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arXiv:1312.2143 · IEEE Conference on Computational Complexity (CCC)

A Composition Theorem for Parity Kill Number

Ryan O’Donnell, John Wright, Yu Zhao, Xiaorui Sun, Li-Yang Tan

Published Jun 1, 2014• 18 citations
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Abstract

In this work, we study the parity complexity measures pCmin[f] and PDT[f]. Pcmin[f] is the parity kill number of f, the fewest number of parities on the input variables one has to fix in order to "kill" f, i.e. To make it constant. PDT[f] is the depth of the shortest emph{parity decision tree} which computes f. These complexity measures have in recent years become increasingly important in the fields of communication complexity [1], [2], [3], [4] and pseudorandomness [5], [6], [7]. Our main result is a composition theorem for pCmin. The k-th power of f, denoted f^{circ k}, is the function which results from composing f with itself k times. We prove that if f is not a parity function, then pCmin[f^{circ k}] geq Omega(Cmin[f]^{k}). In other words, the parity kill number of f is essentially super multiplicative in the normal kill number of f (also known as the minimum certificate complexity). As an application of our composition theorem, we show lower bounds on the parity complexity measures of sort^{circ k} and HI^{circ k}. Here sort is the sort function due to Ambainis [8], and HI is Kushilevitz's hemi-icosahedron function [9]. In doing so, we disprove a conjecture of Montanaro and Osborne [2] which had applications to communication complexity and computational learning theory. In addition, we give new lower bounds for conjectures of [2], [3] and [4].

DOI
10.1109/ccc.2014.22

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One-Way Communication Complexity of Partial XOR Functions

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Boolean function $F(x,y)$ for $x,y \in \{0,1\}^n$ is an XOR function if $F(x,y)=f(x\oplus y)$ for some function $f$ on $n$ input bits, where $\oplus$ is a bit-wise XOR. XOR functions are relevant in communication complexity, partially for allowing Fourier analytic technique. For total XOR functions it is known that deterministic communication complexity of $F$ is closely related to parity decision tree complexity of $f$. Montanaro and Osbourne (2009) observed that one-sided communication complexity $D_{cc}^{\rightarrow}(F)$ of $F$ is exactly equal to nonadaptive parity decision tree complexity $NADT^{\oplus}(f)$ of $f$. Hatami et al. (2018) showed that unrestricted communication complexity of $F$ is polynomially related to parity decision tree complexity of $f$. We initiate the studies of a similar connection for partial functions. We show that in case of one-sided communication complexity whether these measures are equal, depends on the number of undefined inputs of $f$. On the one hand, if $D_{cc}^{\rightarrow}(F)=t$ and $f$ is undefined on at most $O(\frac{2^{n-t}}{\sqrt{n-t}})$, then $NADT^{\oplus}(f)=t$. On the other hand, for a wide range of values of $D_{cc}^{\rightarrow}(F)$ and $NADT^{\oplus}(f)$ (from constant to $n-2$) we provide partial functions for which $D_{cc}^{\rightarrow}(F) < NADT^{\oplus}(f)$. In particular, we provide a function with an exponential gap between the two measures. Our separation results translate to the case of two-sided communication complexity as well, in particular showing that the result of Hatami et al. (2018) cannot be generalized to partial functions. Previous results for total functions heavily rely on Boolean Fourier analysis and the technique does not translate to partial functions. For the proofs of our results we build a linear algebraic framework instead. Separation results are proved through the reduction to covering codes.

