test equivalence of two regular expressions in poly space • Exponentiation operation: 푅푘 = 푅 ↑ 푘 = 푅 ∘ ⋯ ∘ 푅 (푘 times) 퐸푄푹푬푿↑ = {〈푸, 푹〉 ∣ 푸 and R are equivalent regular expressions with exponentiation} 퐸푄푹푬푿↑ is EXPSPACE-complete. 12/5/2019 recognizable 퐸푄푹푬푿↑ Generalized decidable Chess Generalized EXPSPACE Geography EXPTIME PSPACE=NPSPACE NP coNP P CFL regular Motivating question Is there a decidable language which is provably not in P? 12/5/2019 L26.15 Hierarchy Theorems A function 푓: ℕ → ℕ is space constructible if the function that maps the string 1푛 to the binary representation of 푓(푛) is computable in space 푂 푓 푛 . • Fractional functions are rounded down. 2 푛 • Examples: log2 푛 , 푛, 푛 log2 푛, 푛 , 2 , all commonly occurring functions that are Ω(log 푛) 12/5/2019 L26.16 Hierarchy Theorems Space Hierarchy Theorem For any space constructible function 푓, a language exists that is decidable in 푂 푓 푛 space, but not in 표(푓 푛 ) space. • Corollary 1: For all 훼, 훽 ≥ 0, where 훼 < 훽, SPACE(푛훼) is a strict subset of SPACE(푛훽). • Corollary 2: PSPACE is a strict subset of EXPSPACE, where 푛푘 EXPSPACE=⋃푘푆푃퐴퐶퐸(2 ) • Corollary 3: 퐸푄푹푬푿↑ is not in PSPACE. 12/5/2019 L26.17 Hierarchy Theorems A function 푓: ℕ → ℕ is time constructible if the function that maps the string 1푛 to the binary representation of 푓(푛) is computable in time 푂 푓 푛 . • Fractional functions are rounded down. 2 푛 • Examples: 푛 log2 푛, 푛 , 2 , all commonly occurring functions that are Ω(푛 log 푛) 12/5/2019 L26.18 Hierarchy Theorems Time Hierarchy Theorem For any time constructible function 푓, a language exists that is decidable in 푂 푓 푛 time, 푓 푛 but not in 표 time. log 푓 푛 • Corollary 1: For all 훼, 훽 ≥ 0, where 훼 < 훽, TIME(푛훼) is a strict subset of TIME(푛훽). • Corollary 2: P is a strict subset of EXPTIME • Corollary 3: Generalized Chess is not in P. 12/5/2019 L26.19 Review question 3SAT is decidable, but not in polynomial time. A. True B. False C. Unknown D. None of the above 3/17/2016 L29.20 Review question 푬푸푹푬푿↑ is decidable, but not in polynomial time. A. True B. False C. Unknown D. None of the above 3/17/2016 L29.21 Review question 푨푻푴 is decidable, but not in polynomial time. A. True B. False C. Unknown D. None of the above 3/17/2016 L29.22 recognizable 퐸푄푹푬푿↑ 퐸푄푻푴 Generalized decidable Chess 퐴푻푴 Generalized EXPSPACE Geography EXPTIME PSPACE=NPSPACE NP coNP P CFL regular 퐂퐨퐦퐩퐥퐞퐭퐞 recognizable languages decidable 푬푸푹푬푿↑ EXPSPACE 푻푸푩푭 EXPTIME PSPACE=NPSPACE NP coNP P ퟑ푺푨푻 CFL ퟑ푺푨푻 CLIQUE regular 푪푳푰푸푼푬 HamPath 퐇퐚퐦퐏퐚퐭퐡 SubsetSum VertexCover Course parts 1. Automata theory 2. Computability theory 3. Complexity theory 12/5/2019 L30.25.