Algebra Commons^™

Dynamic Sequent Calculus For The Logic Of Epistemic Actions And Knowledge, Giuseppe Greco, Alexander Kurz, Alessandra Palmigiano

Engineering Faculty Articles and Research

"Dynamic Logics (DLs) form a large family of nonclassical logics, and perhaps the one enjoying the widest range of applications. Indeed, they are designed to formalize change caused by actions of diverse nature: updates on the memory state of a computer, displacements of moving robots in an environment, measurements in models of quantum physics, belief revisions, knowledge updates, etc. In each of these areas, DL-formulas express properties of the model encoding the present state of affairs, as well as the pre- and post-conditions of a given action. Actions are semantically represented as transformations of one model into another, encoding the …

Nominal Regular Expressions For Languages Over Infinite Alphabets, Alexander Kurz, Tomoyuki Suzuki, Emilio Tuosto

Engineering Faculty Articles and Research

We propose regular expressions to abstractly model and study properties of resource-aware computations. Inspired by nominal techniques – as those popular in process calculi – we extend classical regular expressions with names (to model computational resources) and suitable operators (for allocation, deallocation, scoping of, and freshness conditions on resources). We discuss classes of such nominal regular expressions, show how such expressions have natural interpretations in terms of languages over infinite alphabets, and give Kleene theorems to characterise their formal languages in terms of nominal automata.

Nominal Computation Theory (Dagstuhl Seminar 13422), Mikołaj Bojanczyk, Bartek Klin, Alexander Kurz, Andrew M. Pitts

Engineering Faculty Articles and Research

This report documents the program and the outcomes of Dagstuhl Seminar 13422 “Nominal Computation Theory”. The underlying theme of the seminar was nominal sets (also known as sets with atoms or Fraenkel-Mostowski sets) and they role and applications in three distinct research areas: automata over infinite alphabets, program semantics using nominal sets and nominal calculi of concurrent processes.

Non-Commutative Stochastic Distributions And Applications To Linear Systems Theory, Daniel Alpay, Guy Salomon

Mathematics, Physics, and Computer Science Faculty Articles and Research

In this paper, we introduce a non-commutative space of stochastic distributions, which contains the non-commutative white noise space, and forms, together with a natural multiplication, a topological algebra. Special inequalities which hold in this space allow to characterize its invertible elements and to develop an appropriate framework of non-commutative stochastic linear systems.

Relation Algebras As Expanded Fl-Algebras, Nikolaos Galatos, Peter Jipsen

Mathematics, Physics, and Computer Science Faculty Articles and Research

This paper studies generalizations of relation algebras to residuated lattices with a unary De Morgan operation. Several new examples of such algebras are presented, and it is shown that many basic results on relation algebras hold in this wider setting. The variety qRA of quasi relation algebras is defined and shown to be a conservative expansion of involutive FL-algebras. Our main result is that equations in qRA and several of its subvarieties can be decided by a Gentzen system, and that these varieties are generated by their finite members.

A Generalized White Noise Space Approach To Stochastic Integration For A Class Of Gaussian Stationary Increment Processes, Daniel Alpay, Alon Kipnis

Mathematics, Physics, and Computer Science Faculty Articles and Research

Given a Gaussian stationary increment processes, we show that a Skorokhod-Hitsuda stochastic integral with respect to this process, which obeys the Wick-Itô calculus rules, can be naturally defined using ideas taken from Hida’s white noise space theory. We use the Bochner-Minlos theorem to associate a probability space to the process, and define the counterpart of the S-transform in this space. We then use this transform to define the stochastic integral and prove an associated Itô formula.

Topological Convolution Algebras, Daniel Alpay, Guy Salomon

Mathematics, Physics, and Computer Science Faculty Articles and Research

In this paper we introduce a new family of topological convolution algebras of the form ⋃p∈NL2(S,μp), where S is a Borel semi-group in a locally compact group G, which carries an inequality of the type ∥f∗g∥p≤Ap,q∥f∥q∥g∥p for p>q+d where d pre-assigned, and Ap,q is a constant. We give a sufficient condition on the measures μp for such an inequality to hold. We study the functional calculus and the spectrum of the elements of these algebras, and present two examples, one in the setting of non commutative stochastic distributions, and the other related to Dirichlet series.

Representation Formulas For Hardy Space Functions Through The Cuntz Relations And New Interpolation Problems, Daniel Alpay, Palle Jorgensen, Izchak Lewkowicz, Itzik Marziano

Mathematics, Physics, and Computer Science Faculty Articles and Research

We introduce connections between the Cuntz relations and the Hardy space H2 of the open unit disk D. We then use them to solve a new kind of multipoint interpolation problem in H2, where for instance, only a linear combination of the values of a function at given points is preassigned, rather than the values at the points themselves.

