P versus NP: Approaches, Rebuttals, and Does It Matter?

Transcription

1 versus N: Approaches, Rebuttals, and Does It Matter? immerman

2 Spike of attention to vs. N problem, Aug Deolalikar claimed that he had tamed the wildness of algorithms and shown that indeed doesnt equal N. Within a few hours of his , the paper got an impressive endorsement: This appears to be a relatively serious claim to have solved versus N, ed Stephen Cook of the University of Toronto, the scientist who had initially formulated the question. That evening, a blogger posted Deolalikar s paper. And the next day, long before researchers had had time to examine the 103-page paper in detail, the recommendation site Slashdot picked it up, sending a fire hose of tens of thousands of readers and dozens of journalists to the paper. Julie Rehmeyer, Science News, Sept. 9, 2010

3 co r.e. co r.e. r.e. r.e. = Recursive DTIME[n k ] k=1 co N co N N N N "truly feasible" co N is a good mathematical wrapper for truly feasible. FO(CFL) FO(REGULAR)

4 NTIME[t(n)]: a mathematical fiction 0 input w w = n s t(n) t(n) 2 b 1 b 2 b 3 b t(n)

5 co r.e. co r.e. r.e. r.e. N = Recursive NTIME[n k ] k=1 co N co N N N N "truly feasible" co N Many optimization problems we want to solve are N. FO(CFL) FO(REGULAR)

6 co r.e. co r.e. r.e. r.e. N = Recursive NTIME[n k ] k=1 co N co N SACE N N N "truly feasible" co N Many optimization problems we want to solve are N. FO(CFL) FO(REGULAR) FO

7 co r.e. co r.e. r.e. r.e. N = Recursive NTIME[n k ] k=1 co N co N EXTIME SACE N N N "truly feasible" co N Many optimization problems we want to solve are N. FO(CFL) FO(REGULAR) FO

8 Descriptive Complexity Query q 1 q 2 q n Computation Answer a 1 a 2 a i a m

9 Descriptive Complexity Query q 1 q 2 q n Computation Answer a 1 a 2 a i a m Restrict attention to the complexity of computing individual bits of the output, i.e., decision problems.

10 Descriptive Complexity Query q 1 q 2 q n Computation Answer a 1 a 2 a i a m S i Restrict attention to the complexity of computing individual bits of the output, i.e., decision problems. How hard is it to check if input has property S?

11 Descriptive Complexity Query q 1 q 2 q n Computation Answer a 1 a 2 a i a m S i Restrict attention to the complexity of computing individual bits of the output, i.e., decision problems. How hard is it to check if input has property S? How rich a language do we need to express property S?

12 Descriptive Complexity Query q 1 q 2 q n Computation Answer a 1 a 2 a i a m S i Restrict attention to the complexity of computing individual bits of the output, i.e., decision problems. How hard is it to check if input has property S? How rich a language do we need to express property S? There is a constructive isomorphism between these two approaches.

13 Interpret Input as Finite Logical Structure Graph G = ({v 1,...,v n },E,s,t) s t Binary A w = ({p 1,...,p 8 },S) String S = {p 2,p 5,p 7,p 8 } w = Vocabularies: τ g = (E 2,s,t), τ s = (S 1 )

14 First-Order Logic input symbols: from τ variables: x,y,z,... boolean connectives:,, quantifiers:, numeric symbols: =,, +,, min, max α x y(e(x,y)) L(τ g ) β x y(x y S(x)) L(τ s ) β S(min) L(τ s )

15 Second-Order Logic Φ 3 color R 1 G 1 B 1 x y ((R(x) G(x) B(x)) (E(x,y) ( (R(x) R(y)) (G(x) G(y)) (B(x) B(y))))) s d t b a c f g e

16 Second-Order Logic Fagin s Theorem: N = SO Φ 3 color R 1 G 1 B 1 x y ((R(x) G(x) B(x)) (E(x,y) ( (R(x) R(y)) (G(x) G(y)) (B(x) B(y))))) s d t b a c f g e

