For an arbitrary similarity type of Boolean Algebras with Operators we define a class of Sahlqvist identities. Sahlqvist identities have two important properties. First, a Sahlqvist identity is valid in a complex algebra if and only if the underlying relational atom structure satisfies a first-order condition which can be effectively read off from the syntactic form of the identity. Second, and as a consequence of the first property, Sahlqvist identities are canonical, that is, their validity is preserved under taking canonical (...) embedding algebras. Taken together, these properties imply that results about a Sahlqvist variety V van be obtained by reasoning in the elementary class of canonical structures of algebras in V. We give an example of this strategy in the variety of Cylindric Algebras: we show that an important identity called Henkin's equation is equivalent to a simpler identity that uses only one variable. We give a conceptually simple proof by showing that the firstorder correspondents of these two equations are equivalent over the class of cylindric atom structures. (shrink)
We study several modal languages in which some (sets of) generalized quantifiers can be represented; the main language we consider is suitable for defining any first order definable quantifier, but we also consider a sublanguage thereof, as well as a language for dealing with the modal counterparts of some higher order quantifiers. These languages are studied both from a modal logic perspective and from a quantifier perspective. Thus the issues addressed include normal forms, expressive power, completeness both of modal systems (...) and of systems in the quantifier tradition, complexity as well as syntactic characterizations of special semantic constraints. Throughout the paper several techniques current in the theory of generalized quantifiers are used to obtain results in modal logic, and conversely. (shrink)
Peirce algebras combine sets, relations and various operations linking the two in a unifying setting. This paper offers a modal perspective on Peirce algebras. Using modal logic as a characterization of the full Peirce algebras is given, as well as a finite axiomatization of their equational theory that uses so-called unorthodox derivation rules. In addition, the expressive power of Peirce algebras is analyzed through their connection with first-order logic and the fragment of first-order logic corresponding to Peirce algebras is described (...) in terms of bisimulations. (shrink)
In  Albert Visser shows that ILP completely axiomatizes all schemata about provability and relative interpretability that are provable in finitely axiomatized theories. In this paper we introduce a system called ILP that completely axiomatizes the arithmetically valid principles of provability in and interpretability over such theories. To prove the arithmetical completeness of ILP we use a suitable kind of tail models; as a byproduct we obtain a somewhat modified proof of Visser's completeness result.
We examine the expressive power of probabilistic context free grammars (PCFGs), with a special focus on the use of probabilities as a mechanism for reducing ambiguity by filtering out unwanted parses. Probabilities in PCFGs induce an ordering relation among the set of trees that yield a given input sentence. PCFG parsers return the trees bearing the maximum probability for a given sentence, discarding all other possible trees. This mechanism is naturally viewed as a way of defining a new class of (...) tree languages. We formalize the tree language thus defined, study its expressive power, and show that the latter is beyond context freeness. While the increased expressive power offered by PCFGs helps to reduce ambiguity, we show that, in general, it cannot be decided whether a PCFG removes all ambiguities. (shrink)