Search results for "Computer Science::Computational Complexity"

Quantum Finite Automata and Logics

2006

The connection between measure once quantum finite automata (MO-QFA) and logic is studied in this paper. The language class recognized by MO-QFA is compared to languages described by the first order logics and modular logics. And the equivalence between languages accepted by MO-QFA and languages described by formulas using Lindstrom quantifier is shown.

A multilinear Phelps' Lemma

2007

We prove a multilinear version of Phelps' Lemma: if the zero sets of multilinear forms of norm one are 'close', then so are the multilinear forms.

On the Class of Languages Recognizable by 1-Way Quantum Finite Automata

2007

It is an open problem to characterize the class of languages recognized by quantum finite automata (QFA). We examine some necessary and some sufficient conditions for a (regular) language to be recognizable by a QFA. For a subclass of regular languages we get a condition which is necessary and sufficient. Also, we prove that the class of languages recognizable by a QFA is not closed under union or any other binary Boolean operation where both arguments are significant.

Quantum Finite Automata and Probabilistic Reversible Automata: R-trivial Idempotent Languages

2011

We study the recognition of R-trivial idempotent (R1) languages by various models of "decide-and-halt" quantum finite automata (QFA) and probabilistic reversible automata (DH-PRA). We introduce bistochastic QFA (MM-BQFA), a model which generalizes both Nayak's enhanced QFA and DH-PRA. We apply tools from algebraic automata theory and systems of linear inequalities to give a complete characterization of R1 languages recognized by all these models. We also find that "forbidden constructions" known so far do not include all of the languages that cannot be recognized by measure-many QFA.

Postselection Finite Quantum Automata

2010

Postselection for quantum computing devices was introduced by S. Aaronson[2] as an excitingly efficient tool to solve long standing problems of computational complexity related to classical computing devices only. This was a surprising usage of notions of quantum computation. We introduce Aaronson's type postselection in quantum finite automata. There are several nonequivalent definitions of quantumfinite automata. Nearly all of them recognize only regular languages but not all regular languages. We prove that PALINDROMES can be recognized by MM-quantum finite automata with postselection. At first we prove by a direct construction that the complement of this language can be recognized this …

Error-Free Affine, Unitary, and Probabilistic OBDDs

2018

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 versions of these models.

Improved constructions of quantum automata

2008

We present a simple construction of quantum automata which achieve an exponential advantage over classical finite automata. Our automata use \frac{4}{\epsilon} \log 2p + O(1) states to recognize a language that requires p states classically. The construction is both substantially simpler and achieves a better constant in the front of \log p than the previously known construction of Ambainis and Freivalds (quant-ph/9802062). Similarly to Ambainis and Freivalds, our construction is by a probabilistic argument. We consider the possibility to derandomize it and present some results in this direction.

2014

Is there a general theorem that tells us when we can hope for exponential speedups from quantum algorithms, and when we cannot? In this paper, we make two advances toward such a theorem, in the black-box model where most quantum algorithms operate. First, we show that for any problem that is invariant under permuting inputs and outputs (like the collision or the element distinctness problems), the quantum query complexity is at least the 9 th root of the classical randomized query complexity. This resolves a conjecture of Watrous from 2002. Second, inspired by recent work of O’Donnell et al. and Dinur et al., we conjecture that every bounded low-degree polynomial has a “highly influential” …

Error-Free Affine, Unitary, and Probabilistic OBDDs

2021

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.

Efficient CNF Encoding of Boolean Cardinality Constraints

2003

In this paper, we address the encoding into CNF clauses of Boolean cardinality constraints that arise in many practical applications. The proposed encoding is efficient with respect to unit propagation, which is implemented in almost all complete CNF satisfiability solvers. We prove the practical efficiency of this encoding on some problems arising in discrete tomography that involve many cardinality constraints. This encoding is also used together with a trivial variable elimination in order to re-encode parity learning benchmarks so that a simple Davis and Putnam procedure can solve them.