Hardness of Function Composition for Semantic Read once Branching. Programs. June 23, 2018

Transcription

1 once for once June 23, 2018

2 P and L once P: Polynomial time computable functions. * * * * L : s computable in logarithmic space. L? P

3 f (x 1, x 2,.., x n ) {0, 1} Definition Deterministic program DAG with a source node and two sinks, 1-sink (for accept) and 0-sink (for reject). x i {0, 1}, i [n] Each non-sink node is labeled by some x i, outdegree 2 with an edge each for x i = 0 and x i = 1. once start, x x 2 0 x 3 0 x 4 0 x x 2 0 x 3 0 x 4 0 x 5 0 1

4 Non-det Definition Non-deterministic program (NBP) Start allow unlabelled guessing nodes and arbitrary out-degree. x 11 x 12 x 13 x 21 x 22 x 23 x 31 x 32 x 33 x 41 x 42 x 43 1 once x 14 x 24 x 34 x 44 The size of a NBP= number of labelled nodes.

5 NBP computing f : {0, 1} n {0, 1} once x 11 x 21 x 31 x 41 Start x 12 x 13 x 14 x 22 x 23 x 24 x 32 x 33 x 34 x 42 x 43 x 44 f (u) = 1 a path from source to accept node that is consistent with input u. 1

6 Program Size and Space Complexity of computing f once BP(f n ) = S(f n ) = min B BP computing f n size (B) min space complexity (T ) T non-uniform TMs computing f n log(bp(f n )) S(f n ) [Cobham 66]

7 Big Picture once It is easy to show functions with high BP(f n ) exist.

8 Big Picture once It is easy to show functions with high BP(f n ) exist. Can we show that some function in P requires exponential size BP? amounts to showing L P.

9 BPs and other Computation Models Formulas Circuits L(f ) BP(f ) 1 3 C(f ) once

10 BPs and other Computation Models Formulas Circuits L(f ) BP(f ) 1 3 C(f ) Ω ( n 3) ( ) Ω n 2 log 2 n Ω (n) Random Restrictions Nechiporuk Gate Elimination once

11 Restricted once Bounded Width: same as NC 1, Barrington s characterization.

12 Restricted once Bounded Width: same as NC 1, Barrington s characterization. Length give Time-Space tradeoffs.

13 Time-Space tradeoffs t cn = s = 2 Ω(n) Jukna 09 once

14 Time-Space tradeoffs t cn = s = 2 Ω(n) Jukna 09 culmination results by Ajtai 99 and Beame,Jayram, Saks 01 once

15 Time-Space tradeoffs t cn = s = 2 Ω(n) Jukna 09 culmination results by Ajtai 99 and Beame,Jayram, Saks 01 We look at: time-space tradeoffs for iterated function composition. once

16 Read Once Syntactic read once: Along any path from source to sink any variable appears atmost once. x 11 x 21 x 31 once Start 1 x 12 x 22 x 32 x 13 x x 33 23

17 Read Once Syntactic read once: Along any path from source to sink any variable appears atmost once. x 11 x 21 x 31 once Start 1 x 12 x 22 x 32 x 13 x 11 x 23 x 21 x 33 x Start 1 x 12 x 22 x x 13 x 23 x Semantic read once: Along any consistent path from source to sink no variable is read more than once.

18 syntactic weaker than semantic The Exact Perfect matching function(epm n ): accept a matrix iff it is a permutation matrix. Jukna and Razborov 98 showed Theorem Every syntactic read once NBP computing EPM n must have size 2 Ω(n). once

19 EPM n semantic read once Theorem (Jukna) EPM n can be solved by a semantic read once NBP of size O(n 3 ) once

20 EPM n semantic read once Theorem (Jukna) EPM n can be solved by a semantic read once NBP of size O(n 3 ). x 11 x 21 x once Start x 12 x 22 x 32 x 13 x 23 x 33

21 EPM n semantic read once Theorem (Jukna) EPM n can be solved by a semantic read once NBP of size O(n 3 ). x 11 x 21 x Start 1 x 12 x 22 x once x 13 x 23 x

22 EPM n semantic read once Theorem (Jukna) EPM n can be solved by a semantic read once NBP of size O(n 3 ). x 11 x 21 x Start 1 x 12 x 22 x once x 13 x 23 Sees only 1s x

23 EPM n semantic read once Theorem (Jukna) EPM n can be solved by a semantic read once NBP of size O(n 3 ). x 11 x 21 x Start 1 x 12 x 22 x once x 13 x 23 Sees only 1s x Sees only 0s

