131 research outputs found
Reversible Kleene lattices
International audienceWe investigate the equational theory of reversible Kleene lattices, that is algebras of languages with the regular operations (union, composition and Kleene star), together with the intersection and mirror image. Building on results by Andréka, Mikulås and Németi from 2011, we construct the free representation of this algebra. We then provide an automaton model to compare representations. These automata are adapted from Petri automata, which we introduced with Pous in 2015 to tackle a similar problem for algebras of binary relations. This allows us to show that testing the validity of equations in this algebra is decidable, and in fact ExpSpace-complete
A Complete Axiomatisation of a Fragment of Language Algebra
We consider algebras of languages over the signature of reversible Kleene lattices, that is the regular operations (empty and unit languages, union, concatenation and Kleene star) together with intersection and mirror image. We provide a complete set of axioms for the equational theory of these algebras. This proof was developed in the proof assistant Coq
A complete axiomatisation of reversible Kleene lattices
We consider algebras of languages over the signature of reversible Kleene lattices, that is the regular operations (empty and unit languages, union, concatenation and Kleene star) together with intersection and mirror image. We provide a complete set of axioms for the equational theory of these algebras. This proof was developed in the proof assistant Coq
Fredkin Gates for Finite-valued Reversible and Conservative Logics
The basic principles and results of Conservative Logic introduced by Fredkin
and Toffoli on the basis of a seminal paper of Landauer are extended to
d-valued logics, with a special attention to three-valued logics. Different
approaches to d-valued logics are examined in order to determine some possible
universal sets of logic primitives. In particular, we consider the typical
connectives of Lukasiewicz and Godel logics, as well as Chang's MV-algebras. As
a result, some possible three-valued and d-valued universal gates are described
which realize a functionally complete set of fundamental connectives.Comment: 57 pages, 10 figures, 16 tables, 2 diagram
A Theory of Computation Based on Quantum Logic (I)
The (meta)logic underlying classical theory of computation is Boolean
(two-valued) logic. Quantum logic was proposed by Birkhoff and von Neumann as a
logic of quantum mechanics more than sixty years ago. The major difference
between Boolean logic and quantum logic is that the latter does not enjoy
distributivity in general. The rapid development of quantum computation in
recent years stimulates us to establish a theory of computation based on
quantum logic. The present paper is the first step toward such a new theory and
it focuses on the simplest models of computation, namely finite automata. It is
found that the universal validity of many properties of automata depend heavily
upon the distributivity of the underlying logic. This indicates that these
properties does not universally hold in the realm of quantum logic. On the
other hand, we show that a local validity of them can be recovered by imposing
a certain commutativity to the (atomic) statements about the automata under
consideration. This reveals an essential difference between the classical
theory of computation and the computation theory based on quantum logic
Undecidability of the Spectral Gap (full version)
We show that the spectral gap problem is undecidable. Specifically, we
construct families of translationally-invariant, nearest-neighbour Hamiltonians
on a 2D square lattice of d-level quantum systems (d constant), for which
determining whether the system is gapped or gapless is an undecidable problem.
This is true even with the promise that each Hamiltonian is either gapped or
gapless in the strongest sense: it is promised to either have continuous
spectrum above the ground state in the thermodynamic limit, or its spectral gap
is lower-bounded by a constant in the thermodynamic limit. Moreover, this
constant can be taken equal to the local interaction strength of the
Hamiltonian.Comment: v1: 146 pages, 56 theorems etc., 15 figures. See shorter companion
paper arXiv:1502.04135 (same title and authors) for a short version omitting
technical details. v2: Small but important fix to wording of abstract. v3:
Simplified and shortened some parts of the proof; minor fixes to other parts.
Now only 127 pages, 55 theorems etc., 10 figures. v4: Minor updates to
introductio
Computational universes
Suspicions that the world might be some sort of a machine or algorithm
existing ``in the mind'' of some symbolic number cruncher have lingered from
antiquity. Although popular at times, the most radical forms of this idea never
reached mainstream. Modern developments in physics and computer science have
lent support to the thesis, but empirical evidence is needed before it can
begin to replace our contemporary world view.Comment: Several corrections of typos and smaller revisions, final versio
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