2,468 research outputs found
The \mu-Calculus Alternation Hierarchy Collapses over Structures with Restricted Connectivity
It is known that the alternation hierarchy of least and greatest fixpoint
operators in the mu-calculus is strict. However, the strictness of the
alternation hierarchy does not necessarily carry over when considering
restricted classes of structures. A prominent instance is the class of infinite
words over which the alternation-free fragment is already as expressive as the
full mu-calculus. Our current understanding of when and why the mu-calculus
alternation hierarchy is not strict is limited. This paper makes progress in
answering these questions by showing that the alternation hierarchy of the
mu-calculus collapses to the alternation-free fragment over some classes of
structures, including infinite nested words and finite graphs with feedback
vertex sets of a bounded size. Common to these classes is that the connectivity
between the components in a structure from such a class is restricted in the
sense that the removal of certain vertices from the structure's graph
decomposes it into graphs in which all paths are of finite length. Our collapse
results are obtained in an automata-theoretic setting. They subsume,
generalize, and strengthen several prior results on the expressivity of the
mu-calculus over restricted classes of structures.Comment: In Proceedings GandALF 2012, arXiv:1210.202
On Modal {\mu}-Calculus over Finite Graphs with Bounded Strongly Connected Components
For every positive integer k we consider the class SCCk of all finite graphs
whose strongly connected components have size at most k. We show that for every
k, the Modal mu-Calculus fixpoint hierarchy on SCCk collapses to the level
Delta2, but not to Comp(Sigma1,Pi1) (compositions of formulas of level Sigma1
and Pi1). This contrasts with the class of all graphs, where
Delta2=Comp(Sigma1,Pi1)
Disjunctive bases: normal forms and model theory for modal logics
We present the concept of a disjunctive basis as a generic framework for
normal forms in modal logic based on coalgebra. Disjunctive bases were defined
in previous work on completeness for modal fixpoint logics, where they played a
central role in the proof of a generic completeness theorem for coalgebraic
mu-calculi. Believing the concept has a much wider significance, here we
investigate it more thoroughly in its own right. We show that the presence of a
disjunctive basis at the "one-step" level entails a number of good properties
for a coalgebraic mu-calculus, in particular, a simulation theorem showing that
every alternating automaton can be transformed into an equivalent
nondeterministic one. Based on this, we prove a Lyndon theorem for the full
fixpoint logic, its fixpoint-free fragment and its one-step fragment, a Uniform
Interpolation result, for both the full mu-calculus and its fixpoint-free
fragment, and a Janin-Walukiewicz-style characterization theorem for the
mu-calculus under slightly stronger assumptions.
We also raise the questions, when a disjunctive basis exists, and how
disjunctive bases are related to Moss' coalgebraic "nabla" modalities. Nabla
formulas provide disjunctive bases for many coalgebraic modal logics, but there
are cases where disjunctive bases give useful normal forms even when nabla
formulas fail to do so, our prime example being graded modal logic. We also
show that disjunctive bases are preserved by forming sums, products and
compositions of coalgebraic modal logics, providing tools for modular
construction of modal logics admitting disjunctive bases. Finally, we consider
the problem of giving a category-theoretic formulation of disjunctive bases,
and provide a partial solution
Model-Checking the Higher-Dimensional Modal mu-Calculus
The higher-dimensional modal mu-calculus is an extension of the mu-calculus
in which formulas are interpreted in tuples of states of a labeled transition
system. Every property that can be expressed in this logic can be checked in
polynomial time, and conversely every polynomial-time decidable problem that
has a bisimulation-invariant encoding into labeled transition systems can also
be defined in the higher-dimensional modal mu-calculus. We exemplify the latter
connection by giving several examples of decision problems which reduce to
model checking of the higher-dimensional modal mu-calculus for some fixed
formulas. This way generic model checking algorithms for the logic can then be
used via partial evaluation in order to obtain algorithms for theses problems
which may benefit from improvements that are well-established in the field of
program verification, namely on-the-fly and symbolic techniques. The aim of
this work is to extend such techniques to other fields as well, here
exemplarily done for process equivalences, automata theory, parsing, string
problems, and games.Comment: In Proceedings FICS 2012, arXiv:1202.317
Zielonka's Recursive Algorithm: dull, weak and solitaire games and tighter bounds
Dull, weak and nested solitaire games are important classes of parity games,
capturing, among others, alternation-free mu-calculus and ECTL* model checking
problems. These classes can be solved in polynomial time using dedicated
algorithms. We investigate the complexity of Zielonka's Recursive algorithm for
solving these special games, showing that the algorithm runs in O(d (n + m)) on
weak games, and, somewhat surprisingly, that it requires exponential time to
solve dull games and (nested) solitaire games. For the latter classes, we
provide a family of games G, allowing us to establish a lower bound of 2^(n/3).
We show that an optimisation of Zielonka's algorithm permits solving games from
all three classes in polynomial time. Moreover, we show that there is a family
of (non-special) games M that permits us to establish a lower bound of 2^(n/3),
improving on the previous lower bound for the algorithm.Comment: In Proceedings GandALF 2013, arXiv:1307.416
Equivalence-Checking on Infinite-State Systems: Techniques and Results
The paper presents a selection of recently developed and/or used techniques
for equivalence-checking on infinite-state systems, and an up-to-date overview
of existing results (as of September 2004)
Making Random Choices Invisible to the Scheduler
When dealing with process calculi and automata which express both
nondeterministic and probabilistic behavior, it is customary to introduce the
notion of scheduler to solve the nondeterminism. It has been observed that for
certain applications, notably those in security, the scheduler needs to be
restricted so not to reveal the outcome of the protocol's random choices, or
otherwise the model of adversary would be too strong even for ``obviously
correct'' protocols. We propose a process-algebraic framework in which the
control on the scheduler can be specified in syntactic terms, and we show how
to apply it to solve the problem mentioned above. We also consider the
definition of (probabilistic) may and must preorders, and we show that they are
precongruences with respect to the restricted schedulers. Furthermore, we show
that all the operators of the language, except replication, distribute over
probabilistic summation, which is a useful property for verification
- …