### Generalized Topological Semantics for First-Order Modal Logic 1

, 2010

"... Abstract. This dissertation provides a new semantics for first-order modal logic. It is philosophically motivated by the epistemic reading of modal operators and, in particular, three desiderata in the analysis of epistemic modalities. (i) The semantic modelling of epistemic modalities, in particula ..."

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Abstract. This dissertation provides a new semantics for first-order modal logic. It is philosophically motivated by the epistemic reading of modal operators and, in particular, three desiderata in the analysis of epistemic modalities. (i) The semantic modelling of epistemic modalities, in particular verifiability and falsifiability, cannot be properly achieved by Kripke’s relational notion of accessibility. It requires instead a more general, topological notion of accessibility. (ii) Also, the epistemic reading of modal operators seems to require that we combine modal logic with fully classical first-order logic. For this purpose, however, Kripke’s semantics for quantified modal logic is inadequate; its logic is free logic as opposed to classical logic. (iii) More importantly, Kripke’s semantics comes with a restriction that is too strong to let us semantically express, for instance, that the identity of Hesperus and Phosphorus, even if metaphysically necessary, can still be a matter of epistemic discovery. To provide a semantics that accommodates the three desiderata, I show, on the one hand, how the desideratum (i) can be achieved with topological semantics, and more generally neighborhood semantics, for propositional modal logic. On the other hand, to achieve (ii) and (iii), it turns out

### Categorial compositionality continued: A category theory explanation for quasi-systematicity

"... universal construction The classical account for systematicity of human cognition supposes: (1) syntactically compositional representations; and (2) processes that are sensitive to their structure. The problem with this account is that there is no explanation as to why these two components must be c ..."

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universal construction The classical account for systematicity of human cognition supposes: (1) syntactically compositional representations; and (2) processes that are sensitive to their structure. The problem with this account is that there is no explanation as to why these two components must be compatible, other than by ad hoc assumption (convention) to exclude nonsystematic variants that, e.g., mix prefix and postfix concatenative compositional schemes. Recently, we proposed an alternative explanation (Phillips & Wilson, 2010) without ad hoc assumptions, using a branch of mathematics, called category theory. In this paper, we extend our explanation to domains that are quasi-systematic (e.g., language), where the domain includes some but not all possible combinations of constituents. The central category-theoretic construct is an adjunction involving pullbacks, where the focus is on the relations between processes, rather than the representations. In so far as cognition is systematic, the basic building blocks of cognitive architecture are adjunctions by our theory.

### An Exposition of Sheaves

, 2009

"... This survey serves to introduce sheaves and some basic properties of sheaves from a categorical perspective. The objective is that if one has a basic understanding of category theory that this approach to defining sheaves is more intuitive than the usual approach of a typical Algebraic Geometry text ..."

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This survey serves to introduce sheaves and some basic properties of sheaves from a categorical perspective. The objective is that if one has a basic understanding of category theory that this approach to defining sheaves is more intuitive than the usual approach of a typical Algebraic Geometry text. We

### On the Topology of Discrete Planning with Uncertainty

, 2011

"... This paper explores the topology of planning with uncertainty in discrete spaces. The paper defines strategy complex as the collection of all plans for accomplishing all tasks specified by goal states in a finite discrete graph. Transitions in the graph may be nondeterministic or stochastic. One key ..."

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This paper explores the topology of planning with uncertainty in discrete spaces. The paper defines strategy complex as the collection of all plans for accomplishing all tasks specified by goal states in a finite discrete graph. Transitions in the graph may be nondeterministic or stochastic. One key result is that a system can attain any state in its graph despite control uncertainty if and only if its strategy complex is homotopic to a sphere of dimension two less than the number of states in the graph.

### Reoccurring Patterns in Hierarchical Protein Materials and Music: The Power of Analogies

, 2011

"... Abstract Complex hierarchical structures composed of simple nanoscale building blocks form the basis of most biological materials. Here, we demonstrate how analogies between seemingly different fields enable the understanding of general principles by which functional properties in hierarchical syste ..."

