52,432 research outputs found
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 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
A note on normal forms for the closed fragment of system IL
In [8], P. Hájek and Švejdar determined normal forms for the system ILF, and showed that we can eliminate the modal operator from --formulas.
The normal form for the closed fragment of the interpretability logic is an open problem (see [13]).
We prove that in some cases we can eliminate the modal operator .
We give an example where it is impossible to eliminate $rhd.
Disjunctive Bases: Normal Forms 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, and a Uniform Interpolation result, for both the full mu-calculus and its fixpoint-free fragment. We also raise the questions, when a disjunctive basis exists, and how disjunctive bases are related to Moss\u27 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. Finally, we consider the problem of giving a category-theoretic formulation of disjunctive
bases, and provide a partial solution
Extensions of modal logic KTB and other topics
This thesis covers four topics. They are the extensions of the modal logic KTB, the use of normal forms in modal logic, automated reasoning in the modal logic S4 and the problem of unavoidable words.
Extensions of KTB: The modal logic KTB is the logic of reflexive and symmetric frames. Dually, KTB-algebras have a unary (normal) operator f that satisfies the identities f (x){u2265}x and {u231D}x{u2264}f ({u231D}f(x)). Extensions of KTB are subvarieties of the algebra KTB. Both of these form a lattice, and we investigate the structure of the bottom of the lattice of subvarieties. The unique atom is known to correspond to the modal logic whose frame is a single reflexive point. Yutaka demonstrated that this atom has a unique cover, corresponding to the frame of the two element chain. We construct covers of this element, and so demonstrate that there are a continuum of such covers.
Normal Forms in Modal Logic: Fine proposed the use of normal forms as an alternative to traditional methods of determining Kripke completeness. We expand on this paper and demonstrate the application of normal forms to a number of traditional modal logics, and define new terms needed to apply normal forms in this situation.
Automated reasoning in 84: History based methods for automated reasoning are well understood and accepted. Pliu{u0161}kevi{u010D}ius & Pliu{u0161}kevi{u010D}ien{u0117} propose a new, potentially revolutionary method of applying marks and indices to sequents. We show that the method is flawed, and empirically compare a different mark/index based method to the traditional methods instead.
Unavoidable words: The unavoidable words problem is concerned with repetition in strings of symbols. There are two main ways to identify a word as unavoidable, one based on generalised pattern matching and one from an algorithm. Both methods are in NP, but do not appear to be in P. We define the simple unavoidable words as a subset of the standard unavoidable words that can be identified by the algorithm in P-time. We define depth separating IX x homomorphisms as an easy way to generate a subset of the unavoidable words using the pattern matching method. We then show that the two simpler problems are equivalent to each other
Focused labeled proof systems for modal logic
International audienceFocused proofs are sequent calculus proofs that group inference rules into alternating positive and negative phases. These phases can then be used to define macro-level inference rules from Gentzen's original and tiny introduction and structural rules. We show here that the inference rules of labeled proof systems for modal logics can similarly be described as pairs of such phases within the LKF focused proof system for first-order classical logic. We consider the system G3K of Negri for the modal logic K and define a translation from labeled modal formulas into first-order polarized formulas and show a strict correspondence between derivations in the two systems, i.e., each rule application in G3K corresponds to a bipole—a pair of a positive and a negative phases—in LKF. Since geometric axioms (when properly polarized) induce bipoles, this strong correspondence holds for all modal logics whose Kripke frames are characterized by geometric properties. We extend these results to present a focused labeled proof system for this same class of modal logics and show its soundness and completeness. The resulting proof system allows one to define a rich set of normal forms of modal logic proofs
Topological Representation of Canonicity for Varieties of Modal Algebras
Thesis (Ph.D.) - Indiana University, Mathematics, 2010The main subject of this dissertation is to approach the question of countable canonicity of varieties of modal algebras from a topological and categorical point of view. The category of coalgebras of the Vietoris functor on the category of Stone spaces provides a class of frames we call sv-frames. We show that the semantic of this frames is equivalent to that of modal algebras so long as we are limited to certain valuations called sv-valuations. We show that the canonical frame of any normal modal logic
which is directly constructed based on the logic is an sv-frame. We then define the notion of canonicity of a logic in terms of varieties and their dual classes. We will then prove that any morphism on the category of coalgebras of the Vietoris functor whose codomain is the canonical frame of the minimal normal modal logic are exactly the ones that are invoked by sv-valuations. We will then proceed to reformulate canonicity of a variety of modal algebras determined by a logic in terms of properties of the class of sv-frames that correspond to that logic. We define ultrafilter extension as an operator on the category of sv-frames, prove a coproduct preservation result followed by some equivalent forms of canonicity. Using Stone duality the notion of co-variety
of sv-frames is defined. The notion of validity of a logic on a frame is presented in terms of ranges of theory maps whose domain is the given frame. Partial equivalent results on co-varieties of sv-frames are proved. We classify theory maps which are
maps invoked by a valuation on a Kripke frame using the classification of sv-theory maps and properties
of ultrafilter extension. A negative categorical result concerning the existence of an adjoint functor for ultrafilter extension is
also proved
- …