Temporal Topological Relationships of Convex Spaces in Space Syntax Theory

GI science development has to be served by an effective GI theory. Development of GI theory requires clear characterisation of GI domain to demonstrate real world effectively and also provide a framework for delineation of rational hypotheses. This requirement is met in recent researches through using mathematics as the basis of GI theory. Axiomatic set theory and formal logic are the foundations of this mathematical approach to study the structures, changes, and spaces in GI domain. While hypotheses require to be modelled into logical structures and be prepared to be evaluated, computer science is also adopted as another foundation of GI theory. One of the recent trends in GI theory development is characterizing the real world as functions and using category theory and algebras as the mathematical basis for handling realities and developing hypotheses. Development of unique and integrated basis for handling static and dynamic GI concepts is one of the hypotheses which are studied in some of these researches. Their outcomes illustrate theoretic feasibility of defining morphisms, known as functors or time liftings, between static and dynamic domains. The time lifting approach was evaluated for some GI models, however, more different models still have to be evaluated. This paper studied time lifting for a topological characterization of convex spaces in real world which is described by Space Syntax theory. This theory illustrates human settlements and societies as a strongly connected space-time relational system between convex spaces. Such a system is represented by a connectivity graph. Also some morphologic analyses are defined for deriving the graph's properties which illustrate how the space and time are overcome by the relational systems and convex spaces. Investigation of temporality in Space Syntax theory has shown that more dynamicity exist among activities in local scale. Then the specific problem of this paper is defined as modelling integrated static and dynamic analyses of an activity based scenario in local scale and studying how effective these activities overcome space and time. The derived model is implemented for analysing a simulated urban public transportation system using a functional programming language known as Haskell. The successful implementation validated the time lifting approach for topological models of convex spaces, as the main aim of this paper. Besides, questions are emerged about mixed usage of static and dynamic data and level of computation's time and memory growth rates. * Corresponding author.