1,848 research outputs found
The Arity Hierarchy in the Polyadic -Calculus
The polyadic mu-calculus is a modal fixpoint logic whose formulas define
relations of nodes rather than just sets in labelled transition systems. It can
express exactly the polynomial-time computable and bisimulation-invariant
queries on finite graphs. In this paper we show a hierarchy result with respect
to expressive power inside the polyadic mu-calculus: for every level of
fixpoint alternation, greater arity of relations gives rise to higher
expressive power. The proof uses a diagonalisation argument.Comment: In Proceedings FICS 2015, arXiv:1509.0282
On P-transitive graphs and applications
We introduce a new class of graphs which we call P-transitive graphs, lying
between transitive and 3-transitive graphs. First we show that the analogue of
de Jongh-Sambin Theorem is false for wellfounded P-transitive graphs; then we
show that the mu-calculus fixpoint hierarchy is infinite for P-transitive
graphs. Both results contrast with the case of transitive graphs. We give also
an undecidability result for an enriched mu-calculus on P-transitive graphs.
Finally, we consider a polynomial time reduction from the model checking
problem on arbitrary graphs to the model checking problem on P-transitive
graphs. All these results carry over to 3-transitive graphs.Comment: In Proceedings GandALF 2011, arXiv:1106.081
Disjunctive form and the modal alternation hierarchy
This paper studies the relationship between disjunctive form, a syntactic
normal form for the modal mu calculus, and the alternation hierarchy. First it
shows that all disjunctive formulas which have equivalent tableau have the same
syntactic alternation depth. However, tableau equivalence only preserves
alternation depth for the disjunctive fragment: there are disjunctive formulas
with arbitrarily high alternation depth that are tableau equivalent to
alternation-free non-disjunctive formulas. Conversely, there are
non-disjunctive formulas of arbitrarily high alternation depth that are tableau
equivalent to disjunctive formulas without alternations. This answers
negatively the so far open question of whether disjunctive form preserves
alternation depth. The classes of formulas studied here illustrate a previously
undocumented type of avoidable syntactic complexity which may contribute to our
understanding of why deciding the alternation hierarchy is still an open
problem.Comment: In Proceedings FICS 2015, arXiv:1509.0282
The Variable Hierarchy for the Games mu-Calculus
Parity games are combinatorial representations of closed Boolean mu-terms. By
adding to them draw positions, they have been organized by Arnold and one of
the authors into a mu-calculus. As done by Berwanger et al. for the
propositional modal mu-calculus, it is possible to classify parity games into
levels of a hierarchy according to the number of fixed-point variables. We ask
whether this hierarchy collapses w.r.t. the standard interpretation of the
games mu-calculus into the class of all complete lattices. We answer this
question negatively by providing, for each n >= 1, a parity game Gn with these
properties: it unravels to a mu-term built up with n fixed-point variables, it
is semantically equivalent to no game with strictly less than n-2 fixed-point
variables
Fixed-point elimination in the intuitionistic propositional calculus
It is a consequence of existing literature that least and greatest
fixed-points of monotone polynomials on Heyting algebras-that is, the algebraic
models of the Intuitionistic Propositional Calculus-always exist, even when
these algebras are not complete as lattices. The reason is that these extremal
fixed-points are definable by formulas of the IPC. Consequently, the
-calculus based on intuitionistic logic is trivial, every -formula
being equivalent to a fixed-point free formula. We give in this paper an
axiomatization of least and greatest fixed-points of formulas, and an algorithm
to compute a fixed-point free formula equivalent to a given -formula. The
axiomatization of the greatest fixed-point is simple. The axiomatization of the
least fixed-point is more complex, in particular every monotone formula
converges to its least fixed-point by Kleene's iteration in a finite number of
steps, but there is no uniform upper bound on the number of iterations. We
extract, out of the algorithm, upper bounds for such n, depending on the size
of the formula. For some formulas, we show that these upper bounds are
polynomial and optimal
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)
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
Model-Checking Process Equivalences
Process equivalences are formal methods that relate programs and system
which, informally, behave in the same way. Since there is no unique notion of
what it means for two dynamic systems to display the same behaviour there are a
multitude of formal process equivalences, ranging from bisimulation to trace
equivalence, categorised in the linear-time branching-time spectrum.
We present a logical framework based on an expressive modal fixpoint logic
which is capable of defining many process equivalence relations: for each such
equivalence there is a fixed formula which is satisfied by a pair of processes
if and only if they are equivalent with respect to this relation. We explain
how to do model checking, even symbolically, for a significant fragment of this
logic that captures many process equivalences. This allows model checking
technology to be used for process equivalence checking. We show how partial
evaluation can be used to obtain decision procedures for process equivalences
from the generic model checking scheme.Comment: In Proceedings GandALF 2012, arXiv:1210.202
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