17 co r.e. FO A (N) Arithmetic Hierarchy r.e. co r.e. FO(N) r.e. Recursive FO (N) E co N SO A EXTIME SACE olynomial Time Hierarchy SO co N N N "truly feasible" co N SO E N FO(CFL) FO(REGULAR) FO

18 Addition is First-Order Q + : STRUC[τ AB ] STRUC[τ s ] A a 1 a 2... a n 1 a n B + b 1 b 2... b n 1 b n S s 1 s 2... s n 1 s n

19 Addition is First-Order Q + : STRUC[τ AB ] STRUC[τ s ] A a 1 a 2... a n 1 a n B + b 1 b 2... b n 1 b n S s 1 s 2... s n 1 s n ( C(i) ( j > i) A(j) B(j) ) ( k.j > k > i)(a(k) B(k))

20 Addition is First-Order Q + : STRUC[τ AB ] STRUC[τ s ] A a 1 a 2... a n 1 a n B + b 1 b 2... b n 1 b n S s 1 s 2... s n 1 s n ( C(i) ( j > i) A(j) B(j) ) ( k.j > k > i)(a(k) B(k)) Q + (i) A(i) B(i) C(i)

21 arallel Machines: CRAM[t(n)] = CRCW-RAM-TIME[t(n)]-HARD[n O(1) ] r 1 Global 2 Memory 3 4 5

22 arallel Machines: Quantifiers are arallel CRAM[t(n)] = CRCW-RAM-TIME[t(n)]-HARD[n O(1) ] Assume array A[x] : x = 1,...,r in memory. r 1 Global 2 Memory 3 4 5

23 arallel Machines: Quantifiers are arallel CRAM[t(n)] = CRCW-RAM-TIME[t(n)]-HARD[n O(1) ] Assume array A[x] : x = 1,...,r in memory. x(a(x)) write(1); proc p i : if (A[i] = 0) then write(0) r 1 Global 2 Memory 3 4 5

24 co r.e. FO A (N) Arithmetic Hierarchy r.e. co r.e. FO(N) r.e. Recursive FO (N) E FO = EXTIME SACE CRAM[1] = AC 0 co N SO A olynomial Time Hierarchy SO co N N N "truly feasible" co N SO E N = Logarithmic-Time Hierarchy FO(CFL) FO(REGULAR) FO Logarithmic Time Hierarchy 0 AC

25 Inductive Definitions E (x,y) x = y E(x,y) z(e (x,z) E (z,y)) s t

26 Inductive Definitions E (x,y) x = y E(x,y) z(e (x,z) E (z,y)) ϕ tc (R,x,y) x = y E(x,y) z(r(x,z) R(z,y)) s t

27 Inductive Definitions E (x,y) x = y E(x,y) z(e (x,z) E (z,y)) ϕ tc (R,x,y) x = y E(x,y) z(r(x,z) R(z,y)) G REACH G = (LFϕ tc )(s,t) s t

28 Inductive Definitions E (x,y) x = y E(x,y) z(e (x,z) E (z,y)) ϕ tc (R,x,y) x = y E(x,y) z(r(x,z) R(z,y)) G REACH G = (LFϕ tc )(s,t) Thus, REACH IND[log n]. s t

29 Inductive Definitions E (x,y) x = y E(x,y) z(e (x,z) E (z,y)) ϕ tc (R,x,y) x = y E(x,y) z(r(x,z) R(z,y)) G REACH G = (LFϕ tc )(s,t) Thus, REACH IND[log n]. s t Next, we ll show that REACH FO[log n].

30 ϕ tc (R, x, y) x = y E(x, y) z (R(x, z) R(z, y)) 1. Dummy universal quantification for base case: ϕ tc (R,x,y) ( z.m 1 )( z)(r(x,z) R(z,y)) M 1 (x = y E(x,y))

31 ϕ tc (R, x, y) x = y E(x, y) z (R(x, z) R(z, y)) 1. Dummy universal quantification for base case: ϕ tc (R,x,y) ( z.m 1 )( z)(r(x,z) R(z,y)) M 1 (x = y E(x,y)) 2. Using, replace two occurrences of R with one: ϕ tc (R,x,y) ( z.m 1 )( z)( uv.m 2 )R(u,v) M 2 (u = x v = z) (u = z v = y)

