XXX-X-XXXX-XXXX-X/XX/$XX.00 ©20XX A Mathematical Language for the Modeling of Geospatial Static Rules Robert Laurini Knowledge Systems Institute, USA, and University of Lyon, France) Roberto.Laurini@gmail.com; ORCID 0000-0003-0426-4030 Abstract— More and more in information technologies, rules are considered as first-class citizens; and many applications in business intelligence are built on rules. But in territorial intelligence and in smart city planning, few works have been done in this direction. The role of this paper is to show the importance of geographic rules and to propose, beyond the modeling in natural language, a mathematical language to model them. This language is primary based on logic and set theory in which some relations and operators coming from topology, computational geometry and operation research are integrated, and will cover only static rules. By mathematical language, one means that it is independent from any software products and applications, and its formal grammar is presented by means of syntactic diagrams. Examples in urban planning are provided. Keywords—geographic knowledge; geographic rules; knowledge engineering; formal grammar; territorial intelligence; smart cities. I. INTRODUCTION According to Graham [6] and Morgan [14], rules must be considered as first-class citizens in information technologies, meaning first that several computer-based activities must be revisited. By definition, a rule is a sequence antecedents- implication-consequents which can be noted by (A)  (B), in which A is a conjunction of conditions. Instead of antecedents, sometimes the expression ‘premise’ is used. In logic, B can be either a disjunction of conditions or a set of assertions. They come from the so-called Horn clauses which are also the basis of logic programming, where it is common to write definite clauses in the form of an implication: (p ∧ q ∧ ... ∧ t)  u. However, in urban planning, rules are essentially coming from laws and by-laws. Moreover, some experts can use other rules in their daily practice, sometimes called best practices. In addition to that more and more specialists in spatial data mining can discover what they call associative rules. In the knowledge society, in territorial intelligence and in smart city management, it is important not only to identify rules, but also to combine them to automate reasoning. Facing this objective, the scope of this paper is to propose a language in order to model rules. Indeed, rules are often made explicit in natural language (i.e. English, French, Spanish, etc.); two main sources will be used, namely rules written in natural language and associative rules as extracted from data mining. This language will be based on mathematics without taking into account practical implementation. In other words, this is not a computer language (code). This paper is organized as follows. First, some elements will be given to explain the role of rules in computing, and especially in geoprocessing. Then the formal grammar of this mathematical language will be detailed. Finally, examples especially in urban planning, will clarify the expressive power of this language. Voluntarily, this paper will not deal with 3D issues, neither with temporal issues: only geographic static rules will be considered. II. ABOUT RULES IN IT In this section, the importance of rules will be emphasized in business intelligence and then in territorial intelligence. In Business intelligence, generally, their implementation is based on two grammatical structures IF-THEN-Fact and IF- THEN-Action (Ross [16]). The first serves above all to involve new facts, that is new objects, attribute values, new relationships between objects. As to the second, it is to involve new actions. But who will be in charge of such new actions? In some cases, the computer itself may run procedures or send a message to other devices; in others, particularly in regulatory contexts, a decision maker (for example, the CEO of a company) must himself initiate the action. Another interpretation could be that the choice of alternatives of an action, for example when a law, in some well-defined contexts, opens many perspectives. Thus, a rule is a basic element of a strategy to build reasoning. In contrast to algorithms, they are expressed declaratively. Among business rules, Dietz [3] distinguishes between three categories:  rejectors typically those related to quality control, that allow a rejection (rejection rules),  producers such as those determining new values (ex VAT calculation); they can be considered as rules of production of information,  and projectors such as those related to the replenishment of stocks. To conclude this section, let us mention that lots of business applications are based on rules, and several computer languages for encoding business rules have been proposed, as XML extensions, such as SWRL (Espinasse, [5]) or RuleML (Boley [1], Boley et al. [2]). The simplest of those extensions is as follows: <Implies> <if> <..> </if> <then> <..> </then> </Implies>