Math 262A Lecture Notes - Nechiporuk s Theorem

Transcription

1 Math 6A Lecture Notes - Nechiporuk s Theore Lecturer: Sa Buss Scribe: Stefan Schneider October, 013 Nechiporuk [1] gives a ethod to derive lower bounds on forula size over the full binary basis B The lower bounds we have seen so far were for the parity function over the {,, } basis The parity function has a trivial linear sized forula over B, hence it is inevitable to study new functions One of the key ingredients in Nechiporuk s Theore is the idea of a restriction, which sets a subset of the variables to constants Definition 1 Restriction Let f : {0, 1} n {0, 1} be an n-ary Boolean function on variable set X = {x 1,, x n } Further let Y = {y 1,, y l } X be a subset of the variable set For Z = X Y, a restriction σ is a apping σ : Z {0, 1} The restriction of f to σ is the function f σ y 1,, y l = fx k1,, x kn where x ki = y j if x ki is the j-th eleent of Y and x ki = σx i if x i Y A subfunction of f on Y is any f σ with doσ = X Y, ie the restriction leaves the variables Y free We soeties alternatively denote a restriction σ as a function σ : X {0, 1, } where σ 1 = Y Theore 1 Nechiporuk s Theore Let f : {0, 1} n {0, 1} be an n-ary Boolean function and let Y 1,, Y be disjoint subsets of the variable set X = {x 1,, x n } Then L B f logs i 4 where s i is the nuber of distinct subfunctions of f on Y i Note that L B f + 1 corresponds to the leaf size of the forula Application 1: Eleent Distinction Before giving the proof of Nechiporuk s Theore, we discuss an application Definition The eleent distinction function on n variables is a function ED n : {0, 1} n {0, 1} where n = log for soe 1 We view the input variables x 1,, x n as blocks of log bits each Then { 1 if the blocks are pairwise distinct ED n x 1,, x n = 0 otherwise 1

2 Proposition 1 n L B ED n = Ω Proof To apply Nechiporuk s Theore we define Y i = x i log +1,, x i+1 log as the i-th block The nuber of values each block can take on is log = Hence the nuber of subfunctions on Y i is s i where we use the fact the subfunction is uniquely defined by the 1 values the other blocks take on and for each setting of the other blocks such that they take on 1 distinct values, the resulting subfunction on Y i is distinct Since n = log we have = Ω and by Nechiporuk s Theore we get n and therefore logs i 1 log + 1 = Ω log = Ωn L B ED n n logs i = Ωn = Ωn = Ω The lower bound above is tight, as we can construct a forula of size O Proposition n L {,, } ED n = O Proof For every 0 i, i 1 with i i we construct the forula log j=1 x i log +j x i log +j which returns true if and only if blocks i and i are identical We then take the disjunction of the forulas and negate the output to get a forula for EDn For two inputs a and b, the function a b requires a constant nuber of gates Furtherore, for every pair i, i the forula contains log 1 conjunctions and the disjunction requires 1 gates The total size of the forula is therefore O log + O n = O log n = O

3 y 1 y y 1 Figure 1: A function restricted to a subtree for Y i = {y 1, y } The set W i is colored red Proof of Nechiporuk s Theore The basic idea of the proof is the look at what happens to the forula when a restriction is applied Proof We prove l i 1 4 logs i where l i is the nuber of occurrences of variables in Y i in a forula ϕ on basis B for f The theore then follows iediately Given ϕ we for a subtree with l i leaves that contains all paths fro any leaf labeled by a variable in Y i to the output gate Let W i be the set of gates that have two inputs in this tree Since a binary tree with l i leaves has exactly l i 1 vertices, we have W i = l i 1 Figure 1 sketches such a subtree Note that an edge in this subtree ight contain an arbitrarily long chain of gates that only have a single input in the subtree Consider a subfunction of ϕ on Y i The forula then siplifies iediately to the subtree described above Furtherore, each path fro an input gate or a gate in W i to the next gate in W i or the output is then a function on one variable Note that there are only 4 Boolean functions on one input, hx = x, hx = x, hx = 1 and hx = 0 Since there is one path originating at each leaf in the subtree and each vertex in W i, the nuber of paths is l i + l i 1 = l i 1 Also, since the functions described by each of the paths fully specify the restricted subfunction, the nuber of subfunctions of f on Y i is bounded by s i 4 li 1 < 4 li = 4li and therefore l i > 1 4 logs i Application : Storage Access Functions In this section we discuss the storage access function or universal function The function was first defined in the sae paper as Nechiporuk s Theore 3

