A nonlinear lower bound on the practical combinational complexity

AbstractAn infinite sequence F = {fn}n = 1∞ of one-output Boolean functions with the following two properties is constructed: 1.(1)fn can be computed by a Boolean circuit with O(n) gates.2.(2)For any positive, nondecreasing, and unbounded function h : N → R, each Boolean circuit having an mh(m) separator requires a nonlinear number Ω(nh(n)) of gates to compute fn (e.g., each planar Boolean circuit requires Ω(n2) gates to compute fn).Thus, one can say that fn has linear combinational complexity and a nonlinear practical combinational complexity because the constant-degree parallel architectures used in practice have separators in O(mlog2 m)

On the limits of the communication complexity technique for proving lower bounds on the size of minimal NFA’s

AbstractIn contrast to the minimization of deterministic finite automata (DFA’s), the task of constructing a minimal nondeterministic finite automaton (NFA) for a given NFA is PSPACE-complete. Moreover, there are no polynomial approximation algorithms with a constant approximation ratio for estimating the number of states of minimal NFA’s.Since one is unable to efficiently estimate the size of a minimal NFA in an efficient way, one should ask at least for developing mathematical proof methods that help to prove good lower bounds on the size of a minimal NFA for a given regular language. Here we consider the robust and most successful lower bound proof technique that is based on communication complexity. In this paper it is proved that even a strong generalization of this method fails for some concrete regular languages.“To fail” is considered here in a very strong sense. There is an exponential gap between the size of a minimal NFA and the achievable lower bound for a specific sequence of regular languages.The generalization of the concept of communication protocols is also strong here. It is shown that cutting the input word into 2O(n1/4) pieces for a size n of a minimal nondeterministic finite automaton and investigating the necessary communication transfer between these pieces as parties of a multiparty protocol does not suffice to get good lower bounds on the size of minimal nondeterministic automata. It seems that for some regular languages one cannot really abstract from the automata model that cuts the input words into particular symbols of the alphabet and reads them one by one using its input head

Lifting query complexity to time-space complexity for two-way finite automata

Time-space tradeoff has been studied in a variety of models, such as Turing machines, branching programs, and finite automata, etc. While communication complexity as a technique has been applied to study finite automata, it seems it has not been used to study time-space tradeoffs of finite automata. We design a new technique showing that separations of query complexity can be lifted, via communication complexity, to separations of time-space complexity of two-way finite automata. As an application, one of our main results exhibits the first example of a language $L$ such that the time-space complexity of two-way probabilistic finite automata with a bounded error (2PFA) is $\widetilde{\Omega}(n^2)$, while of exact two-way quantum finite automata with classical states (2QCFA) is $\widetilde{O}(n^{5/3})$, that is, we demonstrate for the first time that exact quantum computing has an advantage in time-space complexity comparing to classical computing