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Top1. Introduction
The paradigm of Multi Agent Systems is important for designing complex software (Cohen & Levesque, 1990; Glaser, 1997; Grosz & Kraus, 1996; Jennings & Wooldridge, 1998; Marik, Müller & Pechoucek, 2003). Generally, it considers complex problems resolution with multidimensional characteristics such as: reactivity, mobility, dynamicity and adaptation of the system to uncertain or unpredictable factors (Wooldridge, 2009). A Multi agent system may be seen as societies made up of autonomous and independent entities, called agents. These agents interact together in order to resolve a particular problem or to achieve a common task collectively (Kouah, Saïdouni & Ilié, 2013). An agent can be viewed as a computer system situated in some environment; able to execute flexible autonomous actions in this environment in order to meet its design objectives (Jennings & Wooldridge, 1996; Wooldridge & Jennings, 1995).
However, designing complex systems incrementally become a challenge due to the lack of formal models, methods and tools that support both refinement and integration of sub-system specifications at different abstraction levels.
In fact, formal modeling provides rigorous and unambiguous behavior specifications and allows checking specifications requirements, such as safety, liveness and domain dependent properties. In particular, formal refinement is a general strategy of adding details that concerns both environmental constraints and design requirements to system specification. It allows an incremental development and preserves desired properties.
In this paper, we are concerned by formal stepwise designing of multi agent systems. Several labors have been made in this field, such as (Peña, Corchuelo & Arjona, 2003; Graja, Migeon, Maurel, Gleizes, Regayeg, & Kacem, 2013; Pereverzeva, Troubitsyna, & Laibinis, 2012.a; Pereverzeva, Troubitsyna, & Laibinis, 2012.b; Smith & Winter, 2012).
Beside gradual behavior definition; a stepwise designing of multi agent systems needs handling an approximate and uncertain behavior which is the purpose of FLTS and FLTRT models (Kouah & Saidouni, 2014).
In fact, FLTS model provides a complete formal designing system framework that ensures correctness transformations and deals with incomplete information through its fuzziness representation. In FLTS model, fuzzy logic (Azar, 2010; Azar, 2012) is chosen to model incomplete information rather than probability logic, because we need handling imprecision of information not imprecision on events occurrence. While fuzzy logic deals with imprecise information, the information is handled in sound mathematical theory.
An FLTRT structure is a tree (see Figure 1. b) of potential concurrent design trajectories of the corresponding system. By means of bissimulation relations on FLTS, equivalent design trajectories may be easily identified. On the other hand, FLTRT features are recapped as follows:
Figure 1. (a) Example of fuzzy labeled transition system; (b) Example of fuzzy labeled transition refinement tree