Logic and Foundations Commons^™

Many-Valued Coalgebraic Logic: From Boolean Algebras To Primal Varieties, Alexander Kurz, Wolfgang Poiger

Engineering Faculty Articles and Research

We study many-valued coalgebraic logics with primal algebras of truth-degrees. We describe a way to lift algebraic semantics of classical coalgebraic logics, given by an endofunctor on the variety of Boolean algebras, to this many-valued setting, and we show that many important properties of the original logic are inherited by its lifting. Then, we deal with the problem of obtaining a concrete axiomatic presentation of the variety of algebras for this lifted logic, given that we know one for the original one. We solve this problem for a class of presentations which behaves well with respect to a lattice structure …

Semi De Morgan Logic Properly Displayed, Giuseppe Greco, Fei Qin, M. Andrew Moshier, Alessandra Palmigiano

Mathematics, Physics, and Computer Science Faculty Articles and Research

In the present paper, we endow semi De Morgan logic and a family of its axiomatic extensions with proper multi-type display calculi which are sound, complete, conservative, and enjoy cut elimination and subformula property. Our proposal builds on an algebraic analysis of the variety of semi De Morgan algebras, and applies the guidelines of the multi-type methodology in the design of display calculi.

Relation Algebras, Idempotent Semirings And Generalized Bunched Implication Algebras, Peter Jipsen

Mathematics, Physics, and Computer Science Faculty Articles and Research

This paper investigates connections between algebraic structures that are common in theoretical computer science and algebraic logic. Idempotent semirings are the basis of Kleene algebras, relation algebras, residuated lattices and bunched implication algebras. Extending a result of Chajda and Länger, we show that involutive residuated lattices are determined by a pair of dually isomorphic idempotent semirings on the same set, and this result also applies to relation algebras. Generalized bunched implication algebras (GBI-algebras for short) are residuated lattices expanded with a Heyting implication. We construct bounded cyclic involutive GBI-algebras from so-called weakening relations, and prove that the class of weakening …

Quasivarieties And Varieties Of Ordered Algebras: Regularity And Exactness, Alexander Kurz

Engineering Faculty Articles and Research

We characterise quasivarieties and varieties of ordered algebras categorically in terms of regularity, exactness and the existence of a suitable generator. The notions of regularity and exactness need to be understood in the sense of category theory enriched over posets.

We also prove that finitary varieties of ordered algebras are cocompletions of their theories under sifted colimits (again, in the enriched sense).

Features Of Agent-Based Models, Reiko Heckel, Alexander Kurz, Edmund Chattoe-Brown

Engineering Faculty Articles and Research

The design of agent-based models (ABMs) is often ad-hoc when it comes to defining their scope. In order for the inclusion of features such as network structure, location, or dynamic change to be justified, their role in a model should be systematically analysed. We propose a mechanism to compare and assess the impact of such features. In particular we are using techniques from software engineering and semantics to support the development and assessment of ABMs, such as graph transformations as semantic representations for agent-based models, feature diagrams to identify ingredients under consideration, and extension relations between graph transformation systems to …

Foreword: Special Issue On Coalgebraic Logic, Alexander Kurz

Engineering Faculty Articles and Research

The second Dagstuhl seminar on coalgebraic logics took place from October 7-12, 2012, in the Leibniz Forschungszentrum Schloss Dagstuhl, following a successful earlier one in December 2009. From the 44 researchers who attended and the 30 talks presented, this collection highlights some of the progress that has been made in the field. We are grateful to Giuseppe Longo and his interest in a special issue in Mathematical Structures in Computer Science.

The Positivication Of Coalgebraic Logics, Fredrik Dahlqvist, Alexander Kurz

Engineering Faculty Articles and Research

We present positive coalgebraic logic in full generality, and show how to obtain a positive coalgebraic logic from a boolean one. On the model side this involves canonically computing a endofunctor T': Pos->Pos from an endofunctor T: Set->Set, in a procedure previously defined by the second author et alii called posetification. On the syntax side, it involves canonically computing a syntax-building functor L': DL->DL from a syntax-building functor L: BA->BA, in a dual procedure which we call positivication. These operations are interesting in their own right and we explicitly compute posetifications and positivications in the case …

Kolmogorov’S Axioms For Probabilities With Values In Hyperbolic Numbers, Daniel Alpay, M. E. Luna-Elizarrarás, Michael Shapiro

Mathematics, Physics, and Computer Science Faculty Articles and Research

We introduce the notion of a probabilistic measure which takes values in hyperbolic numbers and which satisfies the system of axioms generalizing directly Kolmogorov’s system of axioms. We show that this new measure verifies the usual properties of a probability; in particular, we treat the conditional hyperbolic probability and we prove the hyperbolic analogues of the multiplication theorem, of the law of total probability and of Bayes’ theorem. Our probability may take values which are zero–divisors and we discuss carefully this peculiarity.

