1. Residuated Lattices with iteration 2. Background: Semirings and Kleene algebras 3. A Gentzen system for Residuated Kleene Lattices and some reducts 4. Interpreting Kleene algebras with tests 1. Residuated Lattices with iteration This talk is mostly about Residuated Kleene Lattices, which are defined as noncommutative residuated 0,1-lattices expanded with a unary operation * that satisfies x ...

Many of the new fuzzy structures with complete MV-algebras as value sets, such hesitant, intuitionistic, neutrosophic, or soft can be transformed into one type set values in special algebras, called AMV-algebras. The category AMV-algebras is isomorphic to pairs (R,R∗) commutative semirings and corresponding sets are (R,R∗)-fuzzy sets. We use this theory define relations, lower upper approximati...

We extend the big and p-typical Witt vector functors from commutative rings to commutative semirings. In the case of the big Witt vectors, this is a repackaging of some standard facts about monomial and Schur positivity in the combinatorics of symmetric functions. In the p-typical case, it uses positivity with respect to an apparently new basis of the p-typical symmetric functions. We also give...

Given a first-order sentence, a model-checking computation tests whether the sentence holds true in a given finite structure. Data provenance extracts from this computation an abstraction of the manner in which its result depends on the data items that describe the model. Previous work on provenance was, to a large extent, restricted to the negation-free fragment of first-order logic and showed...

We study weighted trace automata with weights in strong bimonoids. Traces form a generalization of words that allow to model concurrency; strong bimonoids are algebraic structures that can be regarded as “semirings without distributivity”. A very important example for the latter are bounded lattices, especially non-distributive ones. We show that if both operations of the bimonoid are locally f...

Simulations of weighted tree automata (wta) are considered. It is shown how such simulations can be decomposed into simpler functional and dual functional simulations also called forward and backward simulations. In addition, it is shown in several cases (fields, commutative rings, Noetherian semirings, semiring of natural numbers) that all equivalent wta M and N can be joined by a finite chain...

We introduce the branching transitive closure operator on weighted monadic second-order logic formulas where the branching corresponds in a natural way to the branching inherent in trees. For arbitrary commutative semirings, we prove that weighted monadic second order logics on trees is equivalent to the definability by formulas which start with one of the following operators: (i) a branching t...

We study D0L power series over commutative semirings. We show that a sequence (cn)n 0 of nonzero elements of a eld A is the coe cient sequence of a D0L power series if and only if there exist a positive integer k and integers i for 1 i k such that cn+k = c 1 n+k 1c 2 n+k 2 : : : c k n for all n 0. As a consequence we solve the equivalence problem of D0L power series over computable elds. TUCS R...

The framework of document spanners abstracts the task information extraction from text as a function that maps every (a string) into relation over document's spans (intervals identified by their start and end indices). For instance, regular are closure under Relational Algebra (RA) expressions with capture variables, expressive power is precisely captured class VSet-automata -- restricted trans...

We investigate power series on pictures which map pictures to elements of a semiring and provide an extension of two-dimensional languages to a quantitative setting. We will assign weights to devices, ranging from tiling systems and domino systems to picture automata. We prove that, for commutative semirings, the behaviors of weighted picture automata are precisely alphabetic projections of ser...