5,536 research outputs found
Generic Model Checking for Modal Fixpoint Logics in COOL-MC
We report on COOL-MC, a model checking tool for fixpoint logics that is
parametric in the branching type of models (nondeterministic, game-based,
probabilistic etc.) and in the next-step modalities used in formulae. The tool
implements generic model checking algorithms developed in coalgebraic logic
that are easily adapted to concrete instance logics. Apart from the standard
modal -calculus, COOL-MC currently supports alternating-time, graded,
probabilistic and monotone variants of the -calculus, but is also
effortlessly extensible with new instance logics. The model checking process is
realized by polynomial reductions to parity game solving, or, alternatively, by
a local model checking algorithm that directly computes the extensions of
formulae in a lazy fashion, thereby potentially avoiding the construction of
the full parity game. We evaluate COOL-MC on informative benchmark sets.Comment: Full Version of VMCAI 2024 publicatio
Succinct Graph Representations of ?-Calculus Formulas
Many algorithmic results on the modal mu-calculus use representations of formulas such as alternating tree automata or hierarchical equation systems. At closer inspection, these results are not always optimal, since the exact relation between the formula and its representation is not clearly understood. In particular, there has been confusion about the definition of the fundamental notion of the size of a mu-calculus formula.
We propose the notion of a parity formula as a natural way of representing a mu-calculus formula, and as a yardstick for measuring its complexity. We discuss the close connection of this concept with alternating tree automata, hierarchical equation systems and parity games. We show that well-known size measures for mu-calculus formulas correspond to a parity formula representation of the formula using its syntax tree, subformula graph or closure graph, respectively. Building on work by Bruse, Friedmann & Lange we argue that for optimal complexity results one needs to work with the closure graph, and thus define the size of a formula in terms of its Fischer-Ladner closure. As a new observation, we show that the common assumption of a formula being clean, that is, with every variable bound in at most one subformula, incurs an exponential blow-up of the size of the closure.
To realise the optimal upper complexity bound of model checking for all formulas, our main result is to provide a construction of a parity formula that (a) is based on the closure graph of a given formula, (b) preserves the alternation-depth but (c) does not assume the input formula to be clean
and the modal -calculus
For a regular cardinal , a formula of the modal -calculus is
-continuous in a variable x if, on every model, its interpretation as a
unary function of x is monotone and preserves unions of -directed sets.
We define the fragment of the modal -calculus and prove
that all the formulas in this fragment are -continuous. For each
formula of the modal -calculus, we construct a formula such that is -continuous, for some
, if and only if is equivalent to . Consequently, we
prove that (i) the problem whether a formula is -continuous for some
is decidable, (ii) up to equivalence, there are only two fragments
determined by continuity at some regular cardinal: the fragment
studied by Fontaine and the fragment . We
apply our considerations to the problem of characterizing closure ordinals of
formulas of the modal -calculus. An ordinal is the closure
ordinal of a formula if its interpretation on every model converges
to its least fixed-point in at most steps and if there is a model
where the convergence occurs exactly in steps. We prove that
, the least uncountable ordinal, is such a closure ordinal. Moreover
we prove that closure ordinals are closed under ordinal sum. Thus, any formal
expression built from 0, 1, , by using the binary operator
symbol + gives rise to a closure ordinal
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 \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
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
An Effective Tableau System for the Linear Time µ-Calculus
We present a tableau system for the model checking problem of the linear time µ-calculus. It improves the system of Stirling and Walker by simplifying the success condition for a tableau. In our system success for a leaf is determined by the path leading to it, whereas Stirling and Walker's method requires the examination of a potentially infinite number of paths extending over the whole tableau
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