Division by zero in common meadows

Common meadows are fields expanded with a total inverse function. Division by zero produces an additional value denoted with "a" that propagates through all operations of the meadow signature (this additional value can be interpreted as an error element). We provide a basis theorem for so-called common cancellation meadows of characteristic zero, that is, common meadows of characteristic zero that admit a certain cancellation law.Comment: 17 pages, 4 tables; differences with v3: axiom (14) of Mda (Table 2) has been replaced by the stronger axiom (12), this appears to be necessary for the proof of Theorem 3.2.

Functorial Semantics for Petri Nets under the Individual Token Philosophy

Although the algebraic semantics of place/transition Petri nets under the collective token philosophy has been fully explained in terms of (strictly) symmetric (strict) monoidal categories, the analogous construction under the individual token philosophy is not completely satisfactory because it lacks universality and also functoriality. We introduce the notion of pre-net to recover these aspects, obtaining a fully satisfactory categorical treatment centered on the notion of adjunction. This allows us to present a purely logical description of net behaviours under the individual token philosophy in terms of theories and theory morphisms in partial membership equational logic, yielding a complete match with the theory developed by the authors for the collective token view of net

Realms: A Structure for Consolidating Knowledge about Mathematical Theories

Since there are different ways of axiomatizing and developing a mathematical theory, knowledge about a such a theory may reside in many places and in many forms within a library of formalized mathematics. We introduce the notion of a realm as a structure for consolidating knowledge about a mathematical theory. A realm contains several axiomatizations of a theory that are separately developed. Views interconnect these developments and establish that the axiomatizations are equivalent in the sense of being mutually interpretable. A realm also contains an external interface that is convenient for users of the library who want to apply the concepts and facts of the theory without delving into the details of how the concepts and facts were developed. We illustrate the utility of realms through a series of examples. We also give an outline of the mechanisms that are needed to create and maintain realms.Comment: As accepted for CICM 201

Algebraic Models for Contextual Nets

We extend the algebraic approach of Meseguer and Montanari from ordinary place/transition Petri nets to contextual nets, covering both the collective and the individual token philosophy uniformly along the two interpretations of net behaviors

A logic road from special relativity to general relativity

We present a streamlined axiom system of special relativity in first-order logic. From this axiom system we "derive" an axiom system of general relativity in two natural steps. We will also see how the axioms of special relativity transform into those of general relativity. This way we hope to make general relativity more accessible for the non-specialist

Modal logic of planar polygons

We study the modal logic of the closure algebra $P_2$, generated by the set of all polygons in the Euclidean plane $\mathbb{R}^2$. We show that this logic is finitely axiomatizable, is complete with respect to the class of frames we call "crown" frames, is not first order definable, does not have the Craig interpolation property, and its validity problem is PSPACE-complete

Conservation of information and the foundations of quantum mechanics

We review a recent approach to the foundations of quantum mechanics inspired by quantum information theory. The approach is based on a general framework, which allows one to address a large class of physical theories which share basic information-theoretic features. We first illustrate two very primitive features, expressed by the axioms of causality and purity-preservation, which are satisfied by both classical and quantum theory. We then discuss the axiom of purification, which expresses a strong version of the Conservation of Information and captures the core of a vast number of protocols in quantum information. Purification is a highly non-classical feature and leads directly to the emergence of entanglement at the purely conceptual level, without any reference to the superposition principle. Supplemented by a few additional requirements, satisfied by classical and quantum theory, it provides a complete axiomatic characterization of quantum theory for finite dimensional systems.Comment: 11 pages, contribution to the Proceedings of the 3rd International Conference on New Frontiers in Physics, July 28-August 6 2014, Orthodox Academy of Crete, Kolymbari, Cret