Challenge to L is not P Hypothesis by Analysing BP Lower Bounds

Transcription

1 Challenge to L is not P Hypothesis by Analysing BP Lower Bounds Atsuki Nagao (The University of Electro-Communications) Joint work with Kazuo Iwama (Kyoto University)

2 Self Introduction : Under graduate student in Kyoto university. Bayesian Network, User Interface : Master course and doctor course student in Kyoto university. Computational Complexity : Post doctor for ELC (A02) in The university of electro-communication. Algorithms (sorting, graph), combinatorics games etc.

3 Complexity Zoo L NC P NP PH PSPACE

4 Talk Plan Long introduction of the problem Tree evaluation problem Branching programs Introduction of the work of [Cook, McKenzie, Wehr, Braverman, Santhanam 2012] Our motivation, goal and summary of new results Quick mention of related work Basic ideas of proofs

5 Tree Evaluation Problem (TEP) Cook, McKenzie, Wehr, Braverman, Santhanam {1,..., k} k Input: Output: the value of the root node 2 h 1 Input length: k 2 log log log n k n k h

6 How to Compute TEP with Small Memory Pebble-game computation

7 How to Compute TEP with Small Memory Pebble-game computation Space becomes maximum Easy space upper bound: (log k) 2 h 1 Input length: k 2 log log log h n k n k h > log space Both k and h increase unlimitedly

8 Branching Programs A Branching Program: Known as a Decision Diagrams One of computational models underlying one DAG one root node represents one Boolean function {0, 1} n {0, 1} n : the number of input variables m : the number of edges 0-sink root 1-sink BP size is defined as the number of nodes.

9 Branching Programs A Branching Program: Known as a Decision Diagrams One of computational models underlying one DAG one root node represents one Boolean function {0, 1} n {0, 1} n : the number of input variables m : the number of edges 0-sink root 1-sink y = f(x 1, x 2, x ) x 1 x 2 x : y : : : : : : : : 0 n = m = 10 BP size is defined as the number of nodes.

10 Back ground Relation to Turing Machine TM space : O s n BP size : 2O s n root If a problem s.t. BP size : ω s (n) TM space : ω log s n This leads to L P 0-sink 1-sink Known best lower bounds. ω n 2 / log n [Nec 66] (Element distinction function)

11 Back ground Relation to circuits width 5 Branching Program (with polynomial length) can recognize NC 1 (O log n depth And-Or circuit) [Barrinton 86] The reverse also holds. [Barrinton 86] Relation to CNF BP can simulate in the same size. x 1 x 2 x x 1 x x 4 x 1 x 1 x 2 x 2 x x x 4

12 f ( x, x, x, x ) mod ( x x x x ) Boolean case Branching Programs

13 Branching Programs f ( x, x, x, x ) mod ( x x x x ) x i {0,1, 2} All BP s in this this talk are deterministic. See e.g., [Komarath, 0 0Sarma 1] 0 for ND case 0 0 0

14 Cook, McKenzie, Wehr, Braverman, Santhanam 2012 Challenge: separating L and P Using TEP and BP s BP's solving TEP need r( h) k states P L h Ok ( ) is easy, but seems optimal (next slide) In fact we do need ( k ) for h h If BP s are thrifty, we need ( k )(2 nd next slide) Conjecture: Thrifty BP s are optimal

15 k=2 Natural Construction

16 k= 6 2,1,1 2,1,2 2,1, 2,2,1 2,2,2 2,2, 2,,1 2,,2 2,, ,1,1,1,2,1,,2,1,2,,,,1,1,2,1,,2,1,2,2,2,,1,1,,2,2 7 Those BP s are thrifty! 6 6 6,1,1,,,2,,,1,,2,,,2,1,1,2,,1,2,2,,2,1,,2, 1,1,1 1,1,2 1,1, 1,2,1 1,2,2 1,2, 1,,1 1,,2 1,, 1 2

17 Thrifty BP s b a Any computation path with never reaches here. has already been known a b 2i and 2i+1 should have been read before reading i

18 Thrifty BP s i 1 2i 1 2 ( i,1,) 1 2

19 Non-Thrifty BP s 2i ,2, 1 ( i,1,1) 1 2 ( i,1,2) ( i,1,2) ( i,1,) ( i,1,) ( i,1,) ( i,1,) ( i,1,) ( i,1,) 2 2i 1 Conjecture seems OK, but proving it is. 2

20 Another Restriction Thrifty restriction is semantic and specific for TEP There is a popular (syntactic and general) restriction, read-once restriction, for BP s

21 Read-Once BP s 0 2i i ( i,1,) 1 2 i i

22 Talk Plan Long introduction of the problem Tree evaluation problem Branching programs Introduction of the work of [Cook, McKenzie, Wehr, Braverman, Santhanam 2012] Our motivation, goal and summary of new result Quick mention of related work Basic ideas of proofs

23 Our Motivation The natural construction is both thrifty and read-once and this BP is absolutely optimal under the two restrictions [Wehr 1]

