11,425 research outputs found
Topological Connectedness and Behavioral Assumptions on Preferences: A Two-Way Relationship
This paper offers a comprehensive treatment of the question as to whether a
binary relation can be consistent (transitive) without being decisive
(complete), or decisive without being consistent, or simultaneously
inconsistent or indecisive, in the presence of a continuity hypothesis that is,
in principle, non-testable. It identifies topological connectedness of the
(choice) set over which the continuous binary relation is defined as being
crucial to this question. Referring to the two-way relationship as the
Eilenberg-Sonnenschein (ES) research program, it presents four synthetic, and
complete, characterizations of connectedness, and its natural extensions; and
two consequences that only stem from it. The six theorems are novel to both the
economic and the mathematical literature: they generalize pioneering results of
Eilenberg (1941), Sonnenschein (1965), Schmeidler (1971) and Sen (1969), and
are relevant to several applied contexts, as well as to ongoing theoretical
work.Comment: 47 pages, 4 figure
Satisfiability in multi-valued circuits
Satisfiability of Boolean circuits is among the most known and important
problems in theoretical computer science. This problem is NP-complete in
general but becomes polynomial time when restricted either to monotone gates or
linear gates. We go outside Boolean realm and consider circuits built of any
fixed set of gates on an arbitrary large finite domain. From the complexity
point of view this is strictly connected with the problems of solving equations
(or systems of equations) over finite algebras.
The research reported in this work was motivated by a desire to know for
which finite algebras there is a polynomial time algorithm that
decides if an equation over has a solution. We are also looking for
polynomial time algorithms that decide if two circuits over a finite algebra
compute the same function. Although we have not managed to solve these problems
in the most general setting we have obtained such a characterization for a very
broad class of algebras from congruence modular varieties. This class includes
most known and well-studied algebras such as groups, rings, modules (and their
generalizations like quasigroups, loops, near-rings, nonassociative rings, Lie
algebras), lattices (and their extensions like Boolean algebras, Heyting
algebras or other algebras connected with multi-valued logics including
MV-algebras).
This paper seems to be the first systematic study of the computational
complexity of satisfiability of non-Boolean circuits and solving equations over
finite algebras. The characterization results provided by the paper is given in
terms of nice structural properties of algebras for which the problems are
solvable in polynomial time.Comment: 50 page
Taking Mermin's Relational Interpretation of QM Beyond Cabello's and Seevinck's No-Go Theorems
In this paper we address a deeply interesting debate that took place at the
end of the last millennia between David Mermin, Adan Cabello and Michiel
Seevinck, regarding the meaning of relationalism within quantum theory. In a
series of papers, Mermin proposed an interpretation in which quantum
correlations were considered as elements of physical reality. Unfortunately,
the very young relational proposal by Mermin was too soon tackled by specially
suited no-go theorems designed by Cabello and Seevinck. In this work we attempt
to reconsider Mermin's program from the viewpoint of the Logos Categorical
Approach to QM. Following Mermin's original proposal, we will provide a
redefinition of quantum relation which not only can be understood as a
preexistent element of physical reality but is also capable to escape Cabello's
and Seevinck's no-go-theorems. In order to show explicitly that our notion of
ontological quantum relation is safe from no-go theorems we will derive a
non-contextuality theorem. We end the paper with a discussion regarding the
physical meaning of quantum relationalism.Comment: 19 pages, 1 phot
The Intrinsic Structure of Quantum Mechanics
The wave function in quantum mechanics presents an interesting challenge to
our understanding of the physical world. In this paper, I show that the wave
function can be understood as four intrinsic relations on physical space. My
account has three desirable features that the standard account lacks: (1) it
does not refer to any abstract mathematical objects, (2) it is free from the
usual arbitrary conventions, and (3) it explains why the wave function has its
gauge degrees of freedom, something that are usually put into the theory by
hand. Hence, this account has implications for debates in philosophy of
mathematics and philosophy of science. First, by removing references to
mathematical objects, it provides a framework for nominalizing quantum
mechanics. Second, by excising superfluous structure such as overall phase, it
reveals the intrinsic structure postulated by quantum mechanics. Moreover, it
also removes a major obstacle to "wave function realism.
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