4 Definition 3 The storage access function is a Boolean function : {0, 1} n {0, 1} with x 1,, x n = y 0,, y 1, z 1,, z l where l = log and n = +l Let 0 i 1 be the integer encoded by the bits z Then y, z = y i The function is universal in the sense that by restricting the variables y 0,, y 1 we can get an arbitrary function on log inputs Definition 4 The indirect storage access function is a Boolean function I : {0, 1} n {0, 1} with I x 1,, x n = I y 0,, y 1, z 1,, z, u 1,, u k where k = log We log log log view z as blocks of log bits each Let 0 i 1 be the integer encoded by the bits u and let j i be the integer encoded by the i-th block of log bits in z Then Proposition 3 I y, z, u = y ji n L B I = Ω Proof Let Y i be the i-th block of z For a fixed 0 i log 1 consider a restriction such that u encodes i and all but the i-th block of z is restricted to soe arbitrary values Then the vector y is the function table of the subfunction on Y i Hence the nuber of subfunctions is given by s i = Therefore L B I where we use n = + log log / log logs i = for the last step n log = Ω For upper bounds, we consider first A possible naïve approach is to construct a DNF as follows z encodes integer i y i i=0 The check if z = z 1,, z log encodes an integer i is a conjunction of log variables or negated variables Hence the size of the DNF is O log = On To get an upper bound of On we construct a forula based on a decision tree Note that a atching lower bound of Ωn is trivial Definition 5 A decision tree is a binary tree with internal vertices labeled by input variables and the leaves labeled by either 0 or 1 The branching progra of a decision tree starts at the root node If the current vertex is labeled x i, then the variable x i is queried If the result is 0, the coputation continues at the left child, otherwise it continues at the right child The value of the leaf reached is the output of the coputation Proposition 4 L {,, } = On 4

5 Proof Consider a decision tree T for, where T is a coplete binary tree of depth log with every node on level i being labeled by the variable z i Here, the root is level 1 The vertices on the last level are labeled by y i, where i is the integer encoded by the queried values of z The decision tree is a direct ipleentation of the intuition of first reading z and then looking up of the appropriate value y i Figure shows the decision tree described above Converting this decision tree into a forula we get where and T = z 1 T 0 z 1 T 1 T 0 = z T 00 z T 01 T 1 = z T 10 z T 11 Continuing this construction for log levels we eventually set T z = y i, where z encodes the integer i Figure 3 shows the constructed forula The forula contains j leaves labeled z j for every j and one leaf labeled y i for every i Hence the size of this forula is given by log L {,, } + j = O = On j=1 Note that ore generally, any decision tree of depth d can be converted into a forula of leaf size O d Using a siilar technique, we also get a tight upper bound for the indirect storage access function Proposition 5 n L {,, } I = O Proof We construct a decision tree of depth log log + log We first query the variables in u and then use the fact that I y, z, u = y, z j where z j is the j-th block of z and j is the integer encoded by u The decision tree consists of log levels that query the variables in u and at every root log we attach a decision tree with log levels equivalent to the decision tree of y, z j Figure 4 shows the decision tree The depth of the decision tree is log + log and can therefore be converted into a forula of size log L {,, } I = O log log +log n = O = O log 5

6 z 1 z z z l z l z l y 0 y 1 y y 3 y y 1 Figure : A decision tree for z 1 z 1 z z z z z l y 0 z l y 1 Figure 3: A {,, }-forula equivalent to the decision tree in Figure 6

7 u 1 u u u k u k u k Figure 4: A decision tree for I References [1] E I Nechiporuk, On a Boolean function Soviet Math Dokl 74, ,