Pontryagin De Branges-Rovnyak Spaces Of Slice Hyperholomorphic Functions, Daniel Alpay, Fabrizio Colombo, Irene Sabadini

Mathematics, Physics, and Computer Science Faculty Articles and Research

We study reproducing kernel Hilbert and Pontryagin spaces of slice hyperholomorphic functions which are analogs of the Hilbert spaces of analytic functions introduced by de Branges and Rovnyak. In the first part of the paper we focus on the case of Hilbert spaces, and introduce in particular a version of the Hardy space. Then we define Blaschke factors and Blaschke products and we consider an interpolation problem. In the second part of the paper we turn to the case of Pontryagin spaces. We first prove some results from the theory of Pontryagin spaces in the quaternionic setting and, in particular, …

Residuated Frames With Applications To Decidability, Nikolaos Galatos, Peter Jipsen

Mathematics, Physics, and Computer Science Faculty Articles and Research

Residuated frames provide relational semantics for substructural logics and are a natural generalization of Kripke frames in intuitionistic and modal logic, and of phase spaces in linear logic. We explore the connection between Gentzen systems and residuated frames and illustrate how frames provide a uniform treatment for semantic proofs of cut-elimination, the finite model property and the finite embeddability property, which imply the decidability of the equational/universal theories of the associated residuated lattice-ordered groupoids. In particular these techniques allow us to prove that the variety of involutive FL-algebras and several related varieties have the finite model property.

On Discrete Analytic Functions: Products, Rational Functions, And Reproducing Kernels, Daniel Alpay, Palle Jorgensen, Ron Seager, Dan Volok

Mathematics, Physics, and Computer Science Faculty Articles and Research

We introduce a family of discrete analytic functions, called expandable discrete analytic functions, which includes discrete analytic polynomials, and define two products in this family. The first one is defined in a way similar to the Cauchy-Kovalevskaya product of hyperholomorphic functions, and allows us to define rational discrete analytic functions. To define the second product we need a new space of entire functions which is contractively included in the Fock space. We study in this space some counterparts of Schur analysis.

Convex Cones Of Generalized Positive Rational Functions And Nevanlinna-Pick Interpolation, Daniel Alpay, Izchak Lewkowicz

Mathematics, Physics, and Computer Science Faculty Articles and Research

Scalar rational functions with a non-negative real part on the right half plane, called positive, are classical in the study of electrical networks, dissipative systems, Nevanlinna-Pick interpolation and other areas. We here study generalized positive functions, i.e with a non-negative real part on the imaginary axis. These functions form a Convex Invertible Cone, cic in short, and we explore two partitionings of this set: (i) into (infinitely many non-invertible) convex cones of functions with prescribed poles and zeroes in the right half plane and (ii) each generalized positive function can be written as a sum of even and odd parts. …

Relation Lifting, With An Application To The Many-Valued Cover Modality, Marta Bílková, Alexander Kurz, Daniela Petrişan, Jirí Velebil

Engineering Faculty Articles and Research

We introduce basic notions and results about relation liftings on categories enriched in a commutative quantale. We derive two necessary and sufficient conditions for a 2-functor T to admit a functorial relation lifting: one is the existence of a distributive law of T over the “powerset monad” on categories, one is the preservation by T of “exactness” of certain squares. Both characterisations are generalisations of the “classical” results known for set functors: the first characterisation generalises the existence of a distributive law over the genuine powerset monad, the second generalises preservation of weak pullbacks.

The results presented in this paper …

Epistemic Updates On Algebras, Alexander Kurz, Alessandra Palmigiano

Engineering Faculty Articles and Research

We develop the mathematical theory of epistemic updates with the tools of duality theory. We focus on the Logic of Epistemic Actions and Knowledge (EAK), introduced by Baltag-Moss-Solecki, without the common knowledge operator. We dually characterize the product update construction of EAK as a certain construction transforming the complex algebras associated with the given model into the complex algebra associated with the updated model. This dual characterization naturally generalizes to much wider classes of algebras, which include, but are not limited to, arbitrary BAOs and arbitrary modal expansions of Heyting algebras (HAOs). As an application of this dual characterization, we …

Nominal Coalgebraic Data Types With Applications To Lambda Calculus, Alexander Kurz, Daniela Petrişan, Paula Severi, Fer-Jan De Vries

Engineering Faculty Articles and Research

We investigate final coalgebras in nominal sets. This allows us to define types of infinite data with binding for which all constructions automatically respect alpha equivalence. We give applications to the infinitary lambda calculus.