217 research outputs found
On closure ordinals for the modal mu-calculus
The closure ordinal of a formula of modal mu-calculus mu X phi is the least ordinal kappa, if it exists, such that the denotation of the formula and the kappa-th iteration of the monotone operator induced by phi coincide across all transition systems (finite and infinite). It is known that for every alpha < omega^2 there is a formula phi of modal logic such that mu X phi has closure ordinal alpha (Czarnecki 2010). We prove that the closure ordinals arising from the alternation-free fragment of modal mu-calculus (the syntactic class capturing Sigma_2 cap Pi_2) are bounded by omega^2. In this logic satisfaction can be characterised in terms of the existence of tableaux, trees generated by systematically breaking down formulae into their constituents according to the semantics of the calculus. To obtain optimal upper bounds we utilise the connection between closure ordinals of formulae and embedded order-types of the corresponding tableaux
On closure ordinals for the modal mu-calculus
The closure ordinal of a formula of modal mu-calculus mu X phi is the least ordinal kappa, if it exists, such that the denotation of the formula and the kappa-th iteration of the monotone operator induced by phi coincide across all transition systems (finite and infinite). It is known that for every alpha < omega^2 there is a formula phi of modal logic such that mu X phi has closure ordinal alpha (Czarnecki 2010). We prove that the closure ordinals arising from the alternation-free fragment of modal mu-calculus (the syntactic class capturing Sigma_2 cap Pi_2) are bounded by omega^2. In this logic satisfaction can be characterised in terms of the existence of tableaux, trees generated by systematically breaking down formulae into their constituents according to the semantics of the calculus. To obtain optimal upper bounds we utilise the connection between closure ordinals of formulae and embedded order-types of the corresponding tableaux
Succinctness in subsystems of the spatial mu-calculus
In this paper we systematically explore questions of succinctness in modal
logics employed in spatial reasoning. We show that the closure operator,
despite being less expressive, is exponentially more succinct than the
limit-point operator, and that the -calculus is exponentially more
succinct than the equally-expressive tangled limit operator. These results hold
for any class of spaces containing at least one crowded metric space or
containing all spaces based on ordinals below , with the usual
limit operator. We also show that these results continue to hold even if we
enrich the less succinct language with the universal modality
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
Canonical completeness of infinitary mu
AbstractThis paper presents a new model construction for a natural cut-free infinitary version Kω+(μ) of the propositional modal μ-calculus. Based on that the completeness of Kω+(μ) and the related system Kω(μ) can be established directly – no detour, for example through automata theory, is needed. As a side result we also obtain a finite, cut-free sound and complete system for the propositional modal μ-calculus
Fixpoint Games on Continuous Lattices
Many analysis and verifications tasks, such as static program analyses and
model-checking for temporal logics reduce to the solution of systems of
equations over suitable lattices. Inspired by recent work on lattice-theoretic
progress measures, we develop a game-theoretical approach to the solution of
systems of monotone equations over lattices, where for each single equation
either the least or greatest solution is taken. A simple parity game, referred
to as fixpoint game, is defined that provides a correct and complete
characterisation of the solution of equation systems over continuous lattices,
a quite general class of lattices widely used in semantics. For powerset
lattices the fixpoint game is intimately connected with classical parity games
for -calculus model-checking, whose solution can exploit as a key tool
Jurdzi\'nski's small progress measures. We show how the notion of progress
measure can be naturally generalised to fixpoint games over continuous lattices
and we prove the existence of small progress measures. Our results lead to a
constructive formulation of progress measures as (least) fixpoints. We refine
this characterisation by introducing the notion of selection that allows one to
constrain the plays in the parity game, enabling an effective (and possibly
efficient) solution of the game, and thus of the associated verification
problem. We also propose a logic for specifying the moves of the existential
player that can be used to systematically derive simplified equations for
efficiently computing progress measures. We discuss potential applications to
the model-checking of latticed -calculi and to the solution of fixpoint
equations systems over the reals
A Note on Negative Tagging for Least Fixed-Point Formulae
We consider proof systems with sequents of the formU |- F for proving validity of a propositional
modal mu-calculus formula F over a set U of
states in a given model. Such proof systems usually handle
fixed-point formulae through unfolding, thus allowing such formulae
to reappear in a proof. Tagging is a technique originated by Winskel
for annotating fixed-point formulae with information
about the proof states at which these are unfolded. This information
is used later in the proof to avoid unnecessary unfolding, without
having to investigate the history of the proof. Depending on whether
tags are used for acceptance or for rejection of a branch in the proof
tree, we refer to ``positive'' or ``negative'' tagging, respectively.
In their simplest form, tags consist of the sets U at which
fixed-point formulae are unfolded. In this paper, we generalise results
of earlier work by Andersen, Stirling and Winskel which, in the case
of least fixed-point formulae, are applicable to singleton U sets only
Positional Determinacy of Games with Infinitely Many Priorities
We study two-player games of infinite duration that are played on finite or
infinite game graphs. A winning strategy for such a game is positional if it
only depends on the current position, and not on the history of the play. A
game is positionally determined if, from each position, one of the two players
has a positional winning strategy.
The theory of such games is well studied for winning conditions that are
defined in terms of a mapping that assigns to each position a priority from a
finite set. Specifically, in Muller games the winner of a play is determined by
the set of those priorities that have been seen infinitely often; an important
special case are parity games where the least (or greatest) priority occurring
infinitely often determines the winner. It is well-known that parity games are
positionally determined whereas Muller games are determined via finite-memory
strategies.
In this paper, we extend this theory to the case of games with infinitely
many priorities. Such games arise in several application areas, for instance in
pushdown games with winning conditions depending on stack contents.
For parity games there are several generalisations to the case of infinitely
many priorities. While max-parity games over omega or min-parity games over
larger ordinals than omega require strategies with infinite memory, we can
prove that min-parity games with priorities in omega are positionally
determined. Indeed, it turns out that the min-parity condition over omega is
the only infinitary Muller condition that guarantees positional determinacy on
all game graphs
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