24 EPM n semantic read once Theorem (Jukna) EPM n can be solved by a semantic read once NBP of size O(n 3 ). x 11 x 21 x Start 1 x 12 x 22 x once x 13 x 23 Sees only 1s x Sees only 0s

25 Embedded C red {w} C blue [D] n C red D A w D [n] A B w C blue D B A [n] B [n] { } once

26 Embedded C red {w} C blue [D] n C red D A w D [n] A B w C blue D B A [n] B [n] {102101} once

27 KRW Conjecture once an approach to separating NC 1 from NC 2. KRW conjecture that for every random f and g, D(fog) ɛd(f ) + D(g). KRW conjecture on formula size of a composed function fog. L(fog) L(f )L(g)?

28 Understanding once How does space compose?

29 TEP h d Cook et.al 2012 once f : [k] 2 [k] [k] Figure: TEP2 4 that is height 4, degree 2.

30 TEP h d Cook et.al 2012 once Figure: TEP 4 2 f : [k] 2 [k] [k] that is height 4, degree 2. Is BP(TEP h 2 ) = Ω(kh )??

31 TEP h d Cook et.al 2012 once Figure: TEP 4 2 f : [k] 2 [k] [k] that is height 4, degree 2. Is BP(TEP h 2 ) = Ω(kh )?? = L P

32 Boolean TEP once K ɛ density K 1 ɛ Tree F,ɛ ( )

33 Lower Bound for composition Theorem For any h, and k sufficiently large, there exists ɛ and F such that any k-ary nondeterministic semantic read-once branching program for ternary Tree F,ɛ requires size at least ( ) k h. log k once Proof OverView A special query state for each input Special low entropy node in the Tree Two-way Product Sets Conclusion

34 Black White pebbling Upperbound once Proof OverView A special query state for each input Special low entropy node in the Tree Two-way Product Sets Conclusion

35 1 2 (d 1)h + 1 pebbles at this moment once Proof OverView A special query state for each input Special low entropy node in the Tree Two-way Product Sets Conclusion

36 Guess the remaining siblings once Proof OverView A special query state for each input Special low entropy node in the Tree Two-way Product Sets Conclusion

37 Infer the root once Proof OverView A special query state for each input Special low entropy node in the Tree Two-way Product Sets Conclusion

38 Unpebble blacks once Proof OverView A special query state for each input Special low entropy node in the Tree Two-way Product Sets Conclusion

39 Verify Guesses once Proof OverView A special query state for each input Special low entropy node in the Tree Two-way Product Sets Conclusion

40 lower bound uses Invertible functions once Definition A Latin Cube is a function f : [k] 3 [k] such that f is invertible in each of its coordinates. Equivalently, every element of [k] appears exactly once along every row,column and leg in the cube [k] 3. Proof OverView A special query state for each input Special low entropy node in the Tree Two-way Product Sets Conclusion

41 4-invertible function, f : [k] 3 [k] once Definition Any element in [k] appears at most 4 times along any row,column or leg in the cube [k] 3. Proof OverView A special query state for each input Special low entropy node in the Tree Two-way Product Sets Conclusion

42 Proof Overview A small BP for a = Tree F,ɛ accepts a large Tree F,ɛ rectangle of inputs over its leaves once Proof OverView A special query state for each input Special low entropy node in the Tree Two-way Product Sets Conclusion

43 Proof Overview A small BP for a = Tree F,ɛ accepts a large Tree F,ɛ rectangle of inputs over its leaves A large rectangle = a special node v* in the tree over leaves whose F v can be described in few bits. once Proof OverView A special query state for each input Special low entropy node in the Tree Two-way Product Sets Conclusion

44 Proof Overview A small BP for a = Tree F,ɛ accepts a large Tree F,ɛ rectangle of inputs over its leaves A large rectangle = a special node v* in the tree over leaves whose F v can be described in few bits. Show that the distribution on F is rich or sufficiently random looking that one cannot save these bits. once Proof OverView A special query state for each input Special low entropy node in the Tree Two-way Product Sets Conclusion

45 for every accepting input a special query state: once Ah(x) Bh(x, y) = Fh(Ah(x), Bh 1(x, y)ch(y)) Ch(y) Ch 1(y) Ah 1(x) h 2 A white subtree has at least a fraction of leaf node which are white Ah 2(x) Ai(x) A3(x) i 1 vi Ch 2(y) Bi(x, y) = Fi(Ai(x), Bi 1(x, y)ci(y)) Ci(y) i 1 C3(y) h 2 A red subtree has at least a fraction of leaf node which are red Proof OverView A special query state for each input Special low entropy node in the Tree Two-way Product Sets Conclusion A2(x)? A1(x) C1(y) C2(y)

