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COMP4141 Assignment 8 2020 Term 1 Due: Tuesday, 28th April, 17:00 Submission is through WebCMS/give and should be a single pdf file, maximum size 4Mb. Prose should be typed, not handwritten. Use of LATEX is encouraged, but not required. Discussion of assignment material with one other person is permitted, but the work submitted must be your own in line with the University’s plagiarism policy. Problem 1 An alternating finite automaton (AFA) is a tuple A = ( Q, Σ, l, δ, q0 , F ) where (10 marks) • Q is a finite set of states • Σ is a finite input alphabet • l : Q → {∃, ∀} is a labelling of the states • δ ⊆ Q × Σ × Q is the transition relation • q0 ∈ Q is the start state • F ⊆ Q is the set of final states. A word w is accepted from a state q if and only if: • w = e and q ∈ F; or • w = aw0 and – if l (q) = ∃ then there exists q0 such that (q, a, q0 ) ∈ δ and w0 is accepted from q0 ; or – if l (q) = ∀ then for all q0 such that (q, a, q0 ) ∈ δ we have w0 is accepted from q0 . The language accepted by an AFA is defined to be the set of words accepted from q0 . Show that the class of languages accepted by AFAs are precisely the regular languages. Problem 2 Define ARE to be the class of languages accepted by Alternating Turing Machines. (10 marks) Prove or disprove: ARE = RE. (10 marks) Problem 3 (a) Show that if SAT ∈ BPP then SAT ∈ RP. Hint: Try to build a truth assignment using the fact that ϕ( x ) is satisfiable if and only if ϕ(true) or ϕ(false) is satisfiable. (6 marks) (b) Show that if NP ⊆ BPP then RP = NP. Hint: Use (a) 1 (4 marks) COMP4141 Theory of Computation Lecture 14: The Polynomial Time Hierarchy Paul Hunter CSE, UNSW Revision: 2015/05/13 (Credits: R. Van der Meyden, K. Engelhardt, M Sipser, C Papadimitriou, R. van Glabbeek, P Hofner) Outline Relativization Alternation Polynomial Time Hierarchy Outline Relativization Alternation Polynomial Time Hierarchy Relativization Recall Lecture 8: Definition An oracle for a language B is an external device that can determine whether a given word w is a member of B. An oracle (for B) Turing Machine, M B , is a Turing machine that has the additional capacity of querying an oracle for B. If an oracle TM M B decides A then we say that A is decidable relative to B. Language A is Turing reducible to language B (or A T B) if A is decidable relative to B. P vs m vs T A L B ) A P B ) A m B ) A T B As for the other two notions of reduction we have Theorem If A T B and B is decidable, then A is decidable. Corollary If A T B and A is undecidable, then B is undecidable. but whereas m also transferred r.e. this is not the case for T . Example Recall that ATM is r.e. but ATM isn’t. But ATM T ATM and ATM T ATM by simply reversing the oracles’ answers. Oracle complexity classes Definition AB S For complexity classes A, B let = L2B AL be the class of languages decided by an A-computation using an oracle in B. NB We assume each call to an oracle takes O(1) time. Example PO is the class of languages decided by a polynomial-time oracle TM using oracle O. (Similarly for NPO .) NP ✓ PSAT and coNP ✓ PSAT . Example A formula of propositional logic is minimal if there does not exist a shorter formula such that , is valid (true for all assignments). It is not known whether MIN-F 2 NP where MIN-F = { h i | is a minimal Boolean formula } but MIN-F 2 NPSAT as witnessed by the oracle NTM M SAT = “On input h i 1 Non-deterministically guess a smaller formula 2 Ask the oracle whether h¬( , )i 2 SAT and if it accepts, reject; otherwise accept.” . This problem is not known to be in NP, nor in co-NP. Relativization and Diagonalization Any theorem proved about TMs by using only methods based on I string representations of TMs II simulation of one TM by another without much overhead in time/space lifts to oracle machines. Examples