2,440 research outputs found
Monadic Second-Order Logic and Bisimulation Invariance for Coalgebras
Generalizing standard monadic second-order logic for Kripke models, we
introduce monadic second-order logic interpreted over coalgebras for an
arbitrary set functor. Similar to well-known results for monadic second-order
logic over trees, we provide a translation of this logic into a class of
automata, relative to the class of coalgebras that admit a tree-like supporting
Kripke frame. We then consider invariance under behavioral equivalence of
formulas; more in particular, we investigate whether the coalgebraic
mu-calculus is the bisimulation-invariant fragment of monadic second-order
logic. Building on recent results by the third author we show that in order to
provide such a coalgebraic generalization of the Janin-Walukiewicz Theorem, it
suffices to find what we call an adequate uniform construction for the functor.
As applications of this result we obtain a partly new proof of the
Janin-Walukiewicz Theorem, and bisimulation invariance results for the bag
functor (graded modal logic) and all exponential polynomial functors.
Finally, we consider in some detail the monotone neighborhood functor, which
provides coalgebraic semantics for monotone modal logic. It turns out that
there is no adequate uniform construction for this functor, whence the
automata-theoretic approach towards bisimulation invariance does not apply
directly. This problem can be overcome if we consider global bisimulations
between neighborhood models: one of our main technical results provides a
characterization of the monotone modal mu-calculus extended with the global
modalities, as the fragment of monadic second-order logic for the monotone
neighborhood functor that is invariant for global bisimulations
An expressive completeness theorem for coalgebraic modal mu-calculi
Generalizing standard monadic second-order logic for Kripke models, we
introduce monadic second-order logic interpreted over coalgebras for an
arbitrary set functor. We then consider invariance under behavioral equivalence
of MSO-formulas. More specifically, we investigate whether the coalgebraic
mu-calculus is the bisimulation-invariant fragment of the monadic second-order
language for a given functor. Using automatatheoretic techniques and building
on recent results by the third author, we show that in order to provide such a
characterization result it suffices to find what we call an adequate uniform
construction for the coalgebraic type functor. As direct applications of this
result we obtain a partly new proof of the Janin-Walukiewicz Theorem for the
modal mu-calculus, avoiding the use of syntactic normal forms, and bisimulation
invariance results for the bag functor (graded modal logic) and all exponential
polynomial functors (including the "game functor"). As a more involved
application, involving additional non-trivial ideas, we also derive a
characterization theorem for the monotone modal mu-calculus, with respect to a
natural monadic second-order language for monotone neighborhood models.Comment: arXiv admin note: substantial text overlap with arXiv:1501.0721
Toward a probability theory for product logic: states, integral representation and reasoning
The aim of this paper is to extend probability theory from the classical to
the product t-norm fuzzy logic setting. More precisely, we axiomatize a
generalized notion of finitely additive probability for product logic formulas,
called state, and show that every state is the Lebesgue integral with respect
to a unique regular Borel probability measure. Furthermore, the relation
between states and measures is shown to be one-one. In addition, we study
geometrical properties of the convex set of states and show that extremal
states, i.e., the extremal points of the state space, are the same as the
truth-value assignments of the logic. Finally, we axiomatize a two-tiered modal
logic for probabilistic reasoning on product logic events and prove soundness
and completeness with respect to probabilistic spaces, where the algebra is a
free product algebra and the measure is a state in the above sense.Comment: 27 pages, 1 figur
On the Concept of a Notational Variant
In the study of modal and nonclassical logics, translations have frequently been employed as a way of measuring the inferential capabilities of a logic. It is sometimes claimed that two logics are “notational variants” if they are translationally equivalent. However, we will show that this cannot be quite right, since first-order logic and propositional logic are translationally equivalent. Others have claimed that for two logics to be notational variants, they must at least be compositionally intertranslatable. The definition of compositionality these accounts use, however, is too strong, as the standard translation from modal logic to first-order logic is not compositional in this sense. In light of this, we will explore a weaker version of this notion that we will call schematicity and show that there is no schematic translation either from first-order logic to propositional logic or from intuitionistic logic to classical logic
A Gentle Introduction to Epistemic Planning: The DEL Approach
Epistemic planning can be used for decision making in multi-agent situations
with distributed knowledge and capabilities. Dynamic Epistemic Logic (DEL) has
been shown to provide a very natural and expressive framework for epistemic
planning. In this paper, we aim to give an accessible introduction to DEL-based
epistemic planning. The paper starts with the most classical framework for
planning, STRIPS, and then moves towards epistemic planning in a number of
smaller steps, where each step is motivated by the need to be able to model
more complex planning scenarios.Comment: In Proceedings M4M9 2017, arXiv:1703.0173
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