In 1986, Saks and Wigderson conjectured that the largest separation between deterministic and zero-error randomized query complexity for a total Boolean function is given by the function f on n = 2 k bits defined by a complete binary tree of NAND gates of depth k , which achieves R 0 ( f ) = O ( D ( f ) 0.7537… ). We show that this is false by giving an example of a total Boolean function f on n bits whose deterministic query complexity is Ω( n ) while its zero-error randomized query complexity is Õ(√ n ). We further show that the quantum query complexity of the same function is Õ( n 1/4 ), giving the first example of a total function with a super-quadratic gap between its quantum and deterministic query complexities. We also construct a total Boolean function g on n variables that has zero-error randomized query complexity Ω( n / log ( n )) and bounded-error randomized query complexity R ( g ) = Õ(√ n ). This is the first super-linear separation between these two complexity measures. The exact quantum query complexity of the same function is Q E ( g ) = Õ(√ n ). These functions show that the relations D ( f ) = O ( R 1 ( f ) 2 ) and R 0 ( f ) = Õ( R ( f ) 2 ) are optimal, up to polylogarithmic factors. Further variations of these functions give additional separations between other query complexity measures: a cubic separation between Q and R 0 , a 3/2-power separation between Q E and R , and a 4th-power separation between approximate degree and bounded-error randomized query complexity. All of these examples are variants of a function recently introduced by Göös, Pitassi, and Watson, which they used to separate the unambiguous 1-certificate complexity from deterministic query complexity and to resolve the famous Clique versus Independent Set problem in communication complexity.
In 1986, Saks and Wigderson conjectured that the largest separation between deterministic and zero-error randomized query complexity for a total Boolean function is given by the function f on n = 2 k bits defined by a complete binary tree of NAND gates of depth k , which achieves R 0 ( f ) = O ( D ( f ) 0.7537… ). We show that this is false by giving an example of a total Boolean function f on n bits whose deterministic query complexity is Ω( n ) while its zero-error randomized query complexity is Õ(√ n ). We further show that the quantum query complexity of the same function is Õ( n 1/4 ), giving the first example of a total function with a super-quadratic gap between its quantum and deterministic query complexities. We also construct a total Boolean function g on n variables that has zero-error randomized query complexity Ω( n / log ( n )) and bounded-error randomized query complexity R ( g ) = Õ(√ n ). This is the first super-linear separation between these two complexity measures. The exact quantum query complexity of the same function is Q E ( g ) = Õ(√ n ). These functions show that the relations D ( f ) = O ( R 1 ( f ) 2 ) and R 0 ( f ) = Õ( R ( f ) 2 ) are optimal, up to polylogarithmic factors. Further variations of these functions give additional separations between other query complexity measures: a cubic separation between Q and R 0 , a 3/2-power separation between Q E and R , and a 4th-power separation between approximate degree and bounded-error randomized query complexity. All of these examples are variants of a function recently introduced by Göös, Pitassi, and Watson, which they used to separate the unambiguous 1-certificate complexity from deterministic query complexity and to resolve the famous Clique versus Independent Set problem in communication complexity.
In 1986, Saks and Wigderson conjectured that the largest separation between deterministic and zero-error randomized query complexity for a total boolean function is given by the function f on n=2k bits defined by a complete binary tree of NAND gates of depth k, which achieves R0(f) = O(D(f)0.7537…). We show this is false by giving an example of a total boolean function f on n bits whose deterministic query complexity is Ω(n/log(n)) while its zero-error randomized query complexity is Õ(√n). We further show that the quantum query complexity of the same function is Õ(n1/4), giving the first example of a total function with a super-quadratic gap between its quantum and deterministic query complexities.
