The representation of spatial knowledge has been widely studied leading to formal systems such as the Region-Connection Calculus, and various formal accounts of mereotopology. The majority of this work has been in the context of spaces which are continuous and innitely divisible. However, discrete spaces are of practical importance, and arise in several application areas. The present paper contributes to the representation of discrete spatial knowledge on two fronts. Firstly it provides an algebraic calculus for discrete regions, and demonstrates its expressiveness by showing how the concepts of interior, closure, boundary, exterior, and connection can all be formulated in the calculus. The second advance presented is an account of discrete spaces at multiple levels of detail. This is achieved by developing a general theory of coarsening for sets which is then extended to the case of discrete regions.

A ‘Chen space ’ is a set X equipped with a collection of ‘plots ’ — maps from convex sets to X — satisfying three simple axioms. While an individual

A preliminary investigation is made of possible applications in quantum theory of the topos formed by the collection of all M-sets, where M is a monoid. Earlier results on topos aspects of quantum theory can be rederived in this way. However, the formalism also suggests a new way of constructing a ‘neo-realist’ interpretation of quantum theory in which the truth values of propositions are determined by the actions of the monoid of strings of finite projection operators. By these means, a novel topos perspective is gained on the concept of state-vector reduction. 1

Category theory is a very general formalism, but there is a certain special way that physicists use categories which turns out to have close analogues in topology, logic and computation. A category has objects and morphisms, which represent things and ways to go between things. In physics, the objects are often physical systems, and the morphisms are processes turning a state of one physical system into a state of another system — perhaps

operations Concrete operations Exists a / I Exists c / D Theory Discourse universum Discourse universum Figure 2: Our interpretation of Aristote's causal reasoning model ffl Induction is an operation for hypothesis building. It consists in finding a method for achieving a goal. For instance, given a set of objects, inductive learning systems show the underlying structure allowing their classification. if A and B then A ! B In an ontological point of view, this would be expressing an organisation from a set of classes. ffl Abduction choses hypothesis. Given a method and a goal, abduction searches an initial state on wich the method can provide the goal. if B and A ! B then A Abduction on an ontology consists in finding objects on the real world domain that, represented as classes, would provide a domain model useful for solving a problem. ffl Deduction identifies premices. Deductive reasonning consists in applying a given method on a given system's state, and providing a solution. ...

This paper describes the use of diagrams as pictorial, non analogical representations in mathematics. Three interpretations of diagrams, of increasing complexity, are discussed: commutative diagrams, exact sequences, universal diagrams. Typically, reasoning with diagrams (in the case under scrutiny, geometrical structures of arrows) involves three steps: representation, construction, inspection and interpretation. We show how reasoning with diagrams makes a metaphoric use of the properties of the representations and suggest how extensions of the existing paradigms can enrich the emergent domain of hybrid reasoning. We think that much insight on how to uses picture-based reasoning can be gained by analyzing the way diagrammatic reasoning is used by humans in existing fields of research. This study complements the more familiar topic of common-sense reasoning with pictures and diagrams: here, expert knowledge, rather than commonsense knowledge, is involved. 1 Introduction The use of pic...

. Dijkstra's predicate transformer calculus in its extended form gives an axiomatic semantics to program specications including partiality and recursion. However, even the classical theory is based on innitary rst order logic which is needed to guarantee the existence of predicate transformers for weakest (liberal) preconditions. This theory can be generalized to higher-order intuitionistic logic. Such logics can be interpreted in topoi. Then each topos E canonically corresponds to a denitionally complete theory T such that E is equivalent to the topos IE(T ) of denable types over T . Furthermore, each model of T in an arbitrary topos F canonically corresponds to a logical morphism IE(T ) ! F . This correspondence enables the denition of a type specication discipline with a semantics based on topoi such that the predicate transformers in the associated logic give an axiomatic semantics for typed program specications. 1 Motivation The semantics of programs and pr...

We interpret the Holographic Conjecture in terms of quantum bits (qubits). N-qubit states are

Abstract. An elementary theory of strict ∞-categories with application to concrete duality is given. New examples of first and second order concrete duality are presented. 1. Categories, functors, natural transformations, modifications There are two kinds of weakness happenning to ∞-categories. One is changing all occurences of equality = with a weaker equivalence realtion ∼. The other one is a weak naturality condition. The first one is not proper and implies strict category theory. The second one is proper and gives a weak category theory. Below we use ∼ instead of =. It is not necessary but has an advantage to treat directly the classification problem (up to ∼). Definition 1.1. • ∞-precategory is a (big) set L endowed with (1) grading L = ∐ Ln n≥0 (2) unary operations d, c: ∐ (3) unary operation e: ∐ n≥1 L n → ∐ L n → ∐ L n≥0