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CS21 Decidability and Tractability

This lecture discusses the complexity class PSPACE, defines space complexity, and explains PSPACE-complete problems and their examples.

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CS21 Decidability and Tractability

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  1. CS21 Decidability and Tractability Lecture 24 March 8, 2019 CS21 Lecture 24

  2. Outline • The complexity class PSPACE • a PSPACE-complete problem • PSPACE and 2-player games CS21 Lecture 24

  3. Space complexity Definition: the space complexity of a TM M is a function f:N→ N where 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” CS21 Lecture 24

  4. 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) CS21 Lecture 24

  5. PSPACE coNP EXP PSPACE • NP PSPACE, coNPPSPACE (proof?) • PSPACE EXP (proof?) • containments believed to be proper decidable languages P NP CS21 Lecture 24

  6. PSPACE • A PSPACE-complete problem: • Quantified Satisfiability: QSAT = {φ : φ is a 3-CNF, and x1x2x3x4x5… xnφ(x1, x2, x3, … xn) } • example: φ = (x1 x2x3)(x2x3) x1x2x3 φ? YES: x1=T; if x2=T, set x3=F; if x2=F, set x3=T CS21 Lecture 24

  7. PSPACE • A PSPACE-complete problem: • Quantified Satisfiability: QSAT = {φ : φ is a 3-CNF, andx1x2x3x4x5… xnφ(x1, x2, x3, … xn) } • example: φ = (x1x2 x3)(x2) x1x2x3 φ? NO: x1=T; if x2=T…; x1=F; if x2=T… CS21 Lecture 24

  8. QSAT is PSPACE-complete Theorem: QSAT is PSPACE-complete. • Proof: • in PSPACE: x1x2x3… Qxnφ(x1, x2, …, xn)? • “x1”: for both x1 = 0, x1 = 1, recursively solve x2x3 … Qxnφ(x1, x2, …, xn)? • if at least one “yes”, return “yes”; else return “no” • “x1”: for both x1 = 0, x1 = 1, recursively solve x2x3 … Qxnφ(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 CS21 Lecture 24

  9. QSAT is PSPACE-complete • given TM M deciding L PSPACE; input x • 2nkpossible configurations • single START configuration • assume single ACCEPT configuration • define: REACH(X, Y, i) configuration Y reachable from configuration X in at most 2i steps. CS21 Lecture 24

  10. QSAT is PSPACE-complete REACH(X, Y, i) configuration Y reachable from configuration X in at most 2i steps. • Goal: produce 3-CNF φ(w1,w2,w3,…,wm) such that w1w2 …wmφ(w1,…,wm) REACH(START, ACCEPT, nk) CS21 Lecture 24

  11. QSAT is PSPACE-complete • for i = 0, 1, … nk produce quantified Boolean expressionsψi(A, B, W) such that A,B: w1w2 … ψi(A, B, W) REACH(A, B, i) • convert ψnkto 3-CNF φ • add variables V • hardwire A = START, B = ACCEPT w1w2 … V φ(W, V) x L CS21 Lecture 24

  12. QSAT is PSPACE-complete • ψo(A, B) = true iff • A = B or • A yields B in one step of M Boolean expression of size O(nk) … config. A STEP STEP STEP STEP config. B … CS21 Lecture 24

  13. QSAT is PSPACE-complete • Key observation: REACH(A, B, i+1) Z[REACH(A, Z, i) REACH(Z, B, i)] • cannot define ψi+1(A; B; Z, W, W’) to be Z [w1w2 … ψi(A, Z, W)w1’ w2’… ψi(Z, B, W’)] (why?) CS21 Lecture 24

  14. QSAT is PSPACE-complete • Key idea: use quantifiers • couldn’t do ψi+1(A; B; Z, W, W’) = Z [w1w2 … ψi(A, Z, W)w1’ w2’… ψi(Z, B, W’)] • define ψi+1(A; B; Z, X, Y, W) to be ZXY[((X=A Y=Z) (X=Z Y=B)) w1w2 … ψi(X, Y, W)] • ψi(X, Y, W) is preceded by quantifiers • move to front (they don’t involve X,Y,Z,A,B) CS21 Lecture 24

  15. QSAT is PSPACE-complete ψo(A, B) = true iff A = B or A yields B in 1 step ψi+1(A; B; Z, X, Y, W) = ZXY[((X=A Y=Z) (X=Z Y=B)) w1w2 … ψi(X, Y, W)] • |ψ0| = O(nk) • |ψi+1| = O(nk) + |ψi| • total size of ψnkis O(nk)2 = poly(n) • reduction runs in polynomial time CS21 Lecture 24

  16. PSPACE and games QSAT = {φ : φ is a 3-CNF, and x1x2x3x4x5… xnφ(x1, x2, x3, … xn) } • Think of as 2-player game (player 1 trying to satisfy φ; player 2 adversary): • player 1 picks truth value for x1 • player 2 picks truth value for x2 • player 1 picks truth value for x3… • φQSAT iff player 1 can win no matter what player 2 does. CS21 Lecture 24

  17. PSPACE and games • General phenomenon: many 2-player games are PSPACE-complete. • 2 players I, II • alternate pick- ing edges • lose when no unvisited choice pasadena auckland athens davis san francisco oakland • GEOGRAPHY = {(G, s) : G is a directed graph and player I can win from node s} CS21 Lecture 24

  18. PSPACE Theorem: GEOGRAPHY is PSPACE-complete. Proof: • in PSPACE (proof?) • PSPACE-hard. reduction from QSAT. CS21 Lecture 24

  19. GEOGRAPHY is PSPACE-complete • We are reducing from the language: QSAT = {φ : φ is a 3-CNF, and x1x2x3x4x5… xnφ(x1, x2, x3, … xn) } to the language: GEOGRAPHY = {(G, s) : G is a directed graph and player I can win from node s} CS21 Lecture 24

  20. PSPACE x1x2x3x4x5… xnφ(x1, x2, x3, … xn)? clause choice gadget true false variable gadget for xi … C1 C2 Cm CS21 Lecture 24

  21. PSPACE x1x2x3 …(x1x2x3)(x3x1)…(x1x2) I false alternately pick truth assignment true x1 II I II x2 true false I II I I x3 true false pick a clause II II … I I pick a true literal? CS21 Lecture 24

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