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Abstract Complex hierarchical structures composed of simple nanoscale building blocks form the basis of most biological materials. Here, we demonstrate how analogies between seemingly different fields enable the understanding of general principles by which functional properties in hierarchical systems emerge, similar to an analogy learning process. Specifically, natural hierarchical materials like spider silk exhibit properties comparable to classical music in terms of their hierarchical structure and function. As a comparative tool, here, we apply hierarchical ontology logs that follow a rigorous mathematical formulation based on category theory to provide an insightful system representation by expressing knowledge in a conceptual map. We

### A category theory explanation for systematicity

"... Classical and Connectionist theories of cognitive architecture “explain ” systematicity, whereby the capacity for some cognitive behaviors is intrinsically linked to the capacity for others, as a consequence of syntactically and functionally combinatorial representations, respectively. However, both ..."

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Classical and Connectionist theories of cognitive architecture “explain ” systematicity, whereby the capacity for some cognitive behaviors is intrinsically linked to the capacity for others, as a consequence of syntactically and functionally combinatorial representations, respectively. However, both theories depend on ad hoc assumptions to exclude specific architectures—grammars, or Connectionist networks—that do not account for systematicity. By analogy with the Ptolemaic (i.e., geocentric) theory of planetary motion, although either theory can be made to be consistent with the data, both nonetheless fail to explain it (Aizawa, 2003b). Category theory provides an alternative explanation based on the formal concept of adjunction, which consists of a pair of structure preserving maps, called functors. A functor generalizes the notion of a map between representational states to include a map between state transformations (processes). In a formal sense, systematicity is a necessary consequence of a “higher-order ” theory of cognitive architecture, in contrast to the “first-order ” theories derived from Classicism or Connectionism. Category theory offers a re-conceptualization for cognitive science, analogous to the one that Copernicus provided for astronomy, where representational states are no longer the center of the cognitive universe—replaced by the relationships between the maps that transform them.

### AN ALTERNATIVE DEFINITION OF THE COMPLETION OF METRIC SPACES

, 810

"... Abstract. In this article, the author proposes another way to define the completion of a metric space, which is different from the classical one via the dense property, and prove the equivalence between two definitions. This definition is based on considerations from category theory, and can be gene ..."

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Abstract. In this article, the author proposes another way to define the completion of a metric space, which is different from the classical one via the dense property, and prove the equivalence between two definitions. This definition is based on considerations from category theory, and can be generalized to arbitrary categories. 1.

### AMS Short Course on Computational Topology, JMM2011, New Orleans. On the Topology of Discrete Planning with Uncertainty

, 2012

"... Abstract. This chapter explores the topology of planning with uncertainty in discrete spaces. The chapter defines the strategy complex of a finite discrete graph as the collection of all plans for accomplishing all tasks specified by goal states in the graph. Transitions in the graph may be nondeter ..."

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Abstract. This chapter explores the topology of planning with uncertainty in discrete spaces. The chapter defines the strategy complex of a finite discrete graph as the collection of all plans for accomplishing all tasks specified by goal states in the graph. Transitions in the graph may be nondeterministic or stochastic. One key result is that a system can attain any state in its graph despite control uncertainty if and only if its strategy complex is homotopic to a sphere of dimension two less than the number of states in the graph. 1. Planning with Uncertainty in Robotics The goal of Robotics is to animate the inanimate, so as to endow machines with the ability to act purposefully in the world. Roboticists, working in the subfield of planning, create software by which robots reason about future outcomes of potential actions. Using such planning software, robots combine individual actions into collections that together accomplish particular tasks in the world [29, 30]. Two fundamental and intertwined issues confound this seemingly straightforward approach. One is world complexity, the other is uncertainty.

### 2. Given A f

, 2012

"... These are brief informal notes based on lectures I gave at McGill. There is nothing original about them. 1 Basic definitions A category consists of two “collections ” of things called objects and morphisms or arrows or maps. We write C for a category, C0 for the objects and C1 for the morphisms. The ..."

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These are brief informal notes based on lectures I gave at McGill. There is nothing original about them. 1 Basic definitions A category consists of two “collections ” of things called objects and morphisms or arrows or maps. We write C for a category, C0 for the objects and C1 for the morphisms. They satisfy the following conditions: 1. Every morphism f is associated with two objects (which may be the same) called the domain and codomain of f. One can view a morphism as an arrow from one object to another thus forming a directed graph. We sometimes write cod(f) and dom(f) to denote these objects, more often we give them names like A and B. We write f: A − → B or A f −− → B.

### ii ABSTRACT

, 2010

"... complies with the regulations of this University and meets the accepted standards ..."

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complies with the regulations of this University and meets the accepted standards