Undecidability of ATM : {hM, xi : M is an oracle TM for B that accepts x} cannot be decided by an oracle TM for B Time/Space hierarchy theorems ? P = NP and Diagonalization ? That the resolution of P = NP can not be such a theorem follows from: Theorem (Baker, Gill, Solovay 1975) Oracles A and B exist whereby PA = NPA and PB 6= NPB . Proof of 9A PA = NPA A could be QBF: NPQBF ✓ NPSPACE = PSPACE ✓ PQBF ✓ NPQBF by QBF 2 PSPACE by Savitch’s theorem QBF is PSPACE-complete by P ✓ NP Proof of 9B PB 6= NPB Iteratively construct a B (and its complement B 0 ) such that in the end UB 2 NPB \ PB where UB = { 1n | ⌃n \ B 6= ; } . That UB 2 NPB is easy: “On input 1n guess x 2 ⌃n and accept i↵ the oracle confirms x 2 B.” Proof of 9B PB 6= NPB cont. Initially, B = B 0 = ;. For stage i of the construction, let Mi? be the i’th polynomial-time oracle TM running in w.l.o.g. in time ni . Let m exceed the length of all strings in B [ B 0 so far, and also m i < 2m . We’ll ensure that UB and MiB disagree on 1m . 1 2 Simulate Mi? on 1m by answering queries x to the oracle with “yes” if x 2 B, “no” if x 2 B 0 , otherwise we also answer “no” and add x to B 0 . If Mi? accepts 1m then we put all strings of length m into B 0 ; otherwise, we add the first string of length m neither in B nor in B 0 to B. Such a string exists because Mi? can have queried at most mi < 2m strings of length m and none were queried ever before. It follows that no MiB will decide UB and thus UB 2 / PB . Outline Relativization Alternation Polynomial Time Hierarchy Motivation Recall nondeterministic TM’s, accept if some branch of the computation tree accepts (used for NP) co-nondeterministic TM’s, accept if all branches of the computation tree accepts (used for coNP) Alternating Turing machines combine the two acceptance modes into one type of machine ...... ATMs Definition An alternating Turing Machine (ATM) M = (Q, `, ⌃, , , q0 , qaccept , qreject ) consists of: Q, a finite set of states ` : Q ! {8, 9}, a labelling of states as universal or existential ⌃, the input symbol alphabet, t 2 /⌃ ◆ ⌃, the tape symbol alphabet, t 2 :Q⇥ ! 2Q⇥ ⇥{l,r} , the transition function q0 2 Q, the start state qaccept 2 Q, the accept state qreject 2 Q, the reject state L(ATM) An ATM M runs on a word w as if it were an NTM, creating a tree TM (w ) of TM configurations (or a directed graph if we identify nodes with identical configurations). Definition (ATM acceptance) Mark nodes of TM (w ), proceeding from leaves to the root, as follows: 1 every (accepting) configuration xqaccept y is marked, 2 c is marked if `(c) = 9 and c has some marked successor in TM (w ), and 3 c is marked if `(c) = 8 and c all successors in TM (w ) are marked . If the root of TM (w ) (the initial configuration) is marked at the end of this process, then M accepts w . Definitions of time and space complexity need not be changed. Definition Let t : N ! N. The alternating time complexity class, ATIME(t(n)) is the collection of all languages that are decidable by an O(t(n)) time ATM. The alternating space complexity class, ASPACE(t(n)) is the collection of all languages that are decidable by an O(t(n)) space ATM. [ AP = ATIME(nk ) k2N [ APSPACE = ASPACE(nk ) k2N AL = ASPACE(log n) Example Recall that it is not know whether MIN-F = { h i | is a minimal Boolean formula } is in NP or P. All we know so far is that MIN-F 2 NPSAT . “On input h i: 1 Universally select a shorter formula 2 Existentially select an interpretation ⇡ of 3 Evaluate 4 Accept if the results di↵er and reject otherwise.” and . on ⇡ proves that MIN-F 2 AP. Time, Space, and Alternation Theorem (a) (b) 1. ATIME(t(n)) ✓ SPACE(t(n)) ✓ ATIME(t(n)2 ) if t(n) 2. ASPACE(t(n)) = TIME(2O(t(n)) ) if t(n) log n. Corollary AL = P, AP = PSPACE, and APSPACE = EXPTIME. n. Proof Ideas 1.