Szubstrukturális logikák algebrai és bizonyításelméleti vizsgálata = Algebraic and Proof Theoretic Investigations of Substructural Logics

A kutatás fő eredményei: - Az ""Equality"" algebrák bevezetése (Studia Logica). - A ""strongly involutive uninorm"" algebrák bevezetése és osztályozása, valamint a vonatkozó logika komplexitásának vizsgálata (J Logic and Computation). - Az ""involutive FLe-monoid""-ok algebrai vizsgálata; kúp-reprezentáció és egyes véges láncok osztályozása (Archive for Mathematical Logic). - Az involutív uninormák egy osztályának osztályozása, az itt bevezetett ferde-szimmetrizáció segítségével (J Logic and Computation). - A forgatás konstrukció és az Ábel csoportok kapcsolatának vizsgálata (Fuzzy Sets and Systems). - A reziduált hálók geometriai jellegű vizsgálata (Annals of Pure and Applied Logic). Lektorálás alatt: az ""Elnyelő-folytonos, éles, szubreál láncon értelmezett FLe-algebrák osztályozása"" és a ""Pseudo Equality Algebras"" cikkek. | The scientific achievements of the project are the following: - The introducing of equality algebras (Studia Logica). - The introducing and classifying of strongly involutive uninorm algebras along with complexity issues of the related logic (J Logic and Computation). - Algebraic investigation of involutive FLe-monoids, in particular, conic representation and classification of certain finite chains (Archive for Mathematical Logic). - Structural description of a class of involutive uninorms via inrtoducing skew symmetrization (J Logic and Computation). - An investigation of the link between the rotation-construction and Abelian groups (Fuzzy Sets and Systems). - A geometric flavour study of residuated lattices (Annals of Pure and Applied Logic). - ""Classification of absorbent-continuous sharp FLe-algebras over subreal chains"" and Pseudo Equality Algebras"" (two articles under review

Template iterations with non-definable ccc forcing notions

We present a version with non-definable forcing notions of Shelah's theory of iterated forcing along a template. Our main result, as an application, is that, if $\kappa$ is a measurable cardinal and $\theta<\kappa<\mu<\lambda$ are uncountable regular cardinals, then there is a ccc poset forcing $\mathfrak{s}=\theta<\mathfrak{b}=\mu<\mathfrak{a}=\lambda$. Another application is to get models with large continuum where the groupwise-density number $\mathfrak{g}$ assumes an arbitrary regular value.Comment: To appear in the Annals of Pure and Applied Logic, 45 pages, 2 figure

On lengths of proofs in non-classical logics

AbstractWe give proofs of the effective monotone interpolation property for the system of modal logic K, and others, and the system IL of intuitionistic propositional logic. Hence we obtain exponential lower bounds on the number of proof-lines in those systems. The main results have been given in [P. Hrubeš, Lower bounds for modal logics, Journal of Symbolic Logic 72 (3) (2007) 941–958; P. Hrubeš, A lower bound for intuitionistic logic, Annals of Pure and Applied Logic 146 (2007) 72–90]; here, we give considerably simplified proofs, as well as some generalisations

Computable de Finetti measures

We prove a computable version of de Finetti's theorem on exchangeable sequences of real random variables. As a consequence, exchangeable stochastic processes expressed in probabilistic functional programming languages can be automatically rewritten as procedures that do not modify non-local state. Along the way, we prove that a distribution on the unit interval is computable if and only if its moments are uniformly computable.Comment: 32 pages. Final journal version; expanded somewhat, with minor corrections. To appear in Annals of Pure and Applied Logic. Extended abstract appeared in Proceedings of CiE '09, LNCS 5635, pp. 218-23

Capturing k-ary Existential Second Order Logic with k-ary Inclusion-Exclusion Logic

In this paper we analyze k-ary inclusion-exclusion logic, INEX[k], which is obtained by extending first order logic with k-ary inclusion and exclusion atoms. We show that every formula of INEX[k] can be expressed with a formula of k-ary existential second order logic, ESO[k]. Conversely, every formula of ESO[k] with at most k-ary free relation variables can be expressed with a formula of INEX[k]. From this it follows that, on the level of sentences, INEX[k] captures the expressive power of ESO[k]. We also introduce several useful operators that can be expressed in INEX[k]. In particular, we define inclusion and exclusion quantifiers and so-called term value preserving disjunction which is essential for the proofs of the main results in this paper. Furthermore, we present a novel method of relativization for team semantics and analyze the duality of inclusion and exclusion atoms.Comment: Extended version of a paper published in Annals of Pure and Applied Logic 169 (3), 177-21