Multi-Type Display Calculus For Propositional Dynamic Logic, Sabine Frittella, Giuseppe Greco, Alexander Kurz, Alessandra Palmigiano

Engineering Faculty Articles and Research

We introduce a multi-type display calculus for Propositional Dynamic Logic (PDL). This calculus is complete w.r.t. PDL, and enjoys Belnap-style cut-elimination and subformula property.

Multi-Type Display Calculus For Dynamic Epistemic Logic, Sabine Frittella, Giuseppe Greco, Alexander Kurz, Alessandra Palmigiano, Vlasta Sikimić

Engineering Faculty Articles and Research

In the present paper, we introduce a multi-type display calculus for dynamic epistemic logic, which we refer to as Dynamic Calculus. The displayapproach is suitable to modularly chart the space of dynamic epistemic logics on weaker-than-classical propositional base. The presence of types endows the language of the Dynamic Calculus with additional expressivity, allows for a smooth proof-theoretic treatment, and paves the way towards a general methodology for the design of proof systems for the generality of dynamic logics, and certainly beyond dynamic epistemic logic. We prove that the Dynamic Calculus adequately captures Baltag-Moss-Solecki’s dynamic epistemic logic, and enjoys Belnap-style cut …

Tool Support For Reasoning In Display Calculi, Samuel Balco, Sabine Frittella, Giuseppe Greco, Alexander Kurz, Alessandra Palmigiano

Engineering Faculty Articles and Research

We present a tool for reasoning in and about propositional sequent calculi. One aim is to support reasoning in calculi that contain a hundred rules or more, so that even relatively small pen and paper derivations become tedious and error prone. As an example, we implement the display calculus D.EAK of dynamic epistemic logic. Second, we provide embeddings of the calculus in the theorem prover Isabelle for formalising proofs about D.EAK. As a case study we show that the solution of the muddy children puzzle is derivable for any number of muddy children. Third, there is a set of meta-tools, …

Extensions Of Functors From Set To V-Cat, Adriana Balan, Alexander Kurz, Jirí Velebil

Engineering Faculty Articles and Research

We show that for a commutative quantale V every functor Set --> V-cat has an enriched left- Kan extension. As a consequence, coalgebras over Set are subsumed by coalgebras over V-cat. Moreover, one can build functors on V-cat by equipping Set-functors with a metric.

Approximation Of Nested Fixpoints, Alexander Kurz

Engineering Faculty Articles and Research

The question addressed in this paper is how to correctly approximate infinite data given by systems of simultaneous corecursive definitions. We devise a categorical framework for reasoning about regular datatypes, that is, datatypes closed under products, coproducts and fixpoints. We argue that the right methodology is on one hand coalgebraic (to deal with possible nontermination and infinite data) and on the other hand 2-categorical (to deal with parameters in a disciplined manner). We prove a coalgebraic version of Bekic lemma that allows us to reduce simultaneous fixpoints to a single fix point. Thus a possibly infinite object of interest is …

Presenting Distributive Laws, Marcello M. Bonsangue, Helle H. Hansen, Alexander Kurz, Jurriaan Rot

Engineering Faculty Articles and Research

Distributive laws of a monad T over a functor F are categorical tools for specifying algebra-coalgebra interaction. They proved to be important for solving systems of corecursive equations, for the specification of well-behaved structural operational semantics and, more recently, also for enhancements of the bisimulation proof method. If T is a free monad, then such distributive laws correspond to simple natural transformations. However, when T is not free it can be rather difficult to prove the defining axioms of a distributive law. In this paper we describe how to obtain a distributive law for a monad with an equational presentation …

Positive Fragments Of Coalgebraic Logics, Adriana Balan, Alexander Kurz, Jirí Velebil

Engineering Faculty Articles and Research

Positive modal logic was introduced in an influential 1995 paper of Dunn as the positive fragment of standard modal logic. His completeness result consists of an axiomatization that derives all modal formulas that are valid on all Kripke frames and are built only from atomic propositions, conjunction, disjunction, box and diamond. In this paper, we provide a coalgebraic analysis of this theorem, which not only gives a conceptual proof based on duality theory, but also generalizes Dunn's result from Kripke frames to coalgebras for weak-pullback preserving functors. To facilitate this analysis we prove a number of category theoretic results on …

Coalgebraic Semantics Of Reflexive Economics (Dagstuhl Seminar 15042), Samson Abramsky, Alexander Kurz, Pierre Lescanne, Viktor Winschel

Engineering Faculty Articles and Research

This report documents the program and the outcomes of Dagstuhl Seminar 15042 “Coalgebraic Semantics of Reflexive Economics”.

