6,935 research outputs found
Investigating The Algebraic Structure of Dihomotopy Types
This presentation is the sequel of a paper published in GETCO'00 proceedings
where a research program to construct an appropriate algebraic setting for the
study of deformations of higher dimensional automata was sketched. This paper
focuses precisely on detailing some of its aspects. The main idea is that the
category of homotopy types can be embedded in a new category of dihomotopy
types, the embedding being realized by the Globe functor. In this latter
category, isomorphism classes of objects are exactly higher dimensional
automata up to deformations leaving invariant their computer scientific
properties as presence or not of deadlocks (or everything similar or related).
Some hints to study the algebraic structure of dihomotopy types are given, in
particular a rule to decide whether a statement/notion concerning dihomotopy
types is or not the lifting of another statement/notion concerning homotopy
types. This rule does not enable to guess what is the lifting of a given
notion/statement, it only enables to make the verification, once the lifting
has been found.Comment: 28 pages ; LaTeX2e + 4 figures ; Expository paper ; Minor typos
corrections ; To appear in GETCO'01 proceeding
Linear Time Logics - A Coalgebraic Perspective
We describe a general approach to deriving linear time logics for a wide
variety of state-based, quantitative systems, by modelling the latter as
coalgebras whose type incorporates both branching behaviour and linear
behaviour. Concretely, we define logics whose syntax is determined by the
choice of linear behaviour and whose domain of truth values is determined by
the choice of branching, and we provide two equivalent semantics for them: a
step-wise semantics amenable to automata-based verification, and a path-based
semantics akin to those of standard linear time logics. We also provide a
semantic characterisation of the associated notion of logical equivalence, and
relate it to previously-defined maximal trace semantics for such systems.
Instances of our logics support reasoning about the possibility, likelihood or
minimal cost of exhibiting a given linear time property. We conclude with a
generalisation of the logics, dual in spirit to logics with discounting, which
increases their practical appeal in the context of resource-aware computation
by incorporating a notion of offsetting.Comment: Major revision of previous version: Sections 4 and 5 generalise the
results in the previous version, with new proofs; Section 6 contains new
result
Tree-Automatic Well-Founded Trees
We investigate tree-automatic well-founded trees. Using Delhomme's
decomposition technique for tree-automatic structures, we show that the
(ordinal) rank of a tree-automatic well-founded tree is strictly below
omega^omega. Moreover, we make a step towards proving that the ranks of
tree-automatic well-founded partial orders are bounded by omega^omega^omega: we
prove this bound for what we call upwards linear partial orders. As an
application of our result, we show that the isomorphism problem for
tree-automatic well-founded trees is complete for level Delta^0_{omega^omega}
of the hyperarithmetical hierarchy with respect to Turing-reductions.Comment: Will appear in Logical Methods of Computer Scienc
Coalgebraic Infinite Traces and Kleisli Simulations
Kleisli simulation is a categorical notion introduced by Hasuo to verify
finite trace inclusion. They allow us to give definitions of forward and
backward simulation for various types of systems. A generic categorical theory
behind Kleisli simulation has been developed and it guarantees the soundness of
those simulations with respect to finite trace semantics. Moreover, those
simulations can be aided by forward partial execution (FPE)---a categorical
transformation of systems previously introduced by the authors.
In this paper, we give Kleisli simulation a theoretical foundation that
assures its soundness also with respect to infinitary traces. There, following
Jacobs' work, infinitary trace semantics is characterized as the "largest
homomorphism." It turns out that soundness of forward simulations is rather
straightforward; that of backward simulation holds too, although it requires
certain additional conditions and its proof is more involved. We also show that
FPE can be successfully employed in the infinitary trace setting to enhance the
applicability of Kleisli simulations as witnesses of trace inclusion. Our
framework is parameterized in the monad for branching as well as in the functor
for linear-time behaviors; for the former we mainly use the powerset monad (for
nondeterminism), the sub-Giry monad (for probability), and the lift monad (for
exception).Comment: 39 pages, 1 figur
Real-time and Probabilistic Temporal Logics: An Overview
Over the last two decades, there has been an extensive study on logical
formalisms for specifying and verifying real-time systems. Temporal logics have
been an important research subject within this direction. Although numerous
logics have been introduced for the formal specification of real-time and
complex systems, an up to date comprehensive analysis of these logics does not
exist in the literature. In this paper we analyse real-time and probabilistic
temporal logics which have been widely used in this field. We extrapolate the
notions of decidability, axiomatizability, expressiveness, model checking, etc.
for each logic analysed. We also provide a comparison of features of the
temporal logics discussed
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