32 ϕ tc (R, x, y) x = y E(x, y) z (R(x, z) R(z, y)) 1. Dummy universal quantification for base case: ϕ tc (R,x,y) ( z.m 1 )( z)(r(x,z) R(z,y)) M 1 (x = y E(x,y)) 2. Using, replace two occurrences of R with one: ϕ tc (R,x,y) ( z.m 1 )( z)( uv.m 2 )R(u,v) M 2 (u = x v = z) (u = z v = y) 3. Requantify x and y. M 3 (x = u y = v) ϕ tc (R,x,y) [ ( z.m 1 )( z)( uv.m 2 )( xy.m 3 ) ] R(x,y) Every FO inductive definition is equivalent to a quantifier block.

33 QB tc [( z.m 1 )( z)( uv.m 2 )( xy.m 3 )] ϕ tc (R,x,y) [( z.m 1 )( z)( uv.m 2 )( xy.m 3 )]R(x,y)

34 QB tc [( z.m 1 )( z)( uv.m 2 )( xy.m 3 )] ϕ tc (R,x,y) ϕ tc (R,x,y) [( z.m 1 )( z)( uv.m 2 )( xy.m 3 )]R(x,y) [QB tc ]R(x,y)

35 QB tc [( z.m 1 )( z)( uv.m 2 )( xy.m 3 )] ϕ tc (R,x,y) ϕ tc (R,x,y) ϕ r tc ( ) [( z.m 1 )( z)( uv.m 2 )( xy.m 3 )]R(x,y) [QB tc ]R(x,y) [QB tc] r (false)

36 QB tc [( z.m 1 )( z)( uv.m 2 )( xy.m 3 )] ϕ tc (R,x,y) ϕ tc (R,x,y) ϕ r tc ( ) [( z.m 1 )( z)( uv.m 2 )( xy.m 3 )]R(x,y) [QB tc ]R(x,y) [QB tc] r (false) Thus, for any structure A STRUC[τ g ], A REACH A = (LFϕ tc )(s,t) A = ([QB tc ] 1+log A false)(s,t)

37 CRAM[t(n)] = concurrent parallel random access machine; polynomial hardware, parallel time O(t(n)) IND[t(n)] = first-order, depth t(n) inductive definitions FO[t(n)] = t(n) repetitions of a block of restricted quantifiers: QB = [(Q 1 x 1.M 1 ) (Q k x k.m k )]; M i quantifier-free ϕ n = [QB][QB] [QB] }{{} t(n) M 0

38 parallel time = inductive depth = QB iteration Thm: For all constructible, polynomially bounded t(n), CRAM[t(n)] = IND[t(n)] = FO[t(n)] Thm: For all t(n), even beyond polynomial, CRAM[t(n)] = FO[t(n)]

39 co r.e. FO A (N) co r.e. Arithmetic Hierarchy FO(N) Recursive r.e. FO (N) E r.e. For t(n) poly bdd, CRAM[t(n)] = IND[t(n)] = FO[t(n)] co N SO A O(1) FO[n ] FO(LF) O(1) FO[(log n) ] FO[log n] FO(CFL) FO(TC) FO(REGULAR) olynomial Time Hierarchy SO co N N rimitive Recursive N SACE "truly feasible" co N SO E N NC AC 1 NSACE[log n] FO(DTC) DSACE[log n] 1 NC FO Logarithmic Time Hierarchy 0 AC

40 co r.e. FO A (N) Arithmetic Hierarchy r.e. co r.e. FO(N) r.e. Recursive FO (N) E rimitive Recursive For all t(n), CRAM[t(n)] = O(1) n FO[2 ] co N SO A FO(LF) SACE FO(F) olynomial Time Hierarchy SO co N N N co N O(1) FO[n ] "truly feasible" O(1) FO[(log n) ] NC SO E N FO[t(n)] FO[log n] FO(CFL) FO(TC) FO(DTC) FO(REGULAR) 1 AC NSACE[log n] DSACE[log n] 1 NC FO Logarithmic Time Hierarchy 0 AC

41 Thm: For v = 1,2,..., DSACE[n v ] = VAR[v + 1] Number of variables corresponds to amount of hardware. Since variables range over a universe of size n, a constant number of variables can specify a polynomial number of gates: A bounded number of variables corresponds to polynomially much hardware.

