30 research outputs found
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
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
The Lattice of Congruences of a Finite Line Frame
Let be a finite Kripke frame. A
congruence of is a bisimulation of that is also an
equivalence relation on F. The set of all congruences of is a
lattice under the inclusion ordering. In this article we investigate this
lattice in the case that is a finite line frame. We give concrete
descriptions of the join and meet of two congruences with a nontrivial upper
bound. Through these descriptions we show that for every nontrivial congruence
, the interval embeds into the lattice of
divisors of a suitable positive integer. We also prove that any two congruences
with a nontrivial upper bound permute.Comment: 31 pages, 11 figures. Expanded intro, conclusions rewritten. New,
less geometrical, proofs of Lemma 19 and (former) Lemma 3
Varieties with Definable Factor Congruences
We study direct product representations of algebras in varieties. We collect
several conditions expressing that these representations are "definable" in a
first-order-logic sense, among them the concept of Definable Factor Congruences
(DFC). The main results are that DFC is a Mal'cev property and that it is
equivalent to all other conditions formulated; in particular we prove that V
has DFC if and only if V has 0&1 and Boolean Factor Congruences. We also obtain
an explicit first order definition of the kernel of the canonical projections
via the terms associated to the Mal'cev condition for DFC, in such a manner it
is preserved by taking direct products and direct factors. The main tool is the
use of "central elements," which are a generalization of both central
idempotent elements in rings with identity and neutral complemented elements in
a bounded lattice