We show that there exists a Boolean function F which gives the following separations among deterministic query complexity (D(F)), randomized zero error query complexity (R0(F)) and randomized one-sided error query complexity (R1(F)): R1(F) = Õ( √ D(F)) and R0(F) = Õ(D(F))3/4. This refutes the conjecture made by Saks and Wigderson that for any Boolean function f, R0(f) = Ω(D(f))0.753.. This also shows widest separation between R1(f) and D(f) for any Boolean function. The function F was defined by Göös, Pitassi and Watson who studied it for showing a separation between deterministic decision tree complexity and unambiguous nondeterministic decision tree complexity. Independently of us, Ambainis et al proved that different variants of the function F certify optimal (quadratic) separation between D(f) and R0(f), and polynomial separation between R0(f) and R1(f). Viewed as separation results, our results are subsumed by those of Ambainis et al. However, while the functions considered in the work of Ambainis et al are different variants of F, in this work we show that the original function F itself is sufficient to refute the Saks-Wigderson conjecture and obtain widest possible separation between the deterministic and one-sided error randomized query complexity.
We study how large the certificate complexity C(f) of a total Boolean function can be relative to its randomized and quantum query complexities. We construct a total Boolean function f whose randomized query complexity with one-sided error satisfies R1(f)=Θ(C(f)). This separation is optimal even when two-sided error is allowed.
From this construction, we obtain another total Boolean function F with bounded-error quantum query complexity Q(F)=O(C(F)1/4), attaining the general quartic bound up to polylogarithmic factors. For the same function, both exact and zero-error quantum query complexities are O(C(F)). We also prove matching lower bounds for these two measures on F.
We show that there exists a Boolean function F which gives the following separations among deterministic query complexity (D(F)), randomized zero error query complexity (R_0(F)) and randomized one-sided error query complexity (R_1(F)): R_1(F) = ~O(sqrt{D(F)) and R_0(F)=~O(D(F))^3/4. This refutes the conjecture made by Saks and Wigderson that for any Boolean function f, R_0(f)=Omega(D(f))^0.753.. . This also shows widest separation between R_1(f) and D(f) for any Boolean function. The function F was defined by Göös, Pitassi and Watson who studied it for showing a separation between deterministic decision tree complexity and unambiguous non-deterministic decision tree complexity. Independently of us, Ambainis et al proved that different variants of the function F certify optimal (quadratic) separation between D(f) and R_0(f), and polynomial separation between R_0(f) and R_1(f). Viewed as separation results, our results are subsumed by those of Ambainis et al. However, while the functions considered in the work of Ambainis et al are different variants of F, in this work we show that the original function F itself is sufficient to refute the Saks-Wigderson conjecture and obtain widest possible separation between the deterministic and one-sided error randomized query complexity.
The pointer function of G{ö}{ö}s, Pitassi and Watson \cite{DBLP:journals/eccc/GoosP015a} and its variants have recently been used to prove separation results among various measures of complexity such as deterministic, randomized and quantum query complexities, exact and approximate polynomial degrees, etc. In particular, the widest possible (quadratic) separations between deterministic and zero-error randomized query complexity, as well as between bounded-error and zero-error randomized query complexity, have been obtained by considering {\em variants}~\cite{DBLP:journals/corr/AmbainisBBL15} of this pointer function. However, as was pointed out in \cite{DBLP:journals/corr/AmbainisBBL15}, the precise zero-error complexity of the original pointer function was not known. We show a lower bound of Ω(n3/4) on the zero-error randomized query complexity of the pointer function on Θ(nlogn) bits; since an O(n3/4) upper bound is already known \cite{DBLP:conf/fsttcs/MukhopadhyayS15}, our lower bound is optimal up to a factor of $\polylog\, n$.
Saks and Wigderson [in Proceedings of the 27th FOCS, IEEE Computer Society, Los Alamitos, CA, 1986, pp. 29--38] conjectured that R0(f)=Ω(D(f)0.753…) for all Boolean functions f, where R0 denotes the randomized zero-error query complexity and D denotes the deterministic query complexity. We show that for the pointer function GPWr×s defined by Göös, Pitassi, and Watson [in Proceedings of the 56th FOCS, IEEE, Piscataway, NJ, 2015, pp. 1077--1088], the following hold: (a) R1(GPWr×s)=Θ(r+s) and (b) R1(GPWr×s)=Θ(r+rs), where R1 denotes the randomized one-sided error query complexity. These results imply that (i) R0(GPWs2×s)=O(D(GPWs2×s)2/3), thereby refuting the conjecture of Saks and Wigderson, and (ii) R1(GPWs×s)=O(R0(GPWs×s)2/3), thereby providing a polynomial separation between the randomized zero-error and one-sided error query complexity measures.