42 Key Issue: arallel Time versus Amount of Hardware

43 Key Issue: arallel Time versus Amount of Hardware We would love to understand this tradeoff.

44 Key Issue: arallel Time versus Amount of Hardware We would love to understand this tradeoff. Is there such a thing as an inherently sequential problem? No one knows.

45 Key Issue: arallel Time versus Amount of Hardware We would love to understand this tradeoff. Is there such a thing as an inherently sequential problem? No one knows. Same tradeoff as number of variables vs. number of iterations of a quantifier block.

46 Key Issue: arallel Time versus Amount of Hardware We would love to understand this tradeoff. Is there such a thing as an inherently sequential problem? No one knows. Same tradeoff as number of variables vs. number of iterations of a quantifier block. One second-order variable can name 2 n gates.

47 Key Issue: arallel Time versus Amount of Hardware We would love to understand this tradeoff. Is there such a thing as an inherently sequential problem? No one knows. Same tradeoff as number of variables vs. number of iterations of a quantifier block. One second-order variable can name 2 n gates. Thus, SO[t(n)] = CRAM-HARD[t(n),2 no(1) ].

48 co r.e. FO A (N) Arithmetic Hierarchy r.e. co r.e. FO(N) r.e. Recursive FO (N) E rimitive Recursive SO[t(n)] = CRAM-HARD [t(n),2 no(1) ] O(1) n FO[2 ] co N FO[log n] FO(TC) FO(DTC) FO(M) FO SO O(1) n SO[2 ] A FO(LF) O(1) SO[n ] EXTIME SACE olynomial Time Hierarchy SO co N N SO Horn O(1) FO[(log n) ] NC SO Krom N co N FO(F) Logarithmic Time Hierarchy SO(LF) SO SO(TC) O(1) FO[n ] FO(CFL) FO(REGULAR) "truly feasible" E N 1 AC sac 1 NSACE[log n] DSACE[log n] 1 NC 0 ThC 0 AC

49 Recent Breakthroughs in Descriptive Complexity Theorem [Ben Rossman] Any first-order formula with any numeric relations (,+,,...) that means I have a clique of size k must have at least k/4 variables. Creative new proof idea using Håstad s Switching Lemma gives the essentially optimal bound. This lower bound is for a fixed formula, if it were for a sequence of polynomially-sized formulas, i.e., a fixed-point formula, it would follow that CLIQUE and thus N. Best previous bounds: k variables necessary and sufficient without ordering or other numeric relations [I 1980]. Nothing was known with ordering except for the trivial fact that 2 variables are not enough.

50 Recent Breakthroughs in Descriptive Complexity Theorem [Martin Grohe] Fixed-oint Logic with Counting captures olynomial Time on all classes of graphs with excluded minors. Grohe proves that for every class of graphs with excluded minors, there is a constant k such that two graphs of the class are isomorphic iff they agree on all k-variable formulas in fixed-point logic with counting. Using Ehrenfeucht-Fraïssé games, this can be checked in polynomial time, (O(n k (log n))). In the same time we can give a canonical description of the isomorphism type of any graph in the class. Thus every class of graphs with excluded minors admits the same general polynomial time canonization algorithm: we re isomorphic iff we agree on all formulas in C k and in particular, you are isomorphic to me iff your C k canonical description is equal to mine.

51 What We Know Diagonalization: more of the same resource gives us more: DTIME[n] = DTIME[n 2 ], same for DSACE, NTIME, NSACE,...

52 What We Know Diagonalization: more of the same resource gives us more: DTIME[n] = DTIME[n 2 ], same for DSACE, NTIME, NSACE,... Natural Complexity Classes have Natural Complete roblems SAT for N, CVAL for, QSAT for SACE,...