(a) ATIME(t(n)) ✓ SPACE(t(n)): simulate the ATM, do DFS to do the marking. This gives SPACE(t(n)2 ), one t(n) for the recursion depth and one t(n) for the configuration size. Reduce the latter to constant size by merely recording the choices made at each non-deterministic step, and recomputing the configuration from the start when backtracking. Proof Ideas 1.(b) SPACE(t(n)) ✓ ATIME(t 2 (n)): As in Savitch’s theorem, the ATM uses binary search to determine whether the simulated TM could reach the accepting configuration in 2dt(n) steps. Rather than iterating over all possible midpoint configurations cm , the ATM can use one big existential guess of t(n) steps to construct cm , (and then universally branch into two recursive calls). Proof Ideas 2.(“✓”) ASPACE(t(n)) ✓ TIME(2O(t(n)) ): similar to the proof of PSPACE ✓ EXPTIME, on input w , construct the directed acyclic computation graph containing the configurations of the simulated ATM; first mark the accepting nodes and then accept the word according to the definition of ATM acceptance. Proof Ideas cont. 2.(“◆”) ASPACE(t(n)) ◆ TIME(2O(t(n)) ): Simulate a deterministic 2f (n) machine M by an O(f (n)) space ATM S, where f (n) = O(t(n)). W.l.o.g. M accepts after erasing all tape content and moving all the way to the left. On input w , M goes through a sequence of configurations. We can’t even store a single one of those in S because it could be too long! Let’s encode the sequence of configurations as a 2f (|w |) ⇥ 2f (|w |) grid where a cell contains either a tape symbol or a tape symbol and a state if that’s where the head of M is. S then uses a recursive procedure R(i, j, d) where i, j are pointers of size df (|w |)e in binary to a cell and d is a cell contents, to check the bottom left grid cell i = 2f (|w |) , j = 1 has contents d = (qaccept , t). Proof Ideas cont. R = “on input hi, j, di 1 2 if i = 1 then accept if d is consistent with the j’th cell of the initial configuration q0 w in this representation; else reject. 9-guess the contents a, b, c of the parent cells [i 1, j 1], [i 1, j], [i 1, j + 1] 1 2 if the parent cells with contents a, b, c shouldn’t have the child cell [i, j] with contents d, reject 8-recurse into R(i 1, j 1, a), R(i 1, j, b), R(i 1, j + 1, c).” Even though the recursion depth is 2f (|w |) , the ATM can do with O(f (w )) space because it need not store any of the arguments of previous recursive calls: R only ever returns accept or reject. Alternative Proof of PSPACE ✓ AP We show QBF 2 AP by giving a polynomial time ATM. M = “on input Q1 x1 . . . Qk xk (x1 , . . . , xk ) where the Qi 2 {9, 8} 1 for 1  i  k 1 2 Qi -guess a value vi 2 {false, true} for xi evaluate (v1 , . . . , vk ) and accept if it’s true; reject otherwise.” Outline Relativization Alternation Polynomial Time Hierarchy Polynomial Time Hierarchy #1 Definitions A ⌃i -ATM is an ATM whose runs begin at an 9 state, and alternate, i.e., switch the {9, 8}-type of state, at most i 1 times. ⌃i TIME(f (n)) is the class of languages ⌃i -ATMs can decide in O(f (n)) time. S ⌃i P = k ⌃i TIME(nk ). Similarly, define ⇧i -ATMs, ⇧i TIME(f (n)), and ⇧i P by starting with 8 instead of 9. S S PH = i ⌃i P = i ⇧i P is called the polynomial time hierachy. NB NP = ⌃1 P and coNP = ⇧1 P. Polynomial Time Hierarchy #2 Alternatively, we could have defined Definitions 0 P = ⌃0 P = ⇧0 P = P; and for i i+1 P 0 = P ⌃i P ⌃i+1 P = NP⌃i P ⇧i+1 P = coNP⌃i P PSPACE .. . ⌃3 P P 3P = P ⌃2 P ⌃2 P = NPNP 2P ⇧3 P ⇧2 P = coNPNP = PNP = PcoNP ⌃1 P = NP ⇧1 P = coNP P The Polynomial Time Hierarchy Examples MIN-F 2 ⇧2 P Games with a fixed number of moves Open questions Does PH = PSPACE? Is the polynomial time hierarchy strict? The Polynomial Time Hierarchy Examples MIN-F 2 ⇧2 P Games with a fixed number of moves Open questions Does PH = PSPACE? Is the polynomial time hierarchy strict? ...
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