278 research outputs found
On first-order expressibility of satisfiability in submodels
Let be regular cardinals, , let
be a sentence of the language in a given
signature, and let express the fact that holds
in a submodel, i.e., any model in the signature satisfies
if and only if some submodel of satisfies . It was shown in [1] that, whenever is in
in the signature having less than
functional symbols (and arbitrarily many predicate symbols), then
is equivalent to a monadic existential sentence in the
second-order language , and that for any
signature having at least one binary predicate symbol there exists in
such that is not equivalent
to any (first-order) sentence in . Nevertheless, in
certain cases are first-order expressible. In this note,
we provide several (syntactical and semantical) characterizations of the case
when is in and is
or a certain large cardinal
Model Checking Games for the Quantitative mu-Calculus
We investigate quantitative extensions of modal logic and the modal
mu-calculus, and study the question whether the tight connection between logic
and games can be lifted from the qualitative logics to their quantitative
counterparts. It turns out that, if the quantitative mu-calculus is defined in
an appropriate way respecting the duality properties between the logical
operators, then its model checking problem can indeed be characterised by a
quantitative variant of parity games. However, these quantitative games have
quite different properties than their classical counterparts, in particular
they are, in general, not positionally determined. The correspondence between
the logic and the games goes both ways: the value of a formula on a
quantitative transition system coincides with the value of the associated
quantitative game, and conversely, the values of quantitative parity games are
definable in the quantitative mu-calculus
Zero-one laws with respect to models of provability logic and two Grzegorczyk logics
It has been shown in the late 1960s that each formula of first-order logic without constants and function symbols obeys a zero-one law: As the number of elements of finite models increases, every formula holds either in almost all or in almost no models of that size. Therefore, many properties of models, such as having an even number of elements, cannot be expressed in the language of first-order logic. Halpern and Kapron proved zero-one laws for classes of models corresponding to the modal logics K, T, S4, and S5 and for frames corresponding to S4 and S5. In this paper, we prove zero-one laws for provability logic and its two siblings Grzegorczyk logic and weak Grzegorczyk logic, with respect to model validity. Moreover, we axiomatize validity in almost all relevant finite models, leading to three different axiom systems
Partial Awareness
We develop a modal logic to capture partial awareness. The logic has three
building blocks: objects, properties, and concepts. Properties are unary
predicates on objects; concepts are Boolean combinations of properties. We take
an agent to be partially aware of a concept if she is aware of the concept
without being aware of the properties that define it. The logic allows for
quantification over objects and properties, so that the agent can reason about
her own unawareness. We then apply the logic to contracts, which we view as
syntactic objects that dictate outcomes based on the truth of formulas. We show
that when agents are unaware of some relevant properties, referencing concepts
that agents are only partially aware of can improve welfare.Comment: Appears in AAAI-1
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