17,928 research outputs found
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
General Relativity, the Cosmological Constant and Modular Forms
Strong field (exact) solutions of the gravitational field equations of
General Relativity in the presence of a Cosmological Constant are investigated.
In particular, a full exact solution is derived within the inhomogeneous
Szekeres-Szafron family of space-time line element with a nonzero Cosmological
Constant. The resulting solution connects, in an intrinsic way, General
Relativity with the theory of modular forms and elliptic curves. The
homogeneous FLRW limit of the above space-time elements is recovered and we
solve exactly the resulting Friedmann Robertson field equation with the
appropriate matter density for generic values of the Cosmological Constant
%Lambda and curvature constant K. A formal expression for the Hubble constant
is derived. The cosmological implications of the resulting non-linear solutions
are systematically investigated. Two particularly interesting solutions i) the
case of a flat universe K=0, Lambda not= 0 and ii) a case with all three
cosmological parameters non-zero, are described by elliptic curves with the
property of complex multiplication and absolute modular invariant j=0 and 1728,
respectively. The possibility that all non-linear solutions of General
Relativity are expressed in terms of theta functions associated with
Riemann-surfaces is discussed.Comment: LaTeX file, 34 pages plus 9 EPS figures, Accepted for Publication in
Classical and Quantum Gravit
Propagators and Solvers for the Algebra of Modular Systems
To appear in the proceedings of LPAR 21.
Solving complex problems can involve non-trivial combinations of distinct
knowledge bases and problem solvers. The Algebra of Modular Systems is a
knowledge representation framework that provides a method for formally
specifying such systems in purely semantic terms. Formally, an expression of
the algebra defines a class of structures. Many expressive formalism used in
practice solve the model expansion task, where a structure is given on the
input and an expansion of this structure in the defined class of structures is
searched (this practice overcomes the common undecidability problem for
expressive logics). In this paper, we construct a solver for the model
expansion task for a complex modular systems from an expression in the algebra
and black-box propagators or solvers for the primitive modules. To this end, we
define a general notion of propagators equipped with an explanation mechanism,
an extension of the alge- bra to propagators, and a lazy conflict-driven
learning algorithm. The result is a framework for seamlessly combining solving
technology from different domains to produce a solver for a combined system.Comment: To appear in the proceedings of LPAR 2
Move-minimizing puzzles, diamond-colored modular and distributive lattices, and poset models for Weyl group symmetric functions
The move-minimizing puzzles presented here are certain types of one-player
combinatorial games that are shown to have explicit solutions whenever they can
be encoded in a certain way as diamond-colored modular and distributive
lattices. Such lattices can also arise naturally as models for certain
algebraic objects, namely Weyl group symmetric functions and their companion
semisimple Lie algebra representations. The motivation for this paper is
therefore both diversional and algebraic: To show how some recreational
move-minimizing puzzles can be solved explicitly within an order-theoretic
context and also to realize some such puzzles as combinatorial models for
symmetric functions associated with certain fundamental representations of the
symplectic and odd orthogonal Lie algebras
- …