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In this work, we study the parity complexity measures pCmin[f] and PDT[f]. Pcmin[f] is the parity kill number of f, the fewest number of parities on the input variables one has to fix in order to "kill" f, i.e. To make it constant. PDT[f] is the depth of the shortest emph{parity decision tree} which computes f. These complexity measures have in recent years become increasingly important in the fields of communication complexity [1], [2], [3], [4] and pseudorandomness [5], [6], [7]. Our main result is a composition theorem for pCmin. The k-th power of f, denoted f^{circ k}, is the function which results from composing f with itself k times. We prove that if f is not a parity function, then pCmin[f^{circ k}] geq Omega(Cmin[f]^{k}). In other words, the parity kill number of f is essentially super multiplicative in the normal kill number of f (also known as the minimum certificate complexity). As an application of our composition theorem, we show lower bounds on the parity complexity measures of sort^{circ k} and HI^{circ k}. Here sort is the sort function due to Ambainis [8], and HI is Kushilevitz's hemi-icosahedron function [9]. In doing so, we disprove a conjecture of Montanaro and Osborne [2] which had applications to communication complexity and computational learning theory. In addition, we give new lower bounds for conjectures of [2], [3] and [4].
Boolean function F(x,y) for x,y ∈ {0,1}ⁿ is an XOR function if F(x,y) = f(x⊕ y) for some function f on n input bits, where ⊕ is a bit-wise XOR. XOR functions are relevant in communication complexity, partially for allowing the Fourier analytic technique. For total XOR functions, it is known that deterministic communication complexity of F is closely related to parity decision tree complexity of f. Montanaro and Osbourne (2009) observed that one-way communication complexity D_{cc}^{→}(F) of F is exactly equal to non-adaptive parity decision tree complexity NADT^{⊕}(f) of f. Hatami et al. (2018) showed that unrestricted communication complexity of F is polynomially related to parity decision tree complexity of f. We initiate the study of a similar connection for partial functions. We show that in the case of one-way communication complexity whether these measures are equal, depends on the number of undefined inputs of f. More precisely, if D_{cc}^{→}(F) = t and f is undefined on at most O((2^{n-t})/(√{n-t})) inputs, then NADT^{⊕}(f) = t. We also provide stronger bounds in extreme cases of small and large complexity. We show that the restriction on the number of undefined inputs in these results is unavoidable. That is, for a wide range of values of D_{cc}^{→}(F) and NADT^{⊕}(f) (from constant to n-2) we provide partial functions (with more than Ω((2^{n-t})/(√{n-t})) undefined inputs, where t = D_{cc}^{→}) for which D_{cc}^{→}(F) < NADT^{⊕}(f). In particular, we provide a function with an exponential gap between the two measures. Our separation results translate to the case of two-way communication complexity as well, in particular showing that the result of Hatami et al. (2018) cannot be generalized to partial functions. Previous results for total functions heavily rely on the Boolean Fourier analysis and thus, the technique does not translate to partial functions. For the proofs of our results we build a linear algebraic framework instead. Separation results are proved through the reduction to covering codes.
A longstanding open question in Boolean function complexity asks whether approximate degree composes multiplicatively under block composition. Although a general multiplicative upper bound is known, matching lower bounds have previously been established only for restricted classes of functions. We resolve this question for all total Boolean functions by proving the matching lower bound. Together with Sherstov's upper bound, our result shows that, for every pair of total Boolean functions f:{0,1}n→{0,1}f:\{0,1\}^n\to\{0,1\}f:{0,1}n→{0,1} and g:{0,1}m→{0,1}g:\{0,1\}^m\to\{0,1\}g:{0,1}m→{0,1}, deg~(f∘g)=Θ ⁣(deg~(f) deg~(g)), \widetilde{deg}(f\circ g) = Θ\!\left( \widetilde{deg}(f)\,\widetilde{deg}(g) \right), deg​(f∘g)=Θ(deg​(f)deg​(g)), where deg~\widetilde{deg}deg​ denotes constant-error approximate degree. We also consider the analogous question for exact quantum query complexity and give a counterexample showing that multiplicative composition fails in this setting.
