16,345 research outputs found
Completeness theory for the product of finite partial algebras
AbstractA general completeness criterion for the finite product ∏P(ki) of full partial clones P(ki) (composition-closed subsets of partial operations) defined on finite sets E(ki)(|E(ki)|⩾2,i=1,…,n,n⩾2) is considered and a Galois connection between the lattice of subclones of ∏P(ki), called partial n-clones, and the lattice of subalgebras of multiple-base invariant relation algebra, with operations of a restricted quantifier free calculus, is established. This is used to obtain the full description of all maximal partial n-clones via multiple-base invariant relations and, thus, to solve the general completeness problem in ∏P(ki)
The variety generated by all the ordinal sums of perfect MV-chains
We present the logic BL_Chang, an axiomatic extension of BL (see P. H\'ajek -
Metamathematics of fuzzy logic - 1998, Kluwer) whose corresponding algebras
form the smallest variety containing all the ordinal sums of perfect MV-chains.
We will analyze this logic and the corresponding algebraic semantics in the
propositional and in the first-order case. As we will see, moreover, the
variety of BL_Chang-algebras will be strictly connected to the one generated by
Chang's MV-algebra (that is, the variety generated by all the perfect
MV-algebras): we will also give some new results concerning these last
structures and their logic.Comment: This is a revised version of the previous paper: the modifications
concern essentially the presentation. The scientific content is substantially
unchanged. The major variations are: Definition 2.7 has been improved.
Section 3.1 has been made more compact. A new reference, [Bus04], has been
added. There is some minor modification in Section 3.
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
Quasitraces on exact C*-algebras are traces
It is shown that all 2-quasitraces on a unital exact C*-algebra are traces.
As consequences one gets: (1) Every stably finite exact unital C*-algebra has a
tracial state, and (2) if an AW*-factor of type II_1 is generated (as an
AW*-algebra) by an exact C*-subalgebra, then it is a von Neumann II_1-factor.
This is a partial solution to a well known problem of Kaplansky. The present
result was used by Blackadar, Kumjian and R{\o}rdam to prove that RR(A)=0 for
every simple non-commutative torus of any dimension
Convolution, Separation and Concurrency
A notion of convolution is presented in the context of formal power series
together with lifting constructions characterising algebras of such series,
which usually are quantales. A number of examples underpin the universality of
these constructions, the most prominent ones being separation logics, where
convolution is separating conjunction in an assertion quantale; interval
logics, where convolution is the chop operation; and stream interval functions,
where convolution is used for analysing the trajectories of dynamical or
real-time systems. A Hoare logic is constructed in a generic fashion on the
power series quantale, which applies to each of these examples. In many cases,
commutative notions of convolution have natural interpretations as concurrency
operations.Comment: 39 page
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