30 research outputs found
On equivalence, languages equivalence and minimization of multi-letter and multi-letter measure-many quantum automata
We first show that given a -letter quantum finite automata
and a -letter quantum finite automata over
the same input alphabet , they are equivalent if and only if they are
-equivalent where , , are the
numbers of state in respectively, and . By
applying a method, due to the author, used to deal with the equivalence problem
of {\it measure many one-way quantum finite automata}, we also show that a
-letter measure many quantum finite automaton and a
-letter measure many quantum finite automaton are
equivalent if and only if they are -equivalent
where , , are the numbers of state in respectively,
and .
Next, we study the language equivalence problem of those two kinds of quantum
finite automata. We show that for -letter quantum finite automata, the
non-strict cut-point language equivalence problem is undecidable, i.e., it is
undecidable whether
where
and are -letter quantum finite automata.
Further, we show that both strict and non-strict cut-point language equivalence
problem for -letter measure many quantum finite automata are undecidable.
The direct consequences of the above outcomes are summarized in the paper.
Finally, we comment on existing proofs about the minimization problem of one
way quantum finite automata not only because we have been showing great
interest in this kind of problem, which is very important in classical automata
theory, but also due to that the problem itself, personally, is a challenge.
This problem actually remains open.Comment: 30 pages, conclusion section correcte
Минимизация информационных структур с почти коммутативными свойствами
Решение фундаментальных проблем информационных структур с почти коммутативными свойствами, минимизации эквивалентных преобразований и эквивалентности для подклассов информационных структу
Scheduling Transformation and Dependence Tests for Recursive Programs
Scheduling transformations reorder the execution of operations in a program to improve locality and/or parallelism. The polyhedral model provides a general framework for performing instance-wise scheduling transformations for regular programs, reordering the iterations of loops that operate over dense arrays through transformations like tiling. There is no analogous framework for recursive programs—despite recent interest in optimizations like tiling and fusion for recursive applications. This paper presents PolyRec, the first general framework for applying scheduling transformations—like inlining, interchange, and code motion—to nested recursive programs and reasoning about their correctness. We describe the phases of PolyRec—representing dynamic instances, applying transformations, reasoning about correctness—and show that PolyRec is able to apply sophisticated, composed transformations to complex, nested recursive programs and improve performance through enhanced locality
Another approach to the equivalence of measure-many one-way quantum finite automata and its application
In this paper, we present a much simpler, direct and elegant approach to the
equivalence problem of {\it measure many one-way quantum finite automata}
(MM-1QFAs). The approach is essentially generalized from the work of Carlyle
[J. Math. Anal. Appl. 7 (1963) 167-175]. Namely, we reduce the equivalence
problem of MM-1QFAs to that of two (initial) vectors.
As an application of the approach, we utilize it to address the equivalence
problem of {\it Enhanced one-way quantum finite automata} (E-1QFAs) introduced
by Nayak [Proceedings of the 40th Annual IEEE Symposium on Foundations of
Computer Science, 1999, pp.~369-376]. We prove that two E-1QFAs
and over are equivalence if and only if they are
-equivalent where and are the numbers of states in
and , respectively.Comment: V 10: Corollary 3 is deleted, since it is folk. (V 9: Revised in
terms of the referees's comments) All comments, especially the linguistic
comments, are welcom
On Lindenmayerian algebraic sequences
AbstractWe define and study Lindenmayerian algebraic sequences. These sequences are a generalization of algebraic sequences, k-regular sequences and automatic sequences
From the Closed Classical Algorithmic Universe to an Open World of Algorithmic Constellations
This is a draft of the article to be published in Springer book series SAPERE. The final publication will be available a
Elements of computability, decidability, and complexity (Third edition)
These lecture notes are intended to introduce the reader to the
basic notions of computability theory, decidability, and complexity. More
information on these subjects can be found in classical books such as [Cut80,Dav58,Her69,HoU79,Rog67].
The results reported in these notes are taken from those books and in various
parts we closely follow their style of presentation. The reader is encouraged
to look at those books for improving his/her knowledge on these topics. Some
parts of the chapter on complexity are taken from the lecture notes of a
beautiful course given by Prof. Leslie Valiant at Edinburgh University,
Scotland, in 1979. It was, indeed, a very stimulating and enjoyable course.
For the notions of Predicate Calculus we have used in this book the reader
may refer to [Men87].
I would like to thank Dr. Maurizio Proietti at IASI-CNR (Roma, Italy),
my colleagues, and my students at the University of Roma Tor Vergata and,
in particular, Michele Martone. They have been for me a source of continuous
inspiration and enthusiasm.
Finally, I would like to thank Dr. Gioacchino Onorati and Lorenzo Costantini
of the Aracne Publishing Company for their helpful cooperation