Causes of Ineradicable Spurious Predictions in Qualitative Simulation

It was recently proved that a sound and complete qualitative simulator does not exist, that is, as long as the input-output vocabulary of the state-of-the-art QSIM algorithm is used, there will always be input models which cause any simulator with a coverage guarantee to make spurious predictions in its output. In this paper, we examine whether a meaningfully expressive restriction of this vocabulary is possible so that one can build a simulator with both the soundness and completeness properties. We prove several negative results: All sound qualitative simulators, employing subsets of the QSIM representation which retain the operating region transition feature, and support at least the addition and constancy constraints, are shown to be inherently incomplete. Even when the simulations are restricted to run in a single operating region, a constraint vocabulary containing just the addition, constancy, derivative, and multiplication relations makes the construction of sound and complete qualitative simulators impossible

Succinctness of two-way probabilistic and quantum finite automata

We prove that two-way probabilistic and quantum finite automata (2PFA's and 2QFA's) can be considerably more concise than both their one-way versions (1PFA's and 1QFA's), and two-way nondeterministic finite automata (2NFA's). For this purpose, we demonstrate several infinite families of regular languages which can be recognized with some fixed probability greater than ${1/2}$ by just tuning the transition amplitudes of a 2QFA (and, in one case, a 2PFA) with a constant number of states, whereas the sizes of the corresponding 1PFA's, 1QFA's and 2NFA's grow without bound. We also show that 2QFA's with mixed states can support highly efficient probability amplification. The weakest known model of computation where quantum computers recognize more languages with bounded error than their classical counterparts is introduced.Comment: A new version, 21 pages, late