Error-Free Affine, Unitary, and Probabilistic OBDDs

Abstract

We introduce the affine OBDD model and show that zero-error affine OBDDs can be exponentially narrower than bounded-error unitary and probabilistic OBDDs on certain problems. Moreover, we show that Las-Vegas unitary and probabilistic OBDDs can be quadratically narrower than deterministic OBDDs. We also obtain the same results for the automata counterparts of these models.

Improved constructions for succinct affine automata

“…where t < 2k, m > 0, each x i ∈ {0, 1} for 1 ≤ i ≤ m, and m i=1 x i = 1. It was shown [13] that MODXOR k for k > 0 is recognized by a (2k + 1)-state AfA with zero error. Due to Theorem 5, the same results can also be obtained by designing a 2k-state NFA, which accepts each member on a single path.…”

“…The computational power of AfAs and their generalizations have been examined and compared with their probabilistic and quantum counterparts in a series of papers [23,5,9,18,12,24,10,11,13,14]. In the unbounded-error and boundederror language recognition modes, AfAs are more powerful than PFAs and QFAs, where the latter models recognize all and only stochastic and regular languages, respectively [20,19,16,27,17].…”

“…Regarding the state complexity, bounded-error PFAs can be exponentially more succinct than deterministic finite automata (DFAs) [20,19,2]; and, boundederror QFAs can be exponentially more succinctness than bounded-error PFAs [1,4]. On the other hand, bounded-error AfAs can be more succinct than boundederror PFAs and QFAs and the gap can be super-exponential [5,24,13]. A similar gap between DFAs and bounded-error PFAs and between bounded-error PFAs and bounded-error QFAs can only be obtained on promise problems [3,7].…”

Quantum and Classical Log-Bounded Automata for the Online Disjointness Problem

“…One of them is quantum online algorithms with repeated test [22]; another one is quantum online algorithms with restriction on the size of memory [23,24]; quantum algorithms with a buffer [25,26] and others. The question of comparing quantum and classical models was explored for data stream processing models by different researches [27][28][29][30][31][32][33][34][35][36][37][38][39][40].…”