24 k= 6 2,1,1 2,1,2 2,1, 2,2,1 2,2,2 2,2, 2,,1 2,,2 2,, ,1,1,1,2,1,,2,1,2,,,,1,1,2,1,,2,1,2,2,2,,1,1,,2,2 7 Those BP s are thrifty 6 6 6,1,1,, and Read-Once,2,1,1,2,,2,2,,,1,2,2,1,,2,,,1,,2,, 1,1,1 1,1,2 1,1, i 1,2,1 1,2,2 1,2, 1,,1 1,,2 1,, 1 2

25 Our Motivation The natural construction is both thrifty and read-once and this BP is absolutely optimal under the two restrictions [Wehr 1] [CMWBS 12] proved only thrifty is enough Our motivation: What about only read-once restriction? Can remove the restriction on the order of reading variables, essence of thriftiness Would be a nice progress towards the goal

26 Our Results ( k h ) lower bound for read-once BP s Tight within a constant factor Read-once restriction only for leaf leading states

27 ,1,1 2,1,2 2,1, 2,2,1 2,2,2 2,2, 2,,1 2,,2 2,, ,1,1,1,2,1,,2,1,2,2,2,,,1,,2,,,1,1 7,1,2,1,,2,1,2,2,2,,,1,,,,2,1, ,,,2,1,1,2,,1,2,2,,2,1,,2, 1,1,1 1,1,2 1,1, 1,2,1 1,2,2 1,2, 1,,1 1,,2 1,, 1 2

28 Our Results ( k h ) lower bound for read-once BP s Tight within a constant factor Read-once restriction only for leaf leading states An easier proof for the general lower bound for h=

29 Existing Lower Bounds for BP s Branching programs [Masek 76] Relation between BP size and TM space Lower bounds n n) b 2 2 ( log [Neciporuk 66] is still est ( n log n) for TEP of h ( n log n), ( n lo g n) for h 4,5,,, Lower bounds for read-once BPs n log n (2 ) [Zak 84] cn (2 ) [Ajtai86, Wegener 88, Simon-Szegedy 9] Many others (e.g., read-k BP s)

30 Talk Plan Long introduction of the problem Tree evaluation problem Branching programs Introduction of the work of [Cook, McKenzie, Wehr, Braverman, Santhanam 2012] Our motivation, goal and summary of new result Quick mention of related work Basic ideas of proofs

31 First Key Lemma FTh ( ) 1 k h i i, a, b i, a, b BP with p second-leaf reading states h 1 FT ( ) h k i i i BP with p 2 leaf k reading states h i

32 Corollary (Contraposition) of the Lemma If any (not necessarily read-once) BP for FT ( k) needs q leaf ) tates h 2 reading states, then any BP for FTh 1( k needs at least qk s h=2 case 2 1 It is easy to prove that any (general) BP needs k leaf reading states for FT ( k) Easier than [CMWBS12] We can conclude that any (general) BP needs k states for FT ( k)

33 Proof of the Lemma Select a,b such that # of states (i,a,b) is maximum

34 Main Lemma: Cut Configuration (Tool 1) Given: Input I ( f1,..., f15, a16,..., a1) and a leaf node j CC( I, j) ( a,.., a ) (i) (ii) a 1 a i 1 j 1 h 1 is the value of j's sibling CC(I,j) and the value of j fixes the output is the value of the sibling of a's parent i

35 Tool 2: Special Function Class Impose a restriction

36 Tool 2: Special Function Class f a a a a a a a a k 21 2k k1 k 2 kk (i) a11,..., a1 k (first row) is a permutation of (1,2,..., k) (ii) a11,..., a k 1 (first column) is... (iii) a,..., a can be written as j1 l ( a,..., a ) 11 1k 1 2 = 2 jk for some l, where k 1 k k 1 Elementary properties of symmetric groups

37 Counting Last Leaf Reading States j j j j Last leaf-reading states (LLRS) can be defined without denoting computation paths (read-once restriction and special function class)

38 Main Lemma: Final Claim f j FT ( ) h k k : prime j j s Need Need k h 1 f achieves CC's c1 and c2 wrt LLRS s k The number of such f N0 k! h 1 # of LLRS's is k 1 # of total such f k N N k N LLRS's k! for FT k! h f s.t. its CC is unique for all LLRS's h 1 h 1 2 h 2 0 ( k 1 ) ( k ) h 1 h 1 k k k states for FTh 1 A single CC wrt j can accomodate at most k h 1 2 h 1 different leaf inputs j i k h 1 h 1 h h 1 k k ( 1) (=# of all leaf values) 2 1 2

39 Future Work General lower bounds for height-4 TEP Other classes of BP s Read-k BP s Nondeterministic BP s Ambitious goal..

40 Other works Branching Program SAT Problems k-ibdd SAT talk in ELC meeting in 2015 Indexed Binary Decision Diagram A BP which can be divided into k BDDs Read-k-times BP SAT newest result for us A BP whose consistent paths read x i at most k times

41 k-ibdd SAT π 1 π π Check reachability at most k m times

42 In Minato lab I m interested in BDD lower / upper bounds. Lower bounds Height 4 TEP (if you are interested) Upper bounds enumeration, down sizing, optimization, etc. Also BDD SAT Error detecting, width-restricted BDD SAT, etc.