522 research outputs found
Chain models, trees of singular cardinality and dynamic EF games
Let Îș be a singular cardinal. Karp's notion of a chain model of size ? is defined to be an ordinary model of size Îș along with a decomposition of it into an increasing union of length cf(Îș). With a notion of satisfaction and (chain)-isomorphism such models give an infinitary logic largely mimicking first order logic. In this paper we associate to this logic a notion of a dynamic EF-game which gauges when two chain models are chain-isomorphic. To this game is associated a tree which is a tree of size Îș with no Îș-branches (even no cf(Îș)-branches). The measure of how non-isomorphic the models are is reflected by a certain order on these trees, called reduction. We study the collection of trees of size Îș with no Îș-branches under this notion and prove that when cf(Îș) = Ï this collection is rather regular; in particular it has universality number exactly Îș+. Such trees are then used to develop a descriptive set theory of the space cf(Îș)Îș.The main result of the paper gives in the case of Îș strong limit singular an exact connection between the descriptive set-theoretic complexity of the chain isomorphism orbit of a model, the reduction order on the trees and winning strategies in the corresponding dynamic EF games. In particular we obtain a neat analog of the notion of Scott watershed from the Scott analysis of countable models
Complexity Results for Modal Dependence Logic
Modal dependence logic was introduced recently by V\"a\"an\"anen. It enhances
the basic modal language by an operator =(). For propositional variables
p_1,...,p_n, =(p_1,...,p_(n-1);p_n) intuitively states that the value of p_n is
determined by those of p_1,...,p_(n-1). Sevenster (J. Logic and Computation,
2009) showed that satisfiability for modal dependence logic is complete for
nondeterministic exponential time. In this paper we consider fragments of modal
dependence logic obtained by restricting the set of allowed propositional
connectives. We show that satisfibility for poor man's dependence logic, the
language consisting of formulas built from literals and dependence atoms using
conjunction, necessity and possibility (i.e., disallowing disjunction), remains
NEXPTIME-complete. If we only allow monotone formulas (without negation, but
with disjunction), the complexity drops to PSPACE-completeness. We also extend
V\"a\"an\"anen's language by allowing classical disjunction besides dependence
disjunction and show that the satisfiability problem remains NEXPTIME-complete.
If we then disallow both negation and dependence disjunction, satistiability is
complete for the second level of the polynomial hierarchy. In this way we
completely classify the computational complexity of the satisfiability problem
for all restrictions of propositional and dependence operators considered by
V\"a\"an\"anen and Sevenster.Comment: 22 pages, full version of CSL 2010 pape
Regular Ultrapowers at Regular Cardinals
In earlier work by the first and second authors, the equivalence of a finite square principle ⥠finλD with various model-theoretic properties of structures of size λ and regular ultrafilters was established. In this paper we investigate the principle ⥠finλD -and thereby the above model-theoretic properties-at a regular cardinal. By Chang's two-cardinal theorem, ⥠finλD holds at regular cardinals for all regular filters D if we assume the generalized continuum hypothesis (GCH). In this paper we prove in ZFC that, for certain regular filters that we call doubly+ regular, ⥠finλD holds at regular cardinals, with no assumption about GCH. Thus we get new positive answers in ZFC to Open Problems 18 and 19 in Chang and Keisler's book Model Theory
Knowing Values and Public Inspection
We present a basic dynamic epistemic logic of "knowing the value". Analogous
to public announcement in standard DEL, we study "public inspection", a new
dynamic operator which updates the agents' knowledge about the values of
constants. We provide a sound and strongly complete axiomatization for the
single and multi-agent case, making use of the well-known Armstrong axioms for
dependencies in databases
Epistemic Logic with Partial Dependency Operator
In this paper, we introduce dependency modality
into epistemic logic so as to reason about
dependency relationship in Kripke models. The resulted dependence epistemic
logic possesses decent expressivity and beautiful properties. Several
interesting examples are provided, which highlight this logic's practical
usage. The logic's bisimulation is then discussed, and we give a sound and
strongly complete axiomatization for a sub-language of the logic
- âŠ