We study the composition question for bounded-error randomized query complexity: Is R(f circ g) = Omega(R(f)R(g))? We show that inserting a simple function h, whose query complexity is onlyTheta(log R(g)), in between f and g allows us to prove R(f circ h circ g) = Omega(R(f)R(h)R(g)). We prove this using a new lower bound measure for randomized query complexity we call randomized sabotage complexity, RS(f). Randomized sabotage complexity has several desirable properties, such as a perfect composition theorem, RS(f circ g) >= RS(f) RS(g), and a composition theorem with randomized query complexity, R(f circ g) = Omega(R(f) RS(g)). It is also a quadratically tight lower bound for total functions and can be quadratically superior to the partition bound, the best known general lower bound for randomized query complexity. Using this technique we also show implications for lifting theorems in communication complexity. We show that a general lifting theorem from zero-error randomized query to communication complexity implies a similar result for bounded-error algorithms for all total functions.
The communication class UPP cc is a communication analog of the Turing Machine complexity class PP . It is characterized by a matrix-analytic complexity measure called sign-rank (also called dimension complexity), and is essentially the most powerful communication class against which we know how to prove lower bounds. For a communication problem f , let f ∧ f denote the function that evaluates f on two disjoint inputs and outputs the AND of the results. We exhibit a communication problem f with UPP cc ( f ) = O (log n ), and UPP cc ( f ∧ f ) = Θ (log 2 n ). This is the first result showing that UPP communication complexity can increase by more than a constant factor under intersection. We view this as a first step toward showing that UPP cc , the class of problems with polylogarithmic-cost UPP communication protocols, is not closed under intersection. Our result shows that the function class consisting of intersections of two majorities on n bits has dimension complexity n Omega Ω(log n ) . This matches an upper bound of (Klivans, O’Donnell, and Servedio, FOCS 2002), who used it to give a quasipolynomial time algorithm for PAC learning intersections of polylogarithmically many majorities. Hence, fundamentally new techniques will be needed to learn this class of functions in polynomial time.
We prove two new results about the randomized query complexity of composed functions. First, we show that the randomized composition conjecture is false: there are families of partial Boolean functions f and g such that R(f°g) ≪ R(f)R(g). In fact, we show that the left hand side can be polynomially smaller than the right hand side (though in our construction, both sides are polylogarithmic in the input size of f). Second, we show that for all f and g, R(f°g) = Ω(noisyR(f) R(g)), where noisyR(f) is a measure describing the cost of computing f on noisy oracle inputs. We show that this composition theorem is the strongest possible of its type: for any measure M(·) satisfying R(f°g)=Ω(M(f)R(g)) for all f and g, it must hold that noisyR(f)=Ω(M(f)) for all f. We also give a clean characterization of the measure noisyR(f): it satisfies noisyR(f)=Θ(R(f°GapMajn)/R(GapMajn)), where n is the input size of f and GapMajn is the √n-gap majority function on n bits.
Boolean function F(x,y)F(x,y)F(x,y) for x,y∈{0,1}nx,y \in \{0,1\}^nx,y∈{0,1}n is an XOR function if F(x,y)=f(x⊕y)F(x,y)=f(x\oplus y)F(x,y)=f(x⊕y) for some function fff on nnn input bits, where ⊕\oplus⊕ is a bit-wise XOR. XOR functions are relevant in communication complexity, partially for allowing Fourier analytic technique. For total XOR functions it is known that deterministic communication complexity of FFF is closely related to parity decision tree complexity of fff. Montanaro and Osbourne (2009) observed that one-sided communication complexity Dcc→(F)D_{cc}^{\rightarrow}(F)Dcc→​(F) of FFF is exactly equal to nonadaptive parity decision tree complexity NADT⊕(f)NADT^{\oplus}(f)NADT⊕(f) of fff. Hatami et al. (2018) showed that unrestricted communication complexity of FFF is polynomially related to parity decision tree complexity of fff. We initiate the studies of a similar connection for partial functions. We show that in case of one-sided communication complexity whether these measures are equal, depends on the number of undefined inputs of fff. On the one hand, if Dcc→(F)=tD_{cc}^{\rightarrow}(F)=tDcc→​(F)=t and fff is undefined on at most O(2n−tn−t)O(\frac{2^{n-t}}{\sqrt{n-t}})O(n−t​2n−t​), then NADT⊕(f)=tNADT^{\oplus}(f)=tNADT⊕(f)=t. On the other hand, for a wide range of values of Dcc→(F)D_{cc}^{\rightarrow}(F)Dcc→​(F) and NADT⊕(f)NADT^{\oplus}(f)NADT⊕(f) (from constant to n−2n-2n−2) we provide partial functions for which Dcc→(F)<NADT⊕(f)D_{cc}^{\rightarrow}(F) < NADT^{\oplus}(f)Dcc→​(F)<NADT⊕(f). In particular, we provide a function with an exponential gap between the two measures. Our separation results translate to the case of two-sided communication complexity as well, in particular showing that the result of Hatami et al. (2018) cannot be generalized to partial functions. Previous results for total functions heavily rely on Boolean Fourier analysis and the technique does not translate to partial functions. For the proofs of our results we build a linear algebraic framework instead. Separation results are proved through the reduction to covering codes.