Euler, Reader Of Newton: Mechanics And Algebraic Analysis, Sébastien Maronne, Marco Panza

MPP Published Research

We follow two of the many paths leading from Newton’s to Euler’s scientific productions, and give an account of Euler’s role in the reception of some of Newton’s ideas, as regards two major topics: mechanics and algebraic analysis. Euler contributed to a re-appropriation of Newtonian science, though transforming it in many relevant aspects. We study this re-appropriation with respect to the mentioned topics and show that it is grounded on the development of Newton’s conceptions within a new conceptual frame also influenced by Descartes’s views sand Leibniz’s formalism.

A Proof-Theoretic Semantic Analysis Of Dynamic Epistemic Logic, Sabine Frittella, Giuseppe Greco, Alexander Kurz, Alessandra Palmigiano, Vlasta Sikimić

Engineering Faculty Articles and Research

The present paper provides an analysis of the existing proof systems for dynamic epistemic logic from the viewpoint of proof-theoretic semantics. Dynamic epistemic logic is one of the best known members of a family of logical systems which have been successfully applied to diverse scientific disciplines, but the proof theoretic treatment of which presents many difficulties. After an illustration of the proof-theoretic semantic principles most relevant to the treatment of logical connectives, we turn to illustrating the main features of display calculi, a proof-theoretic paradigm which has been successfully employed to give a proof-theoretic semantic account of modal and substructural …

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 …

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.

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.

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.

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 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.

Strongly Complete Logics For Coalgebras, Alexander Kurz, Jiří Rosický

Engineering Faculty Articles and Research

Coalgebras for a functor model different types of transition systems in a uniform way. This paper focuses on a uniform account of finitary logics for set-based coalgebras. In particular, a general construction of a logic from an arbitrary set-functor is given and proven to be strongly complete under additional assumptions. We proceed in three parts.

Part I argues that sifted colimit preserving functors are those functors that preserve universal algebraic structure. Our main theorem here states that a functor preserves sifted colimits if and only if it has a finitary presentation by operations and equations. Moreover, the presentation of the …

Coalgebraic Logics (Dagstuhl Seminar 12411), Ernst-Erich Doberkat, Alexander Kurz

Engineering Faculty Articles and Research

This report documents the program and the outcomes of Dagstuhl Seminar 12411 “Coalgebraic Logics”. The seminar deals with recent developments in the area of coalgebraic logic, a branch of logics which combines modal logics with coalgebraic semantics. Modal logic finds its uses when reasoning about behavioural and temporal properties of computation and communication, coalgebras have evolved into a general theory of systems. Consequently, it is natural to combine both areas for a mathematical description of system specification. Coalgebraic logics are closely related to the broader categories semantics/formal methods and verification/logic.

Completeness For The Coalgebraic Cover Modality, Clemens Kupke, Alexander Kurz, Yde Venema

Engineering Faculty Articles and Research

We study the finitary version of the coalgebraic logic introduced by L. Moss. The syntax of this logic, which is introduced uniformly with respect to a coalgebraic type functor, required to preserve weak pullbacks, extends that of classical propositional logic with a so-called coalgebraic cover modality depending on the type functor. Its semantics is defined in terms of a categorically defined relation lifting operation.

As the main contributions of our paper we introduce a derivation system, and prove that it provides a sound and complete axiomatization for the collection of coalgebraically valid inequalities. Our soundness and completeness proof is algebraic, …

Lagrange's Theory Of Analytical Functions And His Ideal Of Purity Of Method, Giovanni Ferraro, Marco Panza

MPP Published Research

We reconstruct essential features of Lagrange’s theory of analytical functions by exhibiting its structure and basic assumptions, as well as its main shortcomings. We explain Lagrange’s notions of function and algebraic quantity, and we concentrate on power-series expansions, on the algorithm for derivative functions, and the remainder theorem—especially on the role this theorem has in solving geometric and mechanical problems. We thus aim to provide a better understanding of Enlightenment mathematics and to show that the foundations of mathematics did not, for Lagrange, concern the solidity of its ultimate bases, but rather purity of method—the generality and internal organization of …

Relation Liftings On Preorders And Posets, Marta Bílková, Alexander Kurz, Daniela Petrişan, Jiří Velebil

Engineering Faculty Articles and Research

The category Rel(Set) of sets and relations can be described as a category of spans and as the Kleisli category for the powerset monad. A set-functor can be lifted to a functor on Rel(Set) iff it preserves weak pullbacks. We show that these results extend to the enriched setting, if we replace sets by posets or preorders. Preservation of weak pullbacks becomes preservation of exact lax squares. As an application we present Moss’s coalgebraic over posets.