270 research outputs found
Weak MSO: Automata and Expressiveness Modulo Bisimilarity
We prove that the bisimulation-invariant fragment of weak monadic
second-order logic (WMSO) is equivalent to the fragment of the modal
-calculus where the application of the least fixpoint operator is restricted to formulas that are continuous in . Our
proof is automata-theoretic in nature; in particular, we introduce a class of
automata characterizing the expressive power of WMSO over tree models of
arbitrary branching degree. The transition map of these automata is defined in
terms of a logic that is the extension of first-order
logic with a generalized quantifier , where means that there are infinitely many objects satisfying . An
important part of our work consists of a model-theoretic analysis of
.Comment: Technical Report, 57 page
Permutation Games for the Weakly Aconjunctive -Calculus
We introduce a natural notion of limit-deterministic parity automata and
present a method that uses such automata to construct satisfiability games for
the weakly aconjunctive fragment of the -calculus. To this end we devise a
method that determinizes limit-deterministic parity automata of size with
priorities through limit-deterministic B\"uchi automata to deterministic
parity automata of size and with
priorities. The construction relies on limit-determinism to avoid the full
complexity of the Safra/Piterman-construction by using partial permutations of
states in place of Safra-Trees. By showing that limit-deterministic parity
automata can be used to recognize unsuccessful branches in pre-tableaux for the
weakly aconjunctive -calculus, we obtain satisfiability games of size
with priorities for weakly aconjunctive
input formulas of size and alternation-depth . A prototypical
implementation that employs a tableau-based global caching algorithm to solve
these games on-the-fly shows promising initial results
Refinement Modal Logic
In this paper we present {\em refinement modal logic}. A refinement is like a
bisimulation, except that from the three relational requirements only `atoms'
and `back' need to be satisfied. Our logic contains a new operator 'all' in
addition to the standard modalities 'box' for each agent. The operator 'all'
acts as a quantifier over the set of all refinements of a given model. As a
variation on a bisimulation quantifier, this refinement operator or refinement
quantifier 'all' can be seen as quantifying over a variable not occurring in
the formula bound by it. The logic combines the simplicity of multi-agent modal
logic with some powers of monadic second-order quantification. We present a
sound and complete axiomatization of multi-agent refinement modal logic. We
also present an extension of the logic to the modal mu-calculus, and an
axiomatization for the single-agent version of this logic. Examples and
applications are also discussed: to software verification and design (the set
of agents can also be seen as a set of actions), and to dynamic epistemic
logic. We further give detailed results on the complexity of satisfiability,
and on succinctness
The Modal μ-Calculus Hierarchy on Restricted Classes of Transition Systems
We discuss the strictness of the modal µ-calculus hierarchy over some restricted classes of transition systems. First, we show that the hierarchy is strict over reflexive frames. By proving the finite model theorem for reflexive systems the same results holds for finite models. Second, we prove that over transitive systems the hierarchy collapses to the alternation-free fragment. In order to do this the finite model theorem for transitive transition systems is also proved. Further, we verify that if symmetry is added to transitivity the hierarchy collapses to the purely modal fragment
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