1,443 research outputs found
Do Goedel's incompleteness theorems set absolute limits on the ability of the brain to express and communicate mental concepts verifiably?
Classical interpretations of Goedel's formal reasoning imply that the truth
of some arithmetical propositions of any formal mathematical language, under
any interpretation, is essentially unverifiable. However, a language of
general, scientific, discourse cannot allow its mathematical propositions to be
interpreted ambiguously. Such a language must, therefore, define mathematical
truth verifiably. We consider a constructive interpretation of classical,
Tarskian, truth, and of Goedel's reasoning, under which any formal system of
Peano Arithmetic is verifiably complete. We show how some paradoxical concepts
of Quantum mechanics can be expressed, and interpreted, naturally under a
constructive definition of mathematical truth.Comment: 73 pages; this is an updated version of the NQ essay; an HTML version
is available at http://alixcomsi.com/Do_Goedel_incompleteness_theorems.ht
New quantum algorithm for studying NP-complete problems
Ordinary approach to quantum algorithm is based on quantum Turing machine or
quantum circuits. It is known that this approach is not powerful enough to
solve NP-complete problems. In this paper we study a new approach to quantum
algorithm which is a combination of the ordinary quantum algorithm with a
chaotic dynamical system. We consider the satisfiability problem as an example
of NP-complete problems and argue that the problem, in principle, can be solved
in polynomial time by using our new quantum algorithm.Comment: 11 pages, 1 figur
Circular Coloring of Random Graphs: Statistical Physics Investigation
Circular coloring is a constraints satisfaction problem where colors are
assigned to nodes in a graph in such a way that every pair of connected nodes
has two consecutive colors (the first color being consecutive to the last). We
study circular coloring of random graphs using the cavity method. We identify
two very interesting properties of this problem. For sufficiently many color
and sufficiently low temperature there is a spontaneous breaking of the
circular symmetry between colors and a phase transition forwards a
ferromagnet-like phase. Our second main result concerns 5-circular coloring of
random 3-regular graphs. While this case is found colorable, we conclude that
the description via one-step replica symmetry breaking is not sufficient. We
observe that simulated annealing is very efficient to find proper colorings for
this case. The 5-circular coloring of 3-regular random graphs thus provides a
first known example of a problem where the ground state energy is known to be
exactly zero yet the space of solutions probably requires a full-step replica
symmetry breaking treatment.Comment: 19 pages, 8 figures, 3 table
Consistency of circuit lower bounds with bounded theories
Proving that there are problems in that require
boolean circuits of super-linear size is a major frontier in complexity theory.
While such lower bounds are known for larger complexity classes, existing
results only show that the corresponding problems are hard on infinitely many
input lengths. For instance, proving almost-everywhere circuit lower bounds is
open even for problems in . Giving the notorious difficulty of
proving lower bounds that hold for all large input lengths, we ask the
following question: Can we show that a large set of techniques cannot prove
that is easy infinitely often? Motivated by this and related
questions about the interaction between mathematical proofs and computations,
we investigate circuit complexity from the perspective of logic.
Among other results, we prove that for any parameter it is
consistent with theory that computational class , where is one of
the pairs: and , and , and
. In other words, these theories cannot establish
infinitely often circuit upper bounds for the corresponding problems. This is
of interest because the weaker theory already formalizes
sophisticated arguments, such as a proof of the PCP Theorem. These consistency
statements are unconditional and improve on earlier theorems of [KO17] and
[BM18] on the consistency of lower bounds with
Phase Transitions and Backbones of the Asymmetric Traveling Salesman Problem
In recent years, there has been much interest in phase transitions of
combinatorial problems. Phase transitions have been successfully used to
analyze combinatorial optimization problems, characterize their typical-case
features and locate the hardest problem instances. In this paper, we study
phase transitions of the asymmetric Traveling Salesman Problem (ATSP), an
NP-hard combinatorial optimization problem that has many real-world
applications. Using random instances of up to 1,500 cities in which intercity
distances are uniformly distributed, we empirically show that many properties
of the problem, including the optimal tour cost and backbone size, experience
sharp transitions as the precision of intercity distances increases across a
critical value. Our experimental results on the costs of the ATSP tours and
assignment problem agree with the theoretical result that the asymptotic cost
of assignment problem is pi ^2 /6 the number of cities goes to infinity. In
addition, we show that the average computational cost of the well-known
branch-and-bound subtour elimination algorithm for the problem also exhibits a
thrashing behavior, transitioning from easy to difficult as the distance
precision increases. These results answer positively an open question regarding
the existence of phase transitions in the ATSP, and provide guidance on how
difficult ATSP problem instances should be generated
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