6,561 research outputs found
A temporal semantics for Nilpotent Minimum logic
In [Ban97] a connection among rough sets (in particular, pre-rough algebras)
and three-valued {\L}ukasiewicz logic {\L}3 is pointed out. In this paper we
present a temporal like semantics for Nilpotent Minimum logic NM ([Fod95,
EG01]), in which the logic of every instant is given by {\L}3: a completeness
theorem will be shown. This is the prosecution of the work initiated in [AGM08]
and [ABM09], in which the authors construct a temporal semantics for the
many-valued logics of G\"odel ([G\"od32], [Dum59]) and Basic Logic ([H\'aj98]).Comment: 19 pages, 2 table
Varieties of Cost Functions.
Regular cost functions were introduced as a quantitative generalisation of regular languages, retaining many of their equivalent characterisations and decidability properties. For instance, stabilisation monoids play the same role for cost functions as monoids do for regular languages. The purpose of this article is to further extend this algebraic approach by generalising two results on regular languages to cost functions: Eilenberg's varieties theorem and profinite equational characterisations of lattices of regular languages. This opens interesting new perspectives, but the specificities of cost functions introduce difficulties that prevent these generalisations to be straightforward. In contrast, although syntactic algebras can be defined for formal power series over a commutative ring, no such notion is known for series over semirings and in particular over the tropical semiring
Wreath Products of Forest Algebras, with Applications to Tree Logics
We use the recently developed theory of forest algebras to find algebraic
characterizations of the languages of unranked trees and forests definable in
various logics. These include the temporal logics CTL and EF, and first-order
logic over the ancestor relation. While the characterizations are in general
non-effective, we are able to use them to formulate necessary conditions for
definability and provide new proofs that a number of languages are not
definable in these logics
EF+EX Forest Algebras
We examine languages of unranked forests definable using the temporal
operators EF and EX. We characterize the languages definable in this logic, and
various fragments thereof, using the syntactic forest algebras introduced by
Bojanczyk and Walukiewicz. Our algebraic characterizations yield efficient
algorithms for deciding when a given language of forests is definable in this
logic. The proofs are based on understanding the wreath product closures of a
few small algebras, for which we introduce a general ideal theory for forest
algebras. This combines ideas from the work of Bojanczyk and Walukiewicz for
the analogous logics on binary trees and from early work of Stiffler on wreath
product of finite semigroups
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
Admissibility via Natural Dualities
It is shown that admissible clauses and quasi-identities of quasivarieties
generated by a single finite algebra, or equivalently, the quasiequational and
universal theories of their free algebras on countably infinitely many
generators, may be characterized using natural dualities. In particular,
axiomatizations are obtained for the admissible clauses and quasi-identities of
bounded distributive lattices, Stone algebras, Kleene algebras and lattices,
and De Morgan algebras and lattices.Comment: 22 pages; 3 figure
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