16,060 research outputs found
Unprovability of the Logical Characterization of Bisimulation
We quickly review labelled Markov processes (LMP) and provide a
counterexample showing that in general measurable spaces, event bisimilarity
and state bisimilarity differ in LMP. This shows that the logic in Desharnais
[*] does not characterize state bisimulation in non-analytic measurable spaces.
Furthermore we show that, under current foundations of Mathematics, such
logical characterization is unprovable for spaces that are projections of a
coanalytic set. Underlying this construction there is a proof that stationary
Markov processes over general measurable spaces do not have semi-pullbacks.
([*] J. Desharnais, Labelled Markov Processes. School of Computer Science.
McGill University, Montr\'eal (1999))Comment: Extended introduction and comments; extra section on semi-pullbacks;
11 pages Some background details added; extra example on the non-locality of
state bisimilarity; 14 page
Bisimilarity is not Borel
We prove that the relation of bisimilarity between countable labelled
transition systems is -complete (hence not Borel), by reducing the
set of non-wellorders over the natural numbers continuously to it.
This has an impact on the theory of probabilistic and nondeterministic
processes over uncountable spaces, since logical characterizations of
bisimilarity (as, for instance, those based on the unique structure theorem for
analytic spaces) require a countable logic whose formulas have measurable
semantics. Our reduction shows that such a logic does not exist in the case of
image-infinite processes.Comment: 20 pages, 1 figure; proof of Sigma_1^1 completeness added with
extended comments. I acknowledge careful reading by the referees. Major
changes in Introduction, Conclusion, and motivation for NLMP. Proof for Lemma
22 added, simpler proofs for Lemma 17 and Theorem 30. Added references. Part
of this work was presented at Dagstuhl Seminar 12411 on Coalgebraic Logic
Semipullbacks of labelled Markov processes
A labelled Markov process (LMP) consists of a measurable space together
with an indexed family of Markov kernels from to itself. This structure has
been used to model probabilistic computations in Computer Science, and one of
the main problems in the area is to define and decide whether two LMP and
"behave the same". There are two natural categorical definitions of
sameness of behavior: and are bisimilar if there exist an LMP and
measure preserving maps forming a diagram of the shape ; and they are behaviorally equivalent if there exist some
and maps forming a dual diagram .
These two notions differ for general measurable spaces but Doberkat
(extending a result by Edalat) proved that they coincide for analytic Borel
spaces, showing that from every diagram one
can obtain a bisimilarity diagram as above. Moreover, the resulting square of
measure preserving maps is commutative (a "semipullback").
In this paper, we extend the previous result to measurable spaces
isomorphic to a universally measurable subset of a Polish space with the trace
of the Borel -algebra, using a version of Strassen's theorem on common
extensions of finitely additive measures.Comment: 10 pages; v2: missing attribution to Doberka
On curves with one place at infinity
Let be a plane curve. We give a procedure based on Abhyankar's
approximate roots to detect if it has a single place at infinity, and if so
construct its associated -sequence, and consequently its value
semigroup. Also for fixed genus (equivalently Frobenius number) we construct
all -sequences generating numerical semigroups with this given genus.
For a -sequence we present a procedure to construct all curves having
this associated sequence.
We also study the embeddings of such curves in the plane. In particular, we
prove that polynomial curves might not have a unique embedding.Comment: 14 pages, 2 figure
- …