404 research outputs found
Model-theoretic characterization of intuitionistic propositional formulas
Notions of k-asimulation and asimulation are introduced as asymmetric
counterparts to k-bisimulation and bisimulation, respectively. It is proved
that a first-order formula is equivalent to a standard translation of an
intuitionistic propositional formula iff it is invariant with respect to
k-asimulations for some k, and then that a first-order formula is equivalent to
a standard translation of an intuitionistic propositional formula iff it is
invariant with respect to asimulations. Finally, it is proved that a
first-order formula is intuitionistically equivalent to a standard translation
of an intuitionistic propositional formula iff it is invariant with respect to
asimulations between intuitionistic models.Comment: 16 pages, 0 figures. arXiv admin note: substantial text overlap with
arXiv:1202.119
The Expressive Power of Modal Dependence Logic
We study the expressive power of various modal logics with team semantics. We
show that exactly the properties of teams that are downward closed and closed
under team k-bisimulation, for some finite k, are definable in modal logic
extended with intuitionistic disjunction. Furthermore, we show that the
expressive power of modal logic with intuitionistic disjunction and extended
modal dependence logic coincide. Finally we establish that any translation from
extended modal dependence logic into modal logic with intuitionistic
disjunction increases the size of some formulas exponentially.Comment: 19 page
Model-theoretic characterization of predicate intuitionistic formulas
Notions of asimulation and k-asimulation introduced in [Olkhovikov, 2011] are
extended onto the level of predicate logic. We then prove that a first-order
formula is equivalent to a standard translation of an intuitionistic predicate
formula iff it is invariant with respect to k-asimulations for some k, and then
that a first-order formula is equivalent to a standard translation of an
intuitionistic predicate formula iff it is invariant with respect to
asimulations. Finally, it is proved that a first-order formula is equivalent to
a standard translation of an intuitionistic predicate formula over a class of
intuitionistic models (intuitionistic models with constant domain) iff it is
invariant with respect to asimulations between intuitionistic models
(intuitionistic models with constant domain)
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
Complexity of validity for propositional dependence logics
We study the validity problem for propositional dependence logic, modal
dependence logic and extended modal dependence logic. We show that the validity
problem for propositional dependence logic is NEXPTIME-complete. In addition,
we establish that the corresponding problem for modal dependence logic and
extended modal dependence logic is NEXPTIME-hard and in NEXPTIME^NP.Comment: In Proceedings GandALF 2014, arXiv:1408.556
Neighborhood Semantics for Basic and Intuitionistic Logic
In this paper we present a neighborhood semantics for Intuitionistic Propositional Logic (IPL). We show that for each Kripke model of the logic there is a pointwise equivalent neighborhood model and vice versa. In this way, we establish soundness and completeness of IPL with respect to the neighborhood semantics. The relation between neighborhood and topological semantics are also investigated. Moreover, the notions of bisimulation and n-bisimulation between neighborhood models of IPL are defined naturally and some of their basic properties are proved. We also consider Basic Propositional Logic (BPL), a logic weaker than IPL introduced by Albert Visser, and introduce and study its neighborhood models in the same manner
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