53 What We Know Diagonalization: more of the same resource gives us more: DTIME[n] = DTIME[n 2 ], same for DSACE, NTIME, NSACE,... Natural Complexity Classes have Natural Complete roblems SAT for N, CVAL for, QSAT for SACE,... Major Missing Idea: concept of work or conservation of energy in computation, i.e, in order to solve SAT or other hard problem we must do a certain amount of computational work.

54 Strong Lower Bounds on FO[t(n)] for small t(n) [Sipser]: strict first-order alternation hierarchy: FO.

55 Strong Lower Bounds on FO[t(n)] for small t(n) [Sipser]: strict first-order alternation hierarchy: FO. [Beame-Håstad]: hierarchy remains strict up to FO[log n/log log n].

56 Strong Lower Bounds on FO[t(n)] for small t(n) [Sipser]: strict first-order alternation hierarchy: FO. [Beame-Håstad]: hierarchy remains strict up to FO[log n/log log n]. NC 1 FO[log n/log log n] and this is tight.

57 Strong Lower Bounds on FO[t(n)] for small t(n) [Sipser]: strict first-order alternation hierarchy: FO. [Beame-Håstad]: hierarchy remains strict up to FO[log n/log log n]. NC 1 FO[log n/log log n] and this is tight. Does REACH require FO[log n]? This would imply NC 1 NL.

58 Does It Matter? How important is N? Much is known about approximation, e.g., some N problems, e.g., Knapsack, Euclidean TS, can be approximated as closely as we want, others, e.g., Clique, can t be.

59 Does It Matter? How important is N? Much is known about approximation, e.g., some N problems, e.g., Knapsack, Euclidean TS, can be approximated as closely as we want, others, e.g., Clique, can t be. We conjecture that SAT requires DTIME[Ω(2 ǫn )] for some ǫ > 0, but no one has yet proved that it requires more than DTIME[n].

60 Does It Matter? How important is N? Much is known about approximation, e.g., some N problems, e.g., Knapsack, Euclidean TS, can be approximated as closely as we want, others, e.g., Clique, can t be. We conjecture that SAT requires DTIME[Ω(2 ǫn )] for some ǫ > 0, but no one has yet proved that it requires more than DTIME[n]. Basic trade-offs are not understood, e.g., trade-off between time and number of processors. Are any problems inherently sequential? How can we best use mulitcores?

61 Does It Matter? How important is N? Much is known about approximation, e.g., some N problems, e.g., Knapsack, Euclidean TS, can be approximated as closely as we want, others, e.g., Clique, can t be. We conjecture that SAT requires DTIME[Ω(2 ǫn )] for some ǫ > 0, but no one has yet proved that it requires more than DTIME[n]. Basic trade-offs are not understood, e.g., trade-off between time and number of processors. Are any problems inherently sequential? How can we best use mulitcores? SAT solvers are impressive new general purpose problem solvers, e.g., used in model checking, AI planning, code synthesis. How good are current SAT solvers? How much can they be improved?

62 Descriptive Complexity Fact: For constructible t(n), FO[t(n)] = CRAM[t(n)] Fact: For k = 1,2,..., VAR[k + 1] = DSACE[n k ] The complexity of computing a query is closely tied to the complexity of describing the query. = N ThC 0 = N = SACE FO(LF) = SO FO(MAJ) = SO FO(LF) = SO(TC)

63 co r.e. FO A (N) Arithmetic Hierarchy r.e. co r.e. FO(N) r.e. Recursive FO (N) E O(1) n FO[2 ] co N SO O(1) n SO[2 ] A FO(LF) O(1) SO[n ] rimitive Recursive EXTIME SACE FO(F) olynomial Time Hierarchy SO co N N SO Horn N co N SO(LF) SO(TC) O(1) FO[n ] "truly feasible" O(1) FO[(log n) ] NC SO E N FO[log n] FO(CFL) FO(TC) SO Krom FO(DTC) FO(REGULAR) FO(M) FO Logarithmic Time Hierarchy 1 AC sac 1 NSACE[log n] DSACE[log n] 1 NC 0 ThC 0 AC