46 few special states = a large embedded rectangle over leaves v start q v 1 v Most popular Choose a popular labelled path down the tree. Choose a popular red variable for the first red-subtree. Prune the input set. Continue to choose h red variables one for each red-subtree. Similarly for each blue-subtree. Fix the remaining variables in [n]-red-blue to the most popular projection w. once Proof OverView A special query state for each input Special low entropy node in the Tree Two-way Product Sets Conclusion

47 a large embedded rectangle over the leaves. once Ah(x) Bh(x, y) = Fh(Ah(x), Bh 1(x, y)ch(y)) Ch(y) Ch 1(y) Ah 1(x) Ah 2(x) Ai(x) A3(x) i 1 vi Ch 2(y) Bi(x, y) = Fi(Ai(x), Bi 1(x, y)ci(y)) Ci(y) i 1 C3(y) Proof OverView A special query state for each input Special low entropy node in the Tree Two-way Product Sets Conclusion A2(x)? A1(x) C1(y) C2(y)

48 a node v at which leaves in both red and blue trees take a lot of values. once Ah(x) Bh(x, y) = Fh(Ah(x), Bh 1(x, y)ch(y)) Ch(y) Ch 1(y) Ah 1(x) Ah 2(x) Ai(x) A3(x) i 1 vi Ch 2(y) Bi(x, y) = Fi(Ai(x), Bi 1(x, y)ci(y)) Ci(y) i 1 C3(y) Proof OverView A special query state for each input Special low entropy node in the Tree Two-way Product Sets Conclusion A2(x)? A1(x) C1(y) C2(y)

49 a node v at which both red and blue child take a lot of values. once Ah(x) Bh(x, y) = Fh(Ah(x), Bh 1(x, y)ch(y)) Ch(y) Ch 1(y) Ah 1(x) Ah 2(x) Ai(x) A3(x) i 1 vi Ch 2(y) Bi(x, y) = Fi(Ai(x), Bi 1(x, y)ci(y)) Ci(y) i 1 C3(y) Proof OverView A special query state for each input Special low entropy node in the Tree Two-way Product Sets Conclusion A2(x)? A1(x) C1(y) C2(y)

50 a node v with low entropy on a Two-way product set once Ah(x) Bh(x, y) = Fh(Ah(x), Bh 1(x, y)ch(y)) Ch(y) Ch 1(y) Ah 1(x) Ah 2(x) Ai(x) A3(x) i 1 vi Ch 2(y) Bi(x, y) = Fi(Ai(x), Bi 1(x, y)ci(y)) Ci(y) i 1 C3(y) Proof OverView A special query state for each input Special low entropy node in the Tree Two-way Product Sets Conclusion A2(x)? A1(x) C1(y) C2(y)

51 Two-way Product Set at v, in F v () once A C A, C [k] A = C = r << k Proof OverView A special query state for each input Special low entropy node in the Tree Two-way Product Sets Conclusion

52 Two-way Product Set at v, in F v () once A C A, C [k] A = C = r << k S r = {(x, Q(x, y), y) x A, y C} S = r 2 Proof OverView A special query state for each input Special low entropy node in the Tree Two-way Product Sets Conclusion

53 Enropy of spread on Two-way Product Set can t be low A C A, C [k] A = C = r << k S r = {(x, B(x, y), y) x A, y C} S = r 2 Two-way Product Sets S r and target set T ɛ Pr [f (S r ) T ɛ ] 1 f U(All 4-invertible cubes) k ɛr 2 once Proof OverView A special query state for each input Special low entropy node in the Tree Two-way Product Sets Conclusion

54 Label once α h βh h 2 A red subtree has at least one red leaf node α h 1 α h 2 α i α 2 vi β h 1 β h 2 β i β 2 h 2 A white subtree has at least one white leaf node Proof OverView A special query state for each input Special low entropy node in the Tree Two-way Product Sets Conclusion α 1? β1 Figure: This figure depicts a label L F associated with a problem instance Tree F

55 many F that remain unaccounted without such a special label. once F < v, S A, S B.. > Proof OverView A special query state for each input Special low entropy node in the Tree Two-way Product Sets Conclusion

56 More general time space tradeoffs for composition. Exponential lower bound for boolean semantic NBPs for some problem in P. super-quadratic lower bound for BPs via understanding composition fog where g is